COLOR QUANTIZATION BY MORPHOLOGICAL HISTOGRAM

PROCESSING

Franklin C´esar Flores

Department of Informatics, State University of Maring´a, Maring´a, Brazil

Leonardo Bespalhuk Facci

Department of Informatics, State University of Maring´a, Maring´a, Brazil

Roberto de Alencar Lotufo

School of Electrical and Computer Engineering, State University of Campinas, Campinas, Brazil

Keywords:

Color quantization, Morphological histogram processing, Watershed transform.

Abstract:

In a previous paper it was proposed a graylevel quantization method by morphological histogram processing.

This paper introduces the extension of that quantization method to color images. Considering an image under

the RGB color space model, this extension reduces the number of colors in the image by partitioning an 3-D

histogram, similar to the RGB color space, in rectangular parallelepiped regions, through a iterative process.

Such partitioning is done, in each iteration, by application of the graylevel quantization method to the longest

dimension of the current region which has the greatest volume. The ﬁnal classiﬁed color space is used to

quantize the image. This paper also shows the comparison of the proposed method to the classical median cut

one.

1 INTRODUCTION

Color reduction techniques are fundamental in dig-

ital image processing and computer graphics. Im-

age quantization by color reduction (Gonzalez and

Woods, 1992; Heckbert, 1982; Soille, 1996) has

been applied to solve problems of image display, im-

age compression, image simpliﬁcation and segmenta-

tion. (Gomes and Velho, 1994).

Color quantization also provides the reduction of

ﬂat zones (connected regions 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 reduction of ﬂat zones does

not introduct borders in the image, but, by supress-

ing some borders, two of more ﬂat zones may be

joined in one. Flat zone reduction is usually applied to

image compression, image segmentation (Meyer and

Beucher, 1990; Beucher and Meyer, 1992) and in the

reduction of the statistics in an image in order to sim-

plify the number of samples used in pattern recogni-

tion techniques (Hirata Jr. et al., 1999; Flores et al.,

2000; Flores et al., 2002).

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

was proposed a method which givesnot only an image

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 morphological op-

erators 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 distribution in the his-

togram.

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

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

n regional maxima in the processed histogram and

to ﬁlter the other peaks. The chosen maxima will

provide the classiﬁcation of the graylevels in the his-

togram by application of watershed operator. The ma-

jor drawback of the method proposed in (Flores and

Lotufo, 2001) is the choice of the n regional maxima.

In that paper, the chosen regional maxima was the n

highest ones. Note that, by far, it is not the best crite-

rion to choose the regional maxima.

In a following paper (Flores et al., 2006), it

93

Flores F., Bespalhuk Facci L. and de Alencar Lotufo R. (2008).

COLOR QUANTIZATION BY MORPHOLOGICAL HISTOGRAM PROCESSING.

In Proceedings of the Third International Conference on Computer Vision Theory and Applications, pages 93-100

DOI: 10.5220/0001088400930100

Copyright

c

SciTePress

was proposed the application of dynamics (Grimaud,

1992) to select the regional maxima in order to

achieve a better graylevel reduction. Dynamics con-

sists in a valuation of extrema of the image by a mea-

sure of contrast that does not consider the size or

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

markers to morphological segmentation and achieve

hierarchical segmentation (Meyer, 1996). The results

achieved by the application of dynamics as regional

maxima criterion showed itselt far better than the sim-

ple choice of the highest peaks. Both, the visual qual-

ity of the resulting images and the ﬂat zones reduction

are better when dynamics is applied.

This paper introduces the extension of the quan-

tization method by morphological histogram process-

ing to color images. Considering an image under the

RGB color space model, this extension reduces the

number of colors in the image to at most n colors by

partitioning the RGB color space in rectangular par-

allelepiped regions. Such partition is an iterative pro-

cess where, in each iteration, one region is split in at

most k rectangular parallelepiped regions. The split-

ting of a region is done by choosing the longest side

of the parallelepipe (one of the three dimensions, red,

green or blue), computing the histogram along this

side and applying the graylevel quantization method

to this histogram; the result gives where the region

must be split and how many regions are created in

that iteration.

This paper is organized as follows: section 2

presents some preliminar deﬁnitions used in this pa-

per. Section 3 introduces the color quantization

by morphological histogram processing. Section 4

presents some experimental results and section 5 con-

cludes this paper with a brief discussion.

2 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 im-

age 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

(Flores and Lotufo, 2001) of x, x ∈ E. We deﬁne a

path 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]. A ﬂat 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 - recon-

struction 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 (Heij-

mans, 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, be a

threshold function, where τ

i

( f)(x) = 1, if f(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 ad-

jacent to Z (Flores and Lotufo, 2001). 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 con-

nectivity.

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.

2.1 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 hierarchical 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).

VISAPP 2008 - International Conference on Computer Vision Theory and Applications

94

Grimaud (Grimaud, 1992) also deﬁnes the dy-

namics 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. However, 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 histogram func-

tion. So, it will be considered here that Dyn

f

(x, y) =

Dyn

f

(P(x, 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.

Dynamics computation can be implemented by

using 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 minimum:

1. depth of the catchment basin which the minimum

is contained (given by the dynamics of the mini-

mum itself);

2. area of the catchment basin;

3. volume of the catchment basin;

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 mea-

surements. Classiﬁcation of regional minima in an

image can be achieved by application of such metrics.

2.2 Graylevel Quantization by

Morphological Histogram

Processing

For a complete description of the graylevel quan-

tization by morphological histogram processing,

see (Flores et al., 2006; Flores and Lotufo, 2001). It

consists in an application of a set of morphological

operators to the image histogram. Since each object

in the image has a signiﬁcative graylevel distribution,

that is enough to classify its corresponding distribu-

tions to simplify them.

Basically, the method computes all regional max-

ima in the histogram of the graylevel image and ﬁlters

all unnecessary regional maxima located in the signi-

ﬁcative distributions in the histogram (Fig. 2). The

ﬁltered image is negated and the watershed operator

is applied, resulting in a pre-classiﬁcation (Fig. 3).

Figure 1: Histogram.

Given an image with the regional maxima from

the ﬁltered histogram (Fig. 1) labeled with their re-

spective graylevels. Its inf-reconstruction conditioned

to the pre-classiﬁcation achieved by the watershed

operator gives classiﬁcation of all graylevel classes

(Fig. 3). The classiﬁed histogram is used as a look-

up table in order to reduce the graylevels.

Figure 2: Pre-classiﬁcation (ﬁltered histogram and the wa-

tershed result).

As a consequence of such processing we have

a reduction in the graylevels appearing in the im-

age. In other words, the proposed ﬁlter is a mapping

ψ : Fun[E, K

1

] → Fun[E, K

2

], where |K

2

| < |K

1

|.

COLOR QUANTIZATION BY MORPHOLOGICAL HISTOGRAM PROCESSING

95

Figure 3: Classiﬁed graylevels.

2.3 The Use of Dynamics in Graylevel

Quantization by Morphological

Histogram Processing

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 the ﬁltered histogram. 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 sub-

section we will present a way to select the n most sig-

niﬁcant regional maxima of the ﬁltered histogram 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 original histogram was ﬁltered

in order to preserve the highest regional maximum

among a set of regiona maxima belonging to the same

distribution. Let h

f

and η(·) be, respectively, the orig-

inal and the ﬁltered histograms.

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, if µ

max

B

(η)(x) = 1

0, otherwise

,

where ν is the negation operator and Z is one of the re-

gional minima of the negation of η(·). In other words,

if x belongs to a regional maximum in η(·), D

i

(η)(x)

will be equal to the dynamics (see section 2.1) 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}.

(i.e. Q is a sequence of all computed dynamics, ac-

cording to criterion i, in decreasing order).

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

), if x ∈ Q

0, otherwise

.

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 classiﬁca-

tion of n classes (given by application of watershed

operator). The remaining peaks are removed.

The method proposed in subsection 2.2 can be

now extended to reduce an image to n graylevels, by

adding the dynamics step introduced in this subsec-

tion to the framework, before the application of the

watershed operator.

3 THE PROPOSED METHOD

The color quantization method proposed in this paper

receives, as input data:

• The original image to be quantize (under the RGB

color space model);

• A positive integer n : the number of colors the

original image will be reduced to.

• A positive integer k : the maximum number of re-

gions a parallelepiped region will be split in each

iteration. If a region is split without the control

of this parameter, the splitting method may com-

pute many points and it may lead to the splitting

of many regions in a single iteration and to a bad

quantization result. With this parameter,no region

is split in more than k regions in a single iteration

(i.e., it is possible that the method ﬁnds less than

k splitting points in a iteration. If it is the case, the

method uses the points it found).

The starting parallelepiped region is a 3-D his-

togram from the original color image. This 3-D his-

togram is a discrete RGB color space cube, which di-

mensions are related, respectively, to red, green and

blue. Each color appearing in the original image has

a corresponding point inside this cube, and each point

stores how many pixels the color appears in the orig-

inal image. All other points in the cube are valued

zero.

Color quantization by morphological histogram

processing is the iterative process describe below:

VISAPP 2008 - International Conference on Computer Vision Theory and Applications

96

1. Compute the starting region (the 3-D histogram

from the original image);

2. i ← 1 (the current number of regions);

3. Find, among the i current parallelepiped regions,

that one which have the greatest volume (let us

call it R

i

);

4. Take the longest dimension of the parallelepiped

region R

i

;

5. Let L

i

= [a

i

, b

i

] be the interval that deﬁnes the

longest dimension of R

i

;

6. For all l ∈ L

i

, let R

i

(l) be the slice of the region

that contains all the points projected in l (for in-

stance, if the longest dimension of R

i

corresponds

to the red band, R

i

(l) contains all points (l, x, y),

such that (l, x, y) ∈ R

i

);

7. Compute the 1-D histogram along the longest di-

mension of R

i

. The 1-D histogram h

i

: L

i

→ Z

+

is

given by, for all l ∈ L

i

,

h

i

(l) =

∑

x∈R

i

(l)

value(x),

where value(x) is the number of pixels that the

color x appears in the original image.

8. Apply the graylevel quantization by morpholog-

ical histogram processing to h

i

. Reduce it to, at

most, k classes. The choice of the k peaks is done

by computing the most signiﬁcative volume dy-

namics. The classiﬁcation of h

i

gives the points

where R

i

must be split. Let s be the number of

regions that R

i

will be split;

9. Split R

i

;

10. Let i ← i+ s− 1;

11. if i < n, go to Step 3.

Otherwise, stop.

The result of this algorithm is the classiﬁcation

of the RGB color space model in at most n paral-

lelepiped regions. The color to be assigned to a region

is given by the centroid point of all color points that

belongs to the region.

The classiﬁed color space also works as a look-up

table. To quantize a color from the original image,

just check the region where the color belongs to and,

then, change the color by the one assigned to that re-

gion.

Figures 4 and 5 show the application of the color

quantization method to reduce Fig. 4 (a) to, respec-

tively, n = 64 and n = 16 colors. The original image

(Fig. 4 (a)) has 47915 distinct colors.

Figure 4 (b-c) shows the quantization to 64 colors

with parameter k = 2 and k = 4, respectively. The

visual result is a few better in Fig. 4 (b) than the result

shown in Fig. 4 (c).

The quantization of Fig. 4 (a) to 16 colors is shown

in Fig. 5. Figures 5 (b-e), show, respectively, the re-

sults achieved by the following choices of k: 2 (Fig. 5

(b)), 3 (Fig. 5 (c)), 4 (Fig. 5 (d)) and 5 (Fig. 5 (e)). In

this example, the lower the value of k, the better the

visual result.

There is a trade off in the choice of k: the lesser

the value, the better the visual result. However, the

greater the value of k, the faster the method converges

to the results. If there is no much difference in the

visual quality provided by several k values, as in the

case shown in Fig. 4, the higher values may be a good

choice.

4 EXPERIMENTAL RESULTS

The color quantization by morphological histogram

processing is, at a few points, similar to the median

cut algorithm (Heckbert, 1982), a classical quantiza-

tion method in computer graphics and image process-

ing context. Some experiments were done in order to

compare the color quantization method proposed in

this paper to the median cut one.

The results of each experiment will be assessed

qualitatively, by assessing the visual quality of the re-

sulting images, and quantitatively, by analyzing the

results of a quantization error function. The quantiza-

tion error function (Braquelaire and Brun, 1997) used

in this paper is given by,

E =

n

∑

i=1

∑

c∈C

i

f(c)

c− c

i

2

,

where n is the number of colors the image was quan-

tized to, C

i

is the set of colors in the original image

image converted to color c

i

, and f (c) is the number of

pixels in the image which color is c.

The goal in the ﬁrst experiment is to reduce Fig. 6

(a) to n = 256 colors using the proposed method and

the median cut to assess the visual quality of the re-

sults provided by them. Figure 6 (a) has 89648 dis-

tinct colors and 288× 451 = 129888 pixels.

Figure 6 (b-c) show, respectively, the results pro-

vided by the color quantization by morphological his-

togram processing (using k = 5) and the median cut.

Both visual results are very good, but the result pro-

vided by the morphological method (Fig. 6 (b)) still

retained some small details from the original image.

The quantization errors computed to both quan-

tized images are very close to each other. The error

computed from the proposed method was 2.2968· 10

7

COLOR QUANTIZATION BY MORPHOLOGICAL HISTOGRAM PROCESSING

97

(a)

(b)

(c)

Figure 4: Reduction to n = 64 colors: (a) Original Image.

(b) k = 2. (c) k = 4.

(mean error of 176.8342 per pixel). The error com-

puted from the median cut image was 2.4003 · 10

7

(mean error of 184.8007 per pixel). The proposed

method achieved an error smaller than the achived by

the median cut one.

The second experiment consists in to reduce the

same original image (Fig. 6 (a)) to n = 16 colors using

the morphological method proposed in this paper and

the median cut technique. Again, the visual quality of

the results provided by them will be assessed.

The results provided by the color quantization by

morphological histogram processing (using k = 3)

(a)

(b)

(c)

(c)

Figure 5: Reduction to n = 16 colors: (a) k = 2. (b) k = 3.

(c) k = 4. (d) k = 5.

VISAPP 2008 - International Conference on Computer Vision Theory and Applications

98

(a)

(b)

(c)

Figure 6: Reduction to n = 256 colors: (a) Original Im-

age. (b) Color Quantization by Morphological Histogram

Processing (k = 5). (c) Median Cut.

and the median cut are shown, respectively, in Fig. 7

(a-b). Both methods present results with a strong loss

of quality in the visualization, but the result provided

by the quantization method proposed in this paper

(Fig. 7 (a)) has a visual quality far better than the pro-

vided by the median cut (Fig. 7 (b)).

The difference between the quantization errors is

more evident in this experiment. Quantization er-

ror computed from the proposed quantization method

was 1.4460·10

8

(mean error of 1113.2730 per pixel).

Median cut error was 2.2306 · 10

8

(mean error of

(a)

(b)

Figure 7: Reduction to n = 16 colors: (a) Color Quantiza-

tion by Morphological Histogram Processing (k = 3). (b)

Median Cut.

1717.3413 per pixel). In this experiment, the error

achieved by the proposed quantization method was

far lower than the error achieved by the median cut

method.

5 CONCLUSIONS

Color quantization by morphological histogram pro-

cessing, the extension of the graylevel processing pro-

posed in a previous paper, reduces the colors of an

image (under the RGB color space model) in a iter-

ative process where a 3-D histogram computed from

the image is split in at most n regions. This classi-

ﬁed ”color space” is used as a look-up table to do the

image quantization.

The splitting of regions is an iterative process. In

each iteration, one region is split in at most k rect-

angular parallelepiped regions; it is done by choos-

ing the longest side of the parallelepipe region, com-

puting the histogram along this side and applying the

graylevel method to this histogram. It provides how

many regions are created in that iteration and where

the region must be split.

COLOR QUANTIZATION BY MORPHOLOGICAL HISTOGRAM PROCESSING

99

The method introduced in this paper depends on a

parameter k, that is the maximum number of regions

a parallelepipe will be split in each iteration. The de-

pendence of the parameter k is a drawback of the pro-

posed method and some way to impose an automatic

k value should be studied. Some quantization results

using several k values are shown and discussed in the

paper.

Experiments were done in order to compare the

quantization method proposed in this paper with the

classical median cut technique. In the ﬁrst experi-

ment, both methods provided good visual quality re-

sults but the morphological methods still retained a

few details from the original image. The second ex-

periment showed a strong loss of information in the

application of both methods, but the color quantiza-

tion introduced in this paper provided a better visual

result. More, quantitative analysis was done in both

experiments and the quantization error given by the

application of the proposed quantization method was

lower than the error given by the median cut one.

Future works include the choice of new criteria to

choose the most signiﬁcant peaks in the ﬁltered his-

togram and the automatic choice of the k parameter.

ACKNOWLEDGEMENTS

First author is on leave from State University of Mar-

ing´a for doctorate purposes at School of Electrical and

Computer Engineering, State University of Camp-

inas, Brazil.

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