Page 299 - Introduction to Information Optics

284 5. Transformation with Optics
transform of the object slice function itself. Equation (5.40) is referred to as the
central-slice theorem. When rotating the angle </> the ID Fourier transform of
the projection, g(R, 0) gives the entire 2D Fourier transform of the object slice.
At each projection angle, the Fourier transform of the object slice is sampled
by a line passing through the origin (hence, the term central slice).
As a result, the inverse Radon transform with Eq. (5.38) gives the entire 2D
Fourier transform of the object, and the object itself. However, this reconstruc-
tion requires a coordinate transform and an interpolation of the Fourier
spectrum from the polar coordinates to the Cartesian coordinates.
The above-mentioned direct Fourier transform method is the simplest in
concept. Various methods exist in practice for reconstructing an image from
the shadow or Radon transform, such as the filtered back-projection algorithm
and the circular harmonic reconstruction algorithms using the Hankel trans-
form [23,24].

5.10. GEOMETRIC TRANSFORM

The geometric transformation, or coordinate transformation, is primarily
used for simplifying forms of mathematical operations, such as equations and
integrals. For example, it is easier to handle the volume integral of a sphere
body in the spherical coordinate system than in the rectangular coordinate
system. Geometric transformations can be implemented by an optical system
using a phase hologram [25, 26, 27]. The optical geometrical transform is
useful for redistributing light illumination [28], correcting aberrations and
compensate for distortions in imaging systems, beam shaping [29], and for
invariant pattern recognition [30]. Recently, the coordinate transformations
also have been applied to analyze surface-relief and multiplayer gratings by
reforming Maxwell electromagnetic equations [31].

5.10.1. BASIC GEOMETRIC TRANSFORMATIONS

The geometric transformation is a change of the coordinate system and is
defined as
r
x)(5[x -<p~»]</x, (5.41)

l
where f(x) is the input function and (p ~ (u) is defined by a desired mapping
from a coordinate system x to another coordinate system u with

u = (p(x).