a la physique de l'information - Lisa - Université d'Angers

Recommendations

Info

Multidimensional Systems and Signal Processing manuscript No. (will be inserted by the editor) Fractal capacity dimension of three-dimensional histogram from color images Julien Chauveau · David Rousseau · François Chapeau-Blondeau Received: date / Accepted: date Abstract To contribute to the important task of characterizing the complex multidimensional structure of natural images, a fractal characterization is proposed for the colorimetric organization of natural color images. This is realized from their threedimensional RGB color histogram, by applying a box-counting procedure to assess the dimensionality of its support. Non-integer values for the resulting capacity dimension of the histograms emerge for natural color images. This manifests a fractal colorimetric organization with a self-similar structure of the color palette typically composing natural images. Such a fractal characterization complements other previously known fractal properties of natural images, some reported recently in their colorimetric organization, and others reported more classically in their spatial organization. Such fractal multiscale features uncovered in natural images provide helpful clues relevant to image modeling, processing and visual perception. Keywords Color images · Three-dimensional histogram · Fractal dimension · Multicomponent images 1 Introduction Natural images are complex multidimensional information-carrying signals. Their understanding and modeling are essential to many tasks in image processing and vision, and advances are still needed in this direction. Among specific properties that have been found constitutive of the complex structure of natural images are fractal properties. Principally, fractal or self-similarity or scaling properties have been observed in the spatial organization of natural images [23,22,14]. Such spatial fractal properties can be related to the many features and details which usually exist across many spatial scales, in a self-similar way, in natural scenes. Such fractal self-similar structures have also been found to extend to the temporal organization of time-varying sequences of natural images, as perceived by the visual system [6]. These findings of such spatial or temporal fractal scaling properties are helpful to construct more realistic models Laboratoire d’Ingénierie des Systèmes Automatisés (LISA), Université d’Angers, 62 avenue Notre Dame du Lac, 49000 Angers, France. 167/197
2 for natural images, and carry relevance for image coding and processing [21,26,4] and vision systems [16,20]. In the present paper, we will address another, complementary, aspect of the fractal properties of natural images. Beyond the spatial and temporal organizations of natural images, we address here their colorimetric organization. Some evidence has recently been reported that a fractal or self-similar organization also exists in the colorimetric domain for natural images [3,2]. The pixels of natural color images would tend to distribute, over the colorimetric space, in a self-similar fractal arrangement. This has been shown in [3,2] by means of the evaluation of the correlation dimension of the distribution of pixels in the colorimetric space, as represented by the three-dimensional color histogram of the images. Two distinct estimators in [3] and in [2] have been tested for the correlation dimension, which confirm the presence of non-integer correlation dimension manifesting a fractal distribution of the pixels in the colorimetric space. We complement these investigations here by considering another approach to further characterize a fractal colorimetric organization. We evaluate the capacity dimension of the three-dimensional histogram of color images. Such a capacity dimension characterizes the structure of the support of the three-dimensional histogram, which expresses, in the colorimetric space, which colors are present in an image and which are not. In other words, the capacity dimension characterizes the structure of the color palette employed by Nature to compose a given natural image. We show here that such a color palette tends to display a fractal organization, identified by a non-integer capacity dimension. By contrast, the correlation dimension of [3,2] offers a more composite view, as it simultaneously characterizes the color palette and the populations of pixels distributing among these existing colors. These are two complementary approaches on the fractal colorimetric organization of images, which are made possible by the evaluation of the capacity dimension we perform here. 2 Three-dimensional color histogram We consider RGB color images [25], with each component R red, G green and B blue varying among Q possible integer values in [0, Q −1] at each pixel of spatial coordinate (x1, x2). The histogram of such color images is a three-dimensional structure comprising Q 3 colorimetric cells, each of which containing the pixels with this color in the image. For the very common choice Q = 2 8 = 256, the histogram with its Q 3 = 2 24 ≈ 16.8×10 6 cells is a large data structure which can display complex organization. Instead of the full three-dimensional histogram, three marginal one-dimensional histograms, separately for each R, G or B component, are often considered to lead to simpler data handling. In addition, the full three-dimensional histogram can be highly nonuniform. For a typical RGB color image with size 512 × 512 and Q = 256, a total of 512 2 = 2 18 pixels distibute among the Q 3 = 2 24 colorimetric cells. This means that in the three-dimensional color histogram, most of the cells are likely to be empty, while cells corresponding to dominant colors in the image may be highly populated. A typical three-dimensional color histogram is thus a large data structure, highly nonuniform, with possibly large voids, some high local concentrations of pixels, an overall diffuse character, and not so often considered (in place of the three marginal histograms). Such features can be perceived on the two examples of three-dimensional color histograms from natural images shown in Fig. 7. We will implement here a characterization of the three-dimensional color histogram, manifesting the prevalence of a fractal organization. 168/197
Page 1 and 2:
Université d’Angers Laboratoire
Page 3 and 4:
Ce mémoire est dédié à mon épo
Page 5 and 6:
4 Physique de l’information 39 4.
Page 7 and 8:
1.2 Organisation du document L’es
Page 9 and 10:
4/197
Page 11 and 12:
de Rennes 1. Licence et Maîtrise
Page 13 and 14:
2.5 Activités de recherche 2.5.1 B
Page 15 and 16:
2.5.2 Encadrement Thèses Thèse so
Page 17 and 18:
2.5.3 Responsabilités Management d
Page 19 and 20:
[A20] S. BLANCHARD, D. ROUSSEAU, D.
Page 21 and 22:
In 8th Euro-American Workshop on In
Page 23 and 24:
études visaient alors à analyser
Page 25 and 26:
Fig. 3.1 réside dans le fait qu’
Page 27 and 28:
également permis de réaliser qu
Page 29 and 30:
caracteristique effective g eff (u)
Page 31 and 32:
des architectures neuronales contr
Page 33 and 34:
Dans l’ Éq.(3.10), ∆t représe
Page 35 and 36:
quantifieur, le niveau de saturatio
Page 37 and 38:
exemples présentés dans [39], l
Page 39 and 40:
• le type de validation des résu
Page 41 and 42:
présenté de façon condensée dan
Page 43 and 44:
38/197
Page 45 and 46:
pour l’étude de la turbulence pa
Page 47 and 48:
A B C Figure 4.1 : Images de coupe
Page 49 and 50:
atteint la capacité informationnel
Page 51 and 52:
* Figure 4.5 : Échelle optimale d
Page 53 and 54:
• L’“intégrale de corrélati
Page 55 and 56:
4.2.3 Analyse multifractale en imag
Page 57 and 58:
fonction de partition Z 10 30 10 25
Page 59 and 60:
exposant τ(q) 15 10 5 0 −5 −10
Page 61 and 62:
complexité colorimétrique des ima
Page 63 and 64:
dimension fractale généralisée D
Page 65 and 66:
est la moyenne du signal calculée
Page 67 and 68:
A B C Figure 4.17 : Observations in
Page 69 and 70:
A B C Figure 4.20 : Nombre moyen de
Page 71 and 72:
[14] S. Blanchard, D. Rousseau et F
Page 73 and 74:
[43] J. Chauveau, D. Rousseau et F.
Page 75 and 76:
[76] A. Humeau, B. Buard, F. Chapea
Page 77 and 78:
[100] M. D. McDonnell, N. G. Stocks
Page 79 and 80:
[131] D. Rousseau, F. Duan, J. Roja
Page 81 and 82:
76/197
Page 83 and 84:
D. ROUSSEAU and F. CHAPEAU-BLONDEAU
Page 85 and 86:
ROUSSEAU AND CHAPEAU-BLONDEAU: BAYE
Page 87 and 88:
ROUSSEAU AND CHAPEAU-BLONDEAU: BAYE
Page 89 and 90:
D. ROUSSEAU, G. V. ANAND, and F. CH
Page 91 and 92:
quantizers. Classically, the design
Page 93 and 94:
Fig. 2. Nonlinear array used as pre
Page 95 and 96:
probability of error P er 10 0 10 -
Page 97 and 98:
probability of error (expected as s
Page 99 and 100:
[3] N.H. Lu, B.A. Einstein, Detecti
Page 101 and 102:
Abstract Neurocomputing 71 (2007) 3
Page 103 and 104:
3. Assessing nonlinear transmission
Page 105 and 106:
output SNR 250 200 150 100 50 0 N=3
Page 107 and 108:
precisely the scope of the present
Page 109 and 110:
[4] F. Chapeau-Blondeau, G. Chauvet
Page 111 and 112:
Nonlinear SNR amplification of harm
Page 113 and 114:
P.R. BHAT and D. ROUSSEAU and G. V.
Page 115 and 116:
It is assumed that the signal s is
Page 117 and 118:
Let us de£ne the improvement in pe
Page 119 and 120:
F. CHAPEAU-BLONDEAU and D. ROUSSEAU
Page 121 and 122: Contents Raising the noise to impro
Page 123 and 124: Raising the noise to improve perfor
Page 135 and 136: S. BLANCHARD, D. ROUSSEAU, D. GINDR
Page 137 and 138: 1984 OPTICS LETTERS / Vol. 32, No.
Page 139 and 140: F. CHAPEAU-BLONDEAU, D. ROUSSEAU, S
Page 141 and 142: 1288 J. Opt. Soc. Am. A/Vol. 25, No
Page 147 and 148: 36 IEEE SIGNAL PROCESSING LETTERS,
Page 149 and 150: 38 IEEE SIGNAL PROCESSING LETTERS,
Page 151 and 152: A. HISTACE and D. ROUSSEAU. Constru
Page 153 and 154: noises Z i in (3) are chosen Gaussi
Page 155 and 156: Author's personal copy Physica A 38
Page 157 and 158: Author's personal copy F. Chapeau-B
Page 161 and 162: edundancy Author's personal copy F.
Page 163 and 164: model description length Author's p
Page 165 and 166: total description length x 10 4 1.6
Page 171: J. CHAUVEAU and D. ROUSSEAU and F.
Page 175 and 176: 4 Fig. 2 Random RGB color image I2(
Page 177 and 178: 6 log2[ number of boxes N(a) ] 24 2
Page 179 and 180: 8 Fig. 7 Color histogram in the RGB
Page 181 and 182: 10 log2[ number of boxes N(a) ] 8 7
Page 183 and 184: 12 17. Landgrebe, D.: Hyperspectral
Page 185 and 186: Numerical simulation of laser Doppl
Page 187 and 188: esults are meaningful to better app
Page 189 and 190: A. HUMEAU, B. BUARD, F. CHAPEAU-BLO
Page 191 and 192: 618 A Humeau et al 1. Introduction
Page 193 and 194: 620 A Humeau et al (a) (b) (c) Figu
Page 195 and 196: 622 A Humeau et al (a) (b) (c) Figu
Page 197 and 198: 624 A Humeau et al Figure 5. Averag
Page 199 and 200: 626 A Humeau et al Therefore, the m
Page 201 and 202: 628 A Humeau et al peripheral cardi
show all

a la physique de l'information - Lisa - Université d'Angers

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?