Radon Transform and Its Applications

Radon Transform and Its Applications Guoping Zhang Department of Mathematics Morgan State University CCICADA Retreat Baltimore 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

How X-ray CT works

Image Reconstruction

Radon transform • Two dimension Radon transform

Radon transform • Two dimension normal Radon transform

Inverse of Radon Transform

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

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

Explicit GRT • If shapes can be expressed by the explicit form, • Then GRT becomes • The curve to be detected is modeled as

Detection of Curves • Let • Then

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

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.

Generalized Slice Theorem • Case 2: m=n>2, is linear in • Let • are angular coordinates. • Let

Generalized Slice Theorem • Take 1D Fourier transform • Take n dimension inverse Fourier transform

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.

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

Radon Transform and Its Applications