# Book Review
## Numerical Methods for Image
Registration
Jan Modersitzki
Oxford University Press, New York, 2004
199 pp., $95.00 hb
ISBN 0-19-852841-8
Reviewed by Gurusham Sudhir
see all book reviews
The domain of computer vision and
image processing is quite an interesting
one—while it is relatively easy for evolved
species, such as humans, to perform tasks
like object recognition, object matching, or
object registration, it can be extremely difficult
to do the same using the most modern
computers. The primary reason for this difficulty
is the complexity faced in defining
“the problem” sufficiently to look for a
solution. Mathematically, one of the main
reasons is the fact that many of the problems
in this domain can be classified as illposed.
An ill-posed problem is one that
does not meet the following three conditions:
(1) there exists a solution, (2) the
solution is unique, and (3) the solution
depends continuously on the input data.
Solving such ill-posed problems invariably
requires a procedure called “regularization,”
where the mathematical formulation
of the problem contains an extra term—a
regularizer—to make the problem a wellposed
one.
In *Numerical Methods for Image Registration*,
author Jan Modersitzki presents a well-focused
text on the solutions to a particular
problem in the field of computer vision and
image processing—image registration. This
specific problem deals with techniques for
computation of the mapping (transformation)
of an object as observed in a template
image to the same object as observed in a
reference image. Note that even though
both the template and reference images
contain observations of the same object, the
object may have undergone some physical
displacement/motion/deformation between
the observations for many reasons, depending
on the context of the application in
which the two observations are made. Modersitzki
addresses this problem systematically
from a mathematical point of view,
classifying the solution approaches into two
categories: parametric and nonparametric.
The first category addresses solution techniques
that express the desired transformation
(also called registration) in terms of a
finite number of basis functions (so that the
problem is typically well-posed). The author
presents a general theory covering three
state-of-the-art techniques for parametric
registration (landmark-based registration,
principal axes-based registration, and optimal
linear registration, using different intensity-
based distance measures) and discusses
the relative advantages and disadvantages of
these techniques.
In the second (and more
challenging) category, Modersitzki presents a
unified approach, based on his own prior
research, to illustrate how nonparametric
registration techniques can be systematically
designed using mathematical formulations
based on variational techniques. Central to
this unified approach is the characterization
of the desired transformation (solution) to be
the minimizer of a certain functional that
embodies a similarity measure (for registration)
and a smoothness measure (for regularization).
Based on this approach, he illustrates
four different techniques—elastic
registration, fluid registration, diffusion registration,
and curvature registration—and provides
a generic numerical treatment for the
computation of the solutions in a fast, efficient,
and stable manner.
Particularly, Modersitzki
shows that the diffusion registration
formulation yields an
O(N) implementation,
whereas all other nonparametric
techniques
yield an O(N log N)
implementation.
For all the techniques
and throughout
the book, Modersitzki
illustrates the ideas
clearly and presents
the results of the experiments
using a standard
set of two different
images of human
hands. He conveys
essential information
in a succinct yet sufficient
manner. He provides
proofs for most
of the pertinent mathematical
results and
gives appropriate references
for the rest. Even
though Modersitzki
addresses image registration
as the central
problem in the book,
many of the ideas and techniques are relevant
to other areas in the same field. I certainly
recommend the book for researchers
as well as mathematically inclined practitioners
in the field of image processing and
computer vision. Gurusham Sudhir is
a senior vision engineer at Adept Technology
in City of Industry, California. He
works in the field of machine vision and
image processing. |