图像几何描述：地形图视角与应用

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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 1

as Topographic Maps, Lecture Notes in Mathematics 1984,

DOI 10.1007/978-3-642-04611-7

 Springer-Verlag Berlin Heidelberg 2010

4 1 Introduction

δε Superposition

Threshold

decomposition

Fig. 1.4 Opening operator by a disk on a gray-level image through threshold de-

composition 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.

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图像几何描述：地形图视角与应用

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