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Chapter 1
Introduction
This chapter is intended as a gentle and casual introduction to the themes
developed in these lecture notes. It cites only one reference book for each
theme and omits voluntarily all further citations to let it remain an overview
of usually independent but here connected topics: Morse theory, topography
and mathematical morphology. The three have in common the fundamental
role of the (iso)level sets of functions, whose structure is one of the main
themes of the present book. The present introduction serves to locate each
Chapter into this general context.
1.1 Morse Theory and Topography
In his classical treaty on Morse theory, J. Milnor motivates the subject by
discussing the variation of topology of a part of a torus below a plane, as a
function of the height of the plane [75]. This reveals the global topology of
the torus. Now this analysis can be done also for the graph of a function.
The subgraph of a real function defined on the Euclidean plane, called here
level set, reveals interest points of the surface, where a slight variation of level
changes the structure of the level set.
A customary way to represent the topography on a map is to draw the level
lines, that is the lines of constant elevation, as in Fig. 1.1. These level lines, if
dense enough, reveal almost all about the terrain they represent. Apart from
peaks (local maxima) and pits (local minima), a third category of interesting
points emerges as passes (saddle points), which are points where two distinct
level lines merge (see Fig. 1.2).
From a mathematical point of view, this analysis can be done with basic
differential calculus tools if the function is Morse. A Morse function is twice
differentiable and wherever its Jacobian vanishes, its Hessian is invertible.
For such well behaved functions, singular points, i.e., points of null Jacobian,
are isolated, and the signs of the eigenvalues of the Hessian reveal the nature
of the singularity:
V. Caselles and P. Monasse, Geometric Description of Images
as Topographic Maps, Lecture Notes in Mathematics 1984,
c Springer-Verlag Berlin Heidelberg 2010
DOI 10.1007/978-3-642-04611-7 1, 1
2
1 Introduction
Fig. 1.1 Aerial photograph and topographic map of terrain. Images courtesy the
U.S. Geological Survey.
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Fig. 1.2 Some level lines and critical points of a Morse function. M is a maximum,
m a minimum and S a saddle point.
– 2 positive eigenvalues: a minimum;
– 2 negative eigenvalues: a maximum;
– 2 eigenvalues of different sign: a saddle point.
As our interest lies mostly in image processing, where a gray level image is
considered as a function defined on a rectangle of the plane, with the value at
a point indicating the amount of light received (see Fig. 1.3), the Morse model
is ill suited for our purposes because of one overly optimistic assumption: the
invertibility of the Hessian at critical points. Whereas the twice differentiability may be admissible, because a convolution with a Gaussian kernel
of small variance would smooth and reduce the effect of noise, the Hessian
condition forbids the presence of a plateau in the image. Although an approximation by a Morse function is possible, as Morse functions are dense
among continuous functions on a closed rectangle, there is no canonical way
1.2 Mathematical Morphology
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Fig. 1.3
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A gray-level image and its representation as a bivariate function graph.
to do this approximation, and the analysis of a Morse approximation does
not necessarily reveal much about the topology of the original function.
1.2 Mathematical Morphology
Mathematical morphology, founded by G. Matheron in the 1960s and 1970s,
is an image processing theory based on manipulation of sets [69]. The basic
operators of mathematical morphology are erosions and dilations. A dilation
δB by a set B (called structuring element) is an operator that maps a set X
to the set X + B. Its dual operator is the erosion by the symmetric of B,
ε−B , defined by ε(X) = δ(X c )c . Both commute with translations.
families of dilations and
If (δi ) and (εi ) are
erosions, the operators δ and
ε defined by δ(X) = i δi (X) and ε(X) = i εi (X) are also a dilation and
an erosion. In this way, non trivial new operators may be defined. Another
way is to combine a dual erosion/dilation. The opening δ ◦ ε and the closing
ε ◦ δ are idempotent.
Although originally defined for binary images, mathematical morphology
was naturally extended to gray level images by the threshold decomposition
and the superposition principle as illustrated by Fig. 1.4. In other words,
mathematical morphology operates on level sets of the image, whence the
interest of an efficient decomposition of an image into its level sets.
4
1 Introduction
Threshold
δε
decomposition
Superposition
ε
δ
ε
δ
Fig. 1.4 Opening operator by a disk on a gray-level image through threshold decomposition and superposition.
1.3 Inclusion Tree of Level Sets
For a continuously differentiable function, at almost all levels the level lines
are Jordan curves. These have a notion of interior and exterior, and as they
do not cross, they can be organized in a tree driven by the inclusion order
relation: a level line is descendant of another if it is contained in the inner
domain of the latter, as in Fig. 1.5. The regular levels do not include in
particular levels of extrema, and therefore cannot be directly accounted for
in the tree.
In a pioneering work intended to extend the total variation of univariate
real functions to functions of two variables, A. Kronrod pointed out that the
differentiability has no relevance to such an analysis, and that continuity is
sufficient to organize the level sets, defined here as sets of constant level [50].
By defining a kind of quotient topology for the equivalence relation of two
points being equivalent if there is a continuum inside a level set joining them,
the family of level sets is endowed with a topology making it a dendrite, the
topological equivalent of an unrooted tree. Even though the (iso-)level sets
are not Jordan curves, they can still be organized in a tree. But there is
no root for this tree, in other words there is no direction for the dendrite.
Actually it can be directed by the choice of a point at infinity.
In Chap. 2, we show that even the continuity condition can be relaxed
to semicontinuity, and that the appropriate sets to consider are the upper
and lower level sets. These can be organized in a tree because the choice
of a point at infinity distinguishes between internal holes and exterior of a
level set. Whereas it is straightforward to define a tree of upper, or a tree of
lower, level sets, their merging is only enabled by the definition of a point at
infinity. The semicontinuity requirement, instead of continuity, is nice to have,
because it is compatible with the Mumford-Shah approximation of an image
by a piecewise constant image. For example, a piecewise constant model for
the image in Fig. 1.3 is obviously better adapted than a global continuous
one. Moreover, it fits perfectly the order 0 interpolation of a digital image,
that is, the nearest neighbor approximation. This provides the mathematical
foundation for the Fast Level Set Transform (FLST), an algorithm that we
detail in Chap. 6.
1.4 Topological Description and Computation of Topographic Maps
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Fig. 1.5
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Tree of level lines of a Morse function.
1.4 Topological Description and Computation
of Topographic Maps
Going back to the continuity assumption in order to describe in the manner
of Morse theory a topographic map, we cannot guarantee that critical points
are isolated, and no discrete description of events among level sets is possible.
Again, there are many possibilities to approximate the function by another
one not suffering from this drawback, but one category stands out as the most
natural one: the grain filters. They simply remove small scale oscillations of
the function, yielding what will be called a weakly oscillating approximation
of the image. Chapter 3 discusses these filters from the point of view of
mathematical morphology. One of them treats upper and lower level sets in
a symmetric manner and is preferable as it is self-dual in the vocabulary of
mathematical morphology (see Fig. 1.6).
Weakly oscillating functions may be analyzed in a discrete description.
Chapter 4 discusses several notions that can be defined for such functions:
on the one hand the maximal monotone sections (branches of the inclusion
tree) and their limit levels (called critical levels), on the other hand the
signature of level sets (the family of extrema it contains) and the levels at
which the signature changes (called singular levels). These notions are shown
to coincide, and when N = 2 they generalize to a weakly oscillating function
the critical levels of a Morse function. When N = 3, the notion of critical
value reflects the changes in the number of connected components of isolevel
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1 Introduction
Fig. 1.6 An image before (left) and after (right) application of a grain filter, which
removes many small oscillations.
sets [u = λ] as λ ∈ R varies. Let us mention that, in its present form, this
theory has not been published elsewhere.
In Chap. 5 we describe an algorithm to construct the tree of shapes by
fusion of the trees of connected components of upper and lower level sets.
Though the algorithm is less efficient than the algorithm described in Chap. 6,
valid when N = 2, it is adapted to any dimension.
The considerations of Chap. 4 justify the construction of the tree for an
order 1 interpolation of a digital image, i.e., bilinear interpolation. The algorithm to extract it is a variant of the FLST and is presented in Chap. 7. Some
applications may use rather the pixelized version for the complete description
of the discrete data and its flexibility (invariance to contrast change, among
others), while others prefer to use the continuous interpolation for its more
regular level lines.
Several applications relying on the inclusion tree have been developed in
recent years. We present a few examples of some of them in Chap. 8, ranging
from low-level image processing (edge detection, corner extraction) to image
alignment and local scale definition.
1.5 Organization of These Notes
The mathematically inclined reader may be most interested in Chaps. 2
to 6, which generalizes the topological Morse description to continuous or
semicontinuous functions. Mathematical morphologists may consider more
closely Chap. 3 about grain filters, although it is presented in the continuous
setting, not in the discrete topology setting most frequent in mathematical
morphology.
1.5 Organization of These Notes
7
The computer scientist may focus on Chaps. 6 and 7 for algorithmic
considerations, keeping in mind that their full justification are respectively
in Chaps. 2 and 4.
All may find motivation for this work in the image processing applications
presented in Chap. 8, knowing that their full description must be found among
articles referenced in the bibliography.