Curvelets and wave atoms for mirror-extended images

Variants of both the digital curvelet transform, and the digital wave atom transform, which handle the image boundaries by mirror extension are presented, which extend the range of applicability of curvelets and wave atoms to situations where periodization at the boundaries is uncalled for.

Abstract

We present variants of both the digital curvelet transform, and the digital wave atom transform, which handle the image boundaries by mirror extension. Previous versions of these transforms treated image boundaries by periodization. The main ideas of the modifications are 1) to tile the discrete cosine domain instead of the discrete Fourier domain, and 2) to adequately reorganize the in-tile data. In their shift-invariant versions, the new constructions come with no penalty on the redundancy or computational complexity. For shift-variant wave atoms, the penalty is a factor 2 instead of the naive factor 4. These various modifications have been included in the CurveLab and WaveAtom toolboxes, and extend the range of applicability of curvelets (good for edges and bandlimited wavefronts) and wave atoms (good for oscillatory patterns and textures) to situations where periodization at the boundaries is uncalled for. The new variants are dubbed ME-curvelets and ME-wave atoms, where ME stands for mirror-extended.