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Interest Point Detection with Wavelet Maxima Lines Christophe Damerval, Sylvain Meignen

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Interest Point Detection with Wavelet Maxima Lines Christophe Damerval, Sylvain Meignen Interest Point Detection with Wavelet Maxima Lines Christophe Damerval, Sylvain Meignen To cite this version: Christophe Damerval, Sylvain Meignen. Interest Point Detection with Wavelet Maxima Lines. 2007. inria-00171678 HAL Id: inria-00171678 https://hal.inria.fr/inria-00171678 Preprint submitted on 12 Sep 2007 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Interest Point Detection with Wavelet Maxima Lines Christophe Damerval and Sylvain Meignen Laboratoire Jean Kuntzmann, LJK, Grenoble, France Abstract. In this paper, we propose a new approach in image processing for interest point detection. Thanks to the link between linear scale- space and wavelet decompositions, we are able to build a structure in scale-space based on wavelet maxima lines. This framework allows robust detection of objects (or blob-like structures) present in the image, and the computation of their characteristic scale. Eventually the estimation of their shape can be performed. Keywords: scale-space representation, maxima lines of wavelet decompo- sition, blob detection, scale selection. 1 Introduction Interest point detection is an important low-level operation in computer vision, since it is a first step towards more complex tasks such as the determination of local deformations in images, the extraction of scale-invariant interest points or the computation of local descriptors ([4] [6] [8]). So this paper is organized as follows: first, we present the framework of the linear scale-space put forward by Lindeberg in [5] (section 2) ; we also highlight the link with wavelet decomposi- tions, especially with wavelet maxima lines (ML), and explain how scale selection can be performed on the basis of these ML. Then we detail the different steps of our method (section 3), laying the emphasis on ML construction. Besides, we will see how ML can be used to estimate the shape of the detected objects (section 4). Eventually, we will see some applications on geometric images and on natural scenes and also some results on the influence of noise (section 5). 2 Linear Scale Space and Wavelet Maxima Lines 2.1 Linear Scale Space We first recall the definition of linear scale-space in the one-dimensional case. Linear scale-space representation consists of convolutions with Gaussian kernels 2 1 x 1 x at different scales. If we put g(x) = exp( ) and gt(x) = g( ), we √2π − 2 √t √t define as in [5] the linear-scale space by: 1 x u L(x,t) = gt f(x) = g( − )f(u)du. ∗ ZR √t √t Scale selection is then carried out through the study of the appropriately nor- malized derivatives of L. Lindeberg [4] suggests to normalize the derivatives as γm γm m 2 m 2 1 u (m) follows : ∂x,γnormL(x,t) = t ∂x L(x,t) = R t √t g √t f (x u)du γm − 2 (m) R = t f gt(x). In a bidimensional context, the study is very similar to that ∗ 2 2 1 x1+x2 done in one dimension. We now put g(x1,x2) = 2π exp( 2 ) and gt(x1,x2) = 1 x1 x2 − t g( √t , √t ). The bidimensional linear scale-space definition is identical to its one α α α1 α2 α1 α2 dimensional counterpart. Let us define ∂x L = Lx = ∂x1 x2 L = Lx1 x2 , with α = (α1, α2), we consider differentiations of the linear scale-space of the form: I i i L = ciL αi where αi = α + α = M is independent on i. For such a D i=1 x | | 1 2 definitionP of the linear scale-space, the appropriate normalization of the opera- Mγ tor is given by [4]: x,γnormL = t 2 L. The Hessian matrix of L defined by D D D L L = x1x1 x1x2 is a basic tool for the analysis of the characteristic scale of H Lx2x1 Lx2x2 objects in images [8][4]. A simple normalized feature is usually extracted from γ this matrix and is: trace ( γnorm) = t ∆L. Now, when γ = 1, it is interesting H to draw a parallel with wavelet decompositions. Using two integrations by part and because of the fast decay of the Gaussian function and of its derivatives, we get: trace( γnorm) = f (∆g)t so that as ∆g is even, this quantity is exactly the L1-normalizedH wavelet∗ decomposition using the ∆g wavelet at scale √t. 2.2 Wavelet Decompositions and Maxima Lines The normalized Laplacian used in our framework can be written as the L1- normalized bidimensional continuous wavelet transform (CWT) of the image f 2 2 using the wavelet ∆g, ∆g(x,y) = 1 2 + (x2 + x2) exp( x1+x2 ) : 2π − 1 2 − 2 R2 R 1 u x v y (x,y) , s +∗ , f(x,y,s) = f(u,v)∆g( − , − )dudv ∀ ∈ ∀ ∈ W s2 ZR2 s s Since the wavelet ∆g can be expressed analytically in the Fourier domain, the computation of the CWT can be done as follows: first, compute ∆g times the discrete Fourier transform of the image (using a FFT), and then obtain the CWT c by the inverse Fourier transform. Note that the cost of this step is O(NLog2N) (where N is the size of the data), thus providing an efficient computation. Given R a certain scale s +∗ , modulus maxima are defined as local maxima in space of the modulus of∈ the CWT of the image f(., ., s) . Let us recall that the absolute value of the CWT has the property|W of low response| where the image is smooth, whereas it has a high response where there are singularities (boundaries, edges, corners or isolated peaks). So the point of studying modulus maxima is that they are related to singularities present in the image. Note that at the finest scales, modulus maxima arise from isolated singularities, while at coarser scales, modulus maxima result from several singularities since the translated-dilated . x . y wavelet ∆g( −s , −s ) covers a wider area. In particular, as we will see, a modulus maxima at a certain scale can be related to the presence of a significant structure. In our context, the maximum principle (see [9] [1]) ensures that modulus maxima propagate towards finer scales, thus making connected curves in scale-space, called maxima lines (ML). The construction of these ML on the basis of the modulus maxima at each scale is detailed in the next section. So a ML can be viewed as a path in scale-space which does not interrupt when the scale decreases. As the scale s increases, (xl,yl) gives the spatial drift of the ML while f((xl,yl,s) gives the evolution of the response. Denoting sint the scale at |W | R∗ N which the ML is interrupted (sint +), one maxima line Ll(l ) is denoted as: ∈ ∈ Ll = (xl(s),yl(s), s, f(x,y,s))s ]0,sint[, { W ∈ s ]0,sint[, (xl(s),yl(s)) local maximum of f(., ., s) ∀ ∈ |W |} 2.3 Scale Selection As pointed out by Lindeberg in [5], stable features (robust to noise or to slight deformations) can be obtained by considering the extrema of the image decompo- sition using an appropriate operator. In particular, when normalized derivatives of the linear scale-space are used, such features can be related to the actual position and size of the objects appearing in the image. Mainly, our approach differs from classical scale space selection to the extent that instead of determin- ing local maxima in the 3D scale space (denoted Max3D) we build a structure (the skeleton of the maxima lines) made of chains of modulus maxima, and take a certain maximum along some of them (the resulting set is noted MaxML). So as to see the link between these two sets which are subsets of the scale-space 2 S D = R R∗ , let us define: 3 × + (x∗,y∗,s∗) = cube of S D centered at (x∗,y∗,s∗) of size η η η(η > 0) V { 3 × × } Max D = (x∗,y∗,s∗) S D, for a given V (x∗,y∗,s∗), 3 { ∈ 3 ∈ V (x,y,s) V, f(x∗,y∗,s∗) f(x,y,s) ∀ ∈ |W | ≥ |W |} MaxML = (x∗,y∗,s∗) S D, there exists a ML Ll containing (x∗,y∗,s∗) { ∈ 3 such that the response attains a local maximum along Ll : V (x∗,y∗,s∗), (xl(s),yl(s),s) Ll V, ∃ ∈ V ∀ ∈ ∩ W f(x∗,y∗,s∗) f(xl(s),yl(s),s) | | ≥ |W |} Given (x∗,y∗,s∗) Max D, we note that it is a modulus maximum at scale ∈ 3 s∗, which belongs to a certain ML, denoted Ll (modulus maxima propagate to- wards finer scales), so we have (x∗,y∗,s∗) = (xl(s∗),yl(s∗),s∗). According to the definition of Max D, we get : (xl(s),yl(s),s) Ll V, 3 ∀ ∈ ∩ f(xl(s∗),yl(s∗),s∗) = f(x∗,y∗,s∗) f(xl(s),yl(s),s) . So a local maximum|W in 3D scale-space| |W is also a local| maximum ≥ |W along a certain| ML. Con- versely, let us consider (x∗,y∗,s∗) MaxML: V (x∗,y∗,s∗), ∈ ∃ 1 ∈ V (xl(s),yl(s),s) Ll V1, f(xl(s∗),yl(s∗),s∗) f(xl(s),yl(s),s) . We can∀ distinguish between∈ ∩ two|W cases: isolated and non-isolated.| ≥ |W First, considering| a ML Ll that is isolated in scale-space at scale s∗, there exists V (x∗,y∗,s∗) 0 ∈ V such that Ll is the unique ML intersecting V ; taking V = V V , we obtain 0 0 ∩ 1 that (x∗,y∗,s∗) Max D.
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