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