application of the karhunen-loeve transform for natural color images analysis

8/8/2019 Application of the Karhunen-Loeve Transform for Natural Color Images Analysis

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Application of the Karhunen-Loeve Transform for Natural ColorImages Analysis

R.K. Kouassi J .C . Devaux P. Gouton M. PaindavoineLaboratoire dElectronique, dInformatique et dImageUniversit4 de Bourgogne

BP400 21011 Dijon Cedex FRANCEpaindaveu-bourgogne.fr

AbstractTh e current s ys tems of images capture are based

on Red, Green and Blue (R ,G,B) principles . But , thismodel of capturing color zmages is di f ferent from hu-man v i sual s ys tem. So, t o obta in a representat ionwhich approaches the hum an sys tem , one uses the in-tensaty and chrominance space. A s this representa-t ion zs non-lanear, it introduces an instability of color.Thus , to analyze natural color images , we use theKarhunen-Loeue space, which al lows a largest decorre-lut ion of color components , a best definition of colorsa,nd a compress ion rat io increasing for very homoge-neous images .1 Introduction

The calculation of correlation coefficients and thespatial representation of color images allow us toestablish a comparative study between RGB (Red,Green, Blue), HSI (Hue, Saturation, Intensity) andI B

if G < BE l s e {2.2 Transformation from RGB to KL

The vectorial relationship that allows to define theKL space is : K = A( I - ml ) where I , m l , A and Kare respectively the original color image vector , th emean vector of theorigina1 color ima ge, the trans-formation matrix and the transformed vector [a].

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C R G B =

Table 1: Mean correlation coefficients

~ R R RG ~ R B~ G R G G G B~ B R B G B B

The analysis of the table shows that the compo-nents of the images are very correlated in the RGBspace (the values are closed to the unit). In the HSIspace, the correlation coefficients are lower and we ob-tain a quasi total non-correlation with the KL space.3.2

We express in the table 2 bellow the ratios of thevolume occupied by each image in RGB space to itsvolume in K L space.

Storage of color images in KL space

,Planes I 3.2 x l o 6Woman I 2. 6 x l o 6

I Volume KL I T I E,(%)Parrots I 13.1x IO6 I 0.86 1 5.73I0.19 I 9.540.16 I 3.72

1 SDheres I 45.3 x l o 6 I 2.73 I 13.30 I- I I I ITable 2: Volume ratios

T represents this ratio. E, represents the meanquadratic error between the original color image andthe color image obtained after the compression in theKL space.

The analysis of the table 2 shows that it is possibleto reduce the volume of spati al representation by usingthe KL space. We explain the decrease by a concen-tration of dat a around the two main axes ( l i L 1 , K L a )of the KL space [4]. These results are very interest-ing and allow to do compression on images. Indeed,one pixel of color image is initially coded on 24 bitsin the space RGB. In the KL space, this number ofbits can be reduced without debasing the initial colorquality. Here, we have fixed the tota l number a t 19bits t o represent one pixel in the KL space, that givesa compression ratio of 20 %. In some cases, this num-ber can be decreased to obtain acceptable compressionratio between 20% and 30%.

In the next pages, we present, on the left column,original color images and, on the right one, the colorimages obtained after compression. These samplesshow that the compression of color images in the KLspace is interesting : the quality of colors after com-pression is nearly the same to the initia l quality fornatural color images.

For the image SPH ERES , the KL transform leadsto an expansion of color. This result explains the im-portance of the mean quadratic error obtained on thisimage (13 .3%) .As for the image PLANES, we obta in a quadratic

error of 9.54% what is high for an image, that seemsto be visually homogeneous, with very few colors. Wecan explain this result by the fact that colors in thisimage are spatially distant (Blue is far from Whiteand Black, even far from White). For colors that arespatially distant, t he color compression is not possibleby KL transform.

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Parrots Parrots after compression

Planes Planes after compression

Woman

House

Woman after compression

House after compression

Lighthouse Lighthouse after compression

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Pond Pond after compression

Mountains

SpheresPut asides these exceptions, the compression of the

color images in the KL space gives very good resultsfor the images of our study.4 Approximation of the KL space t o

stu dy na tural color imagesThis part presents the approximation of the KL

transform. In fact, some linear transforms (as discretFourier transform, Hadamar transform, Haar trans-form, Discrete Cosine Transform (DCT) or DiscreteEven Sine Transform ( DEST )) are approximations ofthe I


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three orthogonal vectors that define the three imagesplans (th e components of th e color image in the newspace), tha t one can compare to the three images de-fined by the three KL axes. Each vect,ors has threecomponents: A,j with U = 1 , 2 , 3and j = 1 , 2 , 3 .4 .2 Analysis of results

We just examine for KL approximation, naturaland homogeneous images (with mean quadratic errorsinferior to 6%). Thu s, by computing the coefficientsof correlation between components in DCT and DESTspaces, we obtain tables 3 and 4.ParrotsWoman

0.02 14 x l o 6 0.8600.51 6.3 x l o 6 0.384HousePondLighthouse

ItXGmntains j 0.20 j 5.6 x l o 6 1 0.340 10.07 6.1 x l o 6 0.3700.28 2.9 x l o 6 0.1800.47 1. 7 x l o 6 0.120

Table 3:Coefficients of correlation in DCT spaceDEST 1 4 I VolumeDEST 1 TParrots 1 0.14 1.7 x l o 6 I 1.00

8.7 x 10House 9.0 x 107.0 x 10Pond 5.3 x 10Mountains 1 0 .32 I 11.4 x l o 6 [ 0.67

Table 4: Coefficients of correlation in DEST space4 and T represent respectively the mean correla-

tion coefficient between components and the ratio ofthe volume o f the image in the DCT or DEST spacesby i ts volume in the RGB space.

These results confirm the previous analysis in theapproximation of th e KL space for homogeneous andnatural color images. Coefficients of correlation areweak as well as volumic ratios. By comparing thesetwo tables, we can see that DCT space approximatebetter the KL space, than the DEST space.

5 ConclusionThis study shows properties of non-correlation and

storage of data by using the KL space rather thanRGB or HSI spaces. In HSI space, we can not differ-entiate Red and Black in the Hue component. So, th eKL space is important for the compression of colorimages. However, it is necessary t o specify tha t re-sults, on the compression, have been obtained for acompression rati o of 20%. But, it is possible, in somecases, to increase the ratio to 30%, specially when im-ages present very few colors and when these colors arespatially neared.

One can use an approximation of the KL space byDCT or DEST spaces with natur al homogeneous colorimages.

In order to improve this method, we are now exper-imenting the multiresolution approach .References[l] T. Carron, Segmentat ions d images couleur dunsla base Teinte-Luminance-Saturation : approche

numerique et symboligue, Thkse de luniversitk deSavoie, France, 1995.

[a] .A . Ocadiz Luna, Analyse en composantes pr in-cipales dune image couleur , Thkse de 1UniversitP:de Grenoble, France, 1985.

[ 3 ] M. Kunt, Trai t eme nt numerique des images ,Presses Polytechniques et Universitaires Roman-des, collection Glectriciti, v01.2, 1993.

[4] A . Brun Buisson, V. Lattuati, D. Lemoine,Prksegmentation dimages couleur par la trans-formke de Karhunen-Loeve, Quar toz rhme Col-loque G R E T S I , Juan-les-Pins, pp. 743-746, 1993.

[5] M. Unser, On the approximation of the dis-crete Karhunen-Loeve transform for statio naryprocesses , Szgnal Proccsszng Laboratoty,Vol. 7,pp. 743-746, 1984.

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application of the karhunen-loeve transform for natural color images analysis

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