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1 Wavelet based multiscale analysis for feature detection on images C. Ducottet Λ J. Fayolle y M. Chouvellon z T. Fournel x F. Trunde Abstract In this paper, we present afeature detection and characterization algorithm based on multiscale edges. We consider three types of features derived from particular edges: the transitions, the peaks and the lines. For each feature, an amplitude and a smoothing size are defined, which represent respectively the contrast and the fuzziness level of the edge. The proposed algorithm is able to detect edges, to measure their amplitude and smoothing size, and to localize them at an adapted scale. This algorithm results from a study across scales, of wavelet transform maxima of the three types of edges. The large application area of this work is illustrated by three particular problems: the detection of a diesel jet, the analysis of the skin relief, and the cavitation bubbles detection and measurement. Key words: Edge detection, multiscale analysis, wavelet transform, blurred images. 1 Introduction Feature detection is an important issue in signal and image processing. Actually, a number of problems become less complex if an image is summarized to its most important features. Edges are considered as important features to analyze information contained in images [8, 10]. Recently, it has been proved that multiscale edges can be detected from the maxima points of the wavelet transform modulus and that they provide a complete description of images [6, 7]. In this paper, we present a feature detection and characterization algorithm based on multiscale edges. This work is an adaptation of Mallat and Hwang singularity detection [7] for three particular types of edges encountered on images: the transitions, the peaks and the lines. For each edge model, an amplitude and a smoothing size are defined. They represent respectively the contrast and the fuzziness level of the edge. The proposed algorithm allows firstly to determine the type, the amplitude, and the smoothing size of edges, and secondly to segment them at an adapted scale. In section 2 we introduce the concept of multiscale edges, its link with the wavelet transform, and the local regularity analysis. This section constitutes the theoretical context of our work. Section 3 is dedicated to the detection and characterization algorithm. We first present the three edge models (transition, peak and line), then we study their behavior across scales, and finally we expose the principle of the algorithm. Three different applications are given in section 4. They concern the detection of a diesel jet, the analysis of the skin relief, and the cavitation bubble detection and measurement respectively. Λ Laboratoire Traitement du Signal et Instrumentation, 23 rue Dr Paul Michelon, Saint-Etienne Cedex 2, France, ducottet@univst-etienne.fr y Laboratoire Traitement du Signal et Instrumentation, 23 rue Dr Paul Michelon, Saint-Etienne, Cedex 2, France, jacques.fayolle@univ-st-etienne.fr z Laboratoire Traitement du Signal et Instrumentation, 23 rue Dr Paul Michelon, Saint-Etienne Cedex 2, France, chouvell@univst-etienne.fr x Laboratoire Traitement du Signal et Instrumentation, 23 rue Dr Paul Michelon, Saint-Etienne Cedex 2, France, fournel@univst-etienne.fr Laboratoire LISA - CPE Lyon, 43 bd du 11 Novembre 1918, BP 2077, Villeurbanne Cedex, France, trunde@cpe.fr 16th IMACS World Congress ( cfl 2000 IMACS) 1

2 Wavelet based multiscale analysis Singularity and edge detection using wavelets 2.1 Multiscale edges The wavelet theory [4] has allowed some advances in the understanding of multiscale edges. It has been proved [6] that multiscale edges can be detected from the maxima points of the wavelet transform modulus of a signal or an image. If we consider the gray level function f associated with an image, the edge detection can be done by means of the gradient of this function. Edges are determined as the points where the modulus of the gradient vector is locally maximum in the direction towards which the gradient vector points in the image plane. This method to define image edges is known as the Canny edge detection [1]. To control the noise influence and to provide regularization of the digital image, a low pass filter is applied before the differentiation. Then, the whole differentiation is composed of a smoothing step using a filter of a certain size, and a differentiation step. A gradient operator obtained from the differentiation of the smoothing filter allows the combinationofthetwo steps [8, 1]. The size of the smoothing filter defines the analyzing scale of the image. Let us define a Gaussian filter at scale s: (1) The associated gradient operator! rg s is defined as: (2) G s = 1 2ßs 2 e 1 2s 2 (x 2 +y 2 )! r Gs @G As proved in [6], the gradient operator is equivalent to the continuous wavelet transform calculated with a wavelet derived from the gradient of a Gaussian function and defined as: (3)! ψ s = ψ 1 s ψ 2 s = @G The associated continuous wavelet transform! Wf(s; x; y) of a function f is then:! (4)! Wf(s; x; y) =f Λ! ψ s where Λ denotes the convolution product. Edges can be detected from the modulus local maxima of the above wavelet transform, in the direction indicated by the phase of this wavelet transform. 2.2 Local regularity analysis In addition, the study of the wavelet transform modulus across scales enables the measurement of the local Lipschitz regularity [5, 9]. The procedure presented by Mallat and Hwang [7] consists in extracting a maxima curve, following across scales, the wavelet transform maxima of a particular point x 0. Then the logarithmic slope of the curve when the scale tends to 0 gives information about the local regularity of the function at the point x 0. Let us consider three particular types of singularities frequently encountered in images : the transition, the peak and the line. The transition is the step corresponding to the boundary of an object, the peak is an ideal impulsion represented by a 2-D Dirac distribution, and the line is an ideal impulsion along one direction represented by a 1-D Dirac distribution. Then, the logarithmic slope of the wavelet maxima curve when the scale tends to 0 is 0 for the transition, -2 for the peak and -1 for the line. 3 Multiscale detection and characterization of features 3.1 Presentation of the models of smoothed edges The local regularity analysis presented by Mallat and Hwang [7] has to be adapted to edge detection in order to consider typical edges identified on images. To do so, we define three edge models obtained after a smoothing of the

3 Wavelet based multiscale analysis Transition Peak Figure 1: Functions associated to the transition edge and the peak edge for ff = 1 and A =1. three singularities cited above, and represented by three functions: the transition edge, the peak edge and the line edge. Each edge model is characterized by atype, a smoothing size ff and an amplitude A. Let us consider the two dimensional extension H of the Heaviside function: ( 0 if x» 0 (5) H = 1 if x>0 Then, the function associated to the transition edge is defined as the convolution product of the Heaviside function and the Gaussian filter of size ff: (6) T ff =A:H Λ G ff The transition edge corresponds to the boundary of an object in an image, taking into account a transition slope proportional to ff (Figure 1). The peak function P ff associated to the peak edge is a Gaussian function of amplitude A and size ff: (7) P ff =2ßff 2 A:G ff This edge corresponds to a Gaussian circular object, whose size is proportional to ff (Figure 1). The line function L ff associated to the line edge is a 1-D Gaussian function of amplitude A and size ff: (8) L ff = p 2ßff:A:G ff (x; 0) This edge corresponds to a thick line whose cross profile is Gaussian, and whose thickness is proportional to ff. 3.2 Maxima curves of the smoothed edges To perform a multiscale detection and characterization of the three edge models presented above, we have to study their behavior across scales. This behavior is given by the maxima curves calculated for any point of the edge. For that purpose, we first, determine the wavelet transform of the corresponding edge function. Then, we calculate the local maximum in the direction of the phase. The maxima curves we obtain are function of the analyzing scale s, the edge amplitude A and the smoothing size ff. We denote MT ff (s);mp ff (s) andml ff (s) as the maxima curves respectively associated to the transition edge, the peak edge and the line edge. Using equation (4), (6), (7), (8), we obtain after calculation: (9) MT ff (s) = p A s p 2ß s2 + ff 2 (10) MP ff (s) = A p e sff 2 (s 2 + ff 2 ) 3=2 ML ff (s) = p A sff (11) e s 2 + ff 2 The Figure 2 gives the shape of these maxima curves.

4 Wavelet based multiscale analysis Transition Peak Line s Figure 2: : Maxima curves associated to the transition edge, the peak edge and the line edge for ff = 1 and A = The feature detection and characterization algorithm The presented algorithm performs a multiscale edge detection and characterization which consists in determining for each detected point: its localization at an adapted scale, its type (transition, peak or line), its amplitude, and its smoothing scale. The input parameters are: the image to process, the maximum analyzing scale s M and the localization factor k l. The outputs are three images giving the detected edges with a level respectively equal to the type of the edge (1, 2 or 3), its amplitude, and its smoothing size. The principle of this algorithm lies in a comparison between maxima curves detected in the images to process and the three theoretical maxima curves corresponding to the transition edge, the peak edge and the line edge. Using least square fittings, we determine the type of the maxima curve, and its characteristic parameters A and ff. The scale s l adapted to the localization of the corresponding point of the edge is then defined as : (12) s l = k l ff According to this definition, the localization scale is adapted to the smoothing size of the edge: the more blurred is the edge, the greater is the localization scale. This allows to localize the edge in the same way regardless of the smoothing size. If the localization factor is close to 0, it is then very precise but very sensitive to noise, and if the factor is high the localization is less precise but less sensitive to noise. The main steps of the algorithm are: 1. Computation of the wavelet transform for the scales ranging from 0.5 to s M by step of For each maxima curve detected, determination of the edge model giving the best least square fitting, and computation of the corresponding amplitude and smoothing size. 3. Following each maxima curve, registering the position of the point of the curve whose scale is equal to the adapted scale s l. 3.4 Validation on synthetic images The algorithm has been tested on a synthetic set of images obtained after the smoothing of a 64 by 64 pixel base image composed of a peak of 2 by 2 pixels, a line of 2 pixels of thickness and a square of 21 by 21 pixels. The 4 images of the set are obtained after a smoothing of the reference image at scales 0.25, 1, 2 and 3 (see Figure 3, first image). Figure 3 presents the results given by the algorithm: the second image gives the type of edge, the third gives the amplitude and the fourth gives the smoothing size. 4 Applications Three applications on real images are presented. These respectively concern the detection of a diesel jet, the analysis of the skin relief, and the bubble detection and measurement. Diesel jet images have been obtained inside the experimental engine of the Laboratoire de Machines Thermiques de l'ecole Centrale de Lyon (France), with a fast movie camera of images per second. The presented result is the amplitude of edges given by the algorithm for two successive images. The main turbulent structures of the jet are detected, and the evolution of these structures can be studied (Figure 4).

5 Wavelet based multiscale analysis... 5 Figure 3: The synthetic set of images obtained with a smoothing size equal to 0.25, 1, 2 and 3 pixels (first image), and the result of the detection and characterization (images 2, 3, and 4). Image 2 gives the type of edge, image 3 gives the amplitude and image 4 gives the smoothing size. Figure 4: Detection of edges of a turbulent jet inside a diesel engine, for two successive images. A low pressure skin relief image is used to study the performance of cosmetic creams, comparing the relief before and after the treatment. Figure 5 presents the original skin relief and the corresponding amplitude image of edges before and after the treatment. These results show a little decrease of the amplitude after the treatment. Cavitation bubbles inside an ultrasonic reactor can be studied by means of a fast intensified video camera and a long distance microscope [2]. The corresponding images are noisy and the bubbles can have blurred edges. Our edge detection algorithm has been used to estimate the bubble size and fuzziness using respectively the amplitude and smoothing size of the edges. Figure 6 gives the smoothing size image of edges, its average level is equal to 3 pixels which correspond to the average fuzziness. Figure 5: Analysis of the amplitude of the skin relief before and after a cosmetic treatment.

6 Wavelet based multiscale analysis Conclusion Figure 6: Smoothing size of edges of a cavitation bubble, the average size is equal to 3 pixels. We have presented in this paper three edge models used to define image features: the transition, the line and the peak. These edge models have been characterized from the evolution of their wavelet transform modulus maxima across scales. Furthermore, an algorithm to detect edges and to estimate their characteristics have been developed and tested. This algorithm is able to determine the position, the type, the amplitude and the smoothing size of each edge. The results of the tests on synthetic images have proved that both the edge model and the edge characterization algorithm are relevant. The extensive application area of this work has been illustrated by three particular problems: the detection of a diesel jet, the analysis of the skin relief, and the cavitation bubble detection and measurement. Another application to motion determination, based on a one dimensional edge characterization has been presented in [3]. References [1] J. Canny, A computational approach to edge detection, IEEE Trans. Pattern Anal. Machine Intell., 8 (1986), pp [2] M. Chouvellon, C. Ducottet, P. Boldo, Y.Gonthier, Visualization of cavitation bubbles in a high frequency ultrasonic reactor, ECCE2, 2 eme congr es européen de Génie des Procédés, Montpellier,(1999),C [3] J. Fayolle, C. Ducottet, J.P.Schon, Application of Multiscale Characterization of Edges to Motion Determination, IEEE Trans. Signal Process.,46 (1998), pp [4] A. Grossmann, J. Morlet, Decomposition of Hardy functions into square integrable wavelets of constant shape, in Lecture Notes in Mathematics, SIAM J. Math., 15 (1984), pp [5] S. Jaffard, Exposants de Holder en des points donnés et coefficients d'ondelettes, Notes au compte-rendu de l'académie des Sciences, France, 308 ser. I (1989), pp [6] S. Mallat,S. Zhong, Characterization of Signals from Multiscale Edges, IEEE Trans. Pattern Anal. Machine Intell 14 (1992), pp [7] S. Mallat, W.L. Hwang, Singularity Detection and Processing with Wavelets, IEEE Trans. Inform. Theory 38 (1992), pp [8] D. Marr, E. Hildreth, Theory of edge detection, Proc. Roy. Soc. Lon., B207 (1980), pp [9] Y. Meyer, Ondelettes et opérateurs I : Ondetettes, Paris, Herman, [10] A. Witkin, Scale space filtering, Proc. Int. Joint Conf. Artificial Intell.Karlsruhe, West Germany, (1983), pp