Ultrasound Imaging: Lecture 2

• Absorption

• Reflection

• Scatter

• Speed of sound

• Signal modeling

• Signal Processing

• Statistics

Interactions of ultrasound with tissue

Image formation

Jan 14, 2009

• Steering• Focusing• Apodization• Design rules

Beams and Arrays

Anatomy of an ultrasound beam• Near field or Fresnel zone

• Far field or Fraunhofer zone

• Near-to-far field transition, L

2aL

L

Lateral distance (mm)Depth (mm)

Anatomy of an ultrasound beam

• Lateral Resolution (FWHM)

FWHM

Lateral distance (mm)Depth (mm)

numberFR

aFWHM 2

Anatomy of an ultrasound beam

• Depth of Field (DOF)

DOF

Lateral distance (mm)Depth (mm)

2)(7 numberFDOF

Array Geometries

• Schematic of a linear phased array

• Definition of azimuth, elevation

• Scanning angle shown, , in negative scan direction.

ya (elevation)

xa (azimuth)

za (depth)

array pitch

Acoustic beam

t

trtrp

,

,

N

ii ttrhWtrh1

),(,

Some Basic Geometry

• Delay determination:

– simple path length difference

– reference point: phase center

– apply Law of Cosines

– approximate for ASIC implementation

• In some cases, split delay into 2 parts:

– beam steering

– dynamic focusing

x

z

x

r

r

0

rx

crr x

rrrxxc

22 cos21

fs

Far field beam steering

• For beam steering:– far field calculation

particularly easy

– often implemented as a fixed delay

c

xs

sin

x

z

xr

0

Beamformation: Focusing• Basic focusing type beamformation

• Symmetrical delays about phase center.

-40 -30 -20 -10 0 10 20 30 40

-20

-10

0

10

20

point source

wavefronts before correction

transducer elements

delay lines

wavefronts after correction

summing stage

Beamformation: Beam steering

• Beam steering with linear phased arrays.• Asymmetrical delays, long delay lines

-40 -30 -20 -10 0 10 20 30 40

-20

-10

0

10

20

point source

wavefronts before correction

array elements

delay lines

wavefronts after beam steering and focusing

summing stage

Anatomy of an ultrasound beam• Electronic Focusing

Grating Lobes

• Linear array:

– 32 element array

– 3 MHz

– ‘pitch’ l = 0.4 mm

– = 0.51 mm

– L= N l = 13 mm

• How to avoid:

– design for horizon-to-horizon safety

275.14.

51.)(

)(

g

g

Sin

lSin

How many elements?

What Spacing?

gl

g

l

Main Lobe

Grating Lobe

Array design

• Linear array:

– 32 element array

– 3 MHz

– pitch l = 0.4 mm

– = 0.51 mm

– Larray= N l = 13 mm

• How to avoid:

– design for horizon-to-horizon safety

2

l

How many elements?

What Spacing?

Apodization

• Same array:– 32 element array– 3 MHz– pitch l = 0.4 mm– = 0.51 mm– Larray = N l = 13 mm

• With & w/o Hanning wting.• Sidelobes way down.• Mainlobe wider• No effect on grating lobes.

Summary of Beam Processing

• Beam shape is improved by several processing steps:

– Transmit apodization

– Multiple transmit focal locations

– Dynamic focusing

– Dynamic receive apodization

– Post-beamsum processing

• Upper frame: fixed transmit focus

• Lower frame: the above steps.

I INTERACTIONS OF ULTRASOUND WITH TISSUE

Some essentials of linear propagation

Recall the equation of motion

t

v

x

p

0 (1)

Assume a plane progressive wave in the +x direction that satisfies the wave equation

ie)(

0kxtepp

(2)

Substituting 2 into 1 we have

t

vjkep kxtj

0

)(0

Z

p

c

pv

epf

ejk

j

pv

dtejkp

v

kxtj

kxtj

kxtj

0

00

)(

0

0

)(

0

0

2

2

Acoustic impedance

(3)

cZ 0Where

= Characteristic Acoustic Impedance

Define a type of Ohm’s Law for acoustics

Electrical:Acoustical:

Extending this analogy to Intensity we have

vZp

IRV

20

20

2

1

2

1Zv

Z

pI

Propagation at an interface between 2 media

111 cZ 222 cZ

iP

rP

tP

xktj

tt

xktjrr

xktjii

ePp

ePp

ePp

2

1

1

Define Reflection/Transmission Coef

i

t

i

r

p

pT

p

pR ,

You will show:

21

2

12

12 2ZZ

ZT

ZZZZ

R

Example: Fat – Bone interface

38.16.7)6.7(2

38.16.738.16.7

TR

70.0 69.1

(4)

(5)

THE DECIBEL (dB) SCALE

refsBd A

ALogA 10)( 20

Where A = measured amplitudeAref = reference amplitude

In the amplitude domain

6 dB is a factor of 2-6 dB is a factor of .5 (i.e. 6dB down)

20 dB is a factor of 10-20 dB is a factor of .1 (i.e. 20dB down)

(6)

0

-10

-20

-30

-40

-50

Reflection Coefficients

Air/solid or liquidBrass/soft tissue or waterBone/soft tissue or water

Perspex/soft tissue or water

Tendon/fatLens/vitreous or aqueous humourFat/non-fatty soft tissuesWater/muscle

Fat/water

Muscle/blood

Muscle/liver

Kidney/liver, spleen/blood

Liver/spleen, blood/brain

Water/soft tissues

R = 1.0

R = .1

R = .01

Ref

lect

ion

Coe

f. dB

3) ULTRASOUND IMAGING AND SIGNAL PROCESSING

Thus far we have been concerned with the ultrasound transducerand beamformer. Let’s now start considering the signalprocessing aspects of ultrasound imaging.

Begin by considering the sources of information in an ultrasound image

a) Large interfaces, let a = structure dimension

a- specular reflection-

- reflection coefficient12

12ZZZZ

R

where cZ

- strong angle dependance- refraction effects

density speed of sound

b) Small interfaces

a-

- Rayleigh scattering

Cosak

D0

0

0

032

2

33

3

Compressibility Density

and Arp , Dr

eikr(7)

Morse and Ingard Theoretical Acousticsp. 427

SCATTER FROM A RIGID SPHERE Cosr

ac

Ds 3134 32

*

*

SCATTER FROM A RIGID SPHERE (Mie Scatter)

*

ATTENUATION

= absorption component + reflectivity component

xepxp 0

The units of are cm-1 for this equation. However attenuationis usually expressed in dB/cm. A simple conversion is givenby

1686.8 cmcm

dB

Attenuation in Various Tissues

Speed of Sound in Various Tissues

0%

5%

10%

15%

-5%

-10%

Assumed speed of sound = 1540 m/s

SUMMARY ULTRASONIC PROPERTIESTable 1

Material Speed of Sound Impedance Attenuation Frequency

ms-1 Kg m-2 s-1

X 106

At 1 MHz (dB cm-1) Dependency

water 1490 @ 23ºC 1.49 0.002 2

muscle 1585 @ 37ºC 1.70 1.3-3.3 1.2

fat 1420 @ 37ºC 1.38 0.63 1.5-2

liver 1560 @ 37ºC 1.65 0.70 1.2

breast 1500 + 80 @ 37ºC ------ 0.75 1.5

blood 1570 @ 37ºC 1.70 0.18 1.2

skull bone 4080 @ 37ºC 7.60 20.00 1.6

air 331 @ STP 0.0004 12.00 2

PZT 4300 @ STP 33.00 ------ --

smc /1540

2.2 Modeling the signal from a point scatterer

Imagine that we have a transducer radiating into a medium and we wish to know the received signal due to a single point scatterer located at position

By modifying the impulse response equation (Lecture 1 Equ. 25 ) we can write:

r

trhtrhtstgtgt

VktrV rtout ,*,***,*, 2

0

transmit + receiveelectromechanical IR’s

scatterer IR

transmitIR

receiveIR

pulse (t)

trHtpulse

trhtrhtpulsetrV rtout

,*)(

,*,*,

easily measured

Now consider a complex distribution of scatterers

Isochronousvolume

rxri

(1)

(2)

(3)

(4)

At any point in the isochronous volume there exists a transmit –receive path length divided by c for a time, t, such that

c

zt

c

ll

21

zl1

l2

If we look at the four field points shown on the previous pagewe would see the following impulse responses

(1)

(2)

(3)

(4)

The total signal for a given ray position rx is given by

trHrWtpulsetrVout xiN

iiix ,

1*)(,

(9)

scattererstrength

The resultant signal is the coherent sum of signals resultingfrom the group of randomly positioned scatterers that make up the isochronous volume as a function of time.

A useful model of the signal is:

ttCostatytVout 2

Envelope Modulatedcarrier

Phase

Grayscale informationfor B-scan Image

How do we calculate a(t) and (t)?

Velocity informationfor Doppler

(10)

3.3 Hilbert Transform

The Hilbert transform is an unusual form of filtration in which thespectral magnitude of a signal is left unchanged but its phase is altered by for negative frequencies and for positive frequencies

2 2

Definition

)(*1

1

xfx

xdxx

xfxFH

(11)

In the frequency domain

sH FsjxF )sgn(

Consider the Hilbert transform of Cos x

RE RE

(12)

IMIM

xCos sjSgn

The application of two successive Hilbert transforms resultsin the inversion of the signal – we have 2 successive rotations in the negative frequency range and 2 rotations in the positive frequency range. Thus the total shift in each direction is .

2

2

1

II

xfxx

xFx H

111

sFsjsj sgnsgn

xfF

sF

s

1

1

The Hilbert transform is interesting but what good is it?

ANALYTIC SIGNAL THEORY

Consider a real function . Associate with this function another function called the analytic signal defined by:

where = Hilbert Transform

The real part of the analytic signal is the function itself whereas the imaginary part is the Hilbert transform of the function.

Note that the real and imaginary components of the analyticsignal are often called the “in phase”, I, and “quadrature”, Q,components.

tjztytf tz(13)

ty

Just as complex phasors simplify many problems in AC circuit analysis the analytic signal simplifies many signal processing problems.

The Fourier transform of the analytic signal has an interestingproperty.

0,2

0,0

][

sY

s

YsSgnY

YsSgnjjYtjzty

s

s

ss

s sy

0

2

s

Y s

(14)

Equation 14 gives us an easy way to calculate the analyticsignal of a function:

1) Fourier transform function2) Truncate negative frequencies to zero3) Multiply positive frequencies by 24) Inverse Fourier Transform

Recall that our resultant ultrasound signal can be expressedas:

ttCostaty 2

Its analytic signal is then

ttetatf

2 (15)

which on the complex plane looks like:

IM

RE ty

t ta tz

Where

and the phase is given by

tztyta 22

tytz

Tant)(1

(16)

(17)

a(t) envelope

Demodulation: estimate using

1) Analytic signal method using FFT (slow)2) Analytic signal using baseband quadrature approach3) Sampled quadrature

)(),( tta QI ,

Baseband Quadrature Demodulation

X

X

Low Pass

Low Pass

tCos 2

tQ

tSin

2

ty

tIt )Re(

Baseband Inphase Signal

)()Im( tQt

Baseband Quadrature Signal

tCostCosa

tCostCosaI

tt

ttt

22

2

22

Use shift and convolution theorems to calculate spectra

ttnote :

2

2 tjt eAI

(slowly varying)

tjeA 2 2

21

tjt eAI

21

tjtjt eAeAI

2

1

2

1

tt CostaI 21

tjt etas )(

2

1

Similarly

tt SintaQ )(21 Baseband

AnalyticSignalNo carrierPhase preserved

ttt

t

ttt

QIa

a

CosSinaQI

22

2

22222

2

4

1

)(4

1

Thus

)()(

)()(

)(

tItQ

ArcTan

I

QTan

t

t

tt

and

Sampled Quadrature

Begin with the signal of the ultrasound waveform

ttt Cosay 2

Sample with period 1T

* **

)(nTI

Tt

IIIy t

)(nTQ

Tt

IIIy t

Recall that the quadrature signal is the Hilbert Transform of theinphase component of the analytic signal i.e. for a cos wave it is a negative sine wave. Thus we see that . . .

nTIT

tIIIy t

If the inphase and quadrature signals are slowly varyingwe can get the quadrature signal simply by sampling theinphase signal 90º or ¼ period later

Sampling t = nT for I samplest = nT+T/4 for Q sample

142)()(

)2()()(

T

nTTnTCosnTanTQ

nTrTCosnTanTI

let

nTSinntanTnCosnTanTQ

TnCosnTanTnCosnTanTI

)()22()()(

)()()2()()(

(18)

1

Overall Imager Block Diagram

D opplerBeam form er

D ig ita lR eceive

Beam form er

Beam form erC entra lC ontro l

D ig ita lT ransm it

Beam form er

T ransm itD em ux

R eceiveM ux

TransducerC onnectors

SystemC ontro l

Im ageProces-

sing

23 4 5

6

Imaging System Signals

D opplerBeam form er

D ig ita lR eceive

Beam form er

Beam form erC entra lC ontro l

D ig ita lT ransm it

Beam form er

T ransm itD em ux

R eceiveM ux

TransducerC onnectors

SystemC ontro l

Im ageProces-

sing

23 4 5 6

1

Coarse and Fine Beamforming Delays

Ho()e-j/4

Ho()e-j/2

Ho()e-j3/4

Ho()

MUXFIFO

Input fromADC at 20to 40 MHz,8 to 12 bits

Output withdelay accuracyup to 160 MHz

To apodizationand furtherprocessing

CoarseDelay

ControlFine

DelayControl

SIGNAL STATISTICS

Recall that the ultrasound signal is the sum of harmoniccomponents with random phase and amplitude. It can be shownthat the probability density function for such a situation isGaussian with zero mean i.e.

2

2

221

)(

y

eyp

(19)

The quadrature signal will also be Gaussian with the same standard deviation

22

221

)(

z

ezp

(20)

Since p(y) and p(z) are independent random variables the jointprobability density function is given by

2

22

2

2

2

2

22

22

2

1

2

1

2

1),(

zy

zy

e

eezyp

(21)

The probability of a joint event (corresponding to a particularamplitude of the envelope) is the probability that:

)(zp

)( yp

total area = ada2

The probability that a lies betweena and a + da is

222 zya

daea

daapa

22

222

2)(

adad

adad

So that the probability density function for the radiofrequency signal is given by

22

22

a

ea

ap

Rayleigh Prob.Density function

aa

)(ap

few white pixels

many gray pixels

fewblackpixels

The speckle in an ultrasound image is described by thisprobability density function. Let’s define the signal as and the noise as the deviation from this value

arms

21

2212 aaaN Thus

daea

daapaa

a

o

o

2

2

22

2

Recall

2a

Thus:

21

21

22

22

2

22

2

Na

SNR

SNR = 1.91 and is invariant (25)

Note that the SNR in ultrasound imaging is independent of signal level. This is in contrast to x-ray imaging where thenoise is proportional to the square root of the number ofphotons.

Speckle Noise in an Ultrasound Image

a

ias

as

i

00

x

Let’s make several independent measurements of so and si

These measurements will form distributions

i 0

is 0s

The parameter used to define image quality includes boththe observed contrast and the noise due to speckle in the following fashion:

Define Contrast:

Define Normalizedspeckle noise as:

and finally, define our quality factor as the contrast to speckle noise ratio (CSR)

0

0

s

ss i

0

21

220

si

220

0

i

issCSR

(26)

Suggested Ultrasound Book References:

General Biomedical Ultrasound (and physical/mathematical foundations): “Foundations of Biomedical Ultrasound”, RSC Cobbold, Oxford Press 2007.

General Biomedical Ultrasound (bit more applied): “Diagnostic Ultrasound Imaging: inside out” TL Szabo Academic Press 2004.

Ultrasound Blood flow detection/imaging: “Estimation of blood velocities with ultrasound” JA Jensen Cambridge university press 1996

Basic acoustics: “Theoretical Acoustics” PM Morse and KU Ingard, Princeton University Press (many editions).

Bubble behaviour: “The Acoustic bubble” TG Leighton Academic Press 1997.

Nonlinear Acoustics: “Nonlinear Acoustics” Hamilton and Blackstock, Academic Press 1998.