Radon Transform and Its Applications

Guoping Zhang Department of Mathematics

Morgan State University

CCICADA RetreatBaltimore

March 7-8, 2010

Outline

Radon transform and X-ray Tomography Generalized Radon transform Micro-local analysis and Inversion of GRT Some thoughts on DHS research

Radon Transform and CT Radon transform (RT) was named after J.Radon who

showed how to describe a function in terms of its (integral) projection in 1917.

Based on RT, one of the major inventions in last century, CT scanner was invented (1967) by Drs. Cormack and Hounsfield who got the Nobel-prize in Medicine 1979.

RT has been used to detect lines in the image Generalized RT was proposed to shape detection i.e.

detect arbitrary curves, such as circle, hyperbola etc.

Mathematical Model of CT The goal of X-ray computerized tomography (CT) is to get a

picture of internal structure of an object by X-raying the object from many different directions

Physical setting: as X-ray travel on a line L from the X-ray source (emitter) through the object to an X-ray receiver (detector), they are attenuated by the material on the line L. According Beer’s law, the X-rays at a point x are attenuated proportionally to the number of X-ray photons (called the intensity of X-rays) there and the proportionality constant (called linear attenuation coefficient) is proportional to the density of the object if the X-ray is monochromatic.

Mathematical Model

Let be the density function of the object and be the intensity of X-rays at position x on the line L.

The Beer’s law means

)(xf)(xI

)()()(' xIxfxI

)(:)(])(det

)(ln[ LRfdxxf

ectorI

emitterILL

How X-ray CT works

Image Reconstruction

Radon transform

Two dimension Radon transform

( )( , ) ( , )R g p g x px dx

( , ) ( )g x y y px dxdy

( , )p

Radon transform

Two dimension normal Radon transform

( )( , ) ( cos sin , sin cos )R g g s s ds

( , ) ( cos sin )g x y x y dxdy

Inverse of Radon Transform

RsSdxxsxfsRf N

RN

,,)()(),( 1

dxgxgRNS

1

),()(*

)(||))(( FggIF

NN IRR 1*11 )2(2

1

Main applications

1. Image formation 2. Features detection 3. Pattern recognition/Target identification Imaging Devices Invented: CT, MRI, FMRI, some security devices

Detection of Lines

A line is modeled by a delta function

Its Radon transform is

( , ) ( * *)g x y y p x

( )( , ) , *, *;R g p if p p 0, *, *;if p p 1 | * |, *.p p if p p

Generalized Radon Transform

Let be a continuous signal, let denote an m-dimension

parameter vector GRT is defined as follows

Shapes expressed by the parameter form

1( , , )m

( )( ) ( , ) ( ( , ; ))GR g g x y x y dxdy

( , ; ) 0x y

( , )g x y

Explicit GRT If shapes can be expressed by the explicit form,

Then GRT becomes

The curve to be detected is modeled as

( ; )y x

( )( ) ( , ) ( ( ; ))GR g g x y y x dxdy

( , ) ( ( ; *))g x y y x

( , ( ; ))g x x dx

Detection of Curves

Let Then

| ( ; ) ( ; *) | 1, ,ix x x x i I

1

( )( ( ; ) ( ; *))

| ( ; ) ( ; *) |

Ii

i

x xx x

x xx x

1

1( )( )

| ( ; ) ( ; *) |

I

i

GR gx x

x x

Inversion of GRT Central Slice Theorem for normal RT

Is there similar inversion for GRT? Assume that can be solved for one of the parameters, e.g. Let

2( cos , sin ) ( )( , ) iG R g e d cos , sinx yk k

2 ( )( , ) ( , ) x yi x k y kx y x yg x y G k k e dk dk

1 1( ; ) 0, ( , , ), ( , , )n mx x x x

m

1 1( , , ), ( ; )m m x

Inversion of GRT

The GRT of a given function

Take the 1D Fourier transform with respect to

Case 1: m=n=2, is linear in We obtain the central slice theorem.

( )g x

( )( ) ( ) ( ( ; )) , ( , )m mGR g g x x dx m

2 ( ; )( )( , ) ( , ) ( )m

i xmF GR g g x e dx

x

Generalized Slice Theorem Case 2: m=n>2, is linear in

Let are angular coordinates. Let

x

1

( ; ) ( )n

i ii

x x

2

01

( )n

ii

1 1( , , )n 1 1( , , ) ( , , )n nk k k

1 1 2 1 2

3 1 2 3

1 2 1

cos , sin cos ,

sin sin cos , ,

sin sin sinn n

k k

k

k

0

Generalized Slice Theorem

Take 1D Fourier transform

Take n dimension inverse Fourier transform

( ) ( )( , ) ( , )m mG k F GR g

2( ) ( ) i k xg x G k e dk

Nonlinear case If is nonlinear in The inverse transform is extremely difficult to obtain. There are

some existing works about the approximate inverse transform. We generalize the problem and consider the following Fourier

Integral operator

Micro-local analysis theory can be used to obtain the approximate inverse of the above Fourier integral operator.

x

( , )( , ) ( ) ( , , ) i xFIO g g x a x e dx

Some thoughts on DHS research

Watch a video Questions: 1. Has RT been fully utilized in imaging process? 2. How can we identify some hazard/dangerous

targets by detecting their physical parameters such as thermal conductivity and dielectric constant etc?

Acknowledgement

Thanks for the support from CCICADA