Wavelet, active basis, and shape script: a tour in the sparse land

Sparse coding is a key principle that underlies wavelet representation of natural images. In this paper, we explain that the effort of seeking a common wavelet sparse coding of images from the same object category leads to an active basis model, where the images share the same set of selected wavelet elements, which form a linear basis for representing the images. The selected wavelet elements are allowed to perturb their locations and orientations to account for shape deformations, so that the basis becomes active, and the active basis serves as a mathematical representation of a deformable template. We show that a recursive application of the strategy underlying the active basis model leads to a shape script model, which is a composition of shape motifs such as ellipsoids, parallel bars, angles, etc. These shape motifs are allowed to change their locations, orientations, scales and aspect ratios, and the shape motifs themselves are modeled by active bases. Compared to the active basis model, the shape script model is a sparser representation and therefore has stronger generalization power. It can also be considered another layer of sparse coding of the selected wavelet elements that themselves provide sparse coding of the image intensities.