FORTH-ICS / TR-264 December 1999Generalized Scale-SelectionJon Sporring, Christos I. Colios, andPanos E. Trahanias

AbstractThis paper generalizes Lyaponov functionals to the local setting, and it isshown that they may serve as a connection between Florack's et al. pseudo-linear scale-spaces and Koenderink's et al. local soft histograms. A generalizedscale-selection method is proposed based on local Lyaponov functionals andlocal soft histograms, and it is shown that the method includes Lindebergs'sBlob-detector and various generalizations of it. This paper concludes withsome initial experimental results using a statistical blob-detector.Keywords: Linear Scale-Space, Lyaponov Functionals, Pseudo-linear scale-spaces, Locally Disorderly Images, Local Soft Histograms, Scale-Selection,Blob-Detector.

Generalized Scale-Selection�Jon Sporring, Christos I. Colios, and Panos E. TrahaniasInstitute of Computer Science,Foundation for Research and Technology { Hellas,Vassilika Vouton, P.O. Box 1385,GR-71110 Heraklion, Crete, GreeceTechnical Report FORTH-ICS / TR-264 | December 1999c Copyright 1999 by FORTHAbstractThis paper generalizes Lyaponov functionals to the local setting, and it is shown thatthey may serve as a connection between Florack's et al. pseudo-linear scale-spaces andKoenderink's et al. local soft histograms. A generalized scale-selection method is proposedbased on local Lyaponov functionals and local soft histograms, and it is shown that themethod includes Lindebergs's Blob-detector and various generalizations of it. This paperconcludes with some initial experimental results using a statistical blob-detector.Keywords: Linear Scale-Space, Lyaponov Functionals, Pseudo-linear scale-spaces, LocallyDisorderly Images, Local Soft Histograms, Scale-Selection, Blob-Detector.

�This work was supported by EC Contract No. ERBFMRX-CT96-0049(VIRGO http://www.ics.forth.gr/virgo) under the TMR Programme.

1 Image StructureAt least three views of image structure exist in the image processing community: mathemati-cal morphology, in which the shapes of the image isophotes are studied, di�usion scale-spacesthat examine the linear or non-linear mean value of neighboring pixels, and statistical meth-ods that study the full distribution of local neighborhoods. Seemingly far apart, these viewsare currently converging. Florack et al. [1] have shown that morphological erosion anddilation scale-spaces and the linear-di�usion scale-space are related through a non-lineartransformation of the gray-values. Additionally, Koenderink et al. [2] have introduced thenotion of locally disorderly images that facilitates the availability of the simple Gaussianstructure for local statistical analysis. The goal of this paper is to reveal the relation be-tween Florack's gray-value transforms and Koenderink's local soft histograms, and to suggesta generalized scale-selection mechanism.2 Scale-Spaces Simplify StructureScale-spaces seem to be a key tool for studying the structure of images. A scale-space is astack of increasingly simpler and simpler images. A central scale-space of this article is theLinear scale-space, which is given by the heat di�usion equation,@@tL(~x; t) = �L(~x; t) =Xi @2@x2i L(~x; t);L(~x; 0) = I(~x); (1)where I is the original image, ~x = fxig 2 is the vector of space coordinates in the spatialdomain, and t is the scale or time parameter. The Green's function of (1) is the Gaussiankernel, thus L(~x; t) = (4�t)�1 exp (�jj~xjj22=(4t))� I(~x) with \*" the convolution operator andjj � jj2 the Euclidean norm.The notion of simpler structure can be made concrete through Lyaponov Functionals [3],S(t) = X~x2��L(~x; t)�; (2)with � being a convex function, i.e. �00 > 0. The Lyaponov functionals are monotonic inthe linear and in many non-linear scale-spaces in compliance with the gradual reduction ofstructure and they seem to be intimately related to the size of dominating image structure[4].3 Local Lyaponov FunctionalsLyaponov functionals can be used to perform global analysis of images but for local analysisof structure, processing must be restricted to a limited area. As recently advocated [2],Gaussian (soft) windows have the simplest structure, since they uniquely guarantee not to2

introduce spurious detail for increasing window size. For the above mentioned reasons, localLyaponov functionals are de�ned as follows:S(~x; t; w) = X~y2G(~x� ~y;w)��L(~y; t)�= G(~x;w) � ��L(~x; t)�: (3)Although the convexity requirement on � could be retained, it does not seem to be ofparticular advantage. Thus in the following, the basic properties of the local Lyaponovfunctionals will be discussed for the cases of � being convex and � being monotonic.For an image of k pixels and convex �'s, the limits of the local functionals w.r.t. w areessentially the global Lyaponov functionals and �(L):limw!1S(~x; t; w) = 1kS(t); (4)limw!0S(~x; t; w) = ��L(~x; t)�: (5)In contrast to global Lyaponov functionals, local Lyaponov functionals are non-monotonicfor �xed w and increasing t, since structure moves independently of w.Regardless of �, the structure of S(~x; t; w) by w is given by the di�usion equation:@@wS(~x; t; w) = �S(~x; t; w);and since the mean value is invariant under di�usion, so is the sum. Hence the sum of localLyaponov functionals is equal to global Lyaponov functionals:X~x2S(~x; t; w) = S(t):When � is monotonic, the structure of S(~x; t; w) w.r.t. w is essentially given by thepseudo-linear di�usion equation [1]. That is, the `re-normalized' functional L�(~x; t; w) =��1(S(~x; t; w)) evolves as:@@wL�(~x; t; w) = �L�(~x; t; w) + �(L)jjrL�(~x; t; w)jj22; (6)where `r�' is the gradient operator, and �(v) = @2�(v)@v2 =@�(v)@v is the non-linearity parameter.4 Lyaponov Functionals and the HistogramIn this section it will be shown that a speci�c class of local Lyaponov functionals is atransform of local (soft) histograms [2]. The global relation is reviewed, and this is used asa skeleton for the following proof of the local relation.3

Let us consider as an example, �(v) = v� for � = 0 : : :N , where N is the number of gray-values of L (typically 255). Since space and values may be interchanged in the summationof (2), the equation may be rewritten as,S(t) = N�1Xi=0 �(vi)fi;where fi is the number of points in L having the value vi. The function ffig is typicallydenoted the gray-value histogram. With this formulation, all the global Lyaponov functionalsS� may be written as,266664 S0S1...SN�1 377775 = 266664 1 1 : : : 1v1 v2 : : : vN... ... . . . ...vN�11 vN�12 : : : vN�1N 377775 266664 f1f2...fN 377775 ; (7)where the matrix fvi�1i g is a Vandermonde matrix independent on t. Since Vandermondematrices are invertible in principle, it may be concluded that there is a one-to-one relationbetween the class of global Lyaponov functionals, fS�g, and the gray-value histogram.The same one-to-one mapping exists for the class of local Lyaponov functionals and local(soft) histograms [2]. A local histogram H of an image L(~x; t) in Linear scale-space (1) isde�ned as H(v; ~x; t; u; w) = X~y2G(~x� ~y;w)M(v; ~y; t; u); (8)where v denotes the isophote value. The function 0 � M � 1 is the soft indicator function[5], M(v; ~x; t; u) = exp �(L(~x; t)� v)24u ! ; (9)in which each isophote i has width proportional to u. In the special case treated here, u! 0and M 2 f0; 1g become the regular indicator function.A general method to show that the local histogram is a transform of at least one familyof local Lyaponov functionals, is to substitute H(i; ~x; t; 0; w) into (7) in place of fi.Firstly, when substituting H(i; ~x; t; 0; w) into (7) in place of fi and for �xed ~x, t, and wwe get local Lyaponov functionals regardless of �, sinceS�(~x; t; w)= N�1Xi=0 �(vi)H(vi; ~x; t; 0; w)= N�1Xi=0 �(vi)X~y2G(~x� ~y;w)M(vi; ~y; t; 0)= N�1Xi=0 X~y2G(~x� ~y;w)�(vi)M(vi; ~y; t; 0):4

The above equation may be rewritten as a sum over space instead of value, resulting inS�(~x; t; w) = X~y2G(~x� ~y;w)� (L(~y; t)) = S(~x; t; w):This proves that local Lyaponov functionals are obtained directly from local soft histograms.Finally, it is observed that the system matrix in (7) is independent on the soft histogramregardless of �. Hence, when the system matrix is invertible, then there is a one-to-one rela-tion between a family of local Lyaponov functionals and local soft Histograms. As previouslydiscussed, that is the case for ��(v) = v�. This concludes the proof.5 Scale-SelectorsIn this section, a generalization of scale-selection is introduced, that is directly related toLindeberg's Blob-detector [6] and the global scale-selection of Sporring and Weickert [4].It will further be used to introduce the morphological equivalent of blob-detection and adetector of size of statistical stable regions.In view of local Lyaponov functionals, local scale-selection is the process of choosing thepair t and w for a selected point curve ~x. The simplest choice is scale-selection by extremaof the gradient magnitude in the normal scale-parameters log t and logw, henceforth calledthe scale-selector: c(~x; t; w) = vuut @S(~x; t; w)@ log t !2 + @S(~x; t; w)@ logw !2: (10)This formulation is a generalization of well-known scale-selection methods and will furtherbe used to suggest a number of new scale-selectors.5.1 Relations to Lindeberg's Blob-DetectorWhen �(v) = av + b for some constants a and b, the non-linearity parameter (6) is zero,i.e � = 0, and the pseudo-linear scale-space degenerates to a linear scale-space S(~x; t; w) =L(~x; t + w). In this case the function c is simpli�ed to c(~x; t; w) = c(~x; t0) = p2 ���@S(~x;t0)@ log t0 ���,where t0 = t+ w. Maximizing c over ~x and t0 yields Lindeberg's blob-detector [6].Two scenarios of scale-selection with (10) have simple relations with the Blob-Detector,and serve as generalization of the two examples given in [6, pp.326{327].Firstly, consider the one dimensional example of a cosine embedded in the Linear scale-space: f(x; t) = a exp(��2t) cos(�x)with � > 0. In the case of w! 0 and �(v) = vn, the local Lyaponov functional is given as,Sn(~x; t; 0) = f(x; t)n:The function f has a maximum at �x = 0 for all values of t, and in this point the scale-selectoris given as, cn = �p2nan�2te�n�2t:5

Di�erentiation with respect to t reveals a maximum at t = 1=(nv2). Hence the order n of �only scales the selection.Secondly, consider the exponential function,f(x) = exp(�jxj2):In the case of t! 0 and �(v) = vn, the local Lyaponov functional is given as,Sn(~x; 0; w) = G(x;w) � f(x; t0)n= G(x;w) � exp(�njxj2):If n = 1, the semigroup property of the Gaussian ensures that the scale-selector will choosew = 0 in the spatial maximum at x = 0. It is clear that n works as a scaling of the x-axis,hence f(x; t0)n = f(nx; t0), or conversely,Sn(~x; 0; w) = G(x;nw) � f(x; t0):The scale-selector will therefore choose w = 0 independently on n.5.2 Relations to other Blob-DetectorsIn this section a simple relation to a global scale-selection presented earlier will be pre-sented, and further two new methods based on morphology and statistical stability will beintroduced.The family of Lyaponov functionals de�ned by the R�enyi Entropies was studied in [4],R�(t) = 11� � logX~x2 p(~x; t)�;p(~x; t) = L(~x; t)P~y2 L(~y; t) : (11)For convex �, the local Lyaponov Functionals trivially go to the global Lyaponov Functionalsas w ! 1 (5). Hence, for �xed w = 1, maxima of c are identical to the scale-selectionmethod presented in [4].The morphological scale-spaces of dilation and erosion are obtained in the pseudo-linearscale-spaces (6), when �! �1. Hence, for �xed t and appropriate � for which � =1, themaxima of c will be the morphological equivalent of blob-detection.Finally, a statistical blob-detector can be de�ned by examining the local histogram (8).For su�cient regular textured patches, the local histogram will remain unchanged as long asthe window function is smaller than the size of the patch. Hence, a zero order statistical blobcan be found by examining the change of the histogram as a function of logw as illustratedin Figure 1. In these experiments the entropy of the local histogram was traced in a �xedposition and t = 312 for the left and t = 4:5 for the right. Both used u = 32 and w wassampled logarithmically in 16 steps. As can be observed, the area of apples is found in leftand the cherry plus the area of cherries is found in the right.6

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Figure 1: Detecting zero order statistical stability by entropy change. Top row shows twoimages together with the detected statistical blobs. Bottom row shows the entropy changeby logarithmic window size.7

6 SummaryIn this paper we have introduced the notion of local Lyaponov functionals. These functionalsform an extension of global Lyaponov functionals, and it is shown that they are related tothe local soft histograms in the most natural way. Based on the local Lyaponov functionals,a generalized scale-selector is de�ned and shown to be directly related to Lindeberg's blob-detector, Sporring and Weickert's global scale-selectors, the morphological equivalent of theblob-detector, and a detector of statistical stability.References[1] Luc Florack, Robert Maas, and Wiro Niessen. Pseudo-linear scale-space theory. Inter-national Journal of Computer Vision, 31(2/3):247{259, 1999.[2] Jan J. Koenderink and Andrea J. van Doorn. The structure of locally orderless images.International Journal of Computer Vision, 31(2/3):159{168, 1999.[3] J. Weickert. Anisotropic di�usion in image processing. Teubner Verlag, Stuttgart, 1998.[4] Jon Sporring and Joachim Weickert. Information measures in scale-spaces. IEEE Trans.on Information Theory, 45(3):1051{1058, 1999. Special Issue on Multiscale StatisticalSignal Analysis and Its Applications.[5] L. D. Gri�n. Scale-imprecision space. Image and Vision Computing, 15:369{398, 1997.[6] Tony Lindeberg. Edge detection and ridge detection with automatic scale-selection.International Journal of Computer Vision, 30(2):117{156, 1998.

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