THE USE OF DYNAMICS IN GRAYLEVEL QUANTIZATION BY

MORPHOLOGICAL HISTOGRAM PROCESSING

Franklin C

´

esar Flores, Leonardo Bespalhuk Facci

Department of Informatics, State University of Maring

´

a

Av. Colombo, 5790 - Bloco 19 - Zona 07, Zip Code 87020-900 - Maring

´

a - PR - BRAZIL

Roberto de Alencar Lotufo

School of Electrical and Computer Engineering, State University of Campinas

Av. Albert Einstein, 400 - PO Box 6101, Zip Code: 13083-970 - Campinas - SP - BRAZIL

Keywords:

Morphological processing, histogram classiﬁcation, dynamics, watershed.

Abstract:

In a previous paper, it was proposed a method applied to image simpliﬁcation in terms of graylevel and ﬂat

zone reduction, by histogram classiﬁcation via morphological processing. It this method, it is possible to

reduce the number of graylevels of an image to n graylevels by selecting n regional maxima in the processed

histogram and discarding the remaining ones, in other to classify the histogram via application of watershed

operator. In the previous paper, it was proposed the choice of the n highest regional maxima. By far, it is not

the best criterion to choose the regional maxima and other criteria had been were tested in order to obtain a

better histogram classiﬁcation. In this paper we propose the selection of the regional maxima via application

of dynamics, a measurement of contrast usually applied to ﬁnd markers to morphological segmentation.

1 INTRODUCTION

Color quantization is a very important research ﬁeld

in digital image processing and computer graphics,

and among its contributions, it may be found sev-

eral techniques applied to simpliﬁcation of images by

color reduction (Gonzalez and Woods, 1992; Heck-

bert, 1982; Soille, 1996). The importance of image

simpliﬁcation by color reduction is clear when deal-

ing with problems of image display and image com-

pression (Gomes and Velho, 1994).

The application of color reduction techniques may

also provide the reduction of ﬂat zones (connected re-

gions of pixels with constant color) in the image, such

a connected ﬁlter (Crespo et al., 1997; Salembier and

Serra, 1995; Heijmans, 1999; Meyer, 1998). The re-

duction of ﬂat zones does not introduct borders in the

image, but, by supressing some borders, two of more

ﬂat zones may be joined in one.

Flat zone reduction have a great number of appli-

cations. They can be, for example, applied to im-

age compression and image segmentation (Meyer and

Beucher, 1990; Beucher and Meyer, 1992). They are

also applied to reduce the statistics of the image, in

order to simplify the number of attributes used in pat-

tern recognition techniques (Hirata Jr. et al., 1999;

Flores et al., 2000; F. C. Flores and Zuben, 2002).

One simple way to reduce graylevels in an image

is the classical thresholding. Given a thresholding

value, the graylevels are classiﬁed by setting the pixel

value to white (maximum graylevel) if its graylevel is

higher than the thresholding value, or to black (min-

imum graylevel), otherwise. The classical threshold-

ing can be extended by applying a set of n threshold-

ing values, in order to reduce the number of graylevels

of an image to n +1graylevels: if a graylevel value

belongs to an interval given by a pair of thresholding

values, it should be replaced by the graylevel assigned

to that interval.

In a previous paper (Flores and Lotufo, 2001), it

was proposed a method which gives not only an im-

age simpliﬁcation in terms of graylevel reduction but

also in terms of ﬂat zone reduction. The proposed

method is given by application of a set of morpho-

logical operators to the image histogram. The main

motivation behind the project of this operator is that

each object in the image has a signiﬁcative graylevel

distribution. So, to simplify an object in the image,

that is enough to classify its corresponding distribu-

tion in the histogram.

In that paper it was also proposed a method to re-

duce an image to n graylevels. It consists in to choose

the n highest regional maxima in the processed his-

togram and to ﬁlter the other peaks. The chosen max-

ima will provide the classiﬁcation of the graylevels in

the histogram by application of watershed operator.

121

César Flores F., Bespalhuk Facci L. and de Alencar Lotufo R. (2006).

THE USE OF DYNAMICS IN GRAYLEVEL QUANTIZATION BY MORPHOLOGICAL HISTOGRAM PROCESSING.

In Proceedings of the First International Conference on Computer Vision Theory and Applications, pages 121-128

DOI: 10.5220/0001374501210128

Copyright

c

SciTePress

Note that, by far, it is not the best criterion to choose

the regional maxima and other criteria were tested in

order to obtain a better histogram classiﬁcation.

In this paper we propose the application of dynam-

ics (Grimaud, 1992) to select the regional maxima in

order to achieve a better graylevel reduction. Dynam-

ics consists in a valuation of extrema of the image by a

measure of contrast that does not consider the size or

shape of valleys and peaks. It is usually applied to ﬁnd

markers to morphological segmentationand achieve

hierarquical segmentation (Meyer, 1996).

Section 2 presents some preliminar deﬁnitions, and

section 3 presents the dynamics. Section 4 proposes

the technique to connected ﬁltering by graylevel clas-

siﬁcation and a variation when it is possible to choose

the desired number of graylevels in the resulting im-

age. Section 5 presents some experimental results and

in the Section 6 we conclude this paper with a brief

discussion.

2 PRELIMINARY DEFINITIONS

Let E ⊂ Z×Z be a rectangular ﬁnite subset of points.

Let K =[0,k] be a totally ordered set. Denote by

Fun[E, K] the set of all functions f : E → K.An

image is one of these functions (called graylevel func-

tions). Particularly, if K =[0, 1], f is a binary im-

age. An image operator (operator, for simplicity) is

a mapping ψ : Fun[E, K] → Fun[E,K].

Let N(x) be the set containing the neighbourhood

(Hirata Jr., 1997; Flores, 2000) of x, x ∈ E.We

deﬁne a path (Hirata Jr., 1997) from x to y, x, y ∈

E as a sequence P (x, y)=(p

0

,p

1

, ..., p

n

) from E,

where p

0

= x, p

n

= y and ∀i ∈ [0,n − 1],p

i

∈

N(p

i+1

).

A connected subset of E is a subset X ⊂ E such

that, ∀x, y ∈ X, there is a path C entirely inside X.

Let f ∈ Fun[E,K].Aﬂat zone of f is a connected

subset X ⊂ E, such that f(x)=f(y), ∀x, y ∈ X.

Deﬁnition 1 The inf - reconstruction and sup - re-

construction operators are given, respectively, by,

∀f,g ∈ Fun[E, K],

ρ

B,g

(f)=δ

∞

B,g

(f)

ρ

∗

B,g

(f)=ε

∞

B,g

(f)

where B ⊂ E is the structuring element, n ∈ Z

+

and

δ

n

B,g

and ε

n

B,g

are, respectively, the n-conditional di-

lation and the n-conditional erosion operators (Serra,

1982; Heijmans, 1994). δ

∞

B,g

(f) (ε

∞

B,g

(f)) means

that the dilation (erosion) is applied till idempotency.

Let τ

i

: Fun[E, K] → Fun[E, [0, 1]], i ∈ K,bea

threshold function, where τ

i

(f)(x)=1,iff(x) ≥ i,

and τ

i

(f)(x)=0, otherwise.

Deﬁnition 2 Let f ∈ Fun[E,K]. A regional maxi-

mum is a ﬂat zone Z such that f (z) >f(n), z ∈ Z,

n ∈ N,N ∈F

Z

, where F

Z

is a set of all ﬂat

zones adjacent to Z (Flores, 2000). The regional

maxima of f is found by application of a operator

µ

max

B

c

: Fun[E,K] → Fun[E,[0, 1]], given by

µ

max

B

c

(f)=τ

1

(ρ

B

c

,(f+1)

(f)) ∨ τ

k

(f)

where B

c

⊂ E is the structuring element deﬁning

connectivity.

A regional miminum is a ﬂat zone Z such that

f(z) <f(n), z ∈ Z, n ∈ N,N ∈F

Z

, where F

Z

is a set of all ﬂat zones adjacent to Z.

3 DYNAMICS

Dynamics (Grimaud, 1992; Meyer, 1996) is a trans-

formation which valuates the extrema of an image ac-

cording to a contrast measurement. One advantage of

application of dynamics is that, while some methods

such as morphological ﬁlters need a size parameter to

evaluate constrast, the dynamics measurement does

not take in account the size and the shape of image

structures.

The evaluation of constrast of a regional minimum

is a good way to provide markers to application of

watershed operator in the morphological segmenta-

tion framework: an hierarquical segmentation may be

achieved by selecting the regional minima which dy-

namics is higher than a thresholding value and assign-

ing markers to them (Meyer, 1996).

Deﬁnition 3 Let x, y ∈ E. The dynamics Dyn

f

of

a path P (x, y) on an image f ∈ Fun[E, K] is given

by,

Dyn

f

(P (x, y)) = { |f(x

i

)−f(x

j

)| : x

i

,x

j

∈ P (x, y)}

i.e., the dynamics of P (x, y) is given by the difference

in altitude between the points of highest and lowest

altitude of P(x,y).

Figure 1 shows the dynamics of a path between x

and y.

Grimaud (Grimaud, 1992) also deﬁnes the dynam-

ics between two points x, y ∈ E on an image f ∈

Fun[E, K] as

Dyn

f

(x, y)={

Dyn

f

(P (x, y)) : P (x, y), }

where P (x, y) is a path between x and y.How-

ever, it will not be applied here, since the histogram

is an 1-D signal and, therefore, there is only one

path between any two points from the domain of his-

togram function. So, it will be considered here that

Dyn

f

(x, y)=Dyn

f

(P (x, y)).

VISAPP 2006 - IMAGE FORMATION AND PROCESSING

122

Figure 1: Dynamics of the Path Between x and y.

Deﬁnition 4 Let a(Z) ∈ K be the altitude of a re-

gional minimum Z in f . The dynamics of Z is given

by,

Dyn(Z)={ Dyn

f

(x, y),x ∈ Z, y ∈ M : a(M) <a(Z)}.

i.e., the dynamics of Z is given by the dynamics of the

path with the lowest dynamics that links Z to a point

y thats belongs to a catchment basin which regional

minimum has an altitude lower than Z.

Figure 2 illustrates the dynamics of a regional min-

ima Z.

Figure 2: Dynamics of a Regional Minimum.

Dynamics computation can be implemented by us-

ing tree of critical lakes (Meyer, 1996) or based on

ﬂooding simulations algorithms (Grimaud, 1992).

Given the dynamics of a regional minimum Z,

some metrics can be used to evaluate such mini-

mum (da Silva, 2001):

1. depth of the catchment basin which the minimum is

contained (given by the dynamics of the minimum

itself) (Fig. 3 (a));

2. area of the catchment basin (Fig. 3 (b));

3. volume of the catchment basin (Fig. 3 (c));

Let us denote by Dyn

i

(f)(Z) the function that

computes to Z from f an value given by the metric

i ∈{1, 2, 3} introduced above. Dyn

i

(f)(Z) will be

used to evaluate the signiﬁcant distributions in the his-

togram, as will be explained below.

Note that two catchment basin which have the same

depth may have different volume or area measure-

ments. Classiﬁcation of regional minima in an image

can be achieved by application of such metrics.

(a) (b)

(c)

Figure 3: Dynamics: (a) Depth. (b) Area. (c) Volume.

4 THE PROPOSED TECHNIQUE

Let us introduce a new technique applied to con-

nected ﬁltering. One characteristic of the proposed

ﬁlter is that, despite its connecting property, it does

not require a connectivity parameter (4-connect or 8-

connect) because all processing is done in the his-

togram of image.

As a consequence of such processing we have a

reduction in the graylevels appearing in the image.

In other words, the proposed ﬁlter is a mapping ψ :

Fun[E, K

1

] → Fun[E,K

2

], where |K

2

| < |K

1

|.

Let f ∈ Fun[E, K]. Let us consider the histogram

of the image as a function h

f

: K → Z

+

(Fig. 4). De-

spite the domain of h

f

, the morphological operators

used in the graylevel classiﬁcation were applied to in

the same manner. In other words, we consider K as a

subset of E.

Let f ∈ Fun[E,K] and h

f

∈ Fun[K, Z

+

] its

histogram. Let max (h

f

) = max {h

f

(x):x ∈ K}.

Let κ : Fun[K, Z

+

] → Fun[K, Z

+

] be the mapping,

given by, ∀x ∈ K,

κ(h

f

)(x)=

h

f

(x),ifµ

max

B

(h

f

)(x)=1

0, otherwise

,

THE USE OF DYNAMICS IN GRAYLEVEL QUANTIZATION BY MORPHOLOGICAL HISTOGRAM PROCESSING

123

Figure 4: Histogram.

where B ⊂ K is the structuring element (Heijmans,

1994). The mapping κ gives a function with only the

regional maxima of h

f

. Figure 5 shows the regional

maxima of the histogram shown in Fig. 4.

Figure 5: Histogram regional maxima.

Let π : Fun[K, Z

+

] → Fun[K, Z

+

] be the map-

ping, given by, ∀x ∈ K,

π(h

f

)(x)=

max (h

f

),ifκ(h

f

)(x)=0

0, otherwise

.

Let η : Fun[K, Z

+

] → Fun[K, Z

+

] be the sup-

reconstruction of κ, denoted by,

η = ρ

∗

B,κ

(π),

where B ⊂ K is the structuring element.

If exists in κ(h

f

) a sequence of increasing regional

maxima followed by a sequence of decreasing re-

gional maxima, its regional maxima must compose

a curve whose regional maximum is the maximum

among them. The operator η is applied to the func-

tion κ(h

f

), in order to preserve the regional maxima

among the set of regional maxima of κ(h

f

) and con-

struct the curves with the remaining regional maxima

(Fig. 6).

The reason of the processing of the graylevel dis-

tributions in the histogram is that each object in the

image is represented by a signiﬁcative graylevel dis-

tribution. The idea is to ﬁlter the histogram in order

to get new distributions where the objects are simpli-

ﬁed and well represented. These new distributions are

used to get a meaningful classiﬁcation of graylevels

by application of watershed operator introduced be-

low.

Figure 6: Reconstruction by application of η operator.

Let ω : Fun[K, Z

+

] → Fun[K, [0, 1]] be the

watershed operator (Beucher and Meyer, 1992; Vin-

cent and Soille, 1991). Let ν : Fun[K, Z

+

] →

Fun[K, Z

+

] be the negation operator.

Let λ : Fun[K, Z

+

] → Fun[K, Z

+

] be the map-

ping, given by, ∀x ∈ K,

λ(x)=

max (h

f

),ifω(ν(η))(x)=1

0, otherwise

.

The mapping λ gives a preliminary classiﬁcation;

the graylevel classes are separeted but not labeled

(Fig. 7). The labeling of classes is given by the ap-

plication of next steps.

Figure 7: Pre-classiﬁcation.

VISAPP 2006 - IMAGE FORMATION AND PROCESSING

124

Let : K → Z

+

, such that (x)=x, ∀x ∈ K.

Let θ : Fun[K, Z

+

] → Fun[K, Z

+

] be the mapping,

given by, ∀x ∈ K,

θ(x)=

max (h

f

),ifµ

max

B

(η)(x)=1

0, otherwise

.

Let β : Fun[K, Z

+

] → Fun[K, Z

+

] be the mapping,

given by,

β = ∧ θ.

The mapping β gives a function where each point

of the regional maxima of η(·) is labeled by its cor-

responding graylevel. Its reconstruction conditioned

to λ(·) gives the labeling of all graylevel classes.

Figure 8 shows a composition of λ(·) (pre-classiﬁed

graylevels) and β(·) (the labeled peaks inside each

pre-class).

Figure 8: Function β assigning labels to each pre-class.

Let ζ : Fun[K, Z

+

] → Fun[K, K].

ζ = δ

B

(ρ

B,λ

(β)),

where B ⊂ K is the structuring element, and δ

B

(·) is

the dilation operator (Serra, 1982; Heijmans, 1994).

Let |K| be the number of distinct graylevels in K.

We can say that the mapping ζ is a graylevel classiﬁ-

cator. Given the histogram h

f

, f ∈ Fun[E, K], the

classiﬁcator gives a new set of graylevels G, where

|G| < |K|. The processed histogram is used as a

look-up table in order to reduce the graylevels.

Deﬁnition 5 Let f ∈ Fun[E, K

1

], K

1

=[0,k

1

],

and h

f

∈ Fun[K

1

, Z

+

] the histogram of f . The

graylevel reducer by classiﬁcation of regional max-

ima is a mapping ψ : Fun[E, K

1

] → Fun[E,K

2

],

where K

2

, |K

2

| < |K

1

|, is given by,

K

2

= ζ(h

f

).

Since there is a reduction in the number of graylevels

in the image, and it may causes the union of two or

more ﬂat zones, without split any ﬂat zone, it is a con-

nected operator. Figure 9 shows the graylevel classi-

ﬁcation provided by ζ, given the histogram shown in

Fig. 4.

Figure 9: Classiﬁed graylevels.

4.1 Reduction to n Graylevels by

Classiﬁcation of Region Maxima

by Dynamics Metrics

When the operator ψ is applied to an image f , it re-

duces the graylevels appearing in f to the number of

regional maxima of η(·). However, it is possible to

reduce the graylevels to a smaller number, by adding

a parameter n which gives the number of graylevels

to appear in ψ(f ). In this section we will present a

way to select the n most signiﬁcant regional maxima

of η(·) by application of dynamics.

We will denote by ψ

n

: Fun[E, K

1

] →

Fun[E, K

2

], |K

2

| < |K

1

|, |K

2

| = n, the operator

which performs the reduction of the graylevels in the

image to n graylevels.

Remember that the result of η operator is the orig-

inal histogram ﬁltered in a way that the highest re-

gional maximum among a set of regiona maxima be-

longing to the same distribution is preserved. Let

D

i

: Fun[K, Z

+

] → Fun[K, Z

+

] be the function

given by,

D

i

(η)(x)=

Dyn

i

(ν(η))(Z):x ∈ Z,ifµ

max

B

(η)(x)=1

0, otherwise

,

where Z is one of the regional minima of the nega-

tion of η(·). I.e., if x belongs to a regional maximum

in η(·), D

i

(η)(x) will be equal to the dynamics (see

section 3) of the regional minimum where x is located

in the negation of η(·). i is the criterium chosen to

evaluate η(·): depth (1), area (2) or volume (3).

Let m be the number of regional maxima in η(·).

Let Q be the set deﬁned by

Q = {q

i

∈ K : D

i

(η)(q

i

) > 0 and

D

i

(η)(q

i

) ≥ D

i

(η)(q

i+1

)),i=1, ··· ,m− 1}.

Let σ

n

: Fun[K, Z

+

] × n → Fun[K, Z

+

] be the

mapping, given by, ∀x ∈ K, ∀n ∈ Z

+

,

σ

n

(x)=

max (h

f

),ifx ∈ Q

0, otherwise

.

THE USE OF DYNAMICS IN GRAYLEVEL QUANTIZATION BY MORPHOLOGICAL HISTOGRAM PROCESSING

125

Let η

n

: Fun[K, Z

+

] × n → Fun[K, Z

+

] be the

mapping, given by, ∀x ∈ K, ∀n ∈ Z

+

,

η

n

= ν(ρ

∗

B,ν(η)

(σ

n

)).

By applying the operator η

n

, the n regional max-

ima of η(·) which have the highest dynamics are se-

lected. The function η

n

(·) contains just n regional

maxima and they are responsible for the classiﬁcation

of n classes (given by application of watershed oper-

ator). The remaining peaks are removed.

The method proposed in section 4 can be now ex-

tended to reduce an image to n graylevels, by adding

the dynamics step introduced in this subsection to the

framework, before the application of the watershed

operator.

5 EXPERIMENTAL RESULTS

In this section, it will be presented two experimen-

tal results. In both presented experiments, it was ap-

plied the proposed method to graylevel reduction and

it were observed the simpliﬁcation of the image in

terms of ﬂat zones and the quality of the obtained

images. Four method were compared in both experi-

ments: the choice of n highest peaks selection (Flores

and Lotufo, 2001) and the three dynamics proposed

in this paper (depth, area and volume).

In the ﬁrst experiment, it was aimed to simplify

the table tennis player (Fig. 10 (a)), an image with

66574 ﬂat zones and 254 graylevels, to an image with

8 graylevels. It were applied the highest peaks method

and the four dynamics to reduce the image to the de-

sired amount of graylevels. Table 1 shows the amount

of graylevel reduction given by the application of each

criterion and Fig. 10 (b) and Fig. 11 (a-c) show their

respective results.

Note that the results given by the highest peaks and

the depth dynamics are close, both in ﬂat zones reduc-

tion and the quality of image, and the depth dynamics

presented a slight better visual result. Area and vol-

ume dynamics provided the best results, both in the

simpliﬁcation of the image and in the quality of the

resulting image.

The goal in the second experiment was to sim-

plify the foreman image (Fig. 12 (a)) to 10 graylevels.

This original image has 69301 ﬂat zones and 252

graylevels. In this experiment, it was compared the

highest peaks to the volume dynamics. Figure 12 (b)

and (c), show, respectively the quantization provided

by application of both metrics. The highest peaks

provided a reduction to 6575 in the number of ﬂat

zones and the volume dynamics provided a reduction

to 4757 ﬂat zones. Note that the result provided by

the volume dynamics is also visually better: note that

the changes given by the dynamics metric is smoother

than the result given by the highest peaks criterion.

Table 1: Flat Zones Reduction.

Criterion Flat Zones

Original Image 66574

Highest Peaks 4375

Depth Dynamics 4595

Area Dynamics 2138

Volume Dynamics 1994

(a)

(b)

Figure 10: (a) Original Image (66574 Flat Zones). (b) High-

est Peaks (4375 Flat Zones).

6 CONCLUSION

This paper proposes an extension to a method applied

to graylevel quantization, proposed in a previous pa-

per (Flores and Lotufo, 2001). The method consists in

the classiﬁcation of graylevels by application of mor-

phological operators to the histogram. Its result is a

new image with a fewer regions and a simpliﬁed col-

ormap, compared to the original image.

In the previous paper, it was proposed a way to

compute the reduction of graylevels to n levels, con-

sisting in the choice of the highest peaks and discard-

ing the remaining ones. The extension of this method

in given by the application of dynamics as a criterion

to the choice signiﬁcant distributions in the histogram,

instead to choose the highest ones.

VISAPP 2006 - IMAGE FORMATION AND PROCESSING

126

(a)

(b)

(c)

Figure 11: (a) Depth Dynamics (4595 Flat Zones). (b) Area

Dynamics (2138 Flat Zones). (c) Volume Dynamics (1994

Flat Zones).

(a)

(b)

(c)

Figure 12: (a) Original Image (69301 Flat Zones). (b) High-

est Peaks (6575 Flat Zones) (c) Volume Dynamics (4757

Flat Zones).

THE USE OF DYNAMICS IN GRAYLEVEL QUANTIZATION BY MORPHOLOGICAL HISTOGRAM PROCESSING

127

Some experiments were done in order to compare

the highest peaks criterion to the dynamics metrics.

The goal in both experiments presented here was to

analyze the ﬂat zone reduction provides by the met-

rics and the visual quality of the resulting images.

The two experiments showed better results provided

by the dynamics, especially the results given by the

volume dynamics.

Further work includes the investigation of alter-

native metrics to choose the distributions in the his-

togram and the extension of the method proposed in

this paper to color images.

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