Image Processing Reference

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back-propagation technique, and interpolation procedures have been studied

(Kaveh and Soumekh, 1985) and both approaches compared when they were

first proposed (Pan and Kak, 1983). Either procedure discussed above can be

adopted when using the first Born approximation or Rytov approximation.

The basic formulation of inverse scattering is due to Wolf (1969) who stud-

ied the determination of object structure within the first Born approximation.

The earliest and probably the best-known application of an inverse scattering

theory in the first Born approximation was in experiments to study the struc-

ture of crystals and molecules using x-rays. Two major difficulties arise in such

structure determination problems. The first is that the intensity of the scattering

pattern of the object (or more accurately for a periodic structure, diffraction) is

measured rather than the complex scattered field, and the second is that these

intensity data are sampled at a rate determined by the reciprocal of the unit

cell dimensions of the crystal. If the complex scattered field data were available

over the entire (far field) scattering domain, then this rate of sampling would

be reasonable according to Shannon's Sampling Theorem to adequately repre-

sent the scattered field The intensity data should Fourier transform to give the

autocorrelation of the unit cell, but the measured data are undersampled for this

purpose, being sampled at the rate determined by the reciprocal of the unit cell

rather than its autocorrelation function. In addition, physical constraints and

signal-to-noise ratio realities mean that only a limited area of data is measurable.

The consequences of this are that the recovery of object information is clearly

severely ill posed requiring phase retrieval and both interpolation and extrapo-

lation of the scattered field It is for a good reason that many Nobel prizes were

awarded for recovering the image of specific objects such as DNA or hemoglobin.

In practice, improved estimates of the scattered field can be made from

this incomplete data set provided additional
a priori
information about the

symmetry or structure of the unit cell is available. More recently, in the opti-

cal domain, use has been made of (quasi)monochromatic laser sources and

holographic techniques to measure the complex scattered field As was obvi-

ous to Wolf (1969), holograms store information about the three-dimensional

(3-D) structure of a scattering object, and it should be possible to compute

this structure from measurements performed on the hologram. The calcula-

tion of the amplitude and phase of the scattered field from measurements of

the amplitude variations of the hologram was first described by Wolf and is a

much more practical solution today than it was over 40 years ago. This added

layer of difficulty is not ignored here, but our focus is on the inverse scat-

tering procedures themselves and so we restrict our further discussions to

lower frequency radiation sources. Clearly, even with the ultrasound waves

in the megahertz region or microwave region, however, the maintenance of a

precise phase reference while varying the directions of illumination might be

a problem if it is not measured carefully. It is also difficult to measure all the

data required to determine the 3-D structure of the objects, and most of the

examples we show will be 2-D structures, not for any fundamental reason but

purely for convenience of explanation and illustration.

1.3 dIFFRACtIon toMogRAphy

We have emphasized how the inverse scattering problem is the determination

of physical parameters and features of an unknown object from a limited set

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