IntroductiontoMathematicalMorphology资源-CSDN文库

Mathematical

Morphology

3星 · 超过75%的资源需积分: 9 26 浏览量 2011-05-31 16:53:00 上传评论收藏 3.7MB PDF 举报

资源推荐

资源详情

资源评论

COMPUTER VISION, GRAPHICS, AND IMAGE PROCESSING 35, 283-305 (1986)

Introduction to Mathematical Morphology

JEAN SERRA

E. N. S. M. de Paris, Paris, France

Received October 6,1983; revised March 20,1986

1. BACKGROUND

As we saw in the foreword, there are several ways of approaching the description

of phenomena which spread in space, and which exhibit a certain spatial structure.

One such approach is to consider them as objects, i.e., as subsets of their space of

definition. The method which derives from this point of view is. called mathematical

morphology [l, 21. In order to define mathematical morphology, we first require some

background definitions.

Consider an arbitrary space (or set) E. The “objects” of this :space are the subsets

E; therefore, the family that we have to handle theoretically is the set 0 (E) of

all the subsets X of E. The set p(E) is incomparably less arbitrary than E itself;

indeed it is constructured to be a Boolean algebra [3], that is:

(i) p(E) is a complete lattice, i.e., is provided with a partial-ordering relation,

called inclusion, and denoted by “

.” Moreover every (finite or not) family of

members Xi E p(E) has a least upper bound (their union C/Xi) and a greatest

lower bound (their intersection f7 X,) which both belong to p(E);

(ii) The lattice p(E) is distributiue, i.e.,

xu(Ynz)=(XuY)n(xuZ) VX,Y,ZE b(E)

and is complemented, i.e., there exist a greatest set (E itself) and a smallest set 0

(the empty set) such that every X E p(E) possesses a complement Xc defined by the

relationships:

XUXC=E

and

xn xc= 0.

Given two sets X and Y E p(E), the notion of their set diference X/Y derives from

the intersection and the complement, as follows:

X/Y=Xn YC;

X/Y is the part of X which does not belong to Y (see Fig. lc). In brief, given two

sets B and X belonging to p(E), we may have (Fig. 1):

(a) B is included in X (notation: B c X)

(p) B hits X (notation: B fi X) meaning B

n X # 0

(y) B misses X (notation B c Xc) meaning B

n X = $I

and these relationships will apply when B is replaced by every family {B;).

Finally, the structure of a Boolean algebra provides the general framework on

which we shall perform morphological treatments. Mathematical morphology is the

application of lattice theory to spatial structures.

283

0734-189X/86 $3.00

All rights of reproduction in any form resrrwd

284

JEAN SERRA

FIG.

1. (a) B, hits X (B, R X); E, misses X (B2

Xc); B, is included in X (B,

X);

(b) complement X’ of set X; (c) difference X/Y of the two sets X and Y.

These definitions may seem rather abstract and far away from the practical

applications. As a matter of fact, we shall pursue the discussion by concentrating

mainly upon the Euclidean space R” with dimension n = 1,2, or 3, or its digital

version Z” in terms of grids of points. Pedagogically speaking, the Euclidean case is

the first to be considered, since it corresponds to the physical world in which we

live; on the other hand, many images are reasonably binary, at least in a first

approximation (such as microstructures (Fig. 10) biological cells, (Fig. 11) seg-

mented zones in remote sensing, etc.). However, to restrict the approach to the

Euclidean sets would mean ignoring other important domains to which the method

applies. Let us briefly quote, for example:

-functions of W” considered as sets of R

Wn-‘, via their umbrae (see [25]);

-planar graphs, such as the partition of a map into counties [4];

-products of spaces, when several images are defined at each pixel. The

conditional operations introduced in Sections 5 and 6 below illustrate this mor-

phology. They result in context dependent transformations;

-vector spaces, such as propagations, when a range of directions is function of

each pixel [5];

-topological spaces which are necessary for introducing probabilities, and for

treating the questions of robustness [l];

-etc.

All in all, these various extensions indicate that we have to keep in mind two

different levels of generality: the Euclidean (or digital) binary case, which is useful

and gives a good intuition of what is going on, and the Boolean algebra associated

with the general framework of the method.

2. MORPHOLOGICAL TRANSFORMATIONS

2.a. Hit-or-Miss Transformation, Dilation, Erosion

The final goal of computer vision is often to segment images into objects and

textures in accordance with the judgment of the human eye. But this is not always

the case. It may happen that slight nuances escape human perception although they

INTRODUCTION

2X5

FIG. 2. (a) Cat brain cortex after hypoxia. The human eye does not detect the slightest difference with

the normal state. Nevertheless, by investigating one thousand neurons by morphological openings,

significant differences are exhibited [6]. (b) Sandstone rock from the oil reservoir of Hassi-Messaoud

(Sahara): how does one predict the permeability of this material from its texture?

are significant (see Fig. 2a). Moreover, the purpose of an image treatment may not

be to match vision, but to estimate some physio-chemical properties. In the case of

Fig. 2b, the Navier-Stokes equation, which governs the physical process, corre-

sponds to nothing intuitive.

Therefore, we do not look for a computerized substitute for human vision, but for

a coherent framework for describing spatial organization.

To this end, we define the

structure of an object by the set of the relationships existing between the various

parts of the object. We will study the structure experimentally by trying each of the

possible relationships in turn, and examining whether or not it is satisfied. Of

course, such knowledge will greatly depend on the choice made for the system of

relationships considered possible, and this a priori choice determines the relative

worth of the resulting concept of structure.

We saw that the media under study may have no recognizable pattern. This lack

of meaning (for us) leads to the position of probing them systematically, by starting

from the simplest relations that one can imagine. From this comes the idea of a

structuring element.

With each point x of the space

in which we work, we

associate a set B(x) called a structuring element. (Note that B(x) may vary from

one place to another). We can modify every set x E $I(

by some B(x) in several

ways. The most important ones are as follows:

dilation of X: (x: B(x) fl X}

(1)

erosionof X: {x: B(x)

X}.

(2)

The dilation of X by B(x) is the set of all the points x such that B(x) hits X. The

erosion of X by B(x) is the set of all the points x such that B(x) is included in X.

Starting from two structuring elements B’(x) and B2(x) we also define the

286

JEAN SERRA

hit-or-miss transformation (in brief HMT) as being the difference of X eroded by

B’(x) and X dilated by B2(x).

HMT of X: (erosion of X by B’)/(dilation of X by B2).

Dilation and erosion turn out to be particular cases of HMTs.

The class of transformations generated by the (possibly infinite) unions, products,

and complementations of HMTs constitutes, by definition, the

morphological truns-

formations

over p(E). Of course, these operations are not the only pieces of

information; in particular, they can be combined with measures on p(E) (e.g., the

area or the volume of sets in the Euclidean space) or with other types of operations

(convolution, for example).

2. b. Basic Properties

First of all, a general comment: all of the morphological transformations are

non-reversible, (except, in each particular case, for some subclasses of sets such as

the invariant ones). In fact, the idea of restoring the images is quite irrelevant here;

on the contrary, our philosophy will consist of stating that the images under study

exhibit too much information, and that the goal of any morphological treatment is

manage the loss of information

through the successive transformations.

In order to do this, we must play with a few underlying general properties which

are the key to any morphological analysis. We now present the four most important

ones. In what follows, $ is the generic symbol of a morphological transformation.

(i)

Increasing:

4 is increasing when it preserves inclusion, i.e., when

xc y=, 44x) = J/(y)

‘dx, YE b(E).

(4)

For example, the transformation “replace X by its boundary JX” is

not

increasing.

(ii)

Anti-extensiuity:

r/ is anti-extensive when it shrinks X, i.e., when

J/(x) = x

Vx E b(E).

(5)

(iii)

Idempotence:

# is idempotent when the result q(X) remains unchanged if

we reapply the transformation, i.e., when

vwxN = W)

vx E b(E).

(6)

(iv)

Homotopy:

Here the set

is considered as a topological space. However,

for the sake of simplicity, we limit ourselves to the bounded sets of the plane. With

each bounded set

associate its

homotopy tree

whose trunk corresponds to the

background

(i.e., the infinite connected component of

Kc),

the first branches

corresponding to the connected components

adjacent to

and the second

branches to the pores

adjacent to

K,,

etc. (see Fig. 3).

剩余22页未读，继续阅读

评论收藏

内容反馈

tianya10319

2016-08-25

写文章参考，资料不错

jhjun65963296

粉丝: 0
资源: 1

Introduction to Mathematical Morphology

An Introduction to Mathematical Modelling

Introduction to Mathematical Statistics _7th

Mathematical Introduction to Robotic Manipulation

A Mathematical Introduction to Robotic Manipulation

An introduction to mathematical cryptography

an introduction to mathematical cryptography.pdf

Mathematical Morphology and Its Applications to Signal and Image Processing

An Introduction to Mathematical Analysis in Economics

A_Mathematical_Introduction_to_Control_Theory

A Mathematical Introduction to Fluid Mechanics

数理逻辑入门 A Friendly Introduction to Mathematical Logic

solution to introduction to mathematical statistics

Robert V.Hogg, Allen T.Craig - Introduction to Mathematical Statistics (4th Edition)

An Introduction to Mathematical Statistics and Its Applications

Introduction to Mathematical Statistics

A_Mathematical_Introduction_with_OpenGL

Introduction to Mathematical Logic

Introduction to Mathematical Fluid Dynamics

Data Assimilation A Mathematical Introduction

A Mathematical Introduction to Data Science

Theoretical Foundations of Spatially-Variant Mathematical Morphology Part I: Binary Images

Introduction to Mathematical Theory of Control, Bressan 2007.pdf

Introduction to Mathematical Logic, Sixth Edition

Introduction_to_Mathematical_Statistics_and_Its_Applications__An.pdf

An Introduction to Mathematical Cosmology

最新资源