Image Processing Reference
In-Depth Information
varying the direction of illumination. This difficulty and the amount of data
required to determine the 3-D structure are clearly reduced if some assump-
tions about object symmetry can be made. It was concluded that since the
experimental procedure is so complicated even for a simple object, further
experimental or theoretical advances would be required in order to exploit
fully the potential value of inverse scattering in many fields
The most attractive outcome would be the availability of an algorithm to
calculate the phase of the scattered field directly from its measured intensity
distribution without the need for a reference wave. We discuss this next, and
in more detail, the problem of limited Fourier data. The phase retrieval prob-
lem has been the subject of many reviews and can be divided into 1-D prob-
lems and more than 1-D problems.
In either case one looks to the analytic properties of the scattered field to
indicate the range of possible phase ambiguities. In 1-D, a band-limited field
can be represented by an infinite product of linear factors, each describing a
zero of the field in the complex plane. Complex zero locations can be complex
conjugated and leave the magnitude changed but not the phase, thus generat-
ing phase ambiguities. In more than 1-D, depending upon the object, the field
may factor into any number of factors or none at all, making zeros difficult to
characterize and the phase ambiguities unclear. It is generally assumed that
for a function of a continuous variable the phase is unique in 2-D or 3-D, but
with a finite number of samples, there is necessarily nonuniqueness.
Let us assume that a Fourier relation can be written between the scattered
field and an “object function” which is of finite support. This, for example,
is readily satisfied if the measurements of the scattered field are taken in the
Fraunhofer or Fresnel regions.
The scattered field is a band-limited function from a compact structure, and
such functions, in one or more variables, have remarkable properties. A finite
Fourier transform is an entire function of exponential type and so remains ana-
lytic everywhere in the complex plane with well-defined growth properties.
These properties provide a basis on which one can specify the relationship, if
any, between the magnitude and phase of a band-limited function and deter-
mine whether or not the phase function is well defined at all points of interest.
By definition, a band-limited function, F ( x ), may be expressed as the finite
Fourier transform of an object function, f ( t ), as follows:
where a and b define the object support for a 1-D object that may be extended
into the complex plane by analytic continuation to give
=+ =
f te
The Paley-Wiener theorem states that F ( z ) is an entire function of expo-
nential type which, by the Hadamard factorization theorem, means it can be
written as an infinite product of the form
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