Image Processing Reference

In-Depth Information

problems. In the limit of the wavelength becoming relatively small, geometrical

optics or ray-based approximations become reasonable. In the very high fre-

quency limit, for example, when using x-rays, one can assume that the radia-

tion emerging from an object has not been refracted at all, and the measured

data are interpreted as a shadow of the attenuation in the object. The math-

ematics describing this is well established, dating back to Radon (1986). Johann

Radon's original paper was published in 1917 (Radon, 1917). The Fourier trans-

form plays an important role here and throughout this topic (see Appendix A).

The technique of computed tomography, which incorporates a Radon transform

(Wolf, 1969), uses projection data which measures the line integral of an object

parameter, for example, of
f
(
x,y
) in the equation shown below, along straight

lines (
y
-axis in this example). This enables the Fourier Slice Theorem to be used

to build up information about
F
(
k
x
,k
y
) by rotating the illumination direction
.

+∞

+∞

+∞

+∞

∫∫

∫∫

Fk

(,)

0 =

fxye

( ,)

ik x

x

(

)
dd

xy

=

fxyy

(, )

d

e

ik x

x

(

)
d

x

(1.1)

x

−∞

−∞

−∞

−∞

where
k
x
and
k
y
are the spatial frequency variables that have units of recipro-

cal distance, that is,
k
x
x
is dimensionless. When object constitutive param-

eter fluctuations or inhomogeneities such as refractive index fluctuations in

a semitransparent object are comparable in size to the interrogating wave-

length, then scattering or diffraction effects become significant As we shall

see, Fourier data on the object are still obtainable in this situation provided

the Born or Rytov approximations are valid. We will describe these approxi-

mations, which allow inversion algorithms to be formulated, and we will dis-

cuss in detail the criteria for their validity. When inverting Fourier data there

is the question of how to make the best use of the limited set of noisy samples

available. At optical frequencies, there is also an additional problem: that the

phase of the scattered field may only be measured with difficulty Some meth-

ods for phase retrieval are discussed in Appendix B.

Usually, approximations are employed to make the scattered fields (which

can be expressed by Fredholm integral equations of the first kind) more trac-

table for numerical computation. The merits of the Born and the Rytov approxi-

mations, and more sophisticated techniques derived from them, have spawned

a lot of controversy over the years. A principle cause for controversy is that these

approximations are based on the interpretation given when strong inequalities

are met, in order to simplify (or linearize) the governing equation. The physical

interpretation of imposing these inequalities can be rather subjective. It is also

problematic that sometimes these approximations appear to provide reasonably

good images when one might not expect them to. This issue also illustrates one

of the cautionary messages to be conveyed when working with inverse prob-

lems, which is that deliberate or inadvertent inverse crimes can be committed!

These are crimes by which, because of the difficulty of acquiring real data

from known objects with which to test an inversion method, the direct prob-

lem is solved to generate data. Occasionally approximations made in solving

the direct problem are the very ones employed in the inverse method, thereby

increasing the chances that the recovered image will look good. Consequently,

we spend some time in this topic describing the importance of understanding

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