Texture scale and image segmentation using wavelet filters

1. Texture scale and image segmentation using wavelet filters Sylvain Meignen, Valérie Perrier LMC-IMAG Laboratory, Mosaic team Algorithm We link the distance d used in the first two steps of the algorithm (splitting and merging steps) to the study of the stability of the features, as follows: Introduction We are interested in defining a texture scale associated to the decomposition of textures on wavelet filter banks. We derive from the coefficients of the decomposition [4] stable and discriminative features. The study of the eigenvectors and eigenvalues that arouse in The Karhunen-Loeve transform of the coefficients [1] puts forward stable features which we use for the construction of a new distance that allows a good separation in the feature space. We build a segmentation algorithm in three steps: a splitting step, a merging step and a final segmentation step [3]. Once we described the general method we present segmentation results on Brodatz images of textures. The choice of distance d is motivated by the fact that has to be stable within a texture. In such a case, the proximity of the first eigenvalues will help to conclude that the two samples and are identical. To make the distance more discriminative,we add a term corresponding to the comparison of the energy in the subspace orthogonal to the first eigenvector for each texture sample. In the merging step, adjacency matrices are build paying particular attention to the size of the region to be merged. No spatial relation between region is used in order to better test the discriminative power of distance d [3]. At the beginning of the final segmentation step, the texture content of the image is known. The problem can be viewed as a supervised segmentation. Different distances are used depending of the kind of textures. When Vis non Gaussian distance d performs well, but averaged maximum likelihood distance can also be used. It is defined by: Notations Stability of the features Through the study of stability of the eigenvectors and the eigenvalues of S, when the size of the texture sample varies, it appears that the first eigen-vector is stable [2]. Filter used for the determination of V is an ortho gonal frame of of the form[4]: where denotes the covariance matrix computed over texture . When Vis Gaussian, simple maximum likelihood distance is used. The Gaussian character of vector V is inherited from that of the texture itself. A good insight into the Gaussian character of textures is given by the computation of their skewness and kurtosis. In the results section, we present a segmentation of an image of textures. We assume that the number of textures that the number of textures is a priori unknown. In [2] we discussed the determination of appropriate thresholds to get the true number of textures at the end of the merging step. The segmentation step is performed using Kmeans algorithm and distance . Best approximations of orthogonal frames on images are obtained by truncated Battle-Lemarié filters given by their Fourier series: Filter g is the conjugate mirror filter of h Results Segmentation results Example of image of textures Hierarchical splitting step References 1. S. Mallat (1998), A wavelet tour on signal processing, Academic Press 2. S.Meignen and V.Perrier, Texture scale and image segmentation using wavelet filters, submitted to IEEE transactions on image processing. 3. T.Ojala and M.Pietikainen (1999), Unsupervised texture segmentation using feature distribution, Pattern recognition, vol. 32, pp.477-486. 4. M. Unser (1995), Texture classification and segmentation using wavelet frames, IEEE transactions on image processing, vol. 4, no. 11, pp.1549-1560. Conclusion This poster shows that it is possible to build a discriminative distance for texture segmentation using the eigenvalues and the eigenvectors associated to the Karhunen-Loeve transform of the decomposition of images on an orthogonal frame. The basic idea of the construction of the distance is that the first eigenvector of the decomposition is stable. The algorithm is unsupervised, in particular, we assume the number of textures is unknown.