Colton-Kirsch Inverse Scattering Algorithms

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Inverse scattering algorithms have generally fallen into two broad

categories: fast, approximate linear algorithms

(often based on fast Fourier transforms)

or slow, accurate nonlinear algorithms. The linear algorithms often

amount to simply inverting a Fourier transform. By contrast, nonlinear

algorithms, such as the Distorted Born Iterative Method, usually require

some sort of computationally expensive Newton-like search.

Historically, there have been few options available between these two

extremes.

A notable exception is the so-called “regularized linear sampling” method

developed by David Colton, Andreas Kirsch, and their colleagues.

Although the Colton-Kirsch methods have received much attention

in the mathematical literature, they do not seem to be very well known

in the engineering community. This web page seeks to bridge that gap.

Although some immensely complex mathematical theory underlies Colton and

Kirsh’s method, the resulting algorithm is remarkably simple. In the

original method, for each point

**y** in the desired image, one solves a linear

equation **Fg = f(y)** for **g**, where

**f(y)** is a vector of complex exponentials

which depends on the position

of the point and **F** is a matrix formed from the measured data.

The theory dictates that the norm of **g** becomes large at the border

of the scatterer. Hence, to visualize the scatterer surface, one can simply

plot an image of the norms of the **g**‘s.

Since the measurements **F** are typically noisy, the method has been

extended

to incorporate regularization via Morozov’s discrepancy principle.

This principle allows for the fact that the uncertainly lies in the operator

**F**, instead

of the right hand side **f** (as is typical in most applications).

We will refer to this method as CK-A (without regularization) or

CK-AR (with regularization).

Note that although the algorithm involves solving sets of linear equations,

it is represents a highly nonlinear kind of processing. It also does not

assume any linearizing approximations (such as the Born or Kirkoff

approximations which underly most linear reconstruction algorithms).

A remarkable aspect of the method is that no *a priori* knowledge of the

boundary condition is needed.

The exact same perscription holds

for Dirichlet, Neumann, or impedance boundary conditions. This is in

sharp contrast to traditional iterative nonlinear methods, which require

prior knowledge of the boundary conditions. It is also helpful in cases

where the scatterer is a penetrable, inhomogenious medium; in these

situations, the method gives the support of the scatterer. The method has

hence been helpful in biomedical applications, such as

detecting leukemia.

Kirsch

derived a variation on the method which involves solving

**(F^star F)^{1/4} g = f(y)** (please forgive the pseudo-Latex)

at each point,

and again looking for points where **||g||** blows up.

We will refer

to this method as CK-B (without regularization) or CK-BR

(with regularization).

Most of the work on Colton-Kirsch methods has been in two dimensions.

The scatterer is assumed to be an infinitely long cylinder whose profile

does not change along the z-dimension. In the acoustic case, one can

then deal with 2-D Helmholz equation. In the electromagnetic case,

if the field is polarized so that the electric or magnetic fields runs

parallel to the axis of the scatter, Maxwell’s equations reduce to the

2-D Helmholz equation.

Although the method has been extended to the three-dimensional scalar case,

our exposition will focus on two dimensional problems.

## Simulated Data

We simulated scattering data for these airplane shapes:

The shapes are top-down

profiles of the VFY-218 (upper left),

the B-2 (upper right), the F-15 (lower left),

and the YF-23 (lower right).

Remember that we

are assuming the objects are infinitely long cylinders,

and not the three-dimensional shape of the original aircraft.

The VFY-218, F-15, and YF-23 are is approximately 15.5, 19.5, and 20.5

meters long, respectively. Although the real

B-2 is much larger than the other aircraft, we have artifically scaled

our model to make it 15.5 meters wide.

We computed data at 100 MHz and 200 MHz for both TE

and TM polarizations,

with c = 3 times 10E8 meters per second. A full 360-degrees of

incident and observation angles were used, with data computed at three degree

increments, so **F** is a 120 by 120 matrix.

## Reconstruction Results

The images in this section are 100 by 100 pixels, with a spacing of

0.25 meters between pixels. We show reconstruction results using the

CK-AR and CK-BR methods. (We use the regularized methods even though we

have not explicitly added any noise to the data.)

We are displaying **1/||g||** (the reciprocal of the norm of **g**) on

a heated object scale. Hence, large **||g||**,

which we expect approaching the

boundary, are indicated by black. Small **||g||** are indicated by

bright colors like yellow and white. (Interestingly, the outline of

the scatter is most clear where we see yellow – i.e. where **||g||**

is small! We currently do not have a good theoretical explanation as to why.)

Some of

the CK-AR images look a bit dim, since a couple of pixels usually are

of much greater magnitude than the rest – this could be remedied by

adjusting the color scale, but we haven’t done that here.

Here are the results for the YF-23.

Top row corresponds to the TE case, bottom row to the TM case.

From left to right, the columns correspond to: CK-AR at 100 MHz,

CK-AR at 200 Mhz, CK-BR at 100 MHz, and CK-BR at 200 MHz. (Notice that

the TM case results in much clearer reconstructions than the TE case.)

Here’s the results in the same sequence for the F-15:

Here’s the results in the same sequence for the scaled B-2:

Finally, here’s the results in the same sequence for the VFY-218:

### Limited Angle Experiments

Here are some experiments using the VFY-218 shape

in which we tried using a limited aperture.

The incident and observation apertures are coincident and of size

60 degrees (so one is essentially using 20 by 20 submatrices of

**F**). The following images use CK-AR. The top, middle, and bottom

rows correspond to illuminations/observations from the bottom, side, and

top, respectively. From left to right, the columns correspond to:

TE case, 100 MHz; TM case, 100 Mhz, TE case, 200 MHz, TM case, 200 Mhz.

These images are as above, except using CK-BR:

Last updated 8/17/00. Send comments or questions to

lanterma@ifp.uiuc.edu.