{"title": "Self-similarity Properties of Natural Images", "book": "Advances in Neural Information Processing Systems", "page_first": 836, "page_last": 842, "abstract": null, "full_text": "Self-similarity properties of natural images \n\nANTONIO TURIEL; GERMAN MATOt NESTOR PARGA t \n\nDepartamento de Fisica Te6rica. Universidad AutOnoma de Madrid \n\nCantoblanco, 28049 Madrid, Spain \n\nand JEAN-PIERRE N ADAL\u00a7 \n\nLaboratoire de Physique Statistique de I 'E. N.S. , Ecole Normale Superieure \n\n24, rue Lhomond, F-75231 Paris Cedex OS, France \n\nAbstract \n\nScale invariance is a fundamental property of ensembles of nat(cid:173)\nural images [1]. Their non Gaussian properties [15, 16] are less \nwell understood, but they indicate the existence of a rich statis(cid:173)\ntical structure. In this work we present a detailed study of the \nmarginal statistics of a variable related to the edges in the images. \nA numerical analysis shows that it exhibits extended self-similarity \n[3, 4, 5]. This is a scaling property stronger than self-similarity: \nall its moments can be expressed as a power of any given moment. \nMore interesting, all the exponents can be predicted in terms of \na multiplicative log-Poisson process. This is the very same model \nthat was used very recently to predict the correct exponents of \nthe structure functions of turbulent flows [6]. These results allow \nus to study the underlying multifractal singularities. In particular \nwe find that the most singular structures are one-dimensional: the \nmost singular manifold consists of sharp edges. \n\nCategory: Visual Processing. \n\n1 \n\nIntroduction \n\nAn important motivation for studying the statistics of natural images is its relevance \nfor the modeling of the visual system. In particular, the epigenetic development \n\n\u2022 e-mail: amturiel@delta.ft.uam.es \nt e-mail: matog@cab.cnea.edu.ar \n+To whom correspondence should be addressed. e-mail: parga@delta.ft.uam.es \n\u00a7e-mail: nadal@lps.ens.fr \n'Laboratoire associe au C.N.R.S. (U.R.A. 1306), a l'ENS, et aux Universites Paris VI \n\net Paris VII. \n\n\fSelf-similarity PropeT1ies of Natural Images \n\n837 \n\ncould lead to the adaptation of visual processing to the statistical regularities in the \nvisual scenes [8, 9, 10, 11, 12, 13]. Most of these predictions on the development of \nreceptive fields have been obtained using a gaussian description of the environment \ncontrast statistics. However non Gaussian properties like the ones found by [15, 16] \ncould be important. To gain further insight into non Gaussian aspects of natural \nscenes we investigate the self similarity properties of an edge type variable [14]. \n\nScale invariance in natural images is a well-established property. In particular it \nappears as a power law behaviour of the power spectrum of luminosity contrast: \nS(f) ex: IfIL'! (the parameter 1] depends on the particular images that has been \nincluded in the dataset). A more detailed analysis of the scaling properties of the \nluminosity contrast was done by [15, 16]. These authors noted the possible analogy \nbetween the statistics of natural images and turbulent flows. There is however no \nmodel to explain the scaling behaviour that they observed. \nOn the other hand, a large amount of effort has been put to understand the statistics \nof turbulent flows and to develop predictable models (see e.g. [17]). Qualitative \nand quantitative theories of fully developed turbulence elaborate on the original \nargument of Kolmogorov [2]. The cascade of energy from one scale to another is \ndescribed in terms of local energy dissipation per unit mass within a box of linear \nsize r. This quantity, fr, is given by: \n\n(1) \n\nwhere Vi(X) is the ith component of the velocity at point x. This variable has Sel/(cid:173)\nSimilarity (SS) properties that is, there is a range of scales r (called the inertial \nrange) where: \n\n(2) \nhere < f~ > denotes the pth moment of the energy dissipation marginal distribution. \nA more general scaling relation, called Extended Self-Similarity (ESS) has been \nfound to be valid in a much larger scale domain. This relation reads \n\n(3) \n\nwhere p(p, q) is the ESS exponent of the pth moment with respect to the qth mo(cid:173)\nment. Let us notice that if SS holds then Tp = Tqp(p, q). In the following we will \nrefer all the moments to < f; >. \n\n2 The Local Edge Variance \n\nFor images the basic field is the contrast c(x), that we define as the difference \nbetween the luminosity and its average. By analogy with the definition in eq. (1) we \nwill consider a variable that accumulates the value of the variation of the contrast. \nWe choose to study two variables, defined at position x and at scale r. The variable \nfh,r(X) takes contributions from edges transverse to a horizontal segment of size r: \n\nl1xl +r (ac(x/))2 \n-a-\ny \n\nr Xl \n\nfh,r(X) = -\n\ndy \n\nX'={y,X2} \n\n(4) \n\nA vertical variable fv,r(X) is defined similarly integrating along the vertical direction. \nWe will refer to the value of the derivative of the contrast along a given direction \nas an edge transverse to that direction. This is justified in the sense that in the \npresence of borders this derivative will take a great value, and it will almost vanish \n\n\f838 \n\nA. Turiel, G. Mato, N. Parga and l-P. Nadal \n\nif evaluated inside an almost-uniformly illuminated surface. Sharp edges will be the \n( 1 = h, v) is the local \nmaxima ofthis derivative. According to its definition, \u20ac/,r(x) \nlinear edge variance along the direction 1 at scale r. Let us remark that edges are \nwell known to be important in characterizing images. A recent numerical analysis \nsuggests that natural images are composed of statistically independent edges [18]. \n\nWe have analyzed the scaling properties of the local linear edge variances in a set \nof 45 images taken into a forest, of 256 x 256 pixels each (the images have been \nprovided to us by D. Ruderman; see [16] for technical details concerning them). An \nanalysis of the image resolution and of finite size effects indicates the existence of \nupper and lower cut-offs. These are approximately r = 64 and r = 8, respectively. \nFirst we show that SS holds in a range of scales r with exponents Th,p and Tv,p. \nThis is illustrated in Fig. (1) where the logarithm of two moments of horizontal \nand vertical local edge variances are plotted as a function of In r; we see that SS \nholds, but not in the whole range. \nESS holds in the whole considered range; two representative graphs are shown in \n(2). The linear dependence of In < \u20acf,r > vs In < \u20acf,r > is observed in \nFig. \nboth the horizontal (l = h) and the vertical (l = v) directions. This is similar to \nwhat is found in turbulence, where this property has been used to obtain a more \naccurate estimation of the exponents of the structure functions (see e.g. [17] and \nreferences therein) . The exponents Ph(p, 2) and Pv(p,2), estimated with a least \nsquares regression, are shown in Fig. (3) as a function of p. The error bars refer to \nthe statistical dispersion. From figs. (1-3) one sees that the horizontal and vertical \ndirections have similar statistical properties. The SS exponents differ, as can be \nseen in Fig(I); but, surprisingly, ESS not only holds in both directions, but it does \nit with the same ESS exponents, i.e. Ph(P,2) '\" Pv(p, 2). \n\n3 ESS and multiplicative processes \n\nLet us now consider scaling models to predict the Jrdependence of the ESS expo(cid:173)\nnents Pl(p, 2). (Since ESS holds, the SS exponents Tl ,p can be obtained from the \nPl(p, 2)' s by measuring 72,2). The simplest scaling hypothesis is that, for a random \nvariable \u20acr(x) observed at the scale r (such as \u20ac/,r(x)), \nits probability distribution \nPr(\u20acr(x) = \u20ac) can be obtained from any other scale L by \n\nPr(\u20ac) = a(r~ L) PL (a(r~ L)) \n\n(5) \n\nFrom this one derives easily that a(r, L) = [~:~~P/p (for any p) and p(p, 2) ex: p; if \nSS holds, Tp ex: p: for turbulent flows this corresponds to the Kolmogorov prediction \nfor the SS exponents [2] . Fig (3) shows that this naive scaling is violated. \nThis discrepancy becomes more dramatic if eq. \n(5) is expressed in terms of a \nnormalized variable. Taking \u20ac~ = limp -+oo < \u20ac~+l > / < \u20ac~ > ( that can be shown \nto be the maximum value of \u20acr, which in fact is finite) the new variable is defined \nas ir = \u20acr/\u20ac~ ; 0 < ir < 1. If Pr(J) is the distribution of ir, the scaling relation \neq.(5) reads Pr(J) = PL(J) ; this identity does not hold as can be seen in Fig. (4). \nA way to generalize this scaling hypothesis is to say that a is no longer a constant \nas in eq. (5), but an stochastic variable. Thus, one has for Pr(J) : \n\nThis scaling relation has been first introduced in the context of turbulent flows \n[6, 19, 7]. Eq. (6) is an integral representation of ESS with general (not necessarily \n\n(6) \n\n\fSelf-similarity Properties of Natural Images \n\n839 \n\nlinear) exponents: once the kernel GrL is chosen, the p(p, 2)'s can be predicted. \nIt can also be phrased in terms of multiplicative processes [20, 21] : now ir = aiL, \nwhere the factor a itself becomes a stochastic variable determined by the kernel \nGrL (1na). Since the scale L is arbitrary (scale r can be reached from any other \nscale L') the kernel must obey a composition law, GrLI \u00aeG L' L = GrL. Consequently \nir can be obtained through a cascade of infinitesimal processes G6 == Gr ,r+6r' \nSpecific choices of G6 define different models of ESS. The She-Leveque (SL) [6] \nmodel corresponds to a simple process such that a is 1 with probability 1 - sand \nis a constant f3 with probability s. One can see that s = ll,lF In( <~tl;\u00bb and that \nthis stochastic process yields a log-Poisson distribution for a [22]. It also gives ESS \nwith exponents p(p, q) that is expressed in terms of the parameter f3 as follows [6]: \n\nf3P - (1 - (3)p \np(p,q) = 1- f3 q - (1- (3)q \n\n1 -\n\n(7) \n\nWe can now test this models with the ESS exponents obtained with the image data \nset. The resulting fit for the SL model is shown in Fig. (3). Both the vertical and \nhorizontal ESS exponents can be fitted with {3 = 0.50 \u00b1 0.03. \nThe integral representation of ESS can also be directly tested on the probability \ndistributions evaluated from the data. In Fig. (4) we show the prediction for Pr (f) \nobtained from PL(f) using eq. (6) , compared with the actual Pr(f). \nThe parameter f3 allows us to predict all the ESS exponents p(p,2). To obtain the \nSS exponents 7p we need another parameter. This can be chosen e.g. as 72 or as the \nasymptotic exponent ~, given by f~ ex: r-t::., r \u00bb 1; we prefer~. As 7p = 72 p(P, 2), \nthen from the definition of f~ one can see that ~ = -1\":!13' A least square fit of 7p \nwas used to determine ~, obtaining ~h = 0.4 \u00b1 0.2 for the horizontal variable and \n~v = 0.5 \u00b1 0.2. for the vertical one. \n\n4 Multifractal analysis \n\nLet us now partition the image in sets of pixels with the same singularity exponent \nh of the local edge variance: fr ex: rh. This defines a multifractal with dimensions \nD(h) given by the Legendre transform of 7p (see e.g. [17]): D(h) = inip{ph+d-7p}, \nwhere d = 2 is the dimension of the images. We are interested in the most singular \nof these manifolds; let us call Doo its dimension and hmin its singularity exponent. \nSince f~ is the maximum value of the variable fr, the most singular manifold \nis given by the set of points where fr = f~, so hmin = -~. Using again that \n7p = -~ (1- {3) p(P, 2) with p(P, 2) given by the SL model, one has Doo = d- (1~13)' \nFrom our data we obtain Doo,h = 1.3 \u00b1 0.3 and Doo,v = 1.1 \u00b1 0.3. As a result \nwe can say that Doo,h \"\" Doo,v \"\" 1: the most singular structures are almost one(cid:173)\ndimensional. This reflects the fact that the most singular manifold consists of sharp \nedges. \n\n5 Conclusions \n\nWe insist on the main result of this work, which is the existence of non trivial \nscaling properties for the local edge variances. This property appears very similar \nto the one observed in turbulence for the local energy dissipation. In fact, we have \nseen that the SL model predicts all the relevant exponents and that, in particular, \nit describes the scaling behaviour of the sharpest edges in the image ensemble. It \nwould also be interesting to have a simple generative model of images which - apart \n\n\f840 \n\nA. Turiel, G. Mato, N. Parga and J-P. Nadal \n\nfrom having the correct power spectrum as in [23] - would reproduce the self-similar \nproperties found in this work. \n\nAcknowledgements \n\nWe are grateful to Dan Ruderman for giving us his image data base. We warmly \nthank Bernard Castaing for very stimulating discussions and Zhen-Su She for a \ndiscussion on the link between the scaling exponents and the dimension of the most \nsingular structure. We thank Roland Baddeley and Patrick Tabeling for fruitful \ndiscussions. We also acknowledge Nicolas BruneI for his collaboration during the \nearly stages of this work. This work has been partly supported by the French(cid:173)\nSpanish program \"Picasso\" and an E.V. grant CHRX-CT92-0063. \n\nReferences \n\n[1] Field D. J., 1. Opt. Soc. Am. 4 2379-2394 (1987). \n[2] Kolmogorov, Dokl. Akad. Nauk. SSSR 30, 301-305 (1941). \n[3] Benzi R., Ciliberto S., Baudet C., Ruiz Chavarria G. and Tripiccione C., Eu(cid:173)\n\nrophys. Lett. 24 275-279 (1993) \n\n[4] Benzi, Ciliberto, Tripiccione, Baudet, Massaioli, and Succi, Phys. Rev. E 48, \n\nR29 (1993) \n\n[5] Benzi, Ciliberto, Baudet and Chavarria Physica D 80 385-398 (1995) \n[6] She and Leveque, Phys. Rev. Lett. 72,336-339 (1994). \n[7] Castaing, 1. Physique II, France 6, 105-114 (1996) \n[8] Barlow H. B., in Sensory Communication (ed. Rosenblith W.) pp. 217. (M.I.T. \n\nPress, Cambridge MA, 1961). \n\n[9] Laughlin S. B., Z. Naturf. 36 910-912 (1981). \n[10] van Hateren J.H. 1. Compo Physiology A 171157-170,1992. \n[11] Atick J. J. Network 3 213-251, 1992. \n[12] Olshausen B.A. and Field D. J., Nature 381, 607-609 (1996). \n[13] Baddeley R., Cognitive Science, in press (1997). \n[14] Turiel A., Mato G., Parga N. and Nadal J.-P., to appear in Phys. Rev. Lett., \n\n1998. \n\n[15] Ruderman D. and Bialek, Phys. Rev. Lett. 73,814 (1994) \n[16] Ruderman D., Network 5,517-548 (1994) \n[17] Frisch V., Turbulence, Cambridge Vniv. Press (1995). \n[18] Bell and Sejnowski, Vision Research 37 3327-3338 (1997). \n[19] Dubrulle B., Phys. Rev. Lett. 73 959-962 (1994) \n[20] Novikov, Phys. Rev. E 50, R3303 (1994) \n[21] Benzi, Biferale, Crisanti, Paladin, Vergassola and Vulpiani, \n\nPhysica D 65, 352-358 (1993). \n\n[22] She and Waymire, Phys. Rev. Lett. 74, 262-265 (1995). \n[23] Ruderman D., Vision Research 37 3385-3398 (1997). \n\n\fSelf-similarity Properties of Natural Images \n\n841 \n\nIn < \u20ac~ > \n\na \n\nb \n\n.. \n\n. \" \n\n\" \n\nInr \n\n.. \n\nInr \n\nFigure 1: Test of SS. We plot In < \u20acf r > vs. In r for p = 3 and 5; r from 8 to 64 \npixels. a) horizontal direction, l = h. b) vertical direction, l = v. \n\nIn < \u20ac~ > \n\n, \n\na \n\n\u00b7 \n\n....... /...-' \n\n\"\" ....\u2022 ' \n.... \n\n/ \n\n.\" .. , \n\n/\n\"./'\" \n\n././ \nIn < \u20ac~ > ~ \n\u00b7 \n\u00b7 \n.. \", /' \n. / \n\n.,..,/ \n\n, \n\n/ / \n\n... \",/ \n\n.. , ....... \n. \" . ' \n\n,,,,,,--\n\n/ \n\n. ~. \n\n./ \n\nb \n\n/ \n\n/ / \n\n/ / \n\n/ .-/\n\n' \n\n/\n\n'\" \n\n, / \n\n/ / \n\n./ \n-' \n// \n\n, ..... \n.... \n\n.-/ \n\n.... . \" \n/ . / \n\nIn < \u20ac~ > \n\nFigure 2: Test of ESS. We plot In < \u20acf.r > vs. In < \u20ac~,r > for p=3, 5; r from 8 to \nr = 64 pixels. a) horizontal direction, l = h. b) vertical direction, l = v. \n\n\f842 \n\np(p, 2) \n\n12 \n\n1. \n\na \n\np \n\nA. Turiel, G. Mato, N. Parga and J-P. Nadal \n\nb \n\np \n\nFigure 3: ESS exponents p(p, 2), for the vertical and horizontal variables. a) hor(cid:173)\nizontal direction, Ph (P, 2) . b) vertical direction, pv (p, 2). The solid line represents \nthe fit with the SL model. The best fit is obtained with (3v '\" (3h '\" 0.50. \n\nP \n\n18 \n\n16 \n\n14 \n\n12 \n\n10 \n\n8 \n\n6 \n\n4 \n\n2 \n\n++ \n+ \n\n+ \n+ \n+ \n\n+ \n+ \n+ \n+ \n+ \n\n++++ \n\n0 \n\n0 \n\n0.05 \n\n0.1 \n\nf \n\n0.15 \n\n0.2 \n\nFigure 4: Verification of the validity of the integral representation of ESS, eq.(6) \nwith a log-Poisson kernel, for horizontal local edge variance. The largest scale is \nL = 64. Starting from the histogram Pdf) (denoted with crosses), and using a \nlog-Poisson distribution with parameter (3 = 0.50 for the kernel GrL , eq.(6) gives \na prediction for the distribution at the scale r = 16 (squares). This has to be \ncompared with the direct evaluation of Pr (I) (diamonds). Similar results hold for \nother pairs of scales. Although not shown in the figure, the test for vertical case is \nas good as for horizontal variable. \n\n\f", "award": [], "sourceid": 1460, "authors": [{"given_name": "Antonio", "family_name": "Turiel", "institution": null}, {"given_name": "Germ\u00e1n", "family_name": "Mato", "institution": null}, {"given_name": "N\u00e9stor", "family_name": "Parga", "institution": null}, {"given_name": "Jean-Pierre", "family_name": "Nadal", "institution": null}]}