SINGULARITY EXTRACTION IN MULTIFRACTALS

SINGULARITY EXTRACTION IN MULTIFRACTALS: APPLICATIONS IN IMAGE PROCESSING∗ ANTONIO TURIEL† Abstract. The multifractal formalism was introduced to describe the statistical and geometrical properties of chaotic, scale invariant systems such as turbulent flows, geophysical signals, time series and recently images from real world scenes. According to the multifractal formalism, natural images can be decomposed in different fractal sets, each one corresponding to a different singular behaviour. These fractal sets are useful to describe and to code images, and also to implement new tools for image treatment. However, the correct classification of points into the so called fractal components is not straightforward, involving the use of wavelet analysis techniques. In this paper, we will present some theoretical results about the capability of different classes of wavelets for the singularity detection task. We will also discuss their performance over real, experimental data. Key words. Multifractal, singularities , wavelets AMS subject classification. 28A80, 32S99, 65T60

1. Introduction. Edge detection is a classical task in image processing [8]. Intuitively, edges contain the majority of the information carried out by images: it is usually possible to recognise a scene by the sketch of its edges. Many different definitions of “edge” (associated to the corresponding “edge detector” filter) have been proposed, but they are usually rather conventional, having very few to do with human vision [10]. The multifractal framework allows to introduce a natural classification scheme, which gives rise not only to an edge-like set, but also to other texture-like structures [15]. It has been experimentally tested over large ensembles of rather different natural scenes (i.e., real world scenes in which the objects have not been specially arranged) that natural images exhibit multifractal structure [14, 15, 11, 17]. Hence, the points in the image can be classified according to the so-called fractal components. Those fractal components are the set of points over which the image undergoes the same kind transition (or singularity in a general sense). The multifractal formalism has two advantages. First, the presence of an underlying multifractal arrangement is reflected in some statistical properties [7, 14, 15]. In the context of Information Theory [2, 3], any knowledge (a priori or acquired after learning) about the statistical structure allows to improve the strategies for coding. The visual areas of the cortex in mammals work much in this manner [2, 9, 6]: the redundancy in the code is reduced by learning the regularities after the repeated presentation of images. And for instance one application of the multifractal formalism over natural images leads to a minimum redundancy wavelet filter [16] which resembles real neural coding. Secondly, the fractal components are structures of great geometrical relevance, determined by statistical properties of images and not by a conventional definition. It is of particular interest one of those components, the socalled Most Singular Manifold (MSM), which is strongly related to the edges or contours of the objects. For the particular type of multifractals observed in natural images, and under a set of assumptions, it has been recently shown that images can be reconstructed from the information contained in the MSM [13]. The existence of this kernel stresses the importance ∗ This work has been partially funded by the French-Spanish Picasso collaboration program (00-37) and by a French (DGA 96 2557A/DSP) † Air Project - INRIA. Domaine de Voluceau BP105. 78153 Le Chesnay CEDEX, France. e-mail: [email protected]. A. Turiel is financially supported by a post-doctoral fellowship from INRIA.

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of the multifractal classification: extracting the MSM is a good strategy to compress and code images. But the application the multifractal formalism implies to develope techniques, stable enough in practical applications and leading to the correct decomposition of images into their fractal components. The most common technique is that of wavelet analysis [4, 1, 15]. In this paper we will show how to perform wavelet analysis in order to extract the fractal components. We will study the link between the asymptotic behaviour of the analyzing wavelets and the maximum possible value for the detected singularities. We will discuss on which wavelets are the best adapted to the different tasks, providing both theoretical arguments and experimental evidence. The paper is structured as follows: In the following Section we present the theoretical framework for the multifractal formalism. In Section 3 we will prove that any fast decreasing function can be used for the task of singularity detection. We will refine our hypothesis in Section 4 to obtain better performing wavelets using a priori information about the multifractal. The experimental evidence and its discussion is presented in Section 5. Finally, in Section 6 the conclusions are presented. 2. Multifractal measure over natural images. Wavelet projections. We will represent images by the field of their luminosities, I(~x), where ~x ∈ 0 will have a complicated dependence on r. However, it is a well known fact that natural images exhibit multifractal behaviour, that is, (2.3)

µ(Br (~x0 )) = α(~x0 ) rd+h(~x0 ) + o(rd+h(~x0 ) )

(see, for instance, [14, 15]) where d is the dimension of the space, introduced for convenience; d = 2 always for natural images. Eq. (2.3) is usually relaxed to a more suitable, easier to deal with expression, in the way: (2.4)

A(~x0 ) rd+h(~x0 ) ≤ µ(Br (~x0 )) ≤ B(~x0 ) rd+h(~x0 )

1 We consider monochrome images, although the extension for multispectral (for instance, color) images is straightforward

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for appropriate positive constants A(~x0 ), B(~x0 ). It is very usual in Physics to present multifractality in a non-rigorous way, as the verification of eq. (2.3); the theoretical derivations, however, can be worked out only using the expression eq. (2.4). In the following we will sometimes make use of expressions like eq. (2.3), but just as an intuitive, loose notation of eq. (2.4). We call µ “multifractal” because eqs. (2.3) and (2.4) allows decompose the image in multiple fractal sets. First, at each point ~x the singularity exponent2 h(~x) is computed. Then, all the points in the image are arranged according to the value of their singularity exponent. The fractal component Fh0 is defined as: (2.5)

Fh0 ≡ {~x : h(~x) = h0 }

The arrangement of points around fractal components is very revealing about the geometrical structure of the image; for instance, the component associated to the smallest singularity exponent (called the Most Singular Manifold, MSM) has usually a fractal dimension of 1.0 and resembles the edges or contours of the objects present in the image [15], while the other components are correlated in such a way that they can be predicted from the MSM using a linear kernel [13]. The multifractal formalism has been found to hold for a large variety of scenes [11] and also for color images [17]. When dealing with digitized images, the direct application of eq. (2.3) to compute the singularity exponents h(~x) gives a very poor performance, mainly due to the difficulties of interpolating the formula for radii r representing non-integer amounts of pixels. In such cases, it is convenient to consider the projections of the measure µ over an appropriate wavelet Ψ around the point ~x to be studied and at different sizes r. Namely, we define the wavelet projections TΨ µ(~x0 , r) as: Z (2.6)

TΨ µ(~x0 , r) ≡

dµ(~x)

~x0 − ~x 1 Ψ( ) rd r

A smooth wavelet has wavelet projections which vary smoothly with the size r; a wavelet supported over a compact set K will give a rise to wavelet projections close to the measures of the r-dilations of K . For a large class of functions Ψ, the wavelet projections provide the same singularity exponents3 as the measure µ, that is: (2.7)

TΨ µ(~x0 , r) = β(~x0 ) rh(~x0 ) + o(rh(~x0 ) )

In the next Section, we will show under which conditions on the function Ψ the wavelet projections TΨ µ(~x0 , r) verify eq. (2.7). 3. Constraints on the analyzer wavelet. We will consider multifractal measures µ defined over d-dimensional spaces for any value of d. The wavelet Ψ will be assumed to be positive to simplify the analysis. A positive wavelet is not an admissible wavelet, that is, the wavelet projections cannot be used to retrieve the original function [4]. However, it can be still used to analyze the singularities of µ, as we 2 We will call h(~ x) “singularity exponent” even if it can be positive. It actually represents the degree of singularity (if negative) or regularity (if positive) of ∇I [15]. 3 The wavelet projections are normalized by a factor r −d and for that reason the exponents appearing in the power law are shifted by −d with respect to those of µ. That normalization of the wavelet is convenient to make the limit r → 0 stable.

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will see. In our proof, we will also restrict the discussion to the case of a continuous function Ψ and Ψ(0) 6= 0. As it will be shown, Ψ will be required to decrease fast enough. We will assume that for any point ~x and any size r the wavelet projection TΨ µ(~x, r) is finite. If it was not the case, as µ is σ-finite it would be possible to define a sequence of finite measures µn defined over compact supports An such that µ|An = µn ; but Ψ is continuous so the wavelet projections of the µn are finite. The proof for µ would be obtained as a limit case, once the theorem is shown to be valid for the µn ’s, Finally, we will just study the singularity at ~x = 0, the extension for the other points being absolutely trivial. We will denote the singularity at ~x = 0 by h0 , that is, µ(Br (0)) ≈ α0 rd+h0

(3.1)

where the symbol ≈ should be understood in the sense of eq. (2.4). The wavelet projections we are interested in are TΨ µ(0, r), given by: Z TΨ µ(0, r) ≡

(3.2)

dµ(~x)

1 ~x Ψ( ) rd r

The statement of the theorem is as follows:

Theorem: If eq. (3.1) holds, then for any wavelet Ψ belonging to an appropriate class the following relation holds: TΨ µ(0, r) ≈ β0 rh0

(3.3)

We will present the proof in three stages, for three different classes of functions: set functions, compact support functions and fast decreasing functions. 3.1. Set functions. Let Ψ(~x) be the characteristic or set function of the ball of radius l centered around the origin, that is, Ψ(~x) = χBl (0) (~x) where the general expression of χBr (~x0 ) is given by:  χBr (~x0 ) (~x) =

(3.4)

1 |~x − ~x0 | < r 0 |~x − ~x0 | > r

According to eq. (3.2). the wavelet projections of Ψ at ~x = 0 are given by: Z (3.5)

TΨ µ(0, r) =

dµ(~x)

1 ~x 1 χBl (0) ( ) = d d r r r

Z dµ(~x) χBrl (0) (~x)

Hence it follows:

(3.6)

TΨ µ(0, r) =

1 rd

Z dµ(~x) = Brl (0)

µ(Brl (0)) ≈ α0 lh0 +d rh0 rd

so for those simple functions the theorem is trivially verified.

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3.2. Compact support functions. We will just try to find two constants 0 < A < B such that: (3.7)

A rh0 ≤ TΨ µ(0, r) ≤ B rh0

A lower bound constant A is easily obtained by the condition of continuity and Ψ(0) 6= 0, as follows: there exists a finite radius l such that Ψ(~x) 6= 0 ∀~x ∈ Bl (0). Let m > 0 be the minimum value of Ψ(~x) in Bl (0); then, for all ~x the following inequality holds: (3.8)

m χBl (0) (~x) ≤ Ψ(~x)

Let 0 < A0 < B0 be the bounding constants for µ at ~x = 0, that is, such that A0 rd+h0 < µ(Br (0)) < B0 rd+h0 (as µ is multifractal, they exist). It follows: (3.9)

m A0 lh0 +d rh0 ≤ m TχB (0) µ(0, r) ≤ TΨ µ(0, r) l

h0 +d

So, A can be taken as A = mA0 l . To obtain B we will use that Ψ has bounded support; so there exists a finite radius L such that Ψ(x) = 0 ∀~x : |~x| > L. As Ψ is continuous, it possesses a finite maximum M in BL (0). Hence, the following functional inequality holds: (3.10)

M χBL (0) (~x) ≥ Ψ(~x)

and analogously to the case of the lower bound, we conclude that a possible upper bound is given by B = M B0 Lh0 +d . 3.3. Fast decreasing functions. We will search constants 0 < A < B as in the preceeding case. The constant A can be calculated exactly like for compact support functions, so we just need to find B. Let the radius l be defined as in the previous subsection and let us define the sets Ri as follows: Ri = B2i l (0) − B2i−1 l (0) for i ≥ 1 and R0 = Bl (0). Let Mi be the maximum of Ψ over Ri . Hence, the following functional inequality holds:

(3.11)

Φ(~x) ≡

∞ X

Mi χRi (~x) ≥ Ψ(~x)

i=0

where χRi is the characteristic function of Ri , which equals 1 over R1 and 0 outside. We can conclude by proving that there exists an upper bound B for Φ such that TΦ µ(0, r) ≤ Brh0 , which by eq. (3.11) is also an upper bound for Ψ. To obtain this bound, let us suppose that there exists a finite radius L > 0 such that if |~x| > L, for all K > 1 the function Ψ verifies: (3.12)

Ψ(K~x) < Ω(K)Ψ(~x)

where Ω(K) decreases to zero as K goes to infinity faster than any polynomial, that is, (3.13)

lim K N Ω(K) = 0 ∀N > 0

K→∞

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This requirement on Ψ above is a small modification on the condition defining Schwatz’s class. We introduce the radius L to avoid forcing the function to be estrictly decreasing from the origin, which would define a very restrictive class of functions. Let i0 be the least integer greater than log2 (L/l). Following eq. (3.12), for all i0 > i > i0 : 0

Mi0 < Ω(2i −i ) Mi

(3.14)

We will first take r as r = 2−J , for J > 0 integer. We can thus decompose the dilation of Φ by a factor r, Φ(~x/r), as:

(3.15)

Φ(~x/r) =

iX 0 +J

∞ X

Mi χRi (~x/r) +

i=0

Mi χRi (~x/r) = ΦJ (~x) + δJ (~x)

i=i0 +J+1

For i > i0 + J, χRi (~x/r) = χRi (2J ~x) = χRi−J (~x). The residual function δJ (~x) verifies the following bound (obtained applying eq. (3.14))

δJ (~x) < Ω(1/r)

(3.16)

∞ X

Mi χRi (~x) = Ω(1/r) δ0 (~x)

i=i0

Given the dependence of δJ (~x) in r it decays very fast, and this function is neglicible in comparison with any power of r, so we can ignore this part. Hence, TΦ µ(0, r) ≈ TΦJ µ(0, r) for r = 2−J . For a general value of r, we also obtain TΦ µ(0, r) ≈ TΦJ µ(0, r) where J is such that 2−J−1 < r ≤ 2−J . Taking into account that: TχR µ(0, r) ≤ lh0 +d 2(h0 +d)i (B0 − 2−(h0 +d) A0 ) rh0

(3.17)

i

it follows:

(3.18)

TΦJ µ(0, r) ≤ l

h0 +d

(B0 − 2

−(h0 +d)

A0 )

"i +J 0 X

# 2

(h0 +d)i

Mi

rh0

i=0

so the upper bound B is given by:

(3.19)

B = lh0 +d (B0 − 2−(h0 +d) A0 )

∞ X

2(h0 +d)i Mi

i=0

where the series

P∞

i=0

2(h0 +d)i Mi is finite due to the fast decreasing of Mi .

4. Refined choices for the wavelet. The preceeding section gives a proof of the capability as singularity analizers for wavelets Ψ chosen in a very restrictive class of functions. Some of the conditions can be relaxed from the mathematical point of view (for instance, the requirement of continuity could be relaxed to integrability and boundness), although their precise formulation does not change very much the result in practical applications. It is also clear that singularities can be detected just using positive wavelets, but non-positive wavelets could be used as well. So none of those

Singularity extraction in multifractals

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conditions is going to change significantly the main result nor the experimental performance. However, there is a requirement critical to extract singularites and whose correct tunning allows improving significantly the performance; namely the condition of fast decay expressed in eq. (3.13). Any a priori knowledge about the properties of the multifractal measure µ allows enlargening the class of valid wavelets by relaxing the fast decay condition. The most relevant information is the knowledge of bounds on the range of posssible singularity exponents h0 . Let us assume that there exists a maximum singularity exponent hM ; in that case, the fast decay condition can be modified so that eq. (3.13) is only required for N ≤ N0 ≡ d + hM . The argument goes as follows: let us assume that for |~x| large enough, |Ψ(~x)| ∼ |~x|−N0 (in the sense that lim|~x|→∞ |~x|N0 |Ψ(~x)| is not divergent). So, Ω(|~x|) ∝ |~x|−N0 and Mi is then given by Mi ∝ 2−N0 i . The existence of the upper bound B given in eq. (3.19) is limited to the cases in which the series appearing in the definition is finite; but for the wavelet we consider now this series is proportional to ∞ X

(4.1)

2(h0 +d−N0 )i

i=0

which converges if N0 > h0 + d and diverges for N0 ≤ h0 + d. So, it suffices to take N0 > hM + d to assure the convergence. This argument can be refined as follows: let Ψ(~x) a wavelet decaying as |~x|−N0 at the infinity. Then, at any point ~x in which µ(Br (~x)) ≈ α(~x) rh(~x)+d , the wavelet projection is given by TΨ µ(~x, r) ∼ rhΨ (~x) , where:

(4.2)

hΨ (~x) =

  

h(~x)

, h(~x) < N0 − d

N0 − d , h(~x) ≥ N0 − d

As we will see, this result allows designing high performance wavelets at the cost of reducing the range of observed singularities. 5. Experimental applications. In this Section, we will present the results obtained for different wavelets when they are used to isolate singularities over 2D images (d = 2). We compare the experimental outputs with the expected theoretical results. 5.1. Definition and methods. We will consider the three following wavelets: Ψ1 (~x) (5.1)

= (1 + |~x|2 )−1

Ψ2 (~x) = Ψ3 (~x) =

(1 + |~x|2 )−2 −

e

|~ x| 2 2

Ψ1 is just capable to isolate negative singularities (h(~x) < 0); Ψ2 is useful to isolate singularities up to h(~x) = 2 and Ψ3 is capable to isolate every singularity exponent h(~x). We will apply those wavelets over 512× 512 patches of several images from Hans van Hateren’s web database (see [18] for details on the images). Given a wavelet, we compute for each image √ seven wavelet projections at each point ~x, for scales r = [r0 ]i , i = 0, ..., 6 and r0 = 2 pixels. Then we perform a log-log linear

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regression; according to eq. (2.7) the slope equals the singularity exponent h(~x). For each image we represent the experimental distributions of singularity exponents and the MSM, as this set is the most informative of the fractal components. The determination of the minimum exponent h∞ (which defines the MSM) is made as the average of the 1% and the 5% quantils of the distribution of singularity exponents. The dispersion around this average value is conventionally fixed in 0.1. 5.2. Results. In the following figures, we present the results obtained over the images imk00446.imc, imk01171.imc, imk01375.imc and imk01471.imc from van Hateren’s ensemble. Those images provide examples of the typical situations for this kind of analysis.

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Singularity extraction in multifractals

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F IG . 5.1. First row: Central 512 × 512 patch in imk00446.imc Second row: MSM and singularity distribution for Ψ1 ; h∞ = −0.45 ± 0.1 Third row: MSM and singularity distribution for Ψ2 ; h∞ = −0.40 ± 0.1 Fourth row: MSM and singularity distribution for Ψ3 ; h∞ = −0.40 ± 0.1

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F IG . 5.2. First row: Central 512 × 512 patch in imk01171.imc Second row: MSM and singularity distribution for Ψ1 ; h∞ = −0.50 ± 0.1 Third row: MSM and singularity distribution for Ψ2 ; h∞ = −0.45 ± 0.1 Fourth row: MSM and singularity distribution for Ψ3 ; h∞ = −0.50 ± 0.1

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Singularity extraction in multifractals

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F IG . 5.4. First row: Central 512 × 512 patch in imk01471.imc Second row: MSM and singularity distribution for Ψ1 ; h∞ = −0.50 ± 0.1 Third row: MSM and singularity distribution for Ψ2 ; h∞ = −0.50 ± 0.1 Fourth row: MSM and singularity distribution for Ψ3 ; h∞ = −0.50 ± 0.1

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5.3. Discussion. As a general observation, we observe that there is a rough agreement among the MSM’s obtained for the different wavelets. It seems that Ψ1 provides the finest version of that set, while Ψ3 is the coarsest. This fact is somewhat expected, because it seems reasonable that the uncertainty in the determination of the singularities is an increasing function with the size of the range of possible detected exponents. As Ψ1 has the narrowest detection range it provides the best performance, but only for the singularities it is able to isolate. The backdraw is that Ψ1 is unable to correctly classify the singularities above h = 0 (which are all of them detected as h = 0). Thus, there is a trade-off between performance and detection range. Another interesting remark is that those images are known to exhibit multifractal structure of the log-Poisson type [15]. For log-Poisson multifractals, it suffices to know the dimension D∞ and the singularity exponent h∞ of the MSM to define all the range of possible singularities (see for instance [5, 12, 14]). The maximum possible exponent hM is defined in terms of h∞ and D∞ as:

(5.2)

hM = h∞ − ωm (d − D∞ ) ln(1 +

h∞ ) d − D∞

where ωm is defined by the relation ωm (1 − ln ωm ) = −D∞ /(d − D∞ ) (see [15] and references therein for a full discussion on the log-Poisson model). Then, the possible values for the singularity exponents h are those contained in the range [h∞ , hM ], which is finite. We can still impose additional constraints on this range. First, according to the multifractal scheme −1 ≤ h∞ ≤ 0 [15]. Second, the value of D∞ is almost always close to 1, which reflects the fact that the MSM is generally composed by contours (i.e., segments of curves) of the objects present in the image (a fact also reflected in the examples presented in the figures). So, for D∞ = 1 and d = 2 as usual, eq. (5.2) can be reduced to: (5.3)

hM = h∞ − ω0 ln(1 + h∞ )

where ω0 is given by ω0 (1 − ln ω0 ) = −1, so ω0 ≈ 3.60. For the h∞ ’s obtained in our examples, which go from h∞ = −0.40 to h∞ = −0.50, the possible hM ’s go from hM = 1.44 to hM = 2.00. For that reason, Ψ2 is a wavelet with the narrowest range capable to detect all the singularities. It is optimal in the sense that it arrives to a compromise between performance and detection range. For specific tasks, however, other choices can be better tuned (for instance, Ψ1 is better than Ψ2 if we just want to isolate the MSM). To finish with the general remarks, let us observe that, as predicted, the distribution of Ψ1 is truncated above h = 0 for all the images considered. That of Ψ2 is truncated above h = 2, but there is no singularity above h = 2 according to the argument of the preceeding paragraph: the amount of such singularities detected by Ψ3 is less than 0.1 % for any image and can be considered as numerical fluctuations. For that reason the distributions of singularities for Ψ2 and Ψ3 are very similar, that of Ψ3 being slightly broader (as can be noticed comparing the behaviour of both functions in the neighbourhood of h = 1 in the figures). Besides, there is an intrinsic difficulty to detected a smooth behaviour (positive h) in the neighbourhood of a very sharp transition (negative h); Ψ3 is partially able to overcome this problem at the cost of “rounding” the MSM and the other singular structures. Other observations, specific to each image, are in order. For instance, for imk00446.imc (figure 5.1) the performances of Ψ2 and Ψ3 are very similar, as the image consists of well defined, rather straight, long lines, which are easy to detect. The performances of these two wavelets are also very similar for imk01171.imc (figure 5.2) (although the “rounding effect”

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of Ψ3 becomes more evident) because that image consists of large enough objects given rise to well isolated edges. But for image imk01375.imc (figure 5.3) the rounding of Ψ3 spoils too much the extraction of the MSM. Figure 5.4, which corresponds to the analysis of imk01471.imc, is interesting by a different reason: it has a significative deviation from the multifractal model. It has two mixed types of singular structures, namely those of a multifractal structure (the branches of the tree and some rather natural textures of the building) and very sharp, isolated transitions (the inner edges of the windows and the lower part of the tree, which is sharply contrasted with respect to the building). Sharp, isolated transitions correspond to singularity exponents h = −1, and this behaviour is reflected in the excess on the left tail of the singularity distributions. The sharp transitions of type h = −1 lie outside the interval defined by the computed MSM exponent h∞ (which turned out to be h = −0.5 ± 0.1 in the three cases) and for that reason the associated edges are partially or totally absent, even for the wavelet which is the best adapted to detect the MSM, Ψ1 . For the task of isolating the branches, however, Ψ1 is very efficient, Ψ2 is much worse and Ψ3 is really very bad. 6. Conclusions. Natural images consist of fractal sets representing different kinds of singularities, which can be associated to edges and texture-like structures [15]. Those sets are very informative about the geometry of the scenes from which they were obtained, and splitting the image into those fractal components is a prerequisite for several techniques of image processing and coding. For that reason, we have studied which wavelets can be efficiently used for this task. In this paper we have shown that there exists a large family of wavelets which can be used to efficiently detect and isolate the singularity exponents giving rise to the fractal components. We have presented a theoretical proof which shows that even positive functions can be used, provided that they decay fast enough. We have shown that even wavelets which do not decay very fast are still able to detect singularities contained in an appropriate range. We have also shown some examples of the experimental application of the theory. It is observed that, depending on the task, it is convenient to choose one or another type of wavelet. For instance, to isolate the set associated to the most singular exponent, a wavelet with very narrow detection range is the best choice, while for a detection over the full range the best choice will be a wavelet with the smallest detection range containing the expected exponents. A wavelet able to detect any singularity will be always a worse choice than a wavelet fitting the actual range of singularities. So, any a priori knowledge about the multifractal structure is worthful to improve performance. Acknowledgements. I am grateful to Hussein Yahia, Jacopo Grazzini and Jean-Pierre Nadal for their comments in the course of the preparation of this paper. REFERENCES [1] A. A RNEODO , F. A RGOUL , E. BACRY, J. E LEZGARAY, AND J. F. M UZY, Ondelettes, multifractales et turbulence, Diderot Editeur, Paris, France, 1995. [2] H. B. BARLOW, Possible principles underlying the transformation of sensory messages, in Sensory Communication, W. Rosenblith, ed., M.I.T. Press, Cambridge MA, 1961, p. 217. [3] T. M. C OVER AND J. A. T HOMAS, Elements of information theory, John Wiley, New York, 1991. [4] I. DAUBECHIES, Ten lectures on wavelets, CBMS-NSF Series in Ap. Math., Capital City Press, Montpelier, Vermont, 1992. [5] B. D UBRULLE, Intermittency in fully developed turbulence: Log-poisson statistics and generalized scale covariance, Physical Review Letters, 73 (1994), pp. 959–962. [6] D. J. F IELD, Relations between the statistics of natural images and the response properties of cortical cells, J. Opt. Soc. Am., 4 (1987), pp. 2379–2394. [7] U. F RISCH, Turbulence, Cambridge Univ. Press, Cambridge MA, 1995.

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