International Journal of Scientific & Engineering Research Volume 2, Issue 4, April-2011 1

ISSN 2229-5518

Morphological Space And Transform Systems

Ramkumar P.B, Pramod K.V

Abstract— Mathematical Morphology in its original form is a set theoretical approach to image analysis.It studies image transformations with a simple geometrical interpretation and their algebraic decomposition and synthesis in terms of elementary set operations.Mathematical Morphology has taken concepts and tools from different branches of Mathematics like algebra (lattice theory) ,Topology,Discrete geometry ,Integral Geometry,Geometrical Probability,Partial Differential Equations etc.In this paper ,a generalization of Morphological terms is introduced.In connection with algebraic generalization,Morphological operators can easily be defined by using this structure.This can provide information about operators and other tools within the system

Index Terms—morphological space, transform systems, slope transforms, legendre, kernel.

—————————— • ——————————


1.1 Definition: Morphogenetic field

LET X :t <p and W c; P( X ) such that i) , X E W ,
ii) If B E W then its complement B E W iii) If
Bi E W is a sequence of signals defined in X, then

Bi EW.


Let “A= { : W U / (U A ) = V ( A ) & (/\ A ) = /\ ( A )} “ .Then WU is

Exampl 3.If (V, * ) (V, *' ) are groups then (V,

V,/\.*,*' ) is called a bounded lattice ordered group or blog.[11]

Proposition 1. Clodum is an operator space.

Since Clodum is a particular case as mentioned above , we can consider it as an operator space.

Proposition 2. Blog is an operator space.

i i i i

called Morphogenetic field [14] where the family Wu is the set of all image signals defined on the continuous or discrete images Plane X and taking values in a set U .The pair ( Wu, A ) is called an operator space where A is the collection of operators defined on X.

1.2 Definition : Morphological Space

The triplet ( X, Wu, A ) consisting of a set X, a morphoge- netic field Wu and an operator A(or collection of opera- tors) defined on X is called a Morphological space.[14] Example 1.If X = Z2 then it is called Discrete Morphologi-
cal space
Blog is another example for operator space. Similar par- ticular cases exist corresponding to the algebra or geome- try under consideration.

1.3 Definition:Dilation

Letbe a complete lattice, with infimum and mini- mum symbolized by and respectively.[1]

A dilation is any operator that distributes over the supremum and pre-

serves the least element. ,

1.4 Definition: Erosion

An erosion is any operator that distributes over the infimum [1]. = , =U

1.5 Definition:Morphological Adjunctions

Example 2.Let V be a complete lattice.

If X= V and A= (V, V,/\.*,*' ) where *,*' are dilation &


( X ,Wu , A) & (Y ,Wu , A)

be a morphological spaces.
erosion then (V,

V,/\.*,*' ) becomes a commutative

The pair

( A, A)

is called an adjunction iff

complete lattice ordered double monoid or ‘Clodum’ [11]
where (V, * ) (V, *' ) are commutative monoids.

A( X ) Y X A(Y ) where

tor of A.

A is an inverse opera-

IJSER © 2011

International Journal of Scientific & Engineering Research Volume 2, Issue 4, April-2011 2

ISSN 2229-5518

Proposition 3.Let ( X ,Wu , ) & (Y ,Wu , ) be a morpholog-

ical spaces with operators dilation and erosion on A. Then

processed by erosion systems 2) A single valued slope
transform for signals processed by dilation systems 3) A

( X ) Y

X (Y ) .

multi valued transform that results by replacing the su-
prema and infima of signals with the signal values at sta-

Proposition 4(for lattice). Let ( X ,Wu , A) & (Y ,Wu , A) be

tionary points.

a morphological spaces. The pair ( A, A)

is called an adjunc-

2.2 Special Case – Continuous Time Signals

tion iff

u, v E X , an adjunction

(lu , v , mv , u ) on U such

All the three transforms stated above coincide for conti-
nuous-time signals that are convex or concave and have


A( x(u)) =

V mv ,u ( x(v)) and

vE X

an invertible derivative. This become equal to the Legen-

A( y(v)) =

/\ l u ,v ( y(u)) , u, v E X , x, y EWU .[1]


dre transform (irrespective of the difference due to the boundary conditions).

1.6 Definition:Morphological Kernel

The operator

= defines a closure called morpho-

2.3 Morphological Signal Operators

The morphological signal operators are defined by using
logical closure and
morphological kernel.

* =

defines a kernel, called
Lattice Dilation and Erosion of Signals. The morphologi- cal signal operators are parallel or serial inter connections


2.1 Introduction

of morphological dilation and erosions, respectively, de- fined as
Fourier transforms are most useful linear signal transfor-

( f + g )( x) =


yER d

f ( x

y) + g ( y)


mations for quantifying the frequency content of signals
and for analyzing their processing by linear time – inva-

and ( f

+ g )( x) =


yER d

f ( x + y)

g ( y)

riant systems .They enable the analysis and design of li- near time invariant systems (LTI)in the frequency do-
Where V denotes supremum and /\ denotes infimum.

2.4 Legendre Transforms


Slope transforms are a special type of non linear signal

Let the signal


be concave and assume that there


transforms that can quantify the slope content of signals.
exist an invertible derivative

x' =

. Imagine that the


It provide a transform domain for morphological systems.

graph of x,not as a set of points (t, x(t))

but as the low-
They are based on eigen functions of morphological sys- tems that are lines parameterized by their slope. Dilation
er envelope of all its tangent lines . The Legendre trans- form [12] of x is based on this concept .The tangent at a
and Erosions are the fundamental operators in Mathemat-

(t, x(t))

on the graph has slope and intercept
ical Morphology. These operators are defined on lattice algebraic structure also. Based on this, Slope transforms
equal to

X = x(t)

a (t)

are generally divided into three.


(a ) = x[(x') 1 (a )]

a ( x') 1 (a ) where

f 1 de-

They are 1) A single valued slope transform for signals
notes the inverse.

IJSER © 2011

International Journal of Scientific & Engineering Research Volume 2, Issue 4, April-2011 3

ISSN 2229-5518

The function

X L of the tangents intercept versus the


slope is the Legendre transform of x [12]

3.1 Definition:Self Conjugate Operator Space


x(t) = X

L [( X

' ) 1 (

t)) + t( X

' ) 1 (

t )]

If the
An operator space (Wu,
A) is called self conjugate if it has
signal x is convex, then the signal is viewed as the upper envelope of its tangent lines.
a negation.

Example4. A clodum V has conjugate a * for every ‘a’


2.5 Definition:Upper Slope Transform

such that (avb)* = a* /\ b* and (a * b)
= (a* * ‘ b*) [11]
For any signal

x : R R its upper slope transform [12]

Example5. If V is a blog [4] then it becomes self conjugate

by setting
is the function

X V : R R


a 1 , whenV


< a < V



X V (a ) = V x(t)


at,a E R

.The mapping between

V sup,




inf = a

sup = a

the signal and its transform is denoted by AV : x X V .

If there is one to one correspondence between the signal


and its transform, then it is denoted by x(t ) X V (a ) .

Proposition6. Upper Slope Transform of a concave signal is equal to its Legendre Transform.

Proof.Let x(t) be a concave signal .Let it has an invertible

derivative. For each real a , the intercept of the line pass-

Example6. If X is a concave class then A* x (t)= x(-t) where

A* = A /\ (A V ) .

3.2 Definition:Self Conjugate Morphological Space

If the operator space (Wu, A) is self conjugate then the morphological space (X, Wu, A) is called a self conjugate morphological space.

3.3 Definition:Operatable Functions

ing from the point

(t, x(t))

in the signals graph with
Let (X, Wu, A) be a morphological space. The collection K
slope a is given by


at .

(X, W, A) of operatable functions consists of all real va-
For a fixeda , assume that t varies. Let there be a time
lued morphologically operatable functions x(t) defined on
X such that x(t) has finite operatability with respect to A.
instant t *
for which the intercept attains its maximum
A morphologically operatable function xE K
value. The intercept attains its maximum value when the

x E K .ie. iff A( x(a ))

A x(a )

line becomes tangent to the graph.Therefore

x'(t * ) = a

Corresponding to the change in a , the tangent also changes,and the maximum intercept becomes a function
of the slope a .By its definition ,the upper slope trans-

3.4 Definition:Morphological Transform Systems

Let ( X, Wu, A) be a perfect morphological space and K= K ( X,Wu, A) be an operatable space. K is called a mor-
phological transform system if
form [12] is equal to this maximum intercept function.

A[x (t)J = X (a )

T (a )

Thus, if the signal

x(t) is concave and has an invertible

Remark1.Since K is an operatable space,

derivative,then the upper slope transform is equal to its

1) A[x(t) + y(t )J = X (a ) + Y (a )

Legendre transform.Hence the proof.

2) A[x

(t)J = X (a )

T (a )

IJSER © 2011

International Journal of Scientific & Engineering Research Volume 2, Issue 4, April-2011 4

ISSN 2229-5518

3.5 Definition:Morphological Slope Transform


If A= Av in the previous definition,then K is called a Morphological slope transform system where Av is the upper slope transform.
Let (X, Wu, A) be a self conjugate morphological space. If X is a concave class then A * (x(t) )= x(-t) where A*= A /\ (A v) and A /\ is the lower slope transform. Also Av ( V xc )

= 'L AV ( xc )


Proposition 7(Characterization of Slope Transforms).

A Slope transform is an extended real valued function Av (or

A /\ ) defined on a Morphogenetic field Wu such that

1. Av ( )= 0

2. Av (xc) 20 xc EWu

3. Av is countably additive in the sense that if (xc)is any disjoint sequence[or sampling Signal]then Av ( V xc )

= 'L AV ( xc )


Remark 2.Av takes + i.e Av ( xc ) = if x(t) =
phological dilation) into an addition .This is similar to the concept in Fourier transform transforms. In Fourier trans- form a linear convolution changed into a multiplication. Difference between the Fourier transform and its mor- phological counterpart, the slope transform is that the Fourier transform is invertible but the slope transform only has an adjoint. In the sense of adjunctions, this means that the ‘inverse’ of the slope – transformed signal is not the original signal but only an approximation with- in the sub collection. In this paper we made an attempt for generalizing the algebraic structures related to the theory of Signal processing using Mathematical Mor- phology. Morphological operators can be redefined by using these structures. We hope that this will be helpful for finding new applications in Mathematical Morpholo- gy.


[1] John Goustias and Henk J.A.M Heijmans ,Mathematical Mor- phology, , I.O.S Press.

[2] H.J.A.M Heijmans, Morphological Image Operators,Boston,

M.A Academic,1994 .

Av ( a ) >

, a unless x(t) = , t t

[3] J .Serra, Image Analysis and Mathematical Morphology,New

York Academic ,1982.

If x= - then Av =

Proposition 8. Let K be a morphological transform system. Let X be a class of concave functions. Let x(a ) E X with each

[4] P .Maragos and R.W Schafer, ”Morphological system for multi dimensional signal processing” Proc. IEEE,Vol,78,P.D 690-

710,April 1990.

[5] P .Maragos,A representation theory for morphological image and signal processing. IEEE Transactions on Pattern analysis and machine intelligence 11,(1989),586-599.

x(a )

has an invertible derivative.

[6] The Matheron Representation Theorem for Gray Scale Morpho- logical Operators, G. CROMBEZ,Proceedings of the American

Then AV ( x(a )) = L( x(a )) where L is the Legendre trans-

form and Av is the upper slope transform.


The slope transform has emerged as a transform which has similar properties with respect to morphological sig- nal processing. Fourier transform does this with respect to linear signal processing .Main property of slope trans-
form is that it transforms a supremal convolution ( mor-

Mathematical Society Volume 108, Number 3, March


[7] Heijmans, H.J.A.M ,and Ronse,C.The algebraic basis of Mathe- matical Morphology –Part I, Dilations and Erosions ,Computer vision, Graphics and Image Processing,50(1990) 245-295.

[8] Rein Van Den Boomgaard and Henk Heijmans, Morphological scale space- operators.

[9] Javier Vidal& Jos´e Crespo,Sets Matching in Binary Images Using Mathematical Morphology,International Conference of the Chilean Computer Science Society.

[10] Jean Cousty, Laurent Najman and Jean Serra,Some morpholog- ical operators in graph spaces,ISSM-2009.

[11] Petros Maragos,Lattice Image Processing: A Unification of

Morphological and Fuzzy Algebraic Systems,Journal of

Mathematical Imaging and Vision 22:333-353,2005.

IJSER © 2011

International Journal of Scientific & Engineering Research Volume 2, Issue 4, April-2011 5

ISSN 2229-5518

[12] Chu-Song Chen, Yi-Ping Hung,Theoretical Aspects of Vertically Invariant Gray-Level Morphological Operators and Their Ap- plication on Adaptive Signal and Image Filtering, IEEE transac- tions on signal processing, vol. 47, no. 4, april 1999 1049.

[13] Petros Maragos,Slope Transforms:Theory and Application to

Nonlinear Signal Processing,IEEE Transactions on Signal Proc- essing,Vol 43,No.4,April 1995.

[14] K.V Pramod, Ramkumar P.B , Convex Geometry and Mathe- matical Morphology, International Journal of Computer Appli- cations,Vol:8,Page 40-45.


Ramkumar P.B is working as Assistant Professor in Mathematics at Raja- giri School of Engineering& Technology , Mahatma Gandhi University, India, PH-04842432058. E-mail:

Pramod K.V is working as Professor at Department of Computer Applica- tions,Cochin University, India, PH-01123456789.


IJSER © 2011