Support Blob Machines
A novel generalization of linear scale space is presented. The generalization allows for a sparse approximation of the function at a certain scale.
To start with, we first consider the Tikhonov regularization viewpoint on scale space theory . The sparsification is then obtained using ideas from support vector machines  and based on the link between sparse approximation and support vector regression as described in  and .
In regularization theory, an ill-posed problem is solved by searching for a solution having a certain differentiability while in some precise sense the final solution is close to the initial signal. To obtain scale space, a quadratic loss function is used to measure the closeness of the initial function to its scale σ image.
We propose to alter this loss function thus obtaining our generalization of linear scale space. Comparable to the linear ε-insensitive loss function introduced in support vector regression , we use a quadratic ε-insensitive loss function instead of the original quadratic measure. The ε-insensitivity loss allows errors in the approximating function without actual increase in loss. It penalizes errors only when they become larger than the a priory specified constant ε. The quadratic form is mainly maintained for consistency with linear scale space.
Although the main concern of the article is the theoretical connection between the foregoing theories, the proposed approach is tested and exemplified in a small experiment on a single image.
KeywordsSupport Vector Machine Support Vector Loss Function Quadratic Programming Support Vector Regression
- 5.Girosi, F., Jones, M., Poggio, T.: Prior stabilizers and basis functions: From regularization to radial, tensor and additive splines. Technical Report AI Memo 1430, CBCL Paper 75. MIT, Cambridge, MA (1993)Google Scholar
- 6.Griffin, L.D.: Local image structure, metamerism, norms, and natural image statistics. Perception 31(3) (2002)Google Scholar
- 7.ter Haar Romeny, B.M. (ed.): Geometry-Driven Diffusion. Kluwer, Dordrecht (1996)Google Scholar
- 9.Jackway, P.R.: Morphological scale-space. In: 11th IAPR International Conference on Pattern Recognition, pp. 252–255. The Hague, The Netherlands (1992)Google Scholar
- 11.Koenderink, J.J., van Doorn, A.J.: Metamerism in complete sets of image operators. In: Advances in Image Understanding 1996, pp. 113–129 (1996)Google Scholar
- 13.Lindeberg, T.: Scale-Space Theory in Computer Vision. Kluwer Academic Press, Boston (1994)Google Scholar
- 19.Smola, J., Schölkopf, B.: A tutorial on support vector regression. Technical Report NC-TR-98-030, Royal Holloway College, University of London, UK (1998)Google Scholar
- 21.Tikhonov, N., Arseninn, V.Y.: Solution of Ill-Posed Problems. W. H. Winston, Washington (1977)Google Scholar
- 24.Witkin, P.: Scale-space filtering. In: Proceedings of IJCAI, Germany (1983)Google Scholar