A Panorama on Multiscale Geometric Representations, Intertwining Spatial, Directional and Frequency Selectivity

The richness of natural images makes the quest for optimal representations in image processing and computer vision challenging. This fact has not prevented the design of candidates, convenient for rendering smooth regions, contours and textures at the same time, with compromises between efficiency and complexity. The most recent ones, proposed in the past decade, share an hybrid heritage highlighting the multiscale and oriented nature of edges and patterns in images. This paper endeavors a panorama of the aforementioned literature on decompositions in multiscale, multi-orientation bases or dictionaries. They typically exhibit redundancy to improve both the sparsity of the representation, and sometimes its invariance to various geometric deformations. Oriented multiscale dictionaries extend traditional wavelet processing and may offer rotation invariance. Highly redundant dictionaries require specific algorithms to simplify the search for an efficient (sparse) representation. We also discuss the extension of multiscale geometric decompositions to non-Euclidean domains such as the sphere $S^2$ and arbitrary meshed surfaces in $\Rbb^3$. The apt and patented etymology of panorama suggests an overview based on a choice of overlapping "pictures", i.e., selected from a broad set of computationally efficient mathematical tools. We hope this work help enlighten a substantial fraction of the present exciting research in image understanding, targeted to better capture data diversity.