DEFORMABLE KERNELS FOR EARLY VISION

Early vision algorithms often have a first stage of linear-filtering t hat 'extracts' from the image information at multiple scales of resolu tion and multiple orientations. A common difficulty in the design and implementation of such schemes is that one feels compelled to discreti ze coarsely the space of scales and orientations in order to reduce co mputation and storage costs. This discretization produces anisotropies due to a loss of translation-, rotation-, and scaling-invariance that makes early vision algorithms less precise and more difficult to desi gn. This need not be so: one can compute and store efficiently the res ponse of families of linear filters defined on a continuum of orientat ions and scales. A technique is presented that allows 1) computing the best approximation of a given family using linear combinations of a s mall number of 'basis' functions; 2) describing all finite-dimensional families, i.e., the families of filters for which a finite dimensiona l representation is possible with no error. The technique is based on singular value decomposition and may be applied to generating filters in arbitrary dimensions and subject to arbitrary deformations; the rel evant functional analysis results are reviewed and precise conditions for the decomposition to be feasible are stated. Experimental results are presented that demonstrate the applicability of the technique to g enerating multi-orientation multi-scale 2D edge-detection kernels. The implementation issues are also discussed.