Simulation and visualization of ultrasound fields · t in tec hnology.F or exam-ple, astronomers...

UNIVERSITY OF OSLODepartment of Informatics

Simulation andVisualization ofUltrasound Fields

Kapila Epasinghe

Cand Scient Thesis

1st August 1997

Simulation and Visualization

of Ultrasound Fields

c©Kapila Epasinghe

1st August 1997

Acknowledgment

This thesis is written as the practical requirement for the Candidatus Sci-

entiarum (Master of Science) degree at The Division of Signal Processing at

the Department of Information Technology, University of Oslo, Norway. This

work was started in June 1995 and �nished in July 1997. Part of this work

was presented as a research article at the Nordic Symposium on Physical

Acoustics in February 1996.

First, I wish to express my sincere gratitude to my main supervisor, Professor

Sverre Holm, for his patience and understanding throughout my labour, and

that little extra support when the going got tough. A big thank you goes to

my bi-supervisor, System Engineer Dag F. Langmyhr, for his collaboration.

A special note of thanks goes to Roger O. Nordby of the Scienti�c Visu-

alization Laboratory in USIT (University Center for Information Services).

Thank you, Roger, for coming to IFI the saturday after your birthday party.

I also wish to thank So�a, Ranjith and Shiva for helping me with small

details of my thesis. Together with Nihal, Sam, Dilu, Bahee, Hisham and

Chamath, they were always there for me with a little bit of encouragement

when I needed it.

Finally, I wish to thank my parents and my two brothers who always are

understanding and encouraging although they were thousands of miles away.

Oslo, July 1997

Kapila Epasinghe

University of Oslo,

Norway.

Contents

1 Introduction 1

1.1 Ultrasound in medical diagnosis . . . . . . . . . . . . . . . . . 1

1.2 Parallel programming in mathematics . . . . . . . . . . . . . . 2

1.3 Objective of this thesis . . . . . . . . . . . . . . . . . . . . . . 2

2 Acoustic Theory and Ultrasound 5

2.1 Basic acoustic principles . . . . . . . . . . . . . . . . . . . . . 5

2.2 Some properties of acoustic waves . . . . . . . . . . . . . . . . 6

2.3 The wave equation . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3.1 Solution in cartesian coordinates . . . . . . . . . . . . 7

2.3.2 Solution in spherical coordinates . . . . . . . . . . . . . 7

2.4 Acoustic impedance . . . . . . . . . . . . . . . . . . . . . . . . 8

2.5 Re�ection and refraction . . . . . . . . . . . . . . . . . . . . . 9

2.6 Velocity potential . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.7 Rayleigh integral . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.8 Calculating the �eld . . . . . . . . . . . . . . . . . . . . . . . 11

2.8.1 Huygens' principle . . . . . . . . . . . . . . . . . . . . 11

2.8.2 Di�raction impulse response method . . . . . . . . . . 11

2.8.3 Direct calculation of the Rayleigh integral . . . . . . . 12

ii CONTENTS

3 Transducer Design and Theory 13

3.1 Transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1.1 Piezo-electric e�ect . . . . . . . . . . . . . . . . . . . . 13

3.1.2 Matching . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 Transducer apertures . . . . . . . . . . . . . . . . . . . . . . . 15

3.2.1 Continuous apertures . . . . . . . . . . . . . . . . . . . 15

3.2.2 Linear apertures . . . . . . . . . . . . . . . . . . . . . 16

3.2.3 Rectangular apertures . . . . . . . . . . . . . . . . . . 17

3.2.4 Elliptical and circular apertures . . . . . . . . . . . . . 18

3.3 Beam characteristics . . . . . . . . . . . . . . . . . . . . . . . 20

3.4 Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.5 Transducer arrays in medical diagnosis . . . . . . . . . . . . . 22

3.6 Electronic focusing . . . . . . . . . . . . . . . . . . . . . . . . 23

3.7 The ultrasound imaging system . . . . . . . . . . . . . . . . . 24

4 The Octogonal Transducer 27

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 Rectangular area A . . . . . . . . . . . . . . . . . . . . . . . . 28

4.3 Rectangular area B . . . . . . . . . . . . . . . . . . . . . . . . 28

4.4 Triangular area C . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.4.1 Section C1 . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.4.2 Section C2 . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.4.3 Section C3 . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.4.4 Section C4 . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.4.5 Sum of the triangular sections . . . . . . . . . . . . . . 34

4.5 Aperture smoothing function of the transducer . . . . . . . . . 36

5 Simulation of Acoustic Fields 37

5.1 UltraSim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.2 Simulation of the acoustic �eld . . . . . . . . . . . . . . . . . 37

5.3 Visualization of the acoustic �eld . . . . . . . . . . . . . . . . 39

CONTENTS iii

6 Scienti�c Parallel Processing 41

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6.2 Parallel machine models . . . . . . . . . . . . . . . . . . . . . 42

6.2.1 The multicomputer . . . . . . . . . . . . . . . . . . . . 43

6.2.2 The multiprocessor . . . . . . . . . . . . . . . . . . . . 44

6.2.3 The SIMD model . . . . . . . . . . . . . . . . . . . . . 44

6.3 Parallel programming models . . . . . . . . . . . . . . . . . . 45

6.3.1 Tasks and channels . . . . . . . . . . . . . . . . . . . . 45

6.3.2 Message passing . . . . . . . . . . . . . . . . . . . . . . 46

6.4 The SPMD model . . . . . . . . . . . . . . . . . . . . . . . . . 46

6.5 Data distribution . . . . . . . . . . . . . . . . . . . . . . . . . 46

6.5.1 Scatter distribution . . . . . . . . . . . . . . . . . . . . 47

6.5.2 Linear distribution . . . . . . . . . . . . . . . . . . . . 48

6.6 Performance analysis . . . . . . . . . . . . . . . . . . . . . . . 49

6.6.1 Speed-up . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6.6.2 Performance modelling . . . . . . . . . . . . . . . . . . 50

6.7 Parallel routine libraries . . . . . . . . . . . . . . . . . . . . . 51

7 Parallel Response Calculation 53

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

7.2 MATLAB and MEX-programming . . . . . . . . . . . . . . . 53

7.3 Data distribution of observational segements . . . . . . . . . . 54

7.4 Parallel routines . . . . . . . . . . . . . . . . . . . . . . . . . . 55

7.5 Performance analysis . . . . . . . . . . . . . . . . . . . . . . . 56

8 Simulation Results 61

8.1 The Octagonal Transducer . . . . . . . . . . . . . . . . . . . . 61

8.2 Parallel programming results . . . . . . . . . . . . . . . . . . . 63

8.3 3D plot results . . . . . . . . . . . . . . . . . . . . . . . . . . 67

8.3.1 Rectangular aperture . . . . . . . . . . . . . . . . . . . 67

iv CONTENTS

8.3.2 Elliptical aperture . . . . . . . . . . . . . . . . . . . . 67

8.3.3 Super-elliptical aperture . . . . . . . . . . . . . . . . . 67

8.3.4 Octagonal aperture . . . . . . . . . . . . . . . . . . . . 74

8.3.5 Range movie and time movie . . . . . . . . . . . . . . 76

9 Conclusion and Summary 77

9.1 The octagonal transducer . . . . . . . . . . . . . . . . . . . . 77

9.2 The parallel algorithm . . . . . . . . . . . . . . . . . . . . . . 77

9.3 Software and hardware . . . . . . . . . . . . . . . . . . . . . . 78

A User Guide to Parallel Execution 83

A.1 C-MEX routine . . . . . . . . . . . . . . . . . . . . . . . . . . 83

A.1.1 Compilation of C-MEX Routine . . . . . . . . . . . . . 83

A.1.2 Execution . . . . . . . . . . . . . . . . . . . . . . . . . 83

A.2 C-MEX routine for parallel processing . . . . . . . . . . . . . 84

A.2.1 Compilation of Parallel C-Mex Routine . . . . . . . . . 84

A.2.2 Interactive Parallel Execution . . . . . . . . . . . . . . 84

A.2.3 Parallel Execution as a Batch job . . . . . . . . . . . . 86

B User Guide to Volumetric Visualization 89

B.1 Observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

B.2 Slice Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

C Programming Code 95

C.1 C routine for sequential computer -pec.c . . . . . . . . . . . . 95

C.2 C routine for parallel computer - pec.c . . . . . . . . . . . . . 107

C.3 C-Routine ppec_slave.c . . . . . . . . . . . . . . . . . . . . . . 119

C.4 MATLAB Routine for parallel network - retrieve.m . . . . . . 125

C.5 MATLAB routine - peplot3d.m . . . . . . . . . . . . . . . . . . 126

C.6 MATLAB routine - plt3d.m . . . . . . . . . . . . . . . . . . . 134

Chapter 1

Introduction

1.1 Ultrasound in medical diagnosis

The average human ear can detect sound waves with frequencies between 20

Hz and 20,000 Hz (20 KHz). Sound waves with higher frequencies than 20

KHz are called ultrasound. In medical ultrasound, frequencies higher than

1.5 MHz and upto 20MHz are used.

Ultrasound was understood to be a potential imaging utility since the 1940s

[10], borrowed from RADAR1 and SONAR2 technology used during the

World War II. Since then, ultrasound has been applied for various medi-

cal purposes. However its acceptance as a powerful diagnostic tool was in

the 1970s.

Today, ultrasound is the second most utilized diagnostic imaging technol-

ogy used in medicine, after X-Rays. Ultrasound scanning provides a less

harmful but certainly e�ective method for �in vivo� diagnosis. Today, ultra-

sound scanners are used by many medical institutes. Two common uses of

ultrasound imaging are the scanning of the fetus during pregnancy and the

scanning of the heart.

In the late sixties, ultrasound scanners with real-time 2D images, formed by

both mechanical and electronic array scanning, was introduced. During the

next two decades, scanners with better image quality, higher frame rates and

doppler imaging methods were introduced to the market. These scanners

used annular arrays and 1D phased arrays[1].

1RADAR - RAdiowave Detection And Ranging2SONAR - SOund Navigation And Ranging

2 Introduction

To keep up with the new developments in medical industry, and due to the

necessity for better diagnostic methods, new ultrasound scanners provide 3D

imaging capabilities. The �rst 3D scanners used 1D phased arrays. Since

these can only be focused in the azimuth direction, the transducers have to

be moved mechanically in elevation direction, successively scanning parallel

planes to make up a non-real time 3D image. To create a real time 3D

image, a 2D phased array can be used. 2D phased arrays can be electronically

focused to any point in a 3-dimentional segement. Using a 2D array, however,

imposes a heavy burden on computations, due to the large number of array

elements involved - in the range of 256x256. But with the availability of

more powerful and cheaper computer products, this can be alleviated to

some extent.

With the development of new signal processing methods and better equip-

ment, 2D arrays with di�erent footprints, such as annular, rectangular, el-

liptical foot prints, are presently being tested and applied.

1.2 Parallel programming in mathematics

Powerful computers have helped scientists and mathematicians to conduct

simulations and receive speedy results. However, with new technology comes

new applications that demand further development in technology. For exam-

ple, astronomers use larger data sets now than a few decades ago, which takes

many days to process even with the most powerful computers. Further, super

computers that are only a few times faster than a powerful Personal Com-

puter (PC), costs many times more. Therefore applying parallel (concurrent)

programming methods can be a useful alternative. Computer parallalism is

the art of dividing a large program into parts or segments. The two main

models of parallel programming is partition of the program and partition

of the data. Mathematical and scienti�c parallel programming usually in-

volve dividing large data sets and computing them in seperate computers or

processors at the same time.

1.3 Objective of this thesis

The main objective of this thesis is to develop a parallel algorithm to to

simulate the acoustical �eld created by arbitrary transducer shapes and ar-

rays. The performance and e�ciency of the algorithm is presented. A 3-

dimensional visualizing tool is then introduced to study the simulated results

1.3 Objective of this thesis 3

in perspective. In addition, a novel shaped transducer is introduced and its

acoustic qualities and quantities are studied.

The thesis can be divided into three main parts. First, some theory behind

medical acoustics is discussed and the novel transducer aperture is intro-

duced. Then some theory behind parallel processing is discussed and the

parallel algorithm is introduced. Finally the simulations using the algorithm

and plots from the visualizing tool are presented and discussed.

4 Introduction

Chapter 2

Acoustic Theory and Ultrasound

2.1 Basic acoustic principles

Sound, like all longitudinal waves, requires a medium to propagate. A lon-

gitudinal wave is when the motions of the particles in the medium is in the

same direction as the direction of propagation. The wave propagates with the

compression and rarefaction of the medium. The displacement plotted with

respect to time gives a sinusoidal wave. The wavelength, λ, is the distance

between two successive compressions or rarefactions called pressure maxima.

And period, T , is the time between two pressure maxima at a particular

point in space. From this we get the propagation velocity of the sound in

medium as:

c =λ

T

Because T = 1/f where f is the temporal frequency, we can also write:

c =λ

T= λf

The mathematical formula c = λf is true for any wave of any type and

frequency. Angular frequency is given by ω = 2πf . The wave number vector,designated by �k, is a spatial frequency variable which has the direction, �k/|�k|,same as the direction of propagation of the wave. The magnitude of �k is also

know as the spatial frequency of the propagating wave. The slowness vector,

designated by α, is de�ned by �k/ω.

6 Acoustic Theory and Ultrasound

2.2 Some properties of acoustic waves

re�ection - An acoustic wave bounces o� a non-absorbing interface. The

acoustic echo has the properties of any other wave. When the acoustic

ray is re�ected, the incidence angle is equal to the re�ective angle. This

suggests that a properly shaped surface can be used to focus or di�ract

acoustic waves.

refraction - When an acoustic wave is incident upon a boundary between

two di�erent media, some of the wave is re�ected and some passes to

the second medium.

interference - Interference occurs when two similar waves travel in one

medium. When the two waves are out of phase, destructive interference

occurs and when they are in phase, constructive interference occurs.

di�raction - Di�raction is the phenomenon which refracts sound around

corners. This can be explained with Huygens's principle. When a wave

front is at an opening, the part of the wave that contacts the opening

acts as a source and propagates circularly.

superposition - The law of superposition states that the existence of one

wave in a certain space and time does not a�ect the existence or prop-

erties of another wave in the same space and time. However, in regard

to sound, some loss occurs when high amplitude sound waves propagate

through air.

2.3 The wave equation

The wave equation is given by:

∇2p =∂2s

∂x2+∂2s

∂y2+∂2s

∂z2=1

c2∂2s

∂t2(2.1)

where s(x, y, z, t) is a general scalar �eld. In the case of acoustics, s is the

acoustic pressure de�ned in a speci�c point in space and time. Scalar c, inacoustical sense, is the speed of sound. In case of �uids, c is related to �uid

parameters by:

c2 =dP

dρ

where P is the acoustic pressure and ρ is the density of the medium.

2.3 The wave equation 7

2.3.1 Solution in cartesian coordinates

There are a number of solutions to 2.1. Since the wave equation is a partial

di�erential equation, we can assume a separable solution. And for simplicity,

we can assume that s(x, y, z, t) has a complex exponential form

s(x, y, z, t) = A exp {j(ωt− kxx− kyy − kzz)} (2.2)

By substituting 2.2 in 2.1, it can be seen that the above equation is indeed

a solution to the wave equation if the constraint

k2x + k2y + k

2z =

ω2

c2

is satis�ed. This solution is interpreted as a monochromatic plane wave. It

is monocromatic since temporal frequency at a given point is constant. It is

a plane wave since at any given instance, the value of s is the same in all

points lying in a plane given by kxx+ kyy+ kzz = C, where C is a constant.

The monochromatic solution, given in 2.2, can be written as

s(�x, t) = A exp {j(ωt− �k · �x)} (2.3)

where �k is the wave number vector, and �x is the spatial location vector.

Sometimes this solution is expressed with the slowness vector, �α = �k/ω.Then

s(�x, t) = A exp{jω(t− �α · �x)}

Then s(�x, t) can be written as s(�x, t) = s(t − �α · �x) = A exp jωu, whereu = t− �α · �x.

Magnitude of �α is reciprocal to the propagation speed, since �α = �k/ω and

|�k|2 = ω2/c2, which is the reason for its name, the slowness vector.

2.3.2 Solution in spherical coordinates

The wave equation 2.1 can be written in polar coordinates (r, θ, φ), where

x = r cos θ sinφ

y = r sin θ sinφ

z = r cosφ


The graphical presentation of the this transformation can be found in �gure

3.3 in page 17. From these variables, the wave equation can be written in

polar coordinates.

1

r2∂

∂r

(r2∂s

∂r

)+

1

r2 sinφ

∂

∂φ

(sinφ

∂s

∂φ

)+

1

r2 sin2 φ

∂2s

∂φ2=1

c2∂2s

∂t2(2.4)

This equation can be solved by using the method of separation of variables.

However, the spherical coordinate wave equation is generally used in situa-

tions where spherical symmetry is evident[18]. Then the solution s(r, θ, φ, t)does not depend on θ and φ and the equation 2.4 reduces to

1

r2

(r2∂s

∂r

)=1

c2∂2s

∂t2

With some manipulation, this equation can be converted to an easy to handle

one dimensional wave equation

∂2 (rs)

∂r2=1

c2∂2 (rs)

∂t2(2.5)

One solution to this is the monochromatic solution

s(r, t) =A

rexp j(ωt− kr)

This is interpreted as a spherical wave propagating outward from the origin.

A propagating spherical wave of the form

s(r, t) =B

rexp j(ωt+ kr)

is also a solution to the equation 2.5. This can be intepreted as a spherical

wave propagating towards the origin.

2.4 Acoustic impedance

Acoustic Impedance, more formally known as speci�c acoustic impedance, is

de�ned as

2.5 Re�ection and refraction 9

Z =P

u

where P is the acoustic pressure and u is the partical velocity. For a plane

wave travelling in a speci�c direction, characteristic acoustic impedance is

de�ned as

Z0 =Pz

uz= ρc

where the subscript z denotes the direction of particle movement, ρ denotes

the density of the medium and c denotes the velocity of sound in the medium.

2.5 Re�ection and refraction

When an acoustic wave meets a boundary, both re�ection and refraction

occurs. The incident angle, θi, is equal to the re�ected angle, θr, and the

transmitted angle, θt, depends on the acoustic velocities of the two media,

c1 and c2 as well as the incident angle. In case of a monochromatic wave, θiand θt is related by

sinθi

vi=sinθt

vtwhere vi and vt are velocities in the incident medium and transmited medium

respectively.

The pressure re�ection and transmission coe�cients R and T can be deduced

by these conditions and with acoustic impedances of the two media, Z1 andZ2.

R =pr

pi=Z2 cos θi − Z1 cos θtZ2 cos θi + Z1 cos θi

and T =pt

pi=

2Z2 cos θiZ2 cos θi + Z1 cos θi

When the acoustic wave is normal to the boundary, the re�ection and trans-

mission coe�cients reduce to

R =Z2 − Z1Z2 + Z1

and T =2Z2

Z2 + Z1

When Z2 � Z1 the re�ection coe�cientR→ 1. Then the amplitude between

the incident and re�ective waves is only slightly reduced. The particle speed

at the boundary is almost zero. Such a medium is called a rigid ba�e.

When Z2 Z1 the transmission coe�cient T → 1. Then the acoustic

pressure at the boundary is almost zero. Such a medium is called a soft

ba�e or pressure release boundary.


2.6 Velocity potential

For acoustic pressure waves, the scalar �eld s in wave equation in 2.1 can be

expressed as the acoustical pressure p. Therefore the acoustic wave equationcan be written as

∇2p−1

c2∂2p

∂t2= 0 (2.6)

From Eular's equation of particle velocity, ρ0∂�v∂t= −∇p, the acoustic pressure

can be converted to partical velocity. Since ∇×∇p = 0, it can be seen that

∇× �v = 0. Therefore, particle velocity v has a scalar potential function Φ,called the velocity potential function, where

v(�x, t) = ∇Φ(�x, t) (2.7)

Substituting this in Eular's equation, acoustic pressure wave can be expressed

as

p(�x, t) = −ρ0∂Φ(�x, t)

∂t(2.8)

By substituting equation 2.8 in equation 2.6 and integrating with respect to

time, it can be seen that the velocity potential function satis�es the acoustic

wave equation.

2.7 Rayleigh integral

In a acoustically rigid ba�e, the acoustic pressure function p(�r, t) can be

derived from the Green's function.

p(�r, t) =∫ ∫

Sρ

∂∂tvn(�r0, t− �r/c)

2π�rdS

or by using the velocity potential:

Φ(�r, t) =∫ ∫

S

vn(�r0, t− �r/c)

2π�rdS

This equation is known as the Rayleigh Integral. Here, ρ is the density of themedium, vn is the particle velocity at the surface of the transducer normal

to the transducer,

2.8 Calculating the �eld 11

2.8 Calculating the �eld

2.8.1 Huygens' principle

Huygens's principle is used to explain the phenomena of di�raction and beam

pro�les of waves. The principle postulates that each point on a travelling

wavefront could be considered as a secondary source of spherical radiation.

In an ultrasonic transducer, this can be applied to �nd the �eld at a spatial

point. The transducer surface can be considered as an in�nite number of

point sources, each emitting a spherical wave. The superposition of the

spherical wavelets generated by all point sources at a certain point yields the

�eld at that point[24].

2.8.2 Di�raction impulse response method

If the transducer is assumed to be planar, that is lateral dimensions and the

radius of curvature are large compared to the wavelength, the surface velocity

at the boundary can be separated. It is further assumed that the transducer

vibrates in a single mode, usually thickness mode[7]. So the surface velocity

can be written as

vn(�r0, t) = O(r0)v(t)

where O(r0) is apodization. The temporal dependence of v(t− �r/c) can be

written as a convolution.

v(t− �r/c) =∫v(t0)δ(t− �r/c− t0)dt0

This can be inserted in the Rayleigh Integral and by changing the order of

integration ,

Φ(�r, t) =∫tv(t0)

∫ ∫S

O(r0)δ(t− �r/c − t0)

2π�rdSdt0

Let the impulse response, h(�r, t) be de�ned as

h(�r, t) =∫ ∫

δ(t− �r/c − t0)

2π�rdS0

Now the velocity potential Φ(�r, t) can be written as a convolution of h(�r, t)and v(t).

Φ(�r, t) = v(t)⊗ h(�r, t)


In order to �nd the impulse response h(�r, t), two di�erent approaches can be

applied[7]. One method is to calculate the impulse response for a speci�c �eld

point and its temporal evolution is obtained by observing that at t = r/c, onlypoints of the source located at a distance r from the �eld point contributes

to the local �eld i.e only points lying in a circle with its center being the

projection of the �eld point in the plance source. This method is known as

the Stepanishen technique. The other method is to �nd the impulse response

at a �xed time on a speci�c plane.

2.8.3 Direct calculation of the Rayleigh integral

Even though Impulse response method is very e�cient, it has some short-

comings. Its in�exibility with regard to the shape of the source is a main

disadvantages. Another important limitation is the assumption of a homo-

geneous medium[22].

A robust way to compute the �eld is to perform the integration of the

Rayleigh integral numarically[6]. Each �eld point will receive a contribu-

tion from each element of the transducer. This method is simple, robust and

can be used to calculated the �eld created by arbitrary shaped transducers.

However when the number of elements in the transducer is large and the �eld

has to be calculated for planes or volumes, the e�ectiveness of the algorithm

would depend heavily on the performance of the computer. Using a parallel

algorithm to alleviate the strain on time consumption is a part of this thesis

and is presented in a later chapter. This method is also advantages when

calculating the �eld in an aberrating medium[21].

Chapter 3

Transducer Design and Theory

In this chapter, the theory behind transducer design and related �elds are

discussed.

3.1 Transducers

3.1.1 Piezo-electric e�ect

Certain materials, like crystals, change their physical dimensions when an

electrical �eld is applied, and vice versa. This phenomenon is known as the

piezo-electric e�ect and this is used to create ultrasound waves in transducers.

Some natural material such as quartz and tourmaline, and arti�cial materials,

known as polarized ferro electricals pocess piezo-electric characteristics.

Usually, electricity is applied to the piezo electric disc via two silver elec-

trodes, as in �gure 3.1. By considering the two surfaces of the piezoelectric

crystal as two independent vibrators, it can be shown that the resonance

frequency is

f0 =ncp

2Lc

or Lc = nλ/2

with the lowest resonant frequency is when n = 1, cp is the acoustic wave

velocity and Lc is the thickness of the piezoelectric material.

The piezoelectric material for a transducer is chosen according to a number

of factors, such as stability, piezoelectric properties and material strength.

Quartz, for example is stable and therefore useful for accurate measurements,

but requires high electric �elds to obtain high power outputs. Ceramics can

14 Transducer Design and Theory

lambda/2

Electrodes

Figure 3.1: A simple transducer. The piezoelectric disc is half lambda thick

so that resonance occurs.

produce the same output at much lower �elds, but shows less stability and

low electric input impedance at high frequencies.

A typical construction of a single element transducer is in �gure 3.2.

Connector

Backing Material

Piezoelectric elementProtectiveLayer

Electrodes

Housing

Matching Layer

Figure 3.2: Major parts of a medical transducer.

3.1.2 Matching

Transducer matching is an important issue in transducer design. When a

transducer is excited using an electrical source, it rings according to its reso-

nant frequency. In case of continuous waves, it is desirable that most energy

3.2 Transducer apertures 15

is transmitted to the forward direction. A backing material with relatively

low impedance, e.g. air, would have a Re�ection Coe�cient R→ 1, allowingthe waves to be almost completely re�ected at the back interface.

In pulse-echo applications, such backing would lengthen the duration of the

pulse, which is undesirable. It is therefore important that the energy is

transmitted to and absorbed by the backing material. Such lossy materials

with equal impedance would create the ideal backing material for pulse-

echo applications. Assume that ZB , ZT and ZM are the impedances of the

backing, the transducer and the medium respectively. In traditional medical

transducers, where ZM ZT and ZB < ZT , the amplitude of the second

impulse would be considerably greater than the �rst and subsequent damping

would occur due to the lossy backing material. When ZB ZT and ZM = Zt,

the �rst two pulses have equal amplitude but opposite phases. All other

pulses are absorbed by the backing material[17].

3.2 Transducer apertures

A sensor placed at the speci�c location in space gathers the propagating

energy and produces an electrical signal. A transducer acts as a sensor in

addition to be able to create an energy wave from electrical signals. Because

propagating waves vary in space and time, sensors are often designed to

have signi�cant spatial extent, gathering energy propagating from speci�c

directions. Such sensors are said to be directional; able to focus on a speci�c

direction. Radar dishes used in aviation is an example of directional sensors.

An aperture gathers signals over a �nite spatial area. An array of such

apertures, placed in a speci�c order in space make up arrays. The output

of each sensor, enhanced, reduced or inhibited, is combined to produce one

single output.

3.2.1 Continuous apertures

When a �eld f(�x, t) is observed through a �nite continuous aperture, the

output is

z(�x, t) = w(�x)f(�x, t)

where w(�x) is called the aperture function. In many cases w(�x) is equal to 1within the aperture boundaries and 0 outside. However, w(�x) can be used asa weighing function where it can take on any value between 0 and 1 within


the spacial boundaries of the aperture. This kind of aperture weighing is

sometimes referred to as shading, taping or apodization.

By Fourier transformation of z(�x, t), the above equation becomes a convolu-

tion of two functions.

Z(�k, ω) =∫ ∞−∞

W (�k−�l)F (�l, ω)d�l

where

W (�k) =∫ ∞−∞

w(�x)ej�k·�xd�x (3.1)

is known as the aperture smoothing function. The spatial extent of an aper-

ture determines the resolution with which two plane waves are separated. A

large aperture can be focused in any direction.

The aperture smoothing function depends mostly on the shape of the aper-

ture. Common aperture shapes, such as rectangular and annular, have aper-

ture smoothing functions that are easy to express and apply. For novel shaped

apertures, the aperture area has to be handled in special ways to �nd the

aperture smoothing function.

Aperture smoothing function can also be expressed with the direction of the

wave. This is known as the directivity function or the radiation pattern.

Since |�k| = k = 2π/λ, a the wavenumber vector of a plane wave can be

expressed as

�k = {kx, ky} = {k sinφ cos θ, k sinφ sin θ}

= {(2π/λ) sin φ cos θ, (2π/λ) sin φ sin θ} (3.2)

where θ is the angle in XY plane and φ is the normal incident angle. Figure

3.3 shows the angles in 3D space.

3.2.2 Linear apertures

Figure 3.4 shows a linear aperture placed symmetrically on the x-axis. For

simplicity, the aperture function w(�k) is assumed to be regular and equal to

1 inside the aperture.

From equation 3.1, the aperture smoothing function of a linear transducer,

Wx(�k) is

WL(�k) =∫y

∫xej

�k·�xdxdy


phi plane

X

Y

Z

phi

theta

Figure 3.3: Geometry of an incidnet wave.

−a a x

y

Figure 3.4: A linear aperture with length 2a

=∫ a

−aejkxxdx

= ejkxx

jkx

∣∣∣a−a

=ejkxa − e−jkxa

jkx

WL(�k) = 2sin(kxa)

kx(3.3)

3.2.3 Rectangular apertures

Figure 3.5 shows a rectangular aperture placed symmetrically on the XY-

plane. The aperture smoothing function is found by assuming two linear

apertures in X and Y direction respectively and multiplying. From equation

3.3, the aperture smoothing function for linear apertures can be found. Mul-

tiplying, the aperture smoothing function, WR(�k) of a rectangular aperture

is

WR(�k) = 2sin(kxa)

kx2sin(kyb)

ky=4

kxkysin(kxa) sin(kyb) (3.4)


a−a

b

−b

x

y

Figure 3.5: A rectangular aperture with length 2a in X-direction and 2b in

Y-direction.

3.2.4 Elliptical and circular apertures

x

y

a

b

Figure 3.6: An elliptical aperture with a and b as main axes.

X

Y

r

Figure 3.7: A circular aperture with radius r.

Figure 3.6 shows an elliptical aperture, placed symmetrically on XY axes

and �gure 3.7. The long and short axes of the elliptical aperture are aand b respectively, and the radius of the circular aperture is r. By setting


a = b = r, the directivity pattern of the circular aperture can be found from

the directivity pattern of the elliptical aperture. The directivity pattern of

the elliptical aperture is found by using the equations 3.1 and 3.2.

WE(�k) = WE(r, θ, φ) =∫ ∫

Sej(2π/λ)(xsinφ cos θ+ysinφ sin θ)dx dy

New variables ρ and ψ are introduced implicitly such that

x = aρ cosψ, 0 ≤ ρ ≤ 1

y = bρ sinψ, 0 ≤ ψ ≤ 2π

for which

dxdy = |J | dρ dψ

where |J | is the Jacobian of the transformation given by

|J | =

∣∣∣∣∣∂x∂ρ∂y

∂ψ−∂x

∂ψ

∂y

∂ρ

∣∣∣∣∣ = abρ

Then the radiation pattern becomes

WE(r, θ, φ) =∫ 2π0

∫ 10ej(2π/λ) sinφ(a cosψ cos θ+b sinψ sin θ)ρabρ dρ dψ (3.5)

By using

cos θ′ =a cos θ

√a2 cos2 θ + b2 sin2 θ

and

sin θ′ =b sin θ

√a2 cos2 θ + b2 sin2 θ

equation 3.5 can be written as

WE(r, θ, φ) =∫ 2π0

∫ 10ej(2π/λ) sinφ

√a2 cos2 θ+b2 sin2 θ cos(ψ−θ′)ρabρ dρ dψ

By using the integral expression of the zero-order Besel function of the �rst

kind1, the above equation can be reduced to one integral.

WE(r, θ, φ) = 2πab∫ 10J0(2π

λρ√a2 cos2 θ + b2 sin2 θ sinφ)ρdρ

12πJ0(z) =∫ 2π0ejz cosψdψ


Since∫ x0 zJ0(z)dz = xJ1(x), the equation reduces to

WE(r, θ, φ) = abJ1(2π

√a2 cos2 θ + b2 sin2 θ sin φ/λ)

√a2 cos2 θ + b2 sin2 θ sinφ/λ

(3.6)

where J1(·) is the �rst-order Bessel function of the �rst kind.

By setting a = b = r in equation 3.6, the directivity pattern of the circular

aperture, WC is

WC(r, θ, φ) = r2J1(2πr sinφ/λ)

r sin φ/λ

Since |�k| = k = 2π/λ, the above equation can be written as

WC(r, θ, φ) = 2πrJ1(kr sinφ)

k sinφ(3.7)

3.3 Beam characteristics

A transducer surface that vibrates in thickness mode creates an ultrasound

beam that radiates normal to the surface. As explained in section 2.8.1, the

beam is the result of the superposition of sperical waves originating from

minute points in the transducer surface.

The beam can be separated into two main regions, as shown in �gure 3.8 for a

circular aperture. In the near �eld, also known as the freznel zone, the beam

is perceived according to the surface characteristics of the transducer [1].

In the far �eld, the beam consists of a large high amplitude beam called the

mainlobe and low amplitude beams skirting the mainlobe, called sidelobes. In

a circular transducer, the separation of these two regions occur at a distance

z0 = D2/2λ

from the transducer surface, where D is the diameter of the aperture and

λ is the wavelength of the acoustic wave. The Extreme near �eld beam is

an cylindrical extension of the transducer surface, extending approximately

0.8D2/4λ from the surface. In the transition region, the beam �rst becomes

narrower and then widens with sidelobes into the far �eld. The narrowing of

the beam is known as the di�raction focus.

By introducing a curvature to the surface of the transducer, the beam can be

focused to a location in the beam region. This kind of geometrical focusing

3.4 Arrays 21

Main Lobe

Side Lobe

Side Lobe

Near Field Far Field

Extreme near field

Transition region

Figure 3.8: An idealistic pressure pro�le of a circular transducer.

is possible for all footprints, but are mostly used in circular aperture. Figure

3.9 shows the region of focus for a circular aperture. Due to di�raction, the

focuse will not be sharp.

mainlobe

Sidelobe

Sidelobe

F

Lf

Figure 3.9: Beam pro�le of a focused circular array.

3.4 Arrays

An array consists of individual apertures or omni-directional transducers,

which sample the wave-�eld in discrete spatial locations. Regular arrays,

made by placing the sensors on a �regular� grid, are easy to analyze and will

be discussed in this section.


A wave�eld with one spatial dimention, f(x, t), has a wavenumber-frequencyrepresentation

F (k, ω) =∫ ∞−∞

∫ ∞−∞

f(x, t) exp{−j(ωt− kx}dxdt

If the �eld is measured by a one-dimentional equally spaced M sensors sep-

arated by a distance d, the sampled wave�eld's wavenumber-frequency rep-

resentation is

Y (k, ω) =∫ ∞−∞

∑m

ym(t) exp{−j(ωt− kmd}dt

where ym(t) = f(md, t). By spatial sampling methods, it can be shown that

Y (k, ω) =1

d

∞∑−∞

F (k −2πp

d, ω)

Here, it is assumed that F is bandlimited, i.e. F (k, ω) is zero for |k| ≥ π/d,so that aliasing will not occur. Generally, a set of weights for each element is

applied. Among other advantages, weighing allows explaining the absence of

an element at the mth location by setting wm = 0. An observed signal can

be represented as

zm(t) = wmym(t)

In the frequency domain, this becomes a convolution

Z(�k, ω) = W (�k)⊗ Y (�k, ω)

where W (�k) is called the discrete aperture function, given by

W (�k) =∑m

wmejkmd

3.5 Transducer arrays in medical diagnosis

Transducer arrays are novel transducers that house more than one transducer

element, ordered in a speci�c fashion. Such transducers, specially transducers

with 2D arrays of elements, provide the capability of electronically focusing

the ultrasound beam to any point in the diagnosis area. Transducer arrays

are mostly used in 3D imaging equipments. Some transducer arrays that are

commonly used are showed in �gure 3.10.

3.6 Electronic focusing 23

(a) (b) (c)

Piezoelectric elements

Isolating material

Piezoelectric material

Isolating material

Figure 3.10: (a) Linear (b) 2D and (c) annular arrays.

3.6 Electronic focusing

Focusing using a curvature on the transducer surface was discussed in sec-

tion 3.3 in page 20. However, to achieve dynamic focusing and to focus to

an arbitrary point in the observation segement, the above method becomes

inadequate. A certain amount of freedom can be achieved by mechanically

turning the transducer to di�erent angles.

In a transducer arrray, an electronic delay can be employed for elements,

thereby delaying the signal according to the required focus. As an example,

a 1D array as in �gure 3.11 can be considered. The array elements are placed

at a distance of less than λ/2 between centers to avoid high energy sidelobes

called grating lobes[24]. If voltage is �rst applied to the �rst element and

then to the next and so on. Therefore the waves from every element will

have a delay before emission. the resultant wavefront will then be steered

according to the applied time delay. The beam can be focused by applying

varying the time delays between elements. With 2D arrays, the beam can be

steered or focused in any direction.

The concentric rings in an annular array can also be electronically focused by

inducing a spherical delay on each ring. In most imaging systems, the beam

is steered mechanically. By rotating the array, a sector image is created

and by moving linearly, a linear image is created. The focus achieved by an


annular array is symmetric both in scan plane and transverse plane to the

scan plane.

Transducerelements

Backing Backing

SteeredBeam

FocusedBeam

(a) (b)

Figure 3.11: (a) Steering and (b) focusing of the beam using a 1D array.

The 1D array of the above paragraph is called a linear phased array. A linear

phased array consists of thin bars mounted on a backing with a λ/4 matching

layer in front. A linear phased array is usually small, about 1cm wide and

about 3cm long and usually with 32 to 128 elements. A slight varient, called

a linear switched array, is also commonly used. The voltage is applied to

groups of elements in succession, so that the beam �moves� across the face of

the array, which has the same e�ect of mechanically moving the transducer

along the axis. Linear switched arrays, also known as linear sequenced arrays,

are much larger than linear phased arrays, 10 to 15 cm long and with 64 to

256 elements.

3.7 The ultrasound imaging system

Figure 3.12 shows a simpli�ed block diagram of an ultrasound imaging sys-

tem. The ultrasound beam is generated and the backscattered wave is re-

ceived by the RF unit. The focusing and steering of the beam is also done

by the RF unit and therefore it is the beamformer of the ultrasound imaging

system. The real time scanline processing unit receives data from RF unit

and this data is processed using the system processor and the memory bank.

3.7 The ultrasound imaging system 25

The processed data is converted to an RGB signal by the Scanconverter dis-

play unit which is sent to the monitor. The real time image is used to carry

out diagnosis.

RF UnitRealtimeScanlineProcessing

SystemProcessor

DataMemory

ScanconverterDisplay Unit

Monitor

Computer Bus

Figure 3.12: Block diagram of an ultrasound imaging system.

Chapter 4

The Octogonal Transducer

4.1 Introduction

The octogonal aperture is a novel shaped aperture introduced in this project.

The transducer has the lengths −2a to 2a in X direction and −2b to 2b inY direction as shown in �gure 4.1.

b

2b

a 2a

−b

−2b

−a−2aAB1 B2

C1C2

C3 C4

Figure 4.1: The aperture smoothing function is found by dividing the octag-

onal transducer into section and summing over all areas.

Since the aperture has a unconventional footprint, the aperture functions

are calculated for di�erent sections, as divided in �gure 4.1. Assume the

aperture smoothing functions of the sections A,B and C are WA, WB and

WC respectively. Aperture function for the transducer is calculated by adding

aperture functions of these sections together.

W (�k) = WA(�k) +WB(�k) +WC(�k) (4.1)

28 The Octogonal Transducer

4.2 Rectangular area A

From the equation 3.4, the array smoothing function of the rectangular

section A can be calculated, using lengths 2a in X-direction and 4b in Y-

direction.

WA(�k) =4

kxkysin(kxa) sin(2kyb) (4.2)

4.3 Rectangular area B

Aperture smoothing function of Area B,WB, is equal to the sum of aperture

smoothing functions of sections B1 and B2, given by WB1 and WB2 respec-

tively. The sections B1 and B2 can be calculated by assuming each is placed

with its phase center in origo and then shifting them in X-direction, as shown

in �gure 4.2.

−3a/2 3a/2

b

−b

0

Figure 4.2: Sections B1 and B2 of the octagonal transducer.

By using the aperture smoothing function of a rectangle in equation 3.4 with

lengths a and 2b, array smoothing functions of the sections B1 and B2, shifted

to their respective places are

WB1 = e−jkx3a/24

kxkysin(kxa/2) sin(kyb)

WB2 = ejkx3a/24

kxkysin(kxa/2) sin(kyb)

Then the aperture smoothing function WB is the sum of the two sections B1

and B2:

4.4 Triangular area C 29

WB = WB1 +WB2

=4

kxkysin(kxa/2) sin(kyb)(e

jkx3a/2+ e−jkx3a/2)

Simply�ed,

WB =8

kxkycos(3kxa/2) sin(kxa/2) sin(kyb) (4.3)

4.4 Triangular area C

Figure 4.1 shows the positions of the 4 triangular parts of the section C,

labelled C1− C4. Due to the symmetricity of the placement, the aperture

smoothing function expressions show certain similarities. However, the aper-

ture smoothing functions for each triangular parts are calculated seperately

below.

For simplicity each triangle is shifted so that the right angled corner is at

the origo. Then the �nal expression is shifted to the correct position with a

e±jkxae±jkyb term according to its cartesian quadrant.

4.4.1 Section C1

(x,0) (a,0)

(0,b)

(x,b−xb/a)

Figure 4.3: Section C1

Let W x be the aperture smoothing function of the linear aperture in �gure

4.3. The vertical linear aperture then has coordinates [0, b − xb/a], for x ∈[0, a]. Aperture function of this line can be calculated with respect to only ycoordinates.


W x = ejkxx∫ b−xb/a

0ejkyydy

= ejkxx · ejkyy

jky

∣∣∣b−xb/ao

=ejkxx

jky

(ejkyb · e−jkyxb/a − 1

)

=ejkyb

jky· ej(kx−kyb/a)x −

ejkxx

jky

For all such linear apertures in the complete triangle, the aperture smoothing

function, W 1, is

W 1 =ejkyb

jky

∫ a

0ej(kx−kyb/a)xdx−

1

jky

∫ a

0ejkxxdx

=ejkyb

jky

ej(kx−kyb/a)x

j(kx−kyb/a)

∣∣∣a0−1

jky

ejkxx

jkx

∣∣∣a0

= −ejkyb

ky (kx − kyb/a)

(ej(kx−kyb/a)a − 1

)+1

kxky

(ejkxa − 1

)

= −aejkyb

ky(kxa− kyb)

(ejkxae−jkyb − 1

)+1

kxky

(ejkxa − 1

)

= −a

ky(kxa− kyb)

(ejkxa − ejkyb

)+1

kxky

(ejkxa − 1

)

To shift the aperture to its correct position, the aperture smoothing function

is multiplied by ejkxaejkyb. Then the aperture smoothing function of the

section C1 is

WC1 = −aejkxaejkyb

ky(kxa− kyb)

(ejkxa − ejkyb

)+ejkxaejkyb

kxky

(ejkxa − 1

)(4.4)

4.4.2 Section C2


4.4. The verticle linear aperture then has coordinates [0, b + xb/a], for x ∈[−a, 0]. Aperture function of this line can be calculated with respect to only

y coordinates.


(x,0)

(0,b)

(−a,0)

(x,b+xb/a)


W x = ejkxx∫ b+xb/a

0ejkyydy

= ejkxx · ejkyy

jky

∣∣∣b+xb/ao

=ejkxx

jky

(ejkyb · ejkyxb/a − 1

)

=ejkyb

jky· ej(kx+kyb/a)x −

ejkxx

jky


function, W 2, is

W 2 =ejkyb

jky

∫ 0−aej(kx+kyb/a)xdx −

1

jky

∫ 0−aejkxxdx

=ejkyb

jky

ej(kx+kyb/a)x

j(kx+kyb/a)

∣∣∣0−a−1

jkyejkxx

jkx

∣∣∣0−a

= −ejkyb

ky (kx + kyb/a)

(1− e−j(kx+kyb/a)a

)+1

kxky

(1− e−jkxa

)

= −aejkyb

ky(kxa+ kyb)

(1− e−jkxae−jkyb

)−1

kxky

(e−jkxa − 1

)

=a

ky(kxa+ kyb)

(e−jkxa − ejkyb

)−1

kxky

(e−jkxa − 1

)


is multiplied by e−jkxaejkyb. Then the aperture smoothing function of the

section C2 is

WC2 =ae−jkxaejkyb

ky(kxa+ kyb)

(e−jkxa − ejkyb

)−e−jkxaejkyb

kxky

(e−jkxa − 1

)(4.5)


4.4.3 Section C3

(0,−b)

(−a,0) (x,0)

(x,−b−xb/a)



4.5. The verticle linear aperture then has the length [−b − xb/a, 0], forx ∈ [−a, 0]. Aperture function of this line can be calculated with respect to

only y coordinates.

W x = ejkxx∫ 0−b−xb/a

ejkyydy

= ejkxx · ejkyy

jky

∣∣∣0−b−xb/a

=ejkxx

jky

(1− e−jkyb · e−jkyxb/a

)

= −e−jkyb

jky· ej(kx−kyb/a)x +

ejkxx

jky


function, W 3, is

W 3 = −e−jkyb

jky

∫ 0−aej(kx−kyb/a)xdx+

1

jky

∫ 0−aejkxxdx

= −e−jkyb

jky

ej(kx−kyb/a)x

j(kx−kyb/a)

∣∣∣0−a+1

jky

ejkxx

jkx

∣∣∣0−a

=e−jkyb

ky (kx − kyb/a)

(1− e−j(kx−kyb/a)a

)−1

kxky

(1− e−jkxa

)

=ae−jkyb

ky(kxa− kyb)

(1− e−jkxaejkyb

)+1

kxky

(e−jkxa − 1

)

= −a

ky(kxa− kyb)

(e−jkxa − e−jkyb

)+1

kxky

(e−jkxa − 1

)



is multiplied by e−jkxae−jkyb. Then the aperture smoothing function of the

section C3 is

WC3 = −ae−jkxae−jkyb

ky(kxa− kyb)


)+e−jkxae−jkyb

kxky

(e−jkxa − 1

)(4.6)

4.4.4 Section C4

(x,0) (a,0)

(x,−b+xb/a)

(0,−b)



4.6. The verticle linear aperture then has the length [−b + xb/a, 0], forx ∈ [0, a]. Aperture function of this line can be calculated with respect to

only y coordinates.

W x = ejkxx∫ 0−b+xb/a

ejkyydy

= ejkxx · ejkyy

jky

∣∣∣0−b+xb/a

=ejkxx

jky

(1− e−jkyb · ejkyxb/a

)

= −e−jkyb

jky· ej(kx+kyb/a)x +

ejkxx

jky


function, W 4, is

W 4 = −e−jkyb

jky

∫ a

0ej(kx+kyb/a)xdx+

1

jky

∫ a

0ejkxxdx


= −e−jkyb

jky

ej(kx+kyb/a)x

j(kx+kyb/a)

∣∣∣a0+1

jky

ejkxx

jkx

∣∣∣a0

=e−jkyb

ky (kx + kyb/a)

(ej(kx+kyb/a)a − 1

)−1

kxky

(ejkxa − 1

)

=ae−jkyb

ky(kxa+ kyb)

(ejkxaejkyb − 1

)−1

kxky

(ejkxa − 1

)

=a

ky(kxa+ kyb)

(ejkxa − e−jkyb

)−1

kxky

(ejkxa − 1

)


is multiplied by ejkxae−jkyb. Then the aperture smoothing function of the

section C4 is

WC4 =aejkxae−jkyb

ky(kxa+ kyb)

(ejkxa − e−jkyb

)−ejkxae−jkyb

kxky

(ejkxa − 1

)(4.7)

4.4.5 Sum of the triangular sections

By adding the �rst terms of equations 4.4, 4.5, 4.6 and 4.7:

WC1 = −

aejkxaejkyb

ky(kxa− kyb)

(ejkxa − ejkyb

)+

ae−jkxaejkyb

ky(kxa+ kyb)

(e−jkxa − ejkyb

)

−ae−jkxae−jkyb

ky(kxa− kyb)


)+

aejkxae−jkyb

ky(kxa+ kyb)

(ejkxa − e−jkyb

)

= −a(e2jkxaejkyb − ejkxae2jkyb + e−2jkxae−jkyb − e−jkxae−2jkyb

)ky(kxa− kyb)

+a(e−2jkxaejkyb − e−jkxae2jkyb + e2jkxae−jkyb − ejkxae−2jkyb

)ky(kxa+ kyb)

= −a(ej(2kxa+kyb) + e−j(2kxa+kyb) − ej(kxa+2kyb) − e−j(kxa+2kyb)

)ky(kxa− kyb)

+a(ej(2kxa−kyb) + e−j(2kxa−kyb) − ej(kxa−2kyb) − e−j(kxa−2kyb)

)ky(kxa+ kyb)

= −2a

ky(kxa− kyb)(cos(2kxa+ kyb)− cos(kxa+ 2kyb))

+2a

ky(kxa+ kyb)(cos(2kxa− kyb)− cos(kxa− 2kyb))


= −2a

ky(kxa− kyb)

(−2 sin(

3kxa+ 3kyb

2) sin(

kxa− kyb

2)

)

+2a

ky(kxa+ kyb)

(−2 sin(

3kxa− 3kyb

2) sin(

kxa+ kyb

2)

)

Therefore, the �rst terms from each equation add up to:

WC1 =

4a

ky(kxa− kyb)sin(3kxa+ 3kyb

2) sin(

kxa− kyb

2)

−4a

ky(kxa+ kyb)sin(3kxa− 3kyb

2) sin(

kxa+ kyb

2) (4.8)

By adding the second terms of equations 4.4, 4.5, 4.6 and 4.7:

WC2 =

1

kxky

[ejkyb

(e2jkxa − ejkxa

)− ejkb

(e−2jkxa − e−jkxa

)

+e−jkyb(e−2jkxa − e−jkxa

)− e−jkyb

(e2jkxa − ejkxa

)]=

1

kxky

[ejkyb

(e2jkxa − ejkxa − e−2jkxa + e−jkxa

)

+ e−jkyb(e−2jkxa − e−jkxa − e2jkxa + ejkxa

)]=

1

kxky

[ejkyb (2j sin(2kxa)− 2j sin(kxa))

−e−jkyb (2j sin(2kxa)− 2j sin(kxa))]

=2j

kxky

(ejkyb − e−jkyb

)(sin(2kxa)− sin(kxa))

=−4

kxkysin(kyb) (sin(2kxa)− sin(kxa))

=−4

kxkysin(kyb)2 sin(

kxa

2)2 cos(

3kxa

2)

Therefore, the second term from each equation add up to:

WC2 =

−8

kxkysin(kxa/2) sin(kyb) cos(3kxa/2) (4.9)


4.5 Aperture smoothing function of the trans-

ducer

From equations 4.2, 4.3, 4.8 and 4.9 the aperture smoothing function of the

transducer is:

W (�k) =4

kxkysin(kxa) sin(2kyb)

+8

kxkycos(3kxa/2) sin(kxa/2) sin(kyb)

+4a


2) sin(

kxa− kyb

2)

−4a


2) sin(

kxa+ kyb

2)

+−8

kxkysin(kxa/2) sin(kyb) cos(3kxa/2)

=4

kxkysin(kxa) sin(2kyb)

+4a


2) sin(

kxa− kyb

2)

−4a


2) sin(

kxa+ kyb

2) (4.10)

Chapter 5

Simulation of Acoustic Fields

In this chapter, the simulation and imaging of ultrasound waves using Ultra-

Sim is discussed.

5.1 UltraSim

UltraSim is a MATLAB based ultrasound simulation package. UltraSim

serves as a standard platform simulation programs concerning ultrasonic

imaging systems. It provides a tool for transducer and dome design, and

will increase the user's understanding of acoustic wave propagation[15][16].

5.2 Simulation of the acoustic �eld

The simulator UltraSim uses the Huygens' principle and a discrete version of

the Rayleigh Integral, given in section 2.7, to calculate the Impulse response.

The arbitrary shaped transducer is divided into a �nite number of points and

the contribution from each source point is summed to �nd the �eld at every

observation point. The impulse response from one transducer point to one

observation point is given by

h(rn, t) =O(rn)ejωT

rA

where rn is the distance from the phase center to the nth transducer point, Ois the weighing of the transducer surface, T is the observation time and rA is

the distance from the transducer point to the observation point. The distance

38 Simulation of Acoustic Fields

rA is normalized with the reference time tr, de�ned by the user. Since the

ultrasound wave that contributes to the �eld at an observation point at the

observation time is originated at an earlier point in time at the transducer, it

is necessary to locate the exact point in time when the pressure wave leaves

the transducer surface. This depends on the distance to the observational

point from the transducer point as well as electronic focusing adjustments.

The observation time is then given by

T = tdelay − T ime− tS

where T ime is the reference time and tS is the steering and focusing delay,

if electronic focusing is used. The time delay Tdelay is the propagation delay

for the ultrasound wave from the transducer point to the observation point,

and is found by:

tdelay =((X(i)−Xt(n))

2 + (Y (i)− Yt(n))2 + (Z(i)− Zt(n))

2) 12 /c

where X, Y, Z are the coordinates of the observation point, Xt, Yt, Zt are the

coordinates of the transducer point and c is the velocity of propagation. Thevariables that are imported from UltraSim are in table 5.1 and the complete

algorithm is presented in �gure 5.1.

X, Y, Z : Coordinate vectors of observation points.

Xt, Yt, Zt : Coordinate vectors of transducer points

T : The time reference vector. Length = the number of

observation points, value = start time reference

SFD : The vector with Steering and Focusing Delays for

each transducer element

TSV : The vector with time references - of length TLen

O : The weighing of the transducer surface

jω : Angular frequency

Table 5.1: Variables used in Response Calculation.

This algorithm can also be used to simulate the time varying �eld. This is

observed as a movie and therefore called the movie option. The �eld is then

calculated seperately for a list of time references. In movie, the �eld can be

observed for dynamic focusing, i.e. electronic focus changing with time. In

that case, the steering and focusing delays for each element on each reference

time have to be calculated before �eld calculations.

The algorithm is further modi�ed for pulse-echo applications. For ultrasound

pulses, the beam exists only through a certain period and, at a single moment,

5.3 Visualization of the acoustic �eld 39

only a part or none of the transducer points, depending on the observation

time, contribute towards the �eld at a single observation point. Therefore, it

is necessary to locate the observational points that are a�ected by the current

transducer point. If the length of the pulse is tP then the points with time

delay tdelay which ful�ll the criterion below is calculated.

T ime+ tS − tP/2 < tdelay < Time+ tS + tP/2

When the above criterion is satis�ed, the observation point is a�ected at

that speci�c time by at least one transducer point. In addition to alleviating

errors caused by adding contributions when the pulse does not exist, this

method decreases the time consumption to calculate for non-existing waves.

Pulse-echo applications sometimes use a weighing function, usually a cosine

weight as the pulse envelope. The weight W is found by

W (rn) = cos2(πT/tP )

In the simulator, the response is normalized by dividing with the number of

transducer points.

5.3 Visualization of the acoustic �eld

The simulator is expanded by creating a program to visualize the volumetric

�eld. The intensity (power) of the �eld is depicted as variations in color on

the chosen planes or slices in X, Y, Z or X, Y, T volume. The visualizing tool

can carry out the following actions:

Slice view The intensity is presented as a color map on the X,Y and Z or

X,Y and T planes. The user can view upto 5 slice in each axis.

Range Movie The user can study how the �eld changes inside a chosen

volume along the z axis. The slices are visualized one after the other

along the z-axis. Slices on X and Y axes can be present on the plot.

Time Movie The user can study how the �eld evolves over a chosen time,

on the chosen slice positions. The images are visualized one after the

other in the time interval chosen by the user.

40 Simulation of Acoustic Fields

! Primary Loop - over time references

For PNo = 1 : TLen ! If movie, TLen > 1tr = T (1)If movie then

T ime = TSV (PNo) · 1length(X)else

T ime = Tend

min elem = 1max elem = #transducer points

If Dynamic focus and movie then

<Find SFD, min elem and max elem for this TSV (PNo).>end

! Secondary Loop - over transducer points

For n = min elm : max elemts = SFD(n)For i = 1 :#�eld points

tdelay =√(X(i)−Xt(n))2 + (Y (i)− Yt(n))2 + (Z(i)− Zt(n))2/c

If Pulsed wave then

tmax = tS + tP/2tmin = tS + tP/2If tmin ≤ tdelay − T ime(i) ≤ tmax then

t = tdelay − T ime(i)− tSW = cos2(πt/tP)rA = tdelay/trRe(H(i)) = +<O(n)W cos(jωt)/rA>Im(H(i)) = +<O(n)W sin(jωt)/rA>

end

else

t = tdelay − T ime(i)− tSrA = tdelay/trRe(H(i)) = +<O(n) cos(jωt)/rA>Im(H(i)) = +<O(n) sin(jωt)/rA>

end

end

end

end

Figure 5.1: Main Loops of the simulator to calculate the Impulse response.

Chapter 6

Scienti�c Parallel Processing

6.1 Introduction

A parallel computer is a set of processors that are able to work cooperatively

to solve a computational problem. Application of parallelism has helped the

scienti�c community in many ways. Large problems could be solved with

better approximations and less time consumption. Some problems that were

not feasible to be solved have been e�ectively solved using parallel computing

methods[29]. Also, parallel computing is a cost e�ective method to achieve

faster computing.

A simple example can explain the construction of a parallel algorithm, shown

in �gure 6.1. A user requires to solve the equation f(a, b, c, d) = ac+ bd. Se-quentially, the user �rst calculates ac, then calculates bd and �nally adds the

two values. In a parallel algorithm with two processors, the user calculates

ac and bd respectively in process 1 and 2 at the same time and then sums

the result.

Application of computer parallelism is not surprising when new trends in

applications, computer design and networking are considered. Application

of computers to solve mathematical problems are now much more than at

the beginning of the computer era. The need for faster computers is driven

by commercial and scienti�c applications that grow in data and complex-

ity, thereby demanding faster and better computers to match that growth.

Building chips that operate faster are not so feasible and not desirable. Fun-

damental results in VLSI1 design say that it is di�cult to build individual

1VLSI - Very Large Scale Integration

42 Scienti�c Parallel Processing

In Sequential Program:

BEGIN val1:= a*c; val2:= b*d;

f:= val1 + val2;END.

In Parallel Program

Proc 1 Proc2val1:= a*c; val2:=b*d;

f:=val1 + val2;

Figure 6.1: How a simple Parallel Program executes.

components that operate faster. It is therefore desirable to use slower but

more components.

Finally, the development in Networking is changing the face of computing.

New networking technology has provided higher bandwidth, higher reliability

and cheaper services[26]. These trends point to the fact that the potential

exists for the use of parallel computing, not only with super computers, but

also with Personal computers, and networks. Personal computers with multi-

ple processors are already on the market, and future software will be required

to exploit the processors inside the computer as well as processors available

over networks, which implies concurrency or the capability of algorithms and

program structures to perform many operations at once. It is also evident

that the number of processors available will increase. Therefore, a software

application should be able to adapt itself to the increasing number of pro-

cessors, as a requirement for its portability. The resilience to the increase in

processor count, known as scalability, will be an important factor.

6.2 Parallel machine models

The simplest computer model, known as a Von Neumann computer, consists

of a Central Processing Unit(CPU), that executes a stored program that spec-

i�es a sequence of read and write operations on the storage unit(memory).

The Von Neumann model, shown in �gure 6.2, is a simple but robust method

to explain the architecture and functions of a computer. This model is used

to understand the more abstract notions of a parallel machine.

6.2 Parallel machine models 43

CPU Memory

Figure 6.2: The Von Neumann Computer.

6.2.1 The multicomputer

A multicomputer consists of a number of von Neumann machines, or nodes,

connected by a network or a bus. Each computer or node will execute its own

program, access its own memory and receive/send data over the network as

shown in �gure 6.3. This allows each CPU to access local memory of other

CPU units. In an idealized network, the cost of sending a message between

two nodes is independent of both node location and other network tra�c, but

depends on message length. It is safe to assume that accessing local memory

is less expensive than accessing remote memory. So it is a desirable factor

in an parallel algorithm that accesses to local data is more frequent than

accesses to remote data. This property, known as locality, is a fundamental

requirement for parallel computing.

NETWORK

Figure 6.3: The multicomputer. Each node is a von Neumann computer.

A similar model, known as distributed-memoryMIMD (Multiple Instructions

Multiple Data) computer, is also commonly used. As in a multicomputer,

MIMD model's memory is distributed among the CPUs. The main di�erence

is that the cost of sending a message between two nodes need not be inde-

pendent of node location. Two models are given in Figure 6.4. The notation

MIMD means that each node or CPU has its own instruction set or program


accessing its own local memory. This is in contrast to SIMD (Single Instruc-

tion Multiple Data) which is explained later. MIMD machines include IBM

SP, Intel Paragon and Cray T3D[8].

(a) (b)

Figure 6.4: (a) Grid and (b) Ring distributed Memory MIMD Computers.

6.2.2 The multiprocessor

This type of computers is also known as shared-memoryMIMD computers. In

a multiprocessor computer, all processors or nodes share a common memory

area, accessed via a data bus or a hierarchy of buses, as in �gure 6.5. To

diminish the access times to the memory area, each processor may use a

small memory area, known as cache, to store most frequently used data from

the global memory. Therefore locality is maintained to some degree. Silicon

Graphics Challenge computers are of this type[8].

6.2.3 The SIMD model

SIMD, or Single Instruction Multiple Data computers, is perhaps the most

important type in scienti�c parallel computing[20]. Here, all processors ex-

ecute the same set of instructions or program on di�erent data items. Both

hardware and software complexity is reduced so it is speci�cally applied for

some problems. SIMD programs can be e�ectively run on MIMD systems

with a little modi�cation [8]. Since the problem presented in this thesis is of

scienti�c nature, emphasis will be made on SIMD methods hereafter.

6.3 Parallel programming models 45

P

CACHE

P

CACHE

P

CACHE

P

CACHE

B U S

GLOBAL MEMORY

Figure 6.5: A shared memory Multiprocessor.

6.3 Parallel programming models

6.3.1 Tasks and channels

A parallel algorithm introduces more complexity to a program. To maintain

the modularity of a program, it is necessary to introduce new abstractions to

the algorithm. A good model is the task-channel method. A task encapsu-

lates a sequential program, local memory and a set of inports and outports

to interface with its environment. A parallel computation consists of atleast

one task. Tasks are executed concurrently and the number of tasks may vary

during execution. In addition to accessing its local memory, a task performs

four basic actions - send messages, receive messages, create new tasks and

terminate. The inports and outports of tasks are connected by message paths

called channels. Channels are dynamic since they can be created and deleted

during task execution.

Tasks interact using channels regardless of task location and therefore the

computational results do not depend upon the node. This is known as map-

ping independence.

The task is a natural building block for a modular program. The tasks are

constructed seperately and combined to complete the program. Since inter-

actions take place via channels which are well de�ned interfaces, implementa-

tions can be changed without modifying other components. Therefore tasks

provide a modular and robust method to reduce complexity in a parallel pro-

gram. It is also desirable to write deterministic code for parallel algorithms.

Such code is easy to understand. A program is said to be deterministic if it

yields the same output everytime when the same input is given.


6.3.2 Message passing

Message passing is the most widely used parallel programming model. It

is very similar to the task/channel model, except that messages are sent to

the task instead of the channel. Even though message-passing model can be

used to create new tasks or execute more than one task per processor, in

practice it is used with a �xed number of identical tasks created at program

startup. This is known as the SPMD (Single Program Multiple Data) model,

the software equivalent to SIMD.

Further emphasis is only made on Message Passing model since this thesis

requires only message passing techniques.

6.4 The SPMD model

In scienti�c �elds like astronomy, meteorology and signal processing, large

data sets, in vector or matrix form, have to be handled. Using SPMD (Single

Program Multiple Data) methods for such data sets are very useful since

partitioned vector or matrix operations can be carried out using the same

program on di�erent nodes while retaining regularity. The rest of this chapter

will look at the criteria involved in a SPMD model.

6.5 Data distribution

Assume a data set with M items, and P available nodes or processes. This

set has to be partitioned over each process. Assume that the process p,0 ≤ p < P , has Mp items from the data set. Each process is given an index

variable i for its local data array. The range of indices over which the arrays

are declared varies from process to process. In process p, the array indices areelements of the index set Ip, where Ip usually is an index range 0 . . . Ip − 1.

Technically, data distribution is to partition the data set to the P nodes and

it is accomplished by a bijective mapping of the global index m to a pair of

indices (p, i), where p is the process identi�er and i is the local index.

De�nition 1 A P-fold distribution of an index set M is a bijective map

µ(m) = (p, i) such that:

µ ::M→ {(p, i) : 0 ≤ p < P and i ∈ Ip} : m→ (p, i)

6.5 Data distribution 47

and

µ−1 :: {(p, i) : 0 ≤ p < P and i ∈ Ip} →M : (p, i)→ m

Here,M = 0 . . . M−1 is the set of global indices. For a given p ∈ 0 . . . P−1,the subsetMp is the set of global indices mapped to process p by the P -folddata distribution µ. Therefore

Mp =⋃i∈Ip

{µ−1(p, i)}

Because the P subsetsMp with 0 ≤ p < P satisfy

M =P−1⋃p=0

and M⋂Mq = ∅ if p �= q,

they de�ne a partition ofM. Because µ is a bijective map, the number of

elements ofMp equals the number of elements of Ip.

Practically, the distribution of the data set should be done so that the load per

process is more or less the same. There are two main methods to distribute

data: scatter distribution and linear distribution.

6.5.1 Scatter distribution

The scatter distribution allocates consecutive vector components to consec-

utive processes. The distribute the index set M over P processes, µ is set

as

µ(m) = (p, i), where

{p = m mod P

i =⌊mP

⌋ (6.1)

It follows then:

Mp = {m : 0 ≤ m < M and m mod P = p}

Ip = �M + P − p− 1

P�

Ip = {i : 0 ≤ i < Ip}

µ−1(p, i) = iP + p


6.5.2 Linear distribution

The linear distribution allocates consecutive vector components to consecu-

tive local-array entries. When M is an exact multiple of P , i.e. M = PL:


{p = �m

L�

i = m− PL(6.2)

This is called a perfectly load balanced linear data distribution, and it satis�es:

Mp = {m : pL ≤ m < (p+ 1)L}

Ip = L

Ip = {i : 0 ≤ i < L}

µ−1(p, i) = pL + i

But since M cannot always expected to be an exact multiple of P , it is

necessary to apply general load balanced linear distribution. Given M =PL+R, where R is the residual, 0 ≤ R < P :

L =⌊M

P

⌋R = M mod P


{p = max(� m

L+1�, �m−R

L�)

i = m− pL −min(p,R)

This maps L+1 components to processes 0 through R−1, and L components

to the rest of the processes. So, we get other distribution quantities as:

Mp = {m : pL +min(p,R) ≤ m < (p + 1)L+min(p+ 1, R)}

Ip =⌊M + P − p− 1

P

⌋Ip = {i : 0 ≤ i < Ip}

µ−1(p, i) = pL +min(p,R) + i

Table 6.1 shows how a vector of length M = 11 is distributed over P = 4processes.

6.6 Performance analysis 49

There is also another, rarely used, method of distribution where the compo-

nents are mapped by means of a random number generator. This method,

called random distribution method, is helpful in cases where balancing has

to be done statistically. This method is rarely used and is considered only in

very special cases.

Process Local i Linear m Scatter m

0 0 0 0

1 1 4

2 2 8

1 0 3 1

1 4 5

2 5 9

2 0 6 2

1 7 6

2 8 10

3 0 9 3

1 10 7

Table 6.1: Linear and Scatter Distribution for M = 11 and P = 4.

6.6 Performance analysis

Performance Analysis of a parallel program is the study of the execution

times of computations2. Achieving good performance is one of the most

important principles of a parallel computation. However search for good

performance has its limitations. A program is the end result of many criteria

besides performance, such as stability, robustness, range and portability. To

achieve good performance some criteria have to sacri�ced and that should be

decided case by case. Therefore generalized performance analysis is di�cult.

However, study of some parameters can give the user some understanding

of the performance for a particular situation. A space of the computation

is simpli�ed so that only one paramater de�nes the computation. This pa-

rameter is called the study parameter. The other parameters are kept �xed

for simplicity. Failing that, they are separated as it �xed parameters and it

dependent parameters.

2A computation is a program with particular input executed on a given computer at a

certain moment in time.


6.6.1 Speed-up

Since a parallel computer di�ers from a sequential computer by the number

of nodes (i.e. only one node in a sequential computer), the number of nodes,

i.e. p, is taken as the study parameter. It can be assumed that the other

parameters are �xed. Let Tp be the execution time on node p and T ∗p be the

best p-node time. When p = 1, the computer is sequential and T ∗1 is the bestsequential execution time.For these parameters, the following theorem holds:

Theorem 1 On a given multicomputer and for a �xed problem, the best p-

node computation is at most p times faster than the best sequential execution

time, or

T ∗p ≥ T ∗1 /p

The proof is done by contradiction and can be found in [29]. The following

de�nitions can now be considered:

De�nition 2 The speed-up of a p-node computation with execution time Tpis given by:

Sp = T ∗1 /Tp (6.3)

where T ∗1 is the sequential time obtained for the same problem on the same

multicomputer.

De�nition 3 The e�ciency of a p-node computation with speed-up Sp is

given by:

ηp = Sp/p

A good, but not exact, understanding of the performance can be found by

evaluating these parameters.

6.6.2 Performance modelling

For theoritical anaysis of performance, a model computer is assumed. In a

sequential computer, the execution time is considered proportional to the

number of �ops in the computation. If the time used to complete a �op is

τA, the execution time is nτa, where n is the number of �ops. In a concur-

rent computer, the communication should also be considered. The message

exchange time, denoted by τC, is the time it takes to send the message by

6.7 Parallel routine libraries 51

one process and receive the message by another, while another is exchanged

in reverse direction. Message exchange time follows a simple linear equation:

τC(L) = τS + βL (6.4)

where τS is the start-up time or latency, L is the length of the message and

β is the inverse of the network bandwidth. For long messages, TC(L) can be

approximated by LTC(1) where TC(1) is the time taken to send a message of

length 1.

These parameters and de�nitions are adequate to model a very simpli�ed

performance analysis. The execution time for a parallel program on P nodes,

T (P ) is therefore given by

T (P ) = τc(L) + nτA (6.5)

There are some commercial packages that are used to measure performance

analysis quite accurately. Many are used to study the performance via pro-

�les, counters and traces, user-embedded in source code. Commonly used

tools include Paragrhaph, Upshot, Pablo, Gauge and AIMS.

6.7 Parallel routine libraries

In message-passing library approach to parallel programming, a collection

of processes executes programs written in a a standard sequential language

augmented with calls to a library of functions for sending and receiving mes-

sages and data. There are many such libraries available, for example, MPI,

PVM, p4, Express, PARMACS. A parallel program uses atleast one of these

libraries to manage the communication among tasks.

• MPI

MPI, or Message Passing Interface, is a commonly used library which

provides a standard for message passing. MPI programming model

comprises of one or processes that communicate using MPI library rou-

tines to send and to receive messages and data to and from other pro-

cesses. Commonly these processes are created at the intialization of the

program, and one process is created per processor. It is not necessary

that all created processes execute the same program. Therefor MPI

model is referred to as multiple program multiple data (MPMD) but

MPI can be used as a single program multiple data (SPMD) programs

too. MPI can be used in Multiprocessor computers.


• PVM

PVM, Parallel Virtual Machine, is a software system that permits a

network of heterogeneous Unix computers to be used as a single large

parallel computer. Under PVM, a user de�ned collection of serial, par-

allel and vector computers appears as one large distributed-memory

computer which is the reason for PVM being known as the poor man's

parallel computer. The computer which acts as the host from which

all tasks are spawned can be any of computer in the network. PVM

supports heterogeneity at the application, machine, and network level.

In other words, PVM allows application tasks to exploit the architec-

ture best suited to their solution. PVM handles all data conversion

that may be required if two computers use di�erent integer or �oat-

ing point representations. And PVM allows the virtual machine to be

interconnected by a variety of di�erent networks.

The PVM system is composed of two parts. The �rst is a daemon that

resides on all the computers making up the virtual machine. Before the

user executes the parallel programs, the daemon has to be activated

from the Unix prompt. The second part is a library of functions which,

among others, carries out the spawning of tasks and sends and receives

messages. This library is linked with the users programs. PVM can

also be used for Multiprocessor computers.

Chapter 7

Parallel Response Calculation

7.1 Introduction

In the previous chapter, the basics of scienti�c parallel programming was

discussed. Now a parallel algorithm for the response calculation problem is

presented.

7.2 MATLAB and MEX-programming

MATLAB uses a fourth generation language for programming. As expected

from a 4th generation language, MATLAB code require heavy doses of CPU

time, heavily increasing the computational time. However, MATLAB pro-

vides its users with an external programming tool which can be used activate

routines written in C, C++ and Fortran. These dynamically linked subrou-

tines are modelled via the so-called MEX programming techniques and are

compiled via a tool provided with MATLAB, which is given an extension in

accordance with the architecture of the host computer1.

Even though there are many de�ned data types in its interactive environment,

MATLAB works with only one object type - the Matrix. The variables in the

object, some of which are given in Table 7.1. MATLAB stores matrices in

columnwise order in a vector. That is, the vector array pr contains the valuesof the �rst column and then the second column and so on. A MEX program

uses and accesses this object type via a Gateway routine. By using this

1E.g. <test.c> complied gives <test.mex4> for Sun4/SPARC, <test.mexsg> for SGI

54 Parallel Response Calculation

storage method, data distribution when parallel processing can be carried

out with ease.

Name : Name of the variable

M : The number of rows in the Matrix

N : The number of columns in the Matrix

DisplayMode : Indicates whether the values are numeric or ASCII

Storage : Indicates whether the Matrix is full or sparse

pr : Pointer to an array of length NM containing the real

part of the Matrix

pi : As above containing the complex part of the Matrix

Table 7.1: A selected list of variables in object type MATRIX.

AMEX Program has two main parts: The Gateway Routine and the Compu-

tation Routine. The Gateway routine starts with the following parameters:

void mexFunction(

int nlhs, Matrix *plhs[ ]

int nrhs, Matrix *prhs[ ])

The variables nlhs and nrhs are integer variables containing the number ofleft hand side and right hand side variables respectively in the MATLAB

routine. The pointers plhs and prhs point to two arrays of MATLAB matri-

ces on the left hand side and right hand side respectively of the MATLAB

routine.

The computation routine contains the mathematical calculations that use

the variables in the Matrix objects.

7.3 Data distribution of observational segements

As mentioned earlier, MATLAB uses columnwise vectorization to breakdown

a matrix. For a planar observation, the X, Y, Z and T coordinates are placed

in four matrices where each point of observation is placed in the same in-

dex in all four. The pr variable of Matrix-object X contains the columnwise

vectorized values. This vector is extracted in MEX Gateway routine and dis-

tributed using general load balanced linear distribution in the Computation

Routine and sent to the invoked tasks, as shown in �gure 7.1.

7.4 Parallel routines 55

1 2 3 4 5

1

2

3 4 5

1

2

3

4

5

Process 0

Process 1

Process 2

In MATLAB In MEX calculation

inMex

Gateway

Figure 7.1: How a 6× 5 Matrix is distributed to 3 processes.

When simulating Volumetric observations, it is necessary to keep track of

many planes. Instead of 3D matrices, MATLABmappes layers of 2D matrices

above each other to create a long 2D matrix. How a 3D observational data

matrix is converted is shown in �gure 7.2.

7.4 Parallel routines

Since parallel Routines have to be invoked from MATLAB environment,

PVM was chosen as the parallel library. The Master parallel program, with

MEX extensions, is invoked fromMATLAB environment. This program then

invokes the slave programs in a user-speci�ed number of nodes and distributes

data to them. Final computed values are then sent back to Master Program

which places the results in MATLAB. The complete programs can be found

in Appendix C.

For time varying simulations, only the data that changes from one reference

time to another is sent to the nodes. The other data, for example observation

point coordinates and transducer point coordinates, are used from the �rst

time reference.

The Figures 7.3 and 7.4 show the master and slave routines respectively.


x

y

z

flip

Figure 7.2: Conversion of volumetric data to matrix form.

To keep the indices simple, no local index is speci�ed. The start index and

the end index for each node is sent to each node, and these two indices

are received back to the master routine. This way, the master program can

update the �nal rsponse vector as soon as data arrives from any node.

7.5 Performance analysis

The performance is analysed here for a simple simulation model. First, the

pressure wave is taken as continuous, so that the execution does not include

the pulse wave calculations and includes all transducer points. Further, a

perfect load balance is assumed so that the number of observation points

in each node is equal. Finally, simulation is assumed to be for one time

reference.

Assume there are N transducer points and MP observation points per node.

From �gure 7.4 it can be seen that there are 11 �ops in the loop If <Pulse

wave> then ... else ... end and 8 �ops in the tdelay calculation. From all

MP points, the execution time for the loop For i ∈ Ip .....end is therefore

19MpτA. Since there are N transducer points, the execution time for the

loop For n = min elem : max elem is 19NMP τA. From the equation 6.5 in

page 51, the execution time can be approximated as

T (P ) ≈ τC(L) + 19NMP τA

where τC(L) is the time taken to send a message of length L. The main

program, in appendix C and simpli�ed in �gure 7.3, sends altogether P


Program RESPONSE

declare

! Variables imported from ULTRASIM. See �gure 5.1 in page 37

! This pseudo code shows the method of data distribution and communication

! lstart, lend - the start and end indices of the distributed data vector

! ElPt - number of horizontal and vertical element points

initially

!Enrolling in PVM

my_tid = PVM_tid ! Get Task ID for MASTER

broadcast ppec_slave to 0...P − 1‖passign

For PNo = 1 : TLen ! If movie, TLen > 1tr = T (1)If <movie> then

T ime = TSV (PNo) · 1length(X)else

T ime = Tend

min elem = 1max elem =#transducer elements

If <Dynamic focus and movie> then

<Find SFD,min elem,max elem for this PNo>

end

For i ∈ Ip(Xp, Yp, Zp, T imep)[i] = (X, Y, Z, T ime)µ−1(p,i)

end

For p = 0 : P − 1If PNo == 1 then

send lstart, lend, ElPt, PNo,min elem,max elem, SFD, T imep,Xp, Yp, Zp, Xt, Yt, Zt, c, tref , O, ....... to p

else

send lstart, lend, ElPt, PNo,min elem,max elem, SFD, T imep to p

end

For p ≤ Preceive Re(Hp), Im(Hp), lstart, lend from p

end

[Re(H), Im(H)] = [Re(Hp), Im(Hp)]µ(m)end

Figure 7.3: Master routine


0...P − 1‖p program ppec_slave

declare

!Variables received from MASTER

initially

! Enroll in PVM

my tid = PVM tid ! Get Task ID for this slave

ptid = PVM parent ! Get Task ID for master

receive ............, Xp, Yp, Zp, T imep, Xt, Yt, Zt, c, tref , O, SFD from Master

assign

For n = min elem : max elem

ts = SFD(n)For i ∈ Ip

tdelay =√(Xp(i)−Xt(n))2 + (Yp(i)−Yt(n))2 + (Zp(i)− Zt(n))2/c

if <Pulse wave> then

tmax = tS + tP/2tmin = tS + tP/2If tmin ≤ tdelay − T imep(i) ≤ tmax then

t = tdelay − T imep(i)− tSW = cos2(πt/tP)rA = tdelay/trRe(Hp(i)) = +<O(n)W cos(jωt)/rA>Im(Hp(i)) = +<O(n)W sin(jωt)/rA>

end

else

t = tdelay − T imep(i)− tSrA = tdelay/trRe(Hp(i)) = +<O(n) cos(jωt)/rA>Im(Hp(i)) = +<O(n) sin(jωt)/rA>

end

end

end

send Re(Hp), Im(Hp), lstart, lend to MAIN;

Figure 7.4: Slave routine


messages of length (13+3N+4MP ). After the calculations, the main program

receives P messages of length 2Mp + 2. If a negligible latency is assumed,

T (P ) and the special case T (1) can be found

T (P ) ≈ τC((15 + 3N + 6Mp)P ) + 19NMpτA

= P (15 + 3N + 6Mp)τC(1) + 19NMpτA

T (1) ≈ 19NMτA

Speed-up, given by SP = T (1)/T (P ) becomes:

SP =19NMτA

P (15 + 3N + 6Mp)τC(1) + 19NMpτA

SP = P1

15P+3NP+6MPP19NM

τC(1)τA+ MP

M

Since MP =M/P ,

SP =1

15P+3NP+6M19NM

τC(1)τA+ 1

P

=P

P (15P+3NP+6M)19NM

τC(1)τA+ 1

(7.1)

When NM >> P , Sp −→ P . This is usually the case since large sets of

observational points are usually used.

Chapter 8

Simulation Results

8.1 The Octagonal Transducer

The aperture smoothing function found in chapter 4 is visualized here. Fig-

ures 8.1(a) and 8.1(b) show the normalized1 aperture smoothing functions for

a = b = 1λ and a = b = 3λ respectively, over {kx, ky} in the range [−5 : 5].Figure 8.2(a) and 8.2(b) show the normalized aperture smoothing functions

for a = 2λ, b = 1λ and a = 5λ, b = 1λ for the same range.

These �gures illustrate the manner in which the lengths of a and b a�ect thethe aperture function. As the lengths increase, the visible range increases.

The number of zeros and sidelobes also increase and this in turn results in a

narrower main lobe.

To further illustrate the qualities of the octagonal aperture, the directivity

pattern can be considered. Figure 8.3 and 8.4 shows the pattern for θ =0, 30, 60, 90◦ for lengths a = b = 1λ and a = 5λ, b = 1λ respectively.

These �gures suggest that this aperture produces a rectangular aperture

type beam, broad in the θ = 90◦ plane and narrow in the θ = 0◦ plane.Compared to an elliptical aperture of the same dimensions however, the

main lobe is narrower[30]. Another interesting development can be noticed

for the symmetric aperture (i.e. a = b = 1λ) in �gure 8.3. The mainlobe

width for all angles are approximately equal, not unlike a circular aperture.

Figure 8.5 shows the patterns of octagonal and rectangular apertures, with

symmetric dimensions, for angles θ = 0, 30, 45, 60◦ plotted together. The

sidelobe levels of the octagonal aperture is less and mainlobe width is wider

1Normaization is done with respect to the aperture area

62 Simulation Results

−5

0

5

−5

0

5−0.2

0

0.2

0.4

0.6

0.8

1

kxky

aper

ture

sm

ooth

ing

func

tion

(a)

−5

0

5

−5

0

5−0.2

0

0.2

0.4

0.6

0.8

1

kxky

aper

ture

sm

ooth

ing

func

tion

(b)

Figure 8.1: Aperture smoothing function of the octagonal transducer with (a)

a = b = 1λ and (b) a = b = 3λ

than for a rectangular apeture for θ = 0. Figure 8.6 looks at an octagonal

aperture with a = 5λ, b = 1λ compared with a rectangular aperture with

the similar dimensions. In this example, the levels of the �rst sidelobes for

angles θ = 0, 30, 60, 90◦ are less for the octagonal aperture.

8.2 Parallel programming results 63

−5

0

5

−5

0

5−0.2

0

0.2

0.4

0.6

0.8

1

kxky

aper

ture

sm

ooth

ing

func

tion

(a)

−5

0

5

−5

0

5−0.2

0

0.2

0.4

0.6

0.8

1

kxky

aper

ture

sm

ooth

ing

func

tion

(b)

Figure 8.2: Aperture smoothing function of the octagonal transducer with (a)

a = 2λ, b = 1λ and (b) a = 5λ, b = 1λ

8.2 Parallel programming results

In this section, the results from executing an example con�guration on the

parallel system at USIT will be discussed. Since parallel algorithm presented


−80 −60 −40 −20 0 20 40 60 80−100

−80

−60

−40

−20

0

20

Theta = 0

Angle in degrees

Rel

ativ

e po

wer

in d

B

−80 −60 −40 −20 0 20 40 60 80−100

−80

−60

−40

−20

0

20

Theta = 30

Angle in degrees

Rel

ativ

e po

wer

in d

B

−80 −60 −40 −20 0 20 40 60 80−100

−80

−60

−40

−20

0

20

Theta = 60

Angle in degrees

Rel

ativ

e po

wer

in d

B

−80 −60 −40 −20 0 20 40 60 80−100

−80

−60

−40

−20

0

20

Theta = 90

Angle in degrees

Rel

ativ

e po

wer

in d

B

Figure 8.3: Directivity pattern of an octagonal transducer with a = b = 1λfor angles θ = 0, 30, 60, 90

−80 −60 −40 −20 0 20 40 60 80−100

−80

−60

−40

−20

0

20

Theta = 0

Angle in degrees

Rel

ativ

e po

wer

in d

B

−80 −60 −40 −20 0 20 40 60 80−100

−80

−60

−40

−20

0

20

Theta = 30

Angle in degrees

Rel

ativ

e po

wer

in d

B

−80 −60 −40 −20 0 20 40 60 80−100

−80

−60

−40

−20

0

20

Theta = 60

Angle in degrees

Rel

ativ

e po

wer

in d

B

−80 −60 −40 −20 0 20 40 60 80−100

−80

−60

−40

−20

0

20

Theta = 90

Angle in degrees

Rel

ativ

e po

wer

in d

B

Figure 8.4: Directivity pattern of an octagonal transducer with a = 5λ, b = 1λfor angles θ = 0, 30, 60, 90

8.2 Parallel programming results 65

−80 −60 −40 −20 0 20 40 60 80−100

−80

−60

−40

−20

0

20

Theta = 0

Angle in degrees

Rel

ativ

e po

wer

in d

B

Octagonal −Theta = 0

Rectangular ..

−80 −60 −40 −20 0 20 40 60 80−100

−80

−60

−40

−20

0

20

Theta = 30

Angle in degrees

Rel

ativ

e po

wer

in d

B


Rectangular ..

−80 −60 −40 −20 0 20 40 60 80−100

−80

−60

−40

−20

0

20

Theta = 45

Angle in degrees

Rel

ativ

e po

wer

in d

B

Octagonal −

Rectangular ..

Theta = 45

−80 −60 −40 −20 0 20 40 60 80−100

−80

−60

−40

−20

0

20

Theta = 60

Angle in degrees

Rel

ativ

e po

wer

in d

B

Rectangular ..


Figure 8.5: Directivity patterns of octagonal and rectangular apertures with

equal width and heights compared.

−80 −60 −40 −20 0 20 40 60 80−100

−80

−60

−40

−20

0

20

Theta = 0

Angle in degrees

Rel

ativ

e po

wer

in d

B

Octagonal −

Rectangular ..

−80 −60 −40 −20 0 20 40 60 80−100

−80

−60

−40

−20

0

20

Theta = 30

Angle in degrees

Rel

ativ

e po

wer

in d

B

Octagonal −

Rectangular ..

−80 −60 −40 −20 0 20 40 60 80−100

−80

−60

−40

−20

0

20

Theta = 60

Angle in degrees

Rel

ativ

e po

wer

in d

B

Octagonal −

Rectangular ..

−80 −60 −40 −20 0 20 40 60 80−100

−80

−60

−40

−20

0

20

Theta = 90

Angle in degrees

Rel

ativ

e po

wer

in d

B

Octagonal −

Rectangular ..

Figure 8.6: Directivity pattern of octagonal aperture with a = 5λ, b = 1λcompared with a rectangular aperture with the same dimensions.


gives better performance for large sets of observational data as explained in

section 7.5, such a large observation volume is used.

The aperture is rectangular is shape, 10mm in azimuth direction (X axis)

and 6mm in elevation direction (Y axis). There are 48 elements in azimuth

direction and 24 points in elevation direction, which combines to 1152 points.

There is no curvature in the aperture and electronic focusing is set to 20mm

in normal axis. The continuous beam is assumed and the speed of sound is

taken as 1540 m/s.

The observation area is the volume [−10 : 10,−10 : 10, 10 : 22] with 4

�eld points per mm. This volume therefore contains 321,489 points. This

con�guration is then executed on the parallel computer network with varying

number of nodes. Table 8.1 presents the run-time results.

Nodes: 1 2 5 10 20 30

1st 1605.53 1289.40 545.43 195.46 114.97 80.63

2nd 2154.83 900.23 370.95 232.59 103.01 78.28

3rd 1802.25 843.06 382.63 205.82 102.10 83.77

4th 1981.00 938.32 556.96 253.58 124.61 82.69

5th 2038.43 832.96 386.46 232.02 103.02 84.62

6th 2059.42 840.96 344.31 199.27 111.43 85.76

7th 1861.31 911.35 380.66 245.91 113.24 80.62

8th 1593.46 1003.22 590.12 190.10 102.04 77.79

9th 2045.46 906.21 369.57 290.26 123.05 77.65

10th 1712.89 906.59 653.64 179.15 103.83 82.21

Average 1885.46 937.23 458.07 222.42 110.13 81.40

Best 1593.46 832.96 344.31 179.15 102.04 77.65

Sp 1.00 1.91 4.63 8.89 15.61 20.52

Table 8.1: Time Consumption for parallel processing. Bold face shows the

best time achieved.

The single node calculations were done using the sequential program so that

no communication was necessary. However the single node simulations were

done on the interactive node which is heavily loaded. Simulations carried

out as batch jobs will give better results both for parallel and sequential

programs.

The best value for each number of nodes are used to study the performance

parameters.From values of Sp, it is evident that the algorithm provides ad-

equate speed-up. Deviation from the expected SP , which is equal to the

8.3 3D plot results 67

number of nodes, as explained in equation 6.3. The averages can be used

to see relative time consumption for calculations. At the time of execution,

PVM was con�gured to the 32 available nodes. There were 15 users logged

in. There were 12-16 jobs running on 18 servers during the time execution.

8.3 3D plot results

The simulations were carried out using rectangular, elliptical, super-elliptical

and octagonal arrays. All arrays consist of 48 elements in X direction and 24

elements in Y direction. Such large numbers were used so that the transduc-

ers will approximate continuous apertures. A continuous ultrasound wave

with frequency 2 MHz was used, while the aperture was electronically fo-

cused to 20mm from the transducer surface. The observation �eld was set to

the [-10:10,-10:10,1:22] volume segement and each millimeter was discretized

to two portions.

8.3.1 Rectangular aperture

Figures 8.7 and 8.8 show the �eld created by a rectangular aperture with

dimensions 10mm length and 6mm wideth. Figure 8.7 looks at �elds in z-

planes 1mm and 20mm from the surface, the latter being the focal point.

Figure 8.8 show x-plane and y-plane views of the �eld. The mainlobe and

the sidelobes, as shown in �gure 3.8 in page 21 and how the beam narrows

in the transition �eld can be noticed.

8.3.2 Elliptical aperture

Figures 8.9 and 8.10 show the �eld created by an elliptical aperture, with

10mm long axis and 6mm short axis. The plots presented correspond to the

slice positions used in the rectangular aperture.

8.3.3 Super-elliptical aperture

The super-elliptical2 aperture with 10mm long axis and 6mm short axis is

tested here. Figures 8.11 and 8.12 show the slice positions as used in the

previous apertures.

2A super ellipse has the equation(xa

)2.5+(yb

)2.5= 1.


ARRAY−RESPONSE Reference=12.99 [us] 6−AUG−1997 16:08

Theta=0 [deg] Phi=0 [deg] N=48 M=1 f=2 [MHz] pitch=0.2706 osc=Inf

Azimuth : no apodization Elevation : no apodization View: 3D default

05

1015

2025

−10

−5

0

5

10−10

−5

0

5

10

Range in [mm], Azimuth focus=20 [mmAperture (AZ) d=10 [mm]

Ran

ge in

[mm

], E

leva

tion

focu

s=0

[mm

]

(a)ARRAY−RESPONSE Reference=12.99 [us] 6−AUG−1997 16:11



05

1015

2025

−10

−5

0

5

10−10

−5

0

5

10


Ran

ge in

[mm

], E

leva

tion

focu

s=0

[mm

]

(b)

Figure 8.7: Slices on the z-axis, (a)near the sufrace and (b)at the focus of the

rectangular aperture.





05

1015

2025

−10

−5

0

5

10−10

−5

0

5

10


Ran

ge in

[mm

], E

leva

tion

focu

s=0

[mm

]




05

1015

2025

−10

−5

0

5

10−10

−5

0

5

10


Ran

ge in

[mm

], E

leva

tion

focu

s=0

[mm

]

(b)

Figure 8.8: Slices at 0 on (a) the x-axis and (b) the y-axis for the rectangular

aperture.





05

1015

2025

−10

−5

0

5

10−10

−5

0

5

10


Ran

ge in

[mm

], E

leva

tion

focu

s=0

[mm

]




05

1015

2025

−10

−5

0

5

10−10

−5

0

5

10


Ran

ge in

[mm

], E

leva

tion

focu

s=0

[mm

]

(b)

Figure 8.9: Slices on the z-axis, (a) near the sufrace and (b) at the focus of

the elliptical aperture.





05

1015

2025

−10

−5

0

5

10−10

−5

0

5

10


Ran

ge in

[mm

], E

leva

tion

focu

s=0

[mm

]




05

1015

2025

−10

−5

0

5

10−10

−5

0

5

10


Ran

ge in

[mm

], E

leva

tion

focu

s=0

[mm

]

(b)

Figure 8.10: Slices at 0 on (a) the x-axis and (b) the y-axis for the elliptical

aperture.





05

1015

2025

−10

−5

0

5

10−10

−5

0

5

10


Ran

ge in

[mm

], E

leva

tion

focu

s=0

[mm

]




05

1015

2025

−10

−5

0

5

10−10

−5

0

5

10


Ran

ge in

[mm

], E

leva

tion

focu

s=0

[mm

]

(b)

Figure 8.11: Slices on the z-axis, (a) near the sufrace and (b) at the focus of

the super-elliptical aperture.





05

1015

2025

−10

−5

0

5

10−10

−5

0

5

10


Ran

ge in

[mm

], E

leva

tion

focu

s=0

[mm

]




05

1015

2025

−10

−5

0

5

10−10

−5

0

5

10


Ran

ge in

[mm

], E

leva

tion

focu

s=0

[mm

]

(b)

Figure 8.12: Slices at 0 on (a) the x-axis and (b) the y-axis for the super-

elliptical aperture.


8.3.4 Octagonal aperture

The octagonal aperture used has the same width and length as in the other

tested apertures. The slices in �gures 8.13 and 8.14 are the same as in

previous plots.




05

1015

2025

−10

−5

0

5

10−10

−5

0

5

10


Ran

ge in

[mm

], E

leva

tion

focu

s=0

[mm

]




05

1015

2025

−10

−5

0

5

10−10

−5

0

5

10


Ran

ge in

[mm

], E

leva

tion

focu

s=0

[mm

]

(b)

Figure 8.13: Slices at 0 on (a) the x-axis and (b) the y-axis for the octagonal

aperture.





05

1015

2025

−10

−5

0

5

10−10

−5

0

5

10


Ran

ge in

[mm

], E

leva

tion

focu

s=0

[mm

]




05

1015

2025

−10

−5

0

5

10−10

−5

0

5

10


Ran

ge in

[mm

], E

leva

tion

focu

s=0

[mm

]

(b)

Figure 8.14: Slices at 0 on (a) the x-axis and (b) the y-axis for the elliptical

aperture.


8.3.5 Range movie and time movie

The range movie is applied when the user needs to study the �eld along the

z-axis. Some examples can be found in web site [3].

The time movie is used to study how the �eld develops with time in any

slice in any axis. This is useful, for instance, for pulse wave simulations since

the user can study the �eld propagation with time. Figure 8.15 shows the

development of such a pulsed wave. The aperture is the rectangular aper-

ture used earlier. Number of pulses are set to 6 with cosine pulse weighing.

Observation volume is [-10:10,-10:10,5:22]. Plots are provided for 4 equal

placed times between the interval when pulse reaches 5mm and 20mm. This

example can be found in WWW site [3].


Theta=0 [deg] Phi=0 [deg] N=48 M=1 f=2 [MHz] pitch=0.2706 osc=6


510

1520

25

−10

−5

0

5

10−10

−5

0

5

10


Ran

ge in

[mm

], E

leva

tion

focu

s=0

[mm

]




510

1520

25

−10

−5

0

5

10−10

−5

0

5

10


Ran

ge in

[mm

], E

leva

tion

focu

s=0

[mm

]




510

1520

25

−10

−5

0

5

10−10

−5

0

5

10


Ran

ge in

[mm

], E

leva

tion

focu

s=0

[mm

]




510

1520

25

−10

−5

0

5

10−10

−5

0

5

10


Ran

ge in

[mm

], E

leva

tion

focu

s=0

[mm

]

Figure 8.15: Snap shots of a pulse wave propagating throught slices x = 0, y =0 and z = 20.

Chapter 9

Conclusion and Summary

9.1 The octagonal transducer

In the previous chapter, the octagonal transducer was extensively tested

using the somewhat complicated equation found for its aperture function.

However, the results in the previous chapter indicate that this footprint does

indeed show a slight reduction in the �rst sidelobe level compared with an

equal sized rectangular aperture. This could also be an alternative model for

circular and elliptical apertures, as it was seen with the plots of the aperture

smoothing function in �gures 8.1 and 8.2.

9.2 The parallel algorithm

Extensive computational results of the parallel algorithm were given in the

previous chapter. Run-time results in table 8.1 indicate that that the algo-

rithm provides good speed up. However, as the number of nodes increases,

speed up value decreases from the perfect speed up - equal to the number of

nodes.The e�ciency, expressed as a percentage, in table 9.1, also shows the

deviation from perfect values.

nodes: 1 2 5 10 20 30

ηp 100% 95.5% 92.2% 88.9 % 78.05% 68.04%

Table 9.1: E�cieny ηp for the parallel algorithm.

78 Conclusion and Summary

This is a natural development since the speed-up parameters in section 6.6.1

are for a model computer. In reality, tra�c congestion, process sharing and

many more factors a�ect the perfect communication and processing model.

In addition, the node in which the master program was executed, was the

interactive node for the super computing network. This tend to increase the

computer time since its processing power is shared by many logged-in users.

A computation done on another node showed some dramatic improvement

in execution times.

The deterioration of speed up and e�ciency with the increase of the number

of nodes can be explained by looking at the speed up equation 7.1 in page 59.

Speed up, Sp −→ p when NM >> P . With increasing nodes, data points

per node decreases. For larger sets of observation points, better speed up

and e�ciency can be expected.

To decrease the time consumption further, two methods can be suggested.

First, as explained above, a node other than the interactive node can be

used to run ULTRASIM command window. Second, the computations can

be carried out as a batch job. Information can be found in the user guide in

appendix A. These two methods will work for the super computing network

in USIT.

9.3 Software and hardware

The software package UltraSim was run on MATLAB version 4.2c. PVM

version 3 was used as the parallel routine library. CMEX version 3.5 was

used to compile the sequential C program and the parallel master program for

MATLAB. This document was created using LATEXversion 2ε. The researcharticle in [4] was created using Framemaker 4 and simulation videos were

created with visualization tools in USIT Scienti�c Visualization Lab [27].

Testing of the sequential programs were carried out on SUN computers run-

ning SunOS 5.5.1 and SGI computers running IRIX 5.3. Testing of the

parallel routines were made on the Super Computing network in USIT. All

computers are IBM RS6000 super computers with DEC and HP tape and

disk stations. More information of the super computing network can be found

in WWW site [28].

Bibliography

[1] Angelsen B.A.J., �Waves, Signals and Signal Processing in Medical Ul-

trasonics�, Norwegian University of Science and Technology, Norway,

April 1996.

[2] Berg R.E.,Stork D.G., �The Physics of Sound�, Prentice Hall, 2nd edi-

tion,1995,ISBN 0-13-183047-3.

[3] Epasinghe K., �Project Thesis of Kapila Epasinghe�, WWW:

http://www.i�.uio.no/� kapilae/thesis , August 1997.

[4] Epasinghe K., �Simulation of 3D acoustic �elds on a concurrent com-

puter�, Proc. Nordic symposium on Physical Acoustics, February 1996.

[5] Erstad J.O., Holm S., �An Approach to the Design of Sparse Array Sys-

tems�, Department of Informatics, University of Oslo, November 1994.

[6] Erstad J.O., �Design of sparse and non-equally spaced arrays for medical

ultrasound�, Masters Thesis, November 1994.

[7] FinkM.A., Cardoso J., �Di�raction E�ects in Pulse-Echo Measurement�,

Proc. IEEE Trans. Sonics Ultrason., vol. SU-31, pp.313-329, July 1984.

[8] Foster I., �Designing and Building Parallel Programs�, Addison-Wesley

Publishing Company, 1994, ISBN 0-201-57594-9.

[9] Geist A., Beguelin A., Dongarra J., Jiang W., Manchek R., Sunderam

V., �PVM 3 User's Guide and Reference Manual�, Oak Ridge Natinal

Laboratory, 1994.

[10] Goldberg B.B., Kimmelman B.A., �Medical Diagnostic Ultrasound: A

Retrospective on its 40th anniversary�, Kodak, Rochester, New York.

[11] Holm S., �Digital Beamforming in Ultrasound Imaging�, Department of

Informatics, University of Oslo.

80 BIBLIOGRAPHY

[12] Holm S., �Elevation focusing properties of superelliptical Probe�, De-

partment of Informatics, University of Oslo, Norway, September 1995.

[13] Holm S., �Medical Ultrasound Transducers and Beamforming�, Proc.

15th Congress on Acoustics, June 1995.

[14] Holm S., �Simulation of Acoustic Fields from Medical Ultrasound Trans-

ducers of Arbitrary Shape�, Proc. Nordic Syhmposium in Physical

Acoustics, January 1995.

[15] Holm S., Teigen F., Odegaard L., Berre V., Erstad J.O., �ULTRASIM

User's Manual�, Department of Informatics, University of Oslo, Version

2.0, 1996, ISBN 82-7368-133-5.

[16] Holm S., Odegaard L., Halvorsen E., Elgetun B., Teigen F., �ULTRA-

SIMUser's Manual for Advanced Functions�, Department of Informatics,

University of Oslo, Version 2.0, 1996, ISBN 82-7368-134-3.

[17] Hunt J.W., Arditi M., Foster F.S., �Ultrasound Transducers for Pulse-

Echo Medical Imaging�, Proc. IEEE Trans. Biomedical Engineering, vol.

BMU-30, No.8, August 1983.

[18] Johnson D.H., Dudgeon D.E., �Array Signal Processing�, Prentice Hall,

1993, ISBN 0-13-048513-6.

[19] Kino G.S., �Acoustic waves. Devices, Imaging and Analog Signal Pro-

cessing�, Prentice Hall, Englewood Cli�s, 1987.

[20] Lewis T.G., El-Rewini H., �Introduction to Parallel Computing�,Prentice

Hall, 1992, ISBN 0-13-498924-4.

[21] Odegaard L., Holm S., Teigen F., Keveland T., �Acoustic Field Simula-

tion for Arbitrarily Shaped Transducers in a Strati�ed Medium�, Proc.

IEEE Ultrasonics Symposium, November 1994.

[22] Odegaard L., Holm S., Torp H., �Phase Aberration Correction Ap-

plied to Annular Array Transducers when Focusing Through a Strati�ed

Medium�, Proc. IEEE Ultrasonics Symposium, 1993.

[23] Rossing T.D.,Fletcher N.H., �Principles of Vibration and Sound�,

Springer-Verlag, 1994, ISBN 0-387-94304-8.

[24] Shung K.K., Smith M.B., Tsui B., �Principles of Medical Imaging�, Aca-

demic Press, Inc.,1992, ISBN 0-12-640970-6.

BIBLIOGRAPHY 81

[25] Simonsen H.H., �Parallellprogramming på klyngen ved UiO�, 1994,

USIT.

[26] Tanenbaum A.S., �Computer Networks�, Prentice Hall, 1996, ISBN 0-

13-394248-1.

[27] USIT �Super computing at University of Oslo - Visualization�, WWW:

http://http://hpc/OsloCluster/SciVis/, July 1997.

[28] USIT �Super computing at University of Oslo - Hardware�, WWW:

http://hpc/OsloCluster/Hardware , July 1997.

[29] Van De Velde E.F., �Concurrent Scienti�c Computing�, Springer Verlag,

1994, ISBN 0-387-94195-9.

[30] Wu L., Zielinski A., �A Novel Array of Elliptic Ring Radiators�, Proc.

IEEE Journal of Oceanic Engineering, Vol. 18. No.3, July 1993.

82 BIBLIOGRAPHY

Appendix A

User Guide to Parallel Execution

This section describes the implementation of the external routines to UL-

TRASIM. The external routines are implemented for the simulation of acous-

tic �eld in 2D plane and 3D volume. These external routines were introduced

primarily to reduce the computation time.

A.1 C-MEX routine

A.1.1 Compilation of C-MEX Routine

C-MEX function pec.c is placed in ULTRASIMHOME/pe directory, and this

routine has to compiled using the C compilor cmex provided by MATLAB.

It is necessary to compile this C-Mex routine for every architecture available.

A.1.2 Execution

For sequential processing, the user should have a compiled programme of

pec.c. This is the C-MEX function which simulates the �eld, called by CAL-

CULATIONS → 2D RESPONSE → COMPUTER RESPONSE from the

CONFIGURATION window of ULTRASIM. If the C-MEX function is avail-

able, ULTRASIM chooses to execute this instead of pec.m

84 User Guide to Parallel Execution

A.2 C-MEX routine for parallel processing

A.2.1 Compilation of Parallel C-Mex Routine

For parallel computers or parallel networks, the C-MEX �le is extended to

calculate the �eld adopting parallel data processing techniques. Two other

external �les are introduced for parallel execution of which one is a slightly

di�erent version of pec.c.

pec.c This is the C-MEX �le for parallel computation. The �le is compiled

as earlier and should be available in ULTRASIMHOME/pe directory.

ppec_slave.c This is the parallel routine executed in each parallel process.

This is compiled using an available C compilor and the compiled func-

tion must be available in ULTRASIMHOME/pe directory.

These two �les should be compiled and present in MATLAB path. In the

Super Computing Network at the University of Oslo, the compilation is done

as below:

cc -I/local/hpc/lib/pvm3/include -o ppec_slave ppec_slave.c

-L/local/lib -lpvm3 -lm

cmex -I/local/hpc/lib/pvm3/include pec.c -L/local/lib -lpvm3 -lm

A.2.2 Interactive Parallel Execution

The parallel routines use PVM (Parallel Virtual Machine) as the communi-

cation library. Before the parallel routines are called from ULTRASIM, it is

necessary to activate the PVM console. Console is activated as below:

bluemaster 17>pvm hostfile

Here, host�le is the name of the description �le PVM uses to set it's envi-

ronment variables and host names. A line starting with a '∗' is considereda command, while a line starting with a 'D' is discarded. A few important

variables are:

A.2 C-MEX routine for parallel processing 85

ep - path to user executables. This speci�es the path where PVM looks for

the spawining tasks.

dx - location of pvmd. This speci�es the location of PVM libraries.

list of hosts - User speci�es the host names of the parallel network. Each

line should contain only one host name.

Below is an example of a host �le used in IBM Super Computing Network

at University Center for Information Technology, University of Oslo. A user

may have to declare more information in the host �le to suit the parallel

network available.

* ep=$HOME:$HOME/paral:$HOME/matlab:$HOME/matlab/toolbox/ultrasim/pe

sp01

sp02

sp03

sp04

sp05

.

.

.

.

.

sp30

sp31

sp32

Once the PVM console is activated, the user can choose to move PVM to the

background by typing 'quit' in PVM console. Calling PVM console again,

without specifying the host�le, reactivates the console. Command 'halt' will

leave the PVM console.

As in the sequential programme, user calls the parallel calculation by CAL-

CULATIONS → 2D RESPONSE → COMPUTER RESPONSE from the

CONFIGURATION window of ULTRASIM. Parallel processing does not

provide information about the calculation loop.


A.2.3 Parallel Execution as a Batch job

For large simulations, it is recommended that the calculation is carried out

as a batch job.

First the con�guration �le should be saved in the user's own con�guration

directory, and then the batch script is written using the QUEUE, which

prompts the user for necessary details to make the batch script.

Once the batch job is executed the the response matrices are written in a

MAT-�le, which is loaded using the M-�le RETRIEVE. The con�guration

should be loaded before the response matrices are loaded. RETRIEVE loads

the �eld point response matrix, maximum response matrix, minimum re-

sponse matrix and sets the visualization enable �ags.

An example of the batch script, created by QUEUE is below:

#!/bin/ksh

# @ input = /dev/null

# @ output = program.$(Cluster).$(Process).out

# @ error = program.$(Cluster).$(Process).err

# @ class = TEST

cd $HOME

pvm hostfile <<EOF

quit

EOF

cd $HOME/matlab

matlab <<EOF

u

[excitation, observasjon, media, transducer, flagg, ...

option, x, y, z, t, elem_pts, centers, elecdelr, elecdelt, ...

comment, pvec, plength, phase_err_ud, amp_ud, ...

array_resp, max_resp, min_resp]= ...

lusim('/usit/rs1/ifi/kapilae/matlab/toolbox/ultrasim/cnf/test.cnf');

[resp,max_resp,min_resp]=pecomp(excitation,transducer, ...

media, observasjon, option,flagg,x,y,z,t,elem_pts, ...

centers,config, plotfig,usr_resdir);

10

save par_res array_resp max_resp min_resp

quit

A.2 C-MEX routine for parallel processing 87

EOF

pvm <<EOF

halt

EOF

# @ queue

The results computed as a batch job can be transferred to a sequential com-

puter for visualization. This method is recommended for the Super Comput-

ing Network in the University of Oslo.

Appendix B

User Guide to Volumetric

Visualization

This appendix explains the setting up of visualization parameters for volu-

metric simulations. Visualization is done by using MATLAB SLICE routines.

B.1 Observation

Observation �ag is set to either Volume, for observation in (x,y,z) or (x,y,t)volume,

or Volume → Movie, for (x,y,z,t) volume. Observation parameters are set

choosing CONFIGURATION → OBSERVATION from the CONFIGURA-

TION window of ULTRASIM.

When observation �ag is set to VOLUME, the observation parameters are

set as below:

------- OBSERVATION SUBMENU ---------

Option : VOLUME

Coordinates : RECTANGULAR

2) 1. axis : x

3) Start value of x [mm] : -15

4) End value of x [mm] : 15

5) 2. axis : y

6) Start value of y [mm] : -10

7) End value of y [mm] : 10

90 User Guide to Volumetric Visualization

12) 3. axis : z

13) Start value of z [mm] : 10

14) End value of z [mm] : 25

9) Fixed value of t [us] : 12.99

15) # obs.pts /[mm] : 2

CHANGE = "number"

-> Decision (<CR> = exit):

The �rst and the second axes are �xed while the third axis can be either z

or t.

----- Choose axis -----

1) z

2) t

Select a menu number:

When observation �ag is set to VOLUME → MOVIE, the observation pa-

rameters are set as below:

------- OBSERVATION SUBMENU ---------

Option : VOLUME (movie)

Coordinates : RECTANGULAR

2) 1. axis : x

3) Start value of x [mm] : -15

4) End value of x [mm] : 15

5) 2. axis : y

B.2 Slice Plot 91

6) Start value of y [mm] : -10

7) End value of y [mm] : 10

12) 3. axis : z

13) Start value of z [mm] : 10

14) End value of z [mm] : 25

10) Start value of t [us] : 12.99

11) Stop value of t [us] : 15.25

... 8 plots will be produced for movie

15) # obs.pts /[mm] : 2

CHANGE = "number"

-> Decision (<CR> = exit):

B.2 Slice Plot

The volumetric simulations area is visualized using the SLICE function in

MATLAB. The results are plotted in ULTRASIM - PLOT window, and the

SLICE plots are called by CALCULATIONS → 2D RESPONSE → SLICE

PLOT in ULTRASIM - CONFIGURATION - CALCULATION window.

Set the fillowing plot options:

-> envelope / rf [e/r] ?

-> linear /logarithmic [lin/log] ?

Obeservation slices for X Axis

Modify slices [y/n] : y

-> Background display ?? [y/n] :y

Range for the axis X: -15 to 15


Enter the number of slices(default 0, max 5) : 1

Axis X slice 1: Enter value ->0

Obeservation slices for Y Axis

Modify slices [y/n] : y

-> Background display ?? [y/n] :y

Range for the axis Y: -10 to 10

Enter the number of slices(default 0, max 5) : 1

Axis Y slice 1: Enter value ->0

-> Range Movie on Third axis ?? [y/n] :y

Range for Axis Z : 10 to 25

Enter start value :15

Enter end value :20

plot number 1 of totally 11 plots











Movie Options

-> Enter number of movie loops [1..100] :

-> Enter speed in frames/sec [1..20] :

Movie is playing ...

B.2 Slice Plot 93

-> More movie ??? [y/n] : n

Appendix C

Programming Code

The main program code used in the thesis are presented here. For clarity

and for page construtction ease, line breaks are added to some lines.

C.1 C routine for sequential computer -pec.c

#include <stdio.h>

#include <math.h>

#include <string.h>

#include "mat.h"

#include "mex.h"

/*

--------------------------------------------------------------

Routine : pec.c

Description : The Matlab-External Library Routine,

called by PECOMP.M. This C-file shoule be

compiled using the MATLAB C complier

CMEX, for each architecture. The compiled

MEX file can be chosen when PECOMP.M is

called via ULTRASIM

Language : C code, embedding MatLab External Libraries

Written by : University of Oslo, Department of Informatikk,

96 Programming Code

Kapila Epasinghe

Version : 1.0 KE 10.05.96 First Version

Called by : pecomp.m

Calling : focus.m

--------------------------------------------------------------

*/

#define PI 3.14159265358979

#define EPS 2.220446049250313e-16

#define INF 1e308

struct ans {Matrix *rsp;

Matrix *maxrp;

Matrix *minrp;};

/* Function to calculate the non conjugate transpose of a

matrix */

Matrix *non_conj_transpose(mtradr)

Matrix *mtradr;

{

int m,n,i,j;

double *pr,*pi,*tpr,*tpi;

Matrix *temp;

m=mxGetM(mtradr);

n=mxGetN(mtradr);

pr=mxGetPr(mtradr);

pi=mxGetPi(mtradr);

temp=mxCreateFull(n,m,mxIsComplex(mtradr));

tpr=mxGetPr(temp);

tpi=mxGetPi(temp);

for (i=0;i<m;i++)

for (j=0;j<n;j++)

C.1 C routine for sequential computer -pec.c 97

{

tpr[n*i+j]=pr[m*j+i];

if (mxIsComplex(temp))

tpi[n*i+j]=pi[m*j+i];

}

return temp;

}

/* Help function... */

char *num2str(val)

int val;

{

int ch;

char aval[4];

char *sval;

ch=(val/100);

val=val % 100;

aval[0]=ch+48;

ch=(val/10);

val=val % 10;

aval[1]=ch+48;

ch=val;

aval[2]=ch+48;

aval[3]='\0';

sval=aval;

return sval;

}

/* Find absolute value.. */

double abs_val(val)

double val;

{

if (val<0)

val=-1*val;

return val;

}

/* Main function: Calculates the response..*/

struct ans pec(x,y,z,t,FieldSize,elem,ElPt,Apodize,excitation,

98 Programming Code

SteerFocusDelay,TimeStepVector,trans,fla,

option,usr_resdir,values)

double x[],y[],z[],t[],Apodize[],excitation[];

double SteerFocusDelay[],TimeStepVector[],values[];

Matrix *elem,*trans,*fla;

char option[],usr_resdir[];

int FieldSize[],ElPt[];

{

double r,c,PulseLength,jomega,TLen,*transducer,*flagg;

double *elem_pts,*Time,Weight,*resp_pr,*resp_pi,*maxpr,*minpr;

Matrix *resp,*temp,*input,*c_foc,*rhs[5],*lhs[2];

double d,d_max,foc_depth,f_number,*input_pr,*c_foc_pr;

int min_elem,max_elem;

int PNo,FieldLength,n,a,i,finite,testint,k,ulen,set;

double t_ref,x0,y0,z0,maxt,mint,TStFo,TimeDel,w_pulse;

double t_vector,r_attenu,trans_arr_value,F_des;

char *mtrname,*filename;

char mtn[10];

char fn[80];

char *conv,*Pict,*mat;

MATFile *fp;

/* Structure definition used when extracting the three

solution matrices. */

struct ans reply;

/* List of variables imported from pecomp.m, that are used

in the mex file */

if ((option[0]=='m') || (option[0]=='f'))

{

ulen=0;

ulen=strlen(usr_resdir);

Pict="Pict";

mat=".mat";


mtn[0]='P';mtn[1]='i';mtn[2]='c';mtn[3]='t';

mtn[4]='0';mtn[5]='0';mtn[6]='0';mtn[7]='\0';

for (k=0;k<ulen+12;k++){

if (k<ulen)

fn[k]=usr_resdir[k];

else if (k<ulen+7)

fn[k]=mtn[k-ulen];

else

fn[k]=mat[k-ulen-7];

}

filename=fn;

mtrname=mtn;

}

flagg=mxGetPr(fla);

transducer=mxGetPr(trans);

elem_pts=mxGetPr(elem);

r=values[0];

c=values[1];

PulseLength=values[2];

TLen=values[3];

jomega=values[4];

Weight=values[5];

trans_arr_value=values[6];

F_des=values[7];

d=transducer[0];

FieldLength=FieldSize[0]*FieldSize[1];

/* Creation of max_resp and min_resp. Earlier allocations

are neglected. */

reply.maxrp=mxCreateFull(1,2,REAL);

maxpr=mxGetPr(reply.maxrp);

reply.minrp=mxCreateFull(1,2,REAL);

minpr=mxGetPr(reply.minrp);

maxpr[0]=0;maxpr[1]=0;

minpr[0]=0;minpr[1]=0;

100 Programming Code

/* LOOP OVER TIME REFERENCES */

for (PNo=1;PNo<=TLen;PNo++)

{

/* Time reference or distance reference */

if ((option[2]=='y') || (option[2]=='z'))

t_ref=t[0];

else

t_ref=r/c;

/* put time array into a vector. Due to the array

nature of t is in array form, no reshaping is done

when making the Time array. */


{

temp=mxCreateFull(1,FieldLength,REAL);

Time=mxGetPr(temp);

for (i=0;i<FieldLength;i++)

Time[i]=TimeStepVector[PNo-1];

}

else

Time=t;

d_max=d;

min_elem=1;

max_elem=ElPt[1];

if ((flagg[0]==2)

&& ((option[0]=='m') || (option[0]=='f')))

{

foc_depth=c*TimeStepVector[PNo-1];

if (F_des > 0)

{

if (d<foc_depth/F_des)

d_max=d;

else

d_max=foc_depth/F_des;

max_elem=0;

min_elem=ElPt[1];


for (set=0;set<ElPt[1];set++)

if (abs_val(elem_pts[set*5])<=d_max/2)

{

if (max_elem<set+1)

max_elem=set+1;

if (min_elem>set+1)

min_elem=set+1;

}

if (F_des>foc_depth/d_max)

f_number=F_des;

else

f_number=foc_depth/d_max;

printf("f-number: %3.0f, aperture: %5.2f mm,

element rage: %d-%d\n",

f_number,d_max*1000,min_elem,max_elem);

}

else

d_max=d;

input=mxCreateFull(1,7,REAL);

input_pr=mxGetPr(input);

for (i=0;i<7;i++)

input_pr[i]=excitation[i];

input_pr[1]=foc_depth;

c_foc=mxCreateFull(1,1,REAL);

c_foc_pr=mxGetPr(c_foc);

c_foc_pr[0]=excitation[13];

rhs[0]=input;rhs[1]=c_foc;rhs[2]=trans;

rhs[3]=elem;rhs[4]=fla;

mexCallMATLAB(2,lhs,5,rhs,"focus");

SteerFocusDelay=mxGetPr(lhs[0]);

}

/* Initialize */

if (PNo>1)


mxFreeMatrix(resp);

resp=mxCreateFull(1,FieldLength,COMPLEX);

resp_pr=mxGetPr(resp);

resp_pi=mxGetPi(resp);

/* OBSERVATION POINTS */

for (n=min_elem;n<=max_elem;n++)

{

if ( (n % 10)==0)

printf("Loop %d of %3.0f now at %d

of totally %d computations\n",

PNo,TLen,n,ElPt[1]);

/* Actual Analysis points are */

x0=elem_pts[ElPt[0]*(n-1)];

y0=elem_pts[ElPt[0]*(n-1)+1];

z0=elem_pts[ElPt[0]*(n-1)+2];

/* Steering and Focusing TimeDelay for the

current element. */

TStFo=-SteerFocusDelay[n-1];

/* Check if z is finite: To be used when computing

field ponts.*/

finite=0;


if ((z[i]<INF) && (z[i]>-INF))

finite=1;


{

/* Propagation delay from origo to field points

and current time. */


if (finite==1)

TimeDel=(sqrt((x[i]-x0)*(x[i]-x0)+

(y[i]-y0)*(y[i]-y0)+

(z[i]-z0)*(z[i]-z0)))/c;

else

TimeDel=(sqrt((x[i]-x0)*180/PI*(x[i]-x0)*180/PI+

(y[i]-y0)*180/PI*(y[i]-y0)*180/PI+

(z[i]-z0)*(z[i]-z0)))/c;

/* check if PulseLength is Finite */

if (abs_val(PulseLength)<HUGE_VAL)

{

maxt=TStFo + PulseLength/2;

mint=TStFo - PulseLength/2;

/* see if the field point is in the transmitted burst */

if (((TimeDel-Time[i]- EPS)<=maxt) &&

((TimeDel-Time[i]+EPS)>=mint))

{

/* Find the actual time for the response computation */

t_vector=TimeDel - Time[i]-TStFo;

/* compute the weight and the attenuation

correction factor */

if (Weight==1)

w_pulse=(cos(PI*t_vector/PulseLength)

*cos(PI*t_vector/PulseLength));

else if (Weight==0)

w_pulse=1;

/* Attenuation */

r_attenu=TimeDel/t_ref;

/*Compute the response of the curent burst for

the fieldpoint*/

resp_pr[i]=resp_pr[i]+Apodize[n-1]*

w_pulse*cos(jomega*t_vector)/r_attenu;

resp_pi[i]=resp_pi[i]+Apodize[n-1]*

w_pulse*sin(jomega*t_vector)/r_attenu;

}

}

else

{


/* Find the actual time for the response computation */

t_vector=TimeDel-Time[i]-TStFo;

/* Attenuation */


/* Compute the response of the curent burst for the

fieldpoint */

resp_pr[i]=resp_pr[i]+Apodize[n-1]*

cos(jomega*t_vector)/r_attenu;

resp_pi[i]=resp_pi[i]+Apodize[n-1]*

sin(jomega*t_vector)/r_attenu;

}

}

}

printf("Loop %d of %3.0f now at %d of totally %d

computations\n",PNo,TLen,n-1,ElPt[1]);

/* Put the response into a matrix */

mxSetM(resp,FieldSize[0]);

mxSetN(resp,FieldSize[1]);


{

/* Normalize response */

resp_pr[i]=resp_pr[i]*trans_arr_value;

resp_pi[i]=resp_pi[i]*trans_arr_value;

/* Compute max and min of response. Real and

Imaginary prt for both. */

if (maxpr[0]<resp_pr[i])

maxpr[0]=resp_pr[i];

if (maxpr[1]<resp_pi[i])

maxpr[1]=resp_pi[i];

if (minpr[0]>resp_pr[i])

minpr[0]=resp_pr[i];

if (minpr[1]>resp_pi[i])

minpr[1]=resp_pi[i];

}


resp=non_conj_transpose(resp);


{

conv=num2str(PNo);

for (k=4;k<7;k++)

mtn[k]=conv[k-4];

for (k=0;k<7;k++)

fn[k+ulen]=mtn[k];

mxSetName(resp,mtrname);

fp= matOpen(filename,"w");

matPutMatrix(fp,resp);

matClose(fp);

}

}

reply.rsp=resp;

return reply;

}

/* NOTE: This section might need to be modified for

the version of C-compiler available

*/

#if __STDC__

void mexFunction(

int nlhs,

Matrix *plhs[],

int nrhs,

Matrix *prhs[]

)

#else

mexFunction (nlhs,plhs,nrhs,prhs)

int nlhs,nrhs;

Matrix *plhs[],*prhs[];

#endif

{

Matrix *trans,*fla,*elem;

double *x,*y,*z,*t,*Apod,*excit,*temp,*SFD,*TSV;

int FS[2],EP[2],n;

char *opt,*dirpath,*delim,*usr_resdir;


struct ans mtr;

x=mxGetPr(prhs[0]);

FS[0]=mxGetM(prhs[0]);FS[1]=mxGetN(prhs[1]);

y=mxGetPr(prhs[1]);

z=mxGetPr(prhs[2]);

t=mxGetPr(prhs[3]);

elem=prhs[4];

EP[0]=mxGetM(prhs[4]);EP[1]=mxGetN(prhs[4]);

Apod=mxGetPr(prhs[5]);

excit=mxGetPr(prhs[6]);

SFD=mxGetPr(prhs[7]);

TSV=mxGetPr(prhs[8]);

trans=prhs[9];

fla=prhs[10];

n=mxGetN(prhs[11])+1;

opt=mxCalloc(n,sizeof(char));

mxGetString(prhs[11],opt,n);


dirpath=mxCalloc(n,sizeof(char));

mxGetString(prhs[12],dirpath,n);


delim=mxCalloc(n,sizeof(char));

mxGetString(prhs[13],delim,n);

usr_resdir=strcat(dirpath,delim);

temp=mxGetPr(prhs[14]);

mtr=pec(x,y,z,t,FS,elem,EP,Apod,excit,SFD,TSV,

trans,fla,opt,usr_resdir,temp);

plhs[0]=mtr.rsp;

plhs[1]=mtr.maxrp;

plhs[2]=mtr.minrp;

C.2 C routine for parallel computer - pec.c 107

}

C.2 C routine for parallel computer - pec.c

#include <stdio.h>

#include <math.h>

#include <string.h>

#include "mat.h"

#include "mex.h"

/*

-------------------------------------------------------------

Routine : pec.c

Description : The Matlab-External Library Routine for

parallel computer network in USIT, called

by PECOMP.M. This C-file shoule be compiled

using the MATLAB C complier CMEX, according

to the parameters in User Guide. The compiled

MEX file can be chosen when PECOMP.M is

called via ULTRASIM

Language : C code, embedding MatLab External Libraries

Written by : University of Oslo, Department of Informatikk,

Kapila Epasinghe

Version : 1.0 KE 10.05.96 First Version

Called by : pecomp.m

Calling : focus.m

--------------------------------------------------------------

*/


#define PI 3.14159265358979

#define EPS 2.220446049250313e-16

#define INF 1e308

#define MAXTASKS 32

struct ans

{ Matrix *rsp;

Matrix *maxrp;

Matrix *minrp;};

/* Help function to find the non conjugate transpose

of a matrix */

Matrix *non_conj_transpose(mtradr)

Matrix *mtradr;

{

int m,n,i,j;

double *pr,*pi,*tpr,*tpi;

Matrix *temp;

m=mxGetM(mtradr);

n=mxGetN(mtradr);

pr=mxGetPr(mtradr);

pi=mxGetPi(mtradr);

temp=mxCreateFull(n,m,mxIsComplex(mtradr));

tpr=mxGetPr(temp);

tpi=mxGetPi(temp);

for (i=0;i<m;i++)

for (j=0;j<n;j++)

{

tpr[n*i+j]=pr[m*j+i];

if (mxIsComplex(temp))

tpi[n*i+j]=pi[m*j+i];

}

return temp;

}

/* Help function to convert a number to a string */


char *num2str(val)

int val;

{

int ch;

char aval[4];

char *sval;

ch=(val/100);

val=val % 100;

aval[0]=ch+48;

ch=(val/10);

val=val % 10;

aval[1]=ch+48;

ch=val;

aval[2]=ch+48;

aval[3]='\0';

sval=aval;

return sval;

}

/* Help function to get the absolute value of a real number */

double abs_val(val)

double val;

{

if (val<0)

val=-1*val;

return val;

}

/* Main Calculation Routine in Mex file */

struct ans pec(x,y,z,t,FieldSize,elem,ElPt,Apodize,excitation,

SteerFocusDelay,TimeStepVector,trans,fla,

option,usr_resdir,values,cent,tasks)

double x[],y[],z[],t[],Apodize[],excitation[];

double SteerFocusDelay[],TimeStepVector[],values[];

Matrix *elem,*trans,*fla,*cent;

char option[],usr_resdir[];

int FieldSize[],ElPt[],tasks;


{

double r,c,PulseLength,jomega,TLen,*transducer,*flagg;

double *elem_pts,*Time,Weight,*resp_pr,*resp_pi;

double *maxpr,*minpr;

Matrix *resp,*temp,*input,*c_foc,*rhs[6],*lhs[2];

double d,d_max,foc_depth,f_number,*input_pr,*c_foc_pr;

int min_elem,max_elem;

int PNo,FieldLength,n,a,i,finite,testint,k,ulen,set;

double t_ref,x0,y0,z0,maxt,mint,TStFo,TimeDel,w_pulse;

double t_vector,r_attenu,trans_arr_value,F_des;

char *mtrname,*filename;

char mtn[10];

char fn[80];

char *conv,*Pict,*mat;

MATFile *fp;

int numt,tids[MAXTASKS],data_len,residual,cc,id,len,bl;

int mytid,lstart,lend,elems,NTASK;

double *l_pr,*l_pi;

char buf[10];int tata;

/* Structure definition used when extracting the three

solution matrices. */

struct ans reply;

/* List of variables imported from pecomp.m, that are

used in the mex file */

printf("\n** COMPUTING RESPONSE IN PARALLEL SYSTEM **\n\n");

if ((option[0] == 'm') || (option[0] =='f'))

{

Pict="Pict";

mat=".mat";

mtn[0]='P';mtn[1]='i';mtn[2]='c';mtn[3]='t';

mtn[4]='0';mtn[5]='0';mtn[6]='0';mtn[7]='\0';

ulen=0;


ulen=strlen(usr_resdir);

for (k=0;k<ulen+12;k++){

if (k<ulen)

fn[k]=usr_resdir[k];

else if (k<ulen+7)

fn[k]=mtn[k-ulen];

else

fn[k]=mat[k-ulen-7];

}

filename=fn;

mtrname=mtn;

}

flagg=mxGetPr(fla);

transducer=mxGetPr(trans);

elem_pts=mxGetPr(elem);

r=values[0];

c=values[1];

PulseLength=values[2];

TLen=values[3];

jomega=values[4];

Weight=values[5];

trans_arr_value=values[6];

F_des=values[7];

d=transducer[0];

FieldLength=FieldSize[0]*FieldSize[1];

/* Creatingmax_resp and min_resp. */

reply.maxrp=mxCreateFull(1,2,REAL);

maxpr=mxGetPr(reply.maxrp);

reply.minrp=mxCreateFull(1,2,REAL);

minpr=mxGetPr(reply.minrp);

maxpr[0]=0;maxpr[1]=0;

minpr[0]=0;minpr[1]=0;

/* Time reference or distance reference */


if ((option[2]=='y') || (option[2]=='z'))

t_ref=t[0];

else

t_ref=r/c;

NTASK=tasks;

/* Enroll in PVM */

mytid=pvm_mytid();

/* Call PVM : Spawn ppec_slave */

numt=pvm_spawn("ppec_slave",0,0,"",NTASK,&tids);

printf("%d host(s) activated\n",numt);

/* Check if Z is finit */

finite=0;


if ((z[i]<INF) && (z[i]>-INF))

{finite=1;break;}

printf("Starting Calculations....\n");

/* Data distribution parameters */

data_len=(int)FieldLength/numt;

residual=FieldLength-data_len*numt;

/* LOOP OVER TIME REFERENCES */

for (PNo=1;PNo<=TLen;PNo++)

{

/* put time array into a vector. Due to the array


nature of t, no reshaping is done when making

the Time array. */


{

temp=mxCreateFull(1,FieldLength,REAL);

Time=mxGetPr(temp);


Time[i]=TimeStepVector[PNo-1];

}

else

Time=t;

d_max=d;

min_elem=1;

max_elem=ElPt[1];

/* If Dynamic Focus and Movie,

find SteerFocusDelays for time step */

if ((flagg[0]==2) &&

((option[0]=='m') || (option[0]=='f')))

{

foc_depth=c*TimeStepVector[PNo-1];

if (F_des > 0)

{

if (d<foc_depth/F_des)

d_max=d;

else

d_max=foc_depth/F_des;

max_elem=0;

min_elem=ElPt[1];

for (set=0;set<ElPt[1];set++)

if (abs_val(elem_pts[set*5])<=d_max/2)

{

if (max_elem<set+1)

max_elem=set+1;

if (min_elem>set+1)

min_elem=set+1;

}

if (F_des>foc_depth/d_max)

f_number=F_des;


else

f_number=foc_depth/d_max;

printf("f-number: %3.0f, aperture: %5.2f mm,

element rage: %d-%d\n",

f_number,d_max*1000,min_elem,max_elem);

}

else

d_max=d;

/* Call ULTRASIM Routine -focus-. */

input=mxCreateFull(1,7,REAL);

input_pr=mxGetPr(input);

for (i=0;i<7;i++)

input_pr[i]=excitation[i];

input_pr[1]=foc_depth;

c_foc=mxCreateFull(1,1,REAL);

c_foc_pr=mxGetPr(c_foc);

c_foc_pr[0]=excitation[13];

rhs[0]=input;rhs[1]=c_foc;rhs[2]=trans;

rhs[3]=elem;rhs[4]=cent;rhs[5]=fla;

mexCallMATLAB(2,lhs,6,rhs,"focus");

SteerFocusDelay=mxGetPr(lhs[0]);

}

/* Initialize */

if (PNo>1)

mxFreeMatrix(resp);

resp=mxCreateFull(1,FieldLength,COMPLEX);

resp_pr=mxGetPr(resp);

resp_pi=mxGetPi(resp);

/* COMPUTING RESPONSE IN PARALLEL PROGRAMMING. */


/* OBSERVATION POINTS */

if (numt !=1)

{

lend=-1;

for (i=0;i<numt;i++)

{

lstart = lend+1;

if (i<residual)

lend = lstart + data_len;

else

lend = lstart + data_len - 1;

pvm_initsend(0);

pvm_pkint(&lstart,1,1);

pvm_pkint(&lend,1,1);

pvm_pkint(&PNo,1,1);

pvm_pkint(&min_elem,1,1);

pvm_pkint(&max_elem,1,1);

pvm_pkint(&ElPt[0],2,1);

pvm_pkdouble(&SteerFocusDelay[0],ElPt[1],1);

pvm_pkdouble(&Time[lstart],lend-lstart+1,1);

if (PNo == 1)

{

pvm_pkint(&FieldLength,1,1);

pvm_pkint(&finite,1,1);

pvm_pkdouble(&t_ref,1,1);

pvm_pkdouble(&c,1,1);

pvm_pkdouble(&jomega,1,1);

pvm_pkdouble(&Weight,1,1);

pvm_pkdouble(&PulseLength,1,1);

pvm_pkdouble(&elem_pts[0],ElPt[0]*ElPt[1],1);

pvm_pkdouble(&Apodize[0],ElPt[1],1);

pvm_pkdouble(&x[lstart],lend-lstart+1,1);

pvm_pkdouble(&y[lstart],lend-lstart+1,1);

pvm_pkdouble(&z[lstart],lend-lstart+1,1);

}

pvm_send(tids[i],3);

}

}



{

cc=pvm_recv(-1,-1);

pvm_bufinfo(cc,(int*)0,(int*)0,&id);

pvm_upkint(&lstart,1,1);

pvm_upkint(&lend,1,1);

len=lend-lstart+1;

l_pr=(double *)calloc(len,sizeof(double));

l_pi=(double *)calloc(len,sizeof(double));

pvm_upkdouble(l_pr,len,1);

pvm_upkdouble(l_pi,len,1);

for (bl=lstart;bl<=lend;bl++)

{

resp_pr[bl]=l_pr[bl-lstart];

resp_pi[bl]=l_pi[bl-lstart];

}

printf("Calculation completed %d of %d hosts. \r",

i+1,numt);

cfree(l_pr);

cfree(l_pi);

}

printf("\n");

printf("Loop %d of %3.0f now at %d of totally %d

computations\n",

PNo,TLen,max_elem,ElPt[1]);

/* Put the response into a matrix */

mxSetM(resp,FieldSize[0]);

mxSetN(resp,FieldSize[1]);


{

/* Normalize response */

resp_pr[i]=resp_pr[i]*trans_arr_value;

resp_pi[i]=resp_pi[i]*trans_arr_value;

/* Compute max and min of response.*/


if (maxpr[0]<resp_pr[i])

maxpr[0]=resp_pr[i];

if (maxpr[1]<resp_pi[i])

maxpr[1]=resp_pi[i];

if (minpr[0]>resp_pr[i])

minpr[0]=resp_pr[i];

if (minpr[1]>resp_pi[i])

minpr[1]=resp_pi[i];

}

resp=non_conj_transpose(resp);


{

conv=num2str(PNo);

for (k=4;k<7;k++)

mtn[k]=conv[k-4];

for (k=0;k<7;k++)

fn[k+ulen]=mtn[k];

mxSetName(resp,mtrname);

fp= matOpen(filename,"w");

matPutMatrix(fp,resp);

matClose(fp);

}

}

lstart=-1;


{

pvm_initsend(0);

pvm_pkint(&lstart,1,1);

pvm_send(tids[i],3);

}

pvm_exit();

reply.rsp=resp;

return reply;

}


/* MEX Gateway function */

#ifdef __STDC__

void mexFunction(

int nlhs,

Matrix *plhs[],

int nrhs,

Matrix *prhs[]

)

#else

mexFunction (nlhs,plhs,nrhs,prhs)

int nlhs,nrhs;

Matrix *plhs[],*prhs[];

#endif

{

Matrix *trans,*fla,*elem,*cent;

double *x,*y,*z,*t,*Apod,*excit,*temp,*SFD,*TSV;

int FS[2],EP[2],n,task;

char *opt,*dirpath,*delim,*usr_resdir;

struct ans mtr;

x=mxGetPr(prhs[0]);

FS[0]=mxGetM(prhs[0]);FS[1]=mxGetN(prhs[1]);

y=mxGetPr(prhs[1]);

z=mxGetPr(prhs[2]);

t=mxGetPr(prhs[3]);

elem=prhs[4];

EP[0]=mxGetM(prhs[4]);EP[1]=mxGetN(prhs[4]);

Apod=mxGetPr(prhs[5]);

excit=mxGetPr(prhs[6]);

SFD=mxGetPr(prhs[7]);

TSV=mxGetPr(prhs[8]);

trans=prhs[9];

fla=prhs[10];


opt=mxCalloc(n,sizeof(char));

C.3 C-Routine ppec_slave.c 119

mxGetString(prhs[11],opt,n);


dirpath=mxCalloc(n,sizeof(char));

mxGetString(prhs[12],dirpath,n);


delim=mxCalloc(n,sizeof(char));

mxGetString(prhs[13],delim,n);

usr_resdir=strcat(dirpath,delim);

temp=mxGetPr(prhs[14]);

cent=prhs[15];

printf("-> Enter the number of tasks [<=32] : ");

scanf("%d",&task);

mtr=pec(x,y,z,t,FS,elem,EP,Apod,excit,SFD,TSV,trans,

fla,opt,usr_resdir,temp,cent,task);

plhs[0]=mtr.rsp;

plhs[1]=mtr.maxrp;

plhs[2]=mtr.minrp;

}

C.3 C-Routine ppec_slave.c

#include <stdio.h>

#include <math.h>

#include "pvm3.h"

#define PI 3.14159265358979

#define EPS 2.220446049250313e-16

#define INF 1e308


main()

{

int ptid,cc,id,i,mytid,k;

int length,FieldLength,finite,n,ElPt[2];

int lstr,lend,PNo,min_elem,max_elem;

double *x,*y,*z,*Time,*elem_pts,*Apodize;

double *local_pr,*local_pi,*SteerFocusDelay;

double t_ref,c,jomega,Weight,TStFo;

double PulseLength,TimeDel,maxt,mint,t_vector,r_attenu;

double w_pulse,x0,y0,z0;

char buf[10];

mytid=pvm_mytid();

ptid=pvm_parent();

cc=pvm_recv(ptid,-1);

pvm_bufinfo(cc,(int *)0,(int *)0,&id);

pvm_upkint(&lstr,1,1);

while(lstr != -1)

{

pvm_upkint(&lend,1,1);

length=lend-lstr+1;

pvm_upkint(&PNo,1,1);

pvm_upkint(&min_elem,1,1);

pvm_upkint(&max_elem,1,1);

pvm_upkint(&ElPt[0],2,1);

SteerFocusDelay=(double *)calloc(ElPt[1],sizeof(double));

pvm_upkdouble(SteerFocusDelay,ElPt[1],1);

Time=(double *)calloc(length,sizeof(double));

pvm_upkdouble(Time,length,1);

if (PNo==1)

{

pvm_upkint(&FieldLength,1,1);

pvm_upkint(&finite,1,1);

pvm_upkdouble(&t_ref,1,1);

pvm_upkdouble(&c,1,1);

pvm_upkdouble(&jomega,1,1);

pvm_upkdouble(&Weight,1,1);

pvm_upkdouble(&PulseLength,1,1);


elem_pts=(double *)calloc(ElPt[0]*ElPt[1],sizeof(double));

pvm_upkdouble(elem_pts,ElPt[0]*ElPt[1],1);

Apodize=(double *)calloc(ElPt[1],sizeof(double));

pvm_upkdouble(Apodize,ElPt[1],1);

x=(double *)calloc(length,sizeof(double));

pvm_upkdouble(x,length,1);

y=(double *)calloc(length,sizeof(double));

pvm_upkdouble(y,length,1);

z=(double *)calloc(length,sizeof(double));

pvm_upkdouble(z,length,1);

}

local_pr=(double *)calloc(length,sizeof(double));

local_pi=(double *)calloc(length,sizeof(double));

for (n=min_elem;n<=max_elem;n++)

{

for (i=0;i<length;i++)

{

x0=elem_pts[ElPt[0]*(n-1)];

y0=elem_pts[ElPt[0]*(n-1)+1];

z0=elem_pts[ElPt[0]*(n-1)+2];

TStFo=-SteerFocusDelay[n-1];

if (finite==1)

TimeDel=(sqrt((x[i]-x0)*(x[i]-x0)+

(y[i]-y0)*(y[i]-y0)+

(z[i]-z0)*(z[i]-z0)))/c;

else

TimeDel=(sqrt((x[i]-x0)*(x[i]-x0)*180/PI*180/PI+

(y[i]-y0)*(y[i]-y0)*180/PI*180/PI+

(z[i]-z0)*(z[i]-z0)))/c;

if ((PulseLength<HUGE_VAL) && (PulseLength>-HUGE_VAL))

{

maxt=TStFo+PulseLength/2;

mint=TStFo-PulseLength/2;

if (((TimeDel-Time[i]-EPS)<=maxt) &&

((TimeDel-Time[i]+EPS)>=mint))

{



if (Weight==1)

w_pulse=(cos(PI*t_vector/PulseLength)

*cos(PI*t_vector/PulseLength));

else if (Weight==0)

w_pulse=1;


local_pr[i]=local_pr[i]+Apodize[n-1]

*w_pulse*cos(jomega*t_vector)/r_attenu;

local_pi[i]=local_pi[i]+Apodize[n-1]

*w_pulse*sin(jomega*t_vector)/r_attenu;

}

}

else

{



local_pr[i]=local_pr[i]

+Apodize[n-1]*cos(jomega*t_vector)/r_attenu;

local_pi[i]=local_pi[i]

+Apodize[n-1]*sin(jomega*t_vector)/r_attenu;

}

}

}

pvm_initsend(0);

pvm_pkint(&lstr,1,1);

pvm_pkint(&lend,1,1);

pvm_pkdouble(local_pr,length,1);

pvm_pkdouble(local_pi,length,1);

pvm_send(ptid,4);

cfree(local_pr);cfree(local_pi);cfree(Time);

cfree(SteerFocusDelay);


cc=pvm_recv(ptid,-1);

pvm_bufinfo(cc,(int*)0,(int*)0,&id);

pvm_upkint(&lstr,1,1);

}

cfree(x);cfree(y);cfree(z);cfree(Time);cfree(Apodize);

pvm_exit();

}

\section{MATLAB Routine for parallel network - \em{queue.m}}

\begin{verbatim}

%

s=input('Have you saved your configuration in your CNF directory??? ','s');

if s~='y'

fprintf('Please save the configurations and continue..\n');

else

disp(['Available files in ',usr_cnfdir]);

disp(' ');

tmp_dir=pwd;

eval(['cd ',usr_cnfdir]);

eval(['dir *.cnf']);

eval(['cd ',tmp_dir]);

clear tmp_dir;

cnff= input('-> File name : ','s');

if isempty(findstr(cnff,'cnf'));

cnff=[cnff,'.cnf'];

end;

cnff=[usr_cnfdir,separator,cnff];

fn='';

while ( isempty(fn))

fn=input('Enter Job file Name : ','s');

if exist(fn)

fprintf('File Exists.');

hm=input('Overwrite it?? [y/n] : ','s');

if hm~='y'

fn='';

end


end

end

qname=input('Enter Queue(TEST/PARALLEL) Name [t/p] : ','s');

if qname=='p'

qname='PARALLEL';

else

qname='TEST';

end;

fnid=fopen(fn,'w');

fprintf(fnid,'#!/bin/ksh\n# @ input = /dev/null\n');

fprintf(fnid,'# @ output = program.$(Cluster).$(Process).out\n');

fprintf(fnid,'# @ error = program.$(Cluster).$(Process).err\n');

fprintf(fnid,'# @ class = %s\n\n',qname);

fprintf(fnid,'cd $HOME\n');

fprintf(fnid,'pvm hostfile <<EOF\nquit\nEOF\n\n');

fprintf(fnid,'cd $HOME/matlab\nmatlab <<EOF\nu \n');

fprintf(fnid,'[excitation, observasjon, media, transducer, flagg, ...\n');

fprintf(fnid,'option, x, y, z, t, elem_pts, centers, elecdelr, ...\n');

fprintf(fnid,'elecdelt, comment, ...\n');

fprintf(fnid,'pvec, plength, phase_err_ud, amp_ud, ...\n');

fprintf(fnid,'array_resp, max_resp, min_resp]= ...\n');

fprintf(fnid,'lusim(''%s'');\n',cnff);

fprintf(fnid,'[resp,max_resp,min_resp]=');

fprintf(fnid,'pecomp(excitation,transducer,media,observasjon, ...\n');

fprintf(fnid,'option,flagg,x,y,z,t,elem_pts,centers,config, ...\n');

fprintf(fnid,'plotfig,usr_resdir);\n');

numproc=input('Enter number of node processes required [<=32] : ');

if (numproc < 1)

numproc=1;

elseif (numproc >32)

numproc=32;

end;

fprintf(fnid,'%d\n',numproc);

if (flagg(1)==1) | (option(1) ~='m') | (option(1) ~='f')

fprintf('Fixed focus and Not Movie.. No f-number needed\n');

else

C.4 MATLAB Routine for parallel network - retrieve.m 125

fnum=input('Enter f-number for dynamic aperture control: ');

fprintf(fnid,'%d\n',fnum);

end

fprintf(fnid,'save par_res array_resp max_resp min_resp\n');

fprintf(fnid,'quit\nEOF\n\npvm <<EOF\nhalt\nEOF\n\n# @ queue\n');

fclose(fnid);

sbt=input('Submit now?? [y/n] : ','s');

if sbt=='y'

eval(['!llsubmit ',fn]);

fprintf('Job Submitted\n');

end

end

C.4 MATLAB Routine for parallel network -

retrieve.m

if ~exist('par_res.mat')

fprintf('Sorry, process not finished yet..!\n');

else

ask=input('Is the Configuration file loaded??[y/n] : ','s');

if ask == 'y'

load par_res;

usimcnf('update_calcresp',flagg,option);

else

fprintf('Load the configuration for this simulation first..\n');

end

end


C.5 MATLAB routine - peplot3d.m

%---------------------------------------------------------------

% Routine : peplot3d.m ( PE - Plot in 3D )

%

% Description : Plots the results from the PE- analysis

% of Volumes in 'plotfig' window

%

% Language : Matlab V4.1

%

% Written by : Department of Informatics, Kapila Epasinghe

%

% Version no. : 1.0 KE 20.01.96 First version

% 1.1 KE 01.10.96 Added matrix 'visualize'

% 1.2 KE/KI 01.08.97 Added gif animation

%

% Called by : usimcnf

% Calling : cleanscr,setcomap

%---------------------------------------------------------------

function peplot3d(resp, max_resp, min_resp, excitation,transducer, media, ...

observation, option, flagg, x,y,z,t, config, visualize, ...

plotfig, plotfigOrigHandle, usr_resdir)

% make up a vector of analysis times

if option(1) == 'f'

delta_t=1/(1e3*excitation(14)*observation(15));

TimeStepVector = [observation(4):delta_t:observation(8)];

TLen = length(TimeStepVector);

else

TLen = 1;

end

% INITIALIZATION

clc

C.5 MATLAB routine - peplot3d.m 127

cleanscr(plotfig, plotfigOrigHandle);

figure(plotfig);

MeshCont = 'Mesh';

setcomap(plotfig,5);

if (flagg(2) == 1),

n_elem = transducer(2)*transducer(6);

else

n_elem = transducer(2);

end

%

% determine computer-dependent path separator

%

csys=computer;

if (csys(1:2) == 'PC'),

d = '\';

else

d = '/';

end

% Set only envelope option for the time being

if (length(visualize)<=1)

clc

disp('Set the fillowing plot options: ');

disp(' ');

plot_type = input('-> envelope / rf [e/r] ? ','s');

if all(plot_type ~= 'r')

plot_type = 'e'; % default

end

if all(plot_type == 'r')

plot_option = 'lin';

else

plot_option=input('-> linear /logarithmic [lin/log] ? ','s');

if all(plot_option ~= 'log')

plot_option = 'lin'; % default

end


end

else

fprintf('Plot Options set as :\n');

fprintf('\t 1. ');

if visualize(1)=='e'

fprintf('Envelope\n');

else

fprintf('rf\n');

end

fprintf('\t 2. ');

if visualize(2:4)=='lin'

fprintf('Linear\n');

else

fprintf('Logarithmic\n');

end

fprintf('\t 3. Colormap %s\n',visualize(5:10));

com=input('-> Accept ??? [y/n] : ','s');

if com=='n'

visualize=setvisu(option,flagg,visualize);

end;

plot_type=visualize(1);

plot_option=visualize(2:4);

end

vers=version;

if vers(1)=='4'

if (csys(1:2) ~= 'PC') & (csys(1:2)~='MA')

gif_anim = input('-> make gif-animation [n/y] ? ','s');

disp(' ')

end

end

% Does not set view option...! Not very actual on a 3D visualization.

tmp_view=3;

vtxt = [' View: 3D default'];

% Set the text variables

[Xtxt, Ytxt, Ztxt, Ttxt1, Ttxt2, Ttxt3] = ...


petxt(excitation, media, transducer, option, flagg, z,t, ...

plot_type,plot_option, MeshCont);

% Set axis, 3rd axis either time analysis or distance

if option(1) == '3' | option(1) == 'f'

Xx=[observation(1):1/(1e3*observation(15)):observation(5)]*1e3;

Yy=[observation(2):1/(1e3*observation(15)):observation(6)]*1e3;

if option(3) == 't'

Zz=[observation(4):1/(1e6*observation(15)):observation(8)]*1e6;

else

Zz=[observation(3):1/(1e3*observation(15)):observation(7)]*1e3;

end

end

% Get current axes position modify it (plot axes)

figure(plotfig);

handle1 = gca;

axis('off');

v1 = get(handle1,'Position');

v1(4) = v1(4) - 0.1;

set(handle1,'Position',v1);

% Make another axes for text (text axes)

v2= v1;

v2(1) = v2(1) - 0.1;

v2(2) = v2(2) + v2(4) + 0.05;

v2(4) = 0.1;

handle2= axes('Position',v2);

axis('off');

% Use of max min z not needed???

rmovie='n';

slcx=plt3d(rmovie,'X',Xx,Xx(length(Xx)),observation);

slcy=plt3d(rmovie,'Y',Yy,Yy(1),observation);

if option(1)=='3'

rmovie=input('-> Range Movie on Third axis ?? [y/n] :','s');

end

if option(3)=='t'


zsl=plt3d(rmovie,'T',Zz,Zz(length(Zz)),observation);

else

zsl=plt3d(rmovie,'Z',Zz,Zz(length(Zz)),observation);

end

if rmovie=='y'

TLen=length(zsl);

end

if (plot_type == 'e') & (plot_option == 'lin')

resp = abs(resp');

elseif (plot_type == 'e') & (plot_option == 'log')

resp = 20*log10(abs(resp')/n_elem +eps);

elseif (plot_type == 'r') & (plot_option == 'lin')

resp = real(resp');

end

zsize=size(Zz);

% add some text ...

text(0 , 1.0 , Ttxt1);

text(0 , 0.5 , Ttxt2);

tmp = [Ttxt3,vtxt];

text( 0 , 0.0 , tmp);

drawnow;

figure(plotfig);

axes(handle1);

if (TLen > 1)

M = moviein(TLen,handle1);

end

csys=computer;

if (csys(1:2) == 'PC')

d='\';

else

d='/';

end


if (length(visualize)>1)

% setcomap(plotfig,visualize(5:10));

colormap(visualize(5:10));

else

colormap(hot);

end

% Loop over TLen

% Make Java Imaging..!

%java=input('Do you want to plot images for Javascript??? [y/n] ','s');

%if java=='y'

% javapath=input('Where in www_docs shall I write the images?? ','s');

% javapath=[usr_resdir,'/../../../www_docs/',javapath,'/'];

%end

for PNo=1:TLen

if option(1)=='f'

if (PNo<10)

eval(['load ',usr_resdir,d,'Pict00',num2str(PNo),'.mat'])

eval(['resp = Pict00',num2str(PNo),';']);

elseif (PNo >= 10)&(PNo < 100)

eval(['load ',usr_resdir,d,'Pict0',num2str(PNo),'.mat'])

eval(['resp = Pict0',num2str(PNo),';']);

elseif (PNo >= 100)&(PNo < 1000)

eval(['load ',usr_resdir,d,'Pict',num2str(Pno),'.mat'])

eval(['resp = Pict',num2str(PNo),';'])

end

%

% set resp according to plot options

%

if (plot_type == 'e') & (plot_option == 'lin')

resp = abs(resp');

elseif (plot_type == 'e') & (plot_option == 'log')

resp = 20*log10(abs(resp')/n_elem +eps);

elseif (plot_type == 'r') & (plot_option == 'lin')

resp = real(resp');


end

end

if rmovie=='y'

slcz = zsl(PNo);

else

slcz = zsl;

end

if (TLen>1)

fprintf('plot number %3.0f of totally %3.0f plots\n',PNo,TLen);

end

if all(plot_option == 'log')

slice(Zz,Xx,Yy,clipl(resp,-50),slcz,slcx,slcy,zsize(2));

% if java=='y'

% fname=[javapath,'imag',num2str(PNo),'.gif'];

% printgif('-dgif8',fname);

% end

else

slice(Zz,Xx,Yy,resp,slcz,slcx,slcy,zsize(2));

% if java=='y'

% fname=[javapath,'imag',num2str(PNo),'.gif'];

% printgif('-dgif8',fname);

% end

end

shading flat;

M(:,PNo) = getframe;

%

% make gif files: gif.A1, gif.A2, ..., gif.B1, ...gif.Z9

%

if isempty(gif_anim)| (gif_anim == 'n')

gif_anim = 'n'; % default

else

ext=sprintf('%c%d',64+fix((10+PNo)/10),rem(PNo,10));

filename = ['gif.',ext];

printgif( '-dgif8', filename )

end


%end of loop over TLen

end

fprintf('Starting gif animation\n');

if (gif_anim == 'y')

eval(['!gpgconvert'])

end

fprintf('Ending gif animation\n');

%set axis and view

xlabel(Xtxt)

ylabel(Ytxt)

zlabel(Ztxt)

%

% Play movie

%

if (option(1)=='f') | (rmovie=='y')

more = 'y'

while(more =='y')

clc;

disp('Movie Options');

disp(' ');

n=input('-> Enter number of movie loops [1..100] : ');

if (n == '')

n=10;

end

fps=input('-> Enter speed in frames/sec [1..20] : ');

if (fps == '')

fps= 8;

end

% make up a vector that plays the movie with a break after each loop.

% To maintain speed, this is done with a vector N to specify the order

% of the frames

%

last_picture=size(M,2);

N=[n,1:last_picture ,ones(1,fps)*last_picture]; % break for 1 sec


disp(' ');

disp('Movie is playing ...');

figure(plotfig);

movie(M,N,fps);

disp(' ')

more = input('-> More movie ??? [y/n] : ','s');

end

%

% save movie matrix ?

%

% if (input('-> Do you want to save the movie [y/n] ? ','s') == 'y')

% [fname,path]=uiputfile('lastone.mvi','Save movie as ...');

% if ~(fname(1) == 0)

% eval(['save ',path,fname,' M N']);

% disp(' ');

% disp(['Saved movie as ',path,fname]);

% end

% end

% mpgwrite(M,hot,'testmeg.mpeg');

end

C.6 MATLAB routine - plt3d.m

%---------------------------------------------------------------

% Routine : plt3d.m ( PE - Plot in 3D )

%

% Description : Accepts data for slice movie in peplot3d.m

%

%

% Language : Matlab V4.2c-1

%

% Written by : Department of Informatics, Kapila Epasinghe

%

C.6 MATLAB routine - plt3d.m 135

% Version no. : 1.0 KE 20.01.96 First version

%

%

% Called by : peplot3d.m

% Calling :

%---------------------------------------------------------------

function res_vec = plt3d(mv,ax,vect,sl_vec,observation)

if mv=='y'

fprintf('Range for Axis %s : %d to %d',ax,vect(1),vect(length(vect)));

start=input('Enter start value :');

if isempty(start) | (start<vect(1)) | (start>vect(length(vect)))

start=vect(1);

end

finish=input('Enter end value :');

if isempty(finish) | (finish<vect(1)) |

(finish>vect(length(vect)))

finish=vect(length(vect));

end

res_vec=[start:1/observation(15):finish];

else

disp([' ']);

disp(['Obeservation slices for ',ax,' Axis']);

mod_sl=input('Modify slices [y/n] : ','s');

if isempty(mod_sl) | (mod_sl~='y')

mod_sl='n';

end

if mod_sl == 'y'

bkgrd=input('-> Background display ?? [y/n] :','s');

if bkgrd=='n'

sl_vec=[];

end

fprintf('Range for the axis %s: %d to %d',

ax,vect(1),vect(length(vect)));

num=input('Enter the number of slices(default 0, max 5) : ');

if isempty(num) | num>5 | num<0

num=0;

end

for k=1:num


temp_str=['Axis ',ax,' slice ',num2str(k),':

Enter value ->'];

tem_val= input(temp_str);

while (tem_val<vect(1) | tem_val>vect(length(vect))) |

find(sl_vec==tem_val)

temp_str=['Axis ',ax,' slice ',num2str(k),':

Re-enter value ->'];

tem_val=input(temp_str);

end

sl_vec=[tem_val sl_vec];

end

end

res_vec=sl_vec;

end

Simulation and visualization of ultrasound fields · t in tec hnology.F or exam-ple, astronomers...

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Transcript of Simulation and visualization of ultrasound fields · t in tec hnology.F or exam-ple, astronomers...