Image Processing Reference
In-Depth Information
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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