Two-dimensional affine frames for image analysis and synthesis

An affine-group-based design methodology of Gabor-type filter bank is presented for the purpose of image analysis and synthesis. Various tessellations of the combined spatial-feature space are considered. We combine ideas introduced by Daugman [J.G. Daugman, Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters, J. Opt. Soc. Am. 2 (7) (1985) 1160-1169], Lee [T.S. Lee, Image representation using 2D Gabor-wavelets, IEEE Trans. PAMI 18 (10) (1996) 959-971] and Manjunath and Ma [B.S. Manjunath, W.Y. Ma, Texture features browsing and retrieval of image data, IEEE Trans. PAMI 18 (8) (1996) 837-842], and extend them by applying the action of the full affine group on Gaborian-type mother wavelets. In this approach we adopt optimality criteria of minimal spatial-features combined uncertainty, as well as tightness of the frame tessellating this combined space. In this work, scalings in the x and y directions, allowing for independent dilations in these two directions, as well as rotations and translations are allowed. For each discrete set of scalings, rotations and translations the frame bounds are calculated. For frames where the frame operator is well approximated by a multiple of the identity, we use the same set of functions in the analysis and synthesis, as though the frame is equivalent to an orthogonal basis. Moreover, we show that in the case of independent scalings in the x and y directions, the number of dominant (characteristic) orientations of the filter bank may depend on scale. We further show that the orientation bandwidths thus obtained, resemble those attained under the constraint imposed by the uncertainty principle.