Table Of ContentINTERNATIONAL CENTRE FOR MECHANICAL SCIENCES
C 0 U R S E S A N D L E C T U R E S - No. 85
~~a···~)
CISM
~
JULIUSZ KULIKOWSKI
AC:\DEMY OF SCIENCES - WARSAW
ALGEBRAIC METHODS IN PATTERN RECOGNITION
COURSE HELD :\.T THE DEPARTMENT
OF AUTOMATION AND INFORMATION
JULY 1971
Springer-Verlag Wien GmbH 1971
This work is I!Uqect to copynght
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whether the whole or part of the material is concerned
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or similar means, and storage in data banks.
© 1972 by Springer-Verlag Wien
Originally published by Springer-Verlag Wien-New York in 1972
ISBN 978-3-211-81128-3 ISBN 978-3-7091-2884-8 (eBook)
DOI 10.1007/978-3-7091-2884-8
P R E F A C E
During the last years electronic computa
tion and data processing methods have reached a com
paratively high level of maturity. More and more com
plicated forms of input data: numerical, alpha-numer
ical, textual can be handled automatically and pro
cessed by electronic computers. Last time a growing
attention is paid to the picture processing. Graphic
al and pictorial forms of information play an impor
tant role anywhere a human individuum is linked on an
informational system as its element. This is a conse
quence of the fact that a human visual tract is dis
tinguished by its possibility to parallel information
processing and by its flexibility to changing circum
stances. That is why the problem of immediate visual
information exchange between man and computer arises
as a very actual one. The recognition of patterns by
specialized automata or by electronic computers is a
first step toward the solution of this problem. The
pictures being of interest in many practical applica
tions are not only typewritten or handwritten texts,
but also graphics, electronic sahemas, graphs, finger
prints, charts, miarophotos eta. Under
meteorogio~l
certain assumptions they can be divided into less or
more formalized classes to be discriminated automati
cally. It seems to us reasonable to consider the pia-
4 Preface
tures as some expressions of a "planar" sub
language~
jected to a certain number of morphological and
syn~
tactical rules. An identification of the sets of gram
matical rules corresponding to the given classes of
patterns is a problem of interest if a pattern recog
nition algorithm is to be chosen. No universal "planar"
language seems to exist and no universal pattern re
cognition algorithm for a computer seems to be possi
as well. the linguistic approach
ble~ Nevertheless~
turns out as a more effective one when composite pat
terns are dealt with. The other methods based on geo
statistical or functional models of patterns
metrical~
recognition can be included as fragmentary ones in the
structural-linguistic anaZusis of a composite pattern.
A need of an universal meta-language for the descrip
tion and theoretical investigation of "planar" lan
guages that are used in particular situations arised.
In our this can be reached on the basis of a
opinion~
general approach offered by a general theory of rela
tions. This last can be specified in such a way that
any simple or composite picture is considered as a
realization of a generalized relation described on an
ordered family of sets of local input signal values.
This idea developed in the below given lectures. It
will be shown that a kind of Boolean algebra can be
defined on the set of possible relations. The well
known methods of Boolean functions minimization thus
aan be used in order to minimize the length of an ex
pression formally describing a pattern.
5
Preface
The considerations will be illustrated by
numerical examples. However, they could not be con
sidered as recommendations for the solutions of prac
tical problems.
This is my duty and a great pleasure to ex
press here my best thanks to Professor Luigi Sobrero
and to Professor Angelo Marzollo for their initiative
of including the lectures being given below to the
program of CISM Summer School in June, 19?1.
Udine, June 19?1
1. Introductory remarks.
The recognition of patterns will be here consideE
ed as a cybernetical subdiscipline investigating the general
principles of decision making in transmission systems with sta-
tionary information sources, under the assumption that the set
of decisions is enumerable. A general model that will be here
considered is given in Fig. 1. It consists, as usually, of an
"information source", a "coder", a "transmission channel" contai!!
ing a "noise source", a "decoder" and a "decision system". How-
ever, no special assumptions concerning the physical nature of
the signals and the properties of the messages like this one con
-
lnfQrmation s.~l Transmission Decoder Decision y
Z(~
'==s{=o~=}=ce=:::!lr"-L:_J-- charmel ~ ~ system
{z} {v}
' t 1
Noise
source
Fig. 1
cerning the finity of the set of messages or another concerning
the statistical nature of the signals will be made. The problem
of detection of the signals in presence of noise is a particular
case of pattern recognition: it is supposed in this case, that
the set of messages is finite and and the signals can be consid
8 Chap. 1 - Introductory Remarks
ered as some realizations of stochastic processes. A problem of
typewritten text reading and interpretation is an example of
more complicated pattern recognition problem: the statistical
properties of the signals are of less importance in this case,
however, the difficulty is connected with the fact that the sets
of transmitted messages and the sets of decisions are infinite
now. A problem of identification of a speaking individual based
on an analysis of the sound of his voice is a particular case of
the recognition of a pattern, where the signals are stochastic
with a priori unknown statistical properties: the decision alg£
rithm may be based on a sequential non-parametric statistical
technique. A problem of recognition of a geometrical form and di
mensions of a rigid body by touch is a pattern problem having
no formal interpretation, as yet, etc.
Since the year 1958, when the first paper by
Frank Rosenblatt concerning the idea of a "perceptron" occurred,
the pattern recognition has passed a specific evolution. First
of all, two different approaches to the recognition of patterns
came out. The first one underlined a necessity of investigating
the models of neuro-physiological mechanisms of the recognition
processes in cerebral systems. The ideas of F. Rosenblatt played
an initiating role for this kind of investigations, which in
some sense can be also referred to the former concepts of Mc
Culloch and Pitts of artificial neuron-sets. The other approach
is interested in investigations of new principles of pattern
re~
Basic Formulations 9
ognition rather than in modelling the natural visual and aural
tracts. This last concept is relative to the former ideas of op
timum signal detection and signal parameters extraction. Never
theless, the problem of pattern recognition is more general and
difficult; it consists in finding out the algorithms of recogni
tion and classification of type- and handwritten characters,
graphical pictures, symptoms of diseases, microscopic images etc.
The statistical formal models of signal reception are no more
than a particular case of a general formulation of the pattern
recognition problem.
Let us denote by S a message transmitted by the
{S}
source and by a set of all possible transmitted messages,
Z -
by a signal observed at the output of the transmitted chan
Y-
nel, by a decision made by a pattern recognition device. The
space of all possible signals observed at the input of the pat
{Y}
tern recognition device will be denoted by{Z}, while will
denote a set of all possible decisions. The pattern recognition
rule will be formally given in the form of a function
Y = h(Z) (1.1)
{z} {Y}
projecting on in an unique manner. It will be supposed
{Y}
that is enumerable and that Xy is its cardinal number.
L
{S}
Let be an enumerable family of subsets {S}~ C
L .
and let X I denote the cardinal number of It will be suppo~
ed that
(1.2)
10 Chap. 1 - Introductory Remarks
Land
and that there exists some one-to-one projection between
{Y}. Otherwise speaking, if {Sh is a 11pattern11, there exists a
Y {Y}
11decision11 ~ E corresponding to this pattern. The patterns
will be here considered as some classes of messages rather than
as single messages. For example, if a graphical representation
of a given triangle on a plane is a message, then the pattern
can be defined as a set of representation of all possible triag
gles of different form and sizes. However, the graphical
repr~
sentations transmitted through a 11channel11 (no matter what is
the physical nature of this channel) are usually disturbed by
the influence of some external processes. That is why pattern
transmissions can be formally described as a projection of {S}
{z},
into in general ambiguous and irreversive. Let {z}~ be a
subset of signals at the input of the pattern recognition device
.
corresponding to the pattern {s}~ The pattern recognition proE
lem will be called simple if
( 1. 3)
for any ~ = c1- , where 0 is an empty set, otherwise it will be cal
led a complex one. The complex pattern recognition problems are
typical in the case of stochastic disturbances or additive noise.
A general theoretical model of pattern
recogni~
tion, which can be considered as a 11classical11 one, was based
on the following geometrical interpretation. Let us suppose,
that {z} is a multidimensional Euclidean space E. It will be
supposed that it is partitioned into a finite family of non-over
