
SpatiallyVariant Mathematical Morphology:
Theory and Applications
Initially, the focus
of mathematical morphology was devoted to translationinvariant
operators. However, the translationinvariance assumption
is not appropriate for many applications like traffic control
analysis, range imagery, adaptive morphological filtering,
shape representation, adaptive image decomposition and minimal
image representation for coding and compression. These applications
and many more clearly illustrate the need to develop a spatiallyvariant
(i.e., nonnecessarily translationinvariant) mathematical
morphology theory that unifies all of the algorithms proposed
thus far into a comprehensive mathematical framework.
We proposed a general theory of SpatiallyVariant (SV) mathematical
morphology. This theory preserves the concept of structuring
element that is crucial in the design of geometrical signal
and image processing applications. We defined the elementary
SV morphological operators and studied their properties.
We subsequently derived a kernel representation of a large
class of nonlinear and nonnecessarily translationinvariant
operators in terms of the elementary spatiallyvariant operators.
The latter representation is a generalization of Matheron’s
representation theorem for increasing and translationinvariant
operators. The SV kernel representation is redundant, in
the sense that a smaller subset of the SV kernel is sufficient
for the representation of the SV operators. Thus, we provided
sufficient conditions for the existence of the minimal basis
of the kernel in terms of uppersemicontinuity in the hitormiss
topology. The latter minimal basis representation is a generalization
of Maragos’s minimal basis representation for increasing
and translationinvariant operators. We used the theory
of spatiallyvariant (SV) mathematical morphology to extend
and analyze many important image processing applications
like morphological image restoration and skeleton representation
of binary images. In particular, we developed an algorithm
to implement the SV morphological skeleton, which has a
storage capacity gain twice as high as its translationinvariant
counterpart.
Related publications:
N. Bouaynaya, M. CharifChefchaouni
and D. Schonfeld, “SpatiallyVariant Morphological
Restoration and Skeleton Representation” , IEEE
Transactions on Image Processing, vol. 15, no. 11, pp. 3579  3591, November 2006.

N. Bouaynaya and
D. Schonfeld, "SpatiallyVariant morphological image
processing: theory and applications", in Proceedings
of SPIE Visual Communications and Image Processing (VCIP'06),
vol. 6077, January 2006. (BEST
STUDENT PAPER AWARD).

N. Bouaynaya, M.
CharifChefchaouni and D. Schonfeld, “
Theoretical Foundations of SpatiallyVariant Mathematical Morphology Part I: Binary Images”,
IEEE Transactions on Pattern Analysis and Machine Intelligence,
vol. 30, no. 5, May 2008.

N. Bouaynaya and D. Schonfeld, “
Theoretical Foundations of SpatiallyVariant Mathematical Morphology Part II: GrayLevel Images”,
IEEE Transactions on Pattern Analysis and Machine Intelligence,
vol. 30, no. 5, May 2008.

