Sparse Mathematical Morphology Using Non-negative Matrix Factorization
Sparse modelling involves constructing a succinct representation of initial data as a linear combination of a few typical atoms of a dictionary. This paper deals with the use of sparse representations to introduce new nonlinear operators which efficiently approximate the dilation/erosion. Non-negative matrix factorization (NMF) is a dimensional reduction (i.e., dictionary learning) paradigm particularly adapted to the nature of morphological processing. Sparse NMF representations are studied to introduce pseudo-morphological binary dilations/erosions. The basic idea consists in processing exclusively the image dictionary and then, the result of processing each image is approximated by multiplying the processed dictionary by the coefficient weights of the current image. These operators are then extended to grey-level images by means of the level-set decomposition. The performance of the present method is illustrated using families of binary shapes and face images.
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- 2.Donoho, D., Stodden, V.: When does non-negative matrix factorization give a correct decomposition into parts? In: Advances in Neural Information Processing 16 (Proc. NIPS 2003). MIT Press, Cambridge (2004)Google Scholar
- 7.Lee, D.D., Seung, H.S.: Algorithms for non-negative matrix factorization. In: Advances in Neural Information Processing 13 (Proc. NIPS 2000). MIT Press, Cambridge (2001)Google Scholar
- 8.Li, S.Z., Hou, X., Zhang, H., Cheng, Q.: Learning spatially localized parts-based representations. In: Proc. IEEE Conf. on Computer Vision and Pattern Recognition (CVPR), Hawaii, USA, vol. I, pp. 207–212 (2001)Google Scholar
- 10.Ronse, C.: Bounded variation in posets, with applications in morphological image processing. In: Passare, M. (ed.) Proceedings of the Kiselmanfest 2006, Acta Universitatis Upsaliensis, vol. 86, pp. 249–281 (2009)Google Scholar
- 11.Serra, J.: Image Analysis and Mathematical Morphology. Image Analysis and Mathematical Morphology, vol. I. Theoretical Advances, vol. II. Academic Press, London (1982) (1988)Google Scholar
- 14.Yu, G., Sapiro, G., Mallat, S.: Solving Inverse Problems with Piecewise Linear Estimators: From Gaussian Mixture Models to Structured Sparsity. IEEE Trans. on Image Processing (2011)Google Scholar