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. The latter observation has not prevented the design of image representations, which trade off between efficiency and complexity, while achieving accurate rendering of smooth regions as well as reproducing faithful contours and textures. 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 presents a panorama of the aforementioned literature on decompositions in multiscale, multi-orientation bases or dictionaries. They typically exhibit redundancy to improve sparsity in the transformed domain and sometimes its invariance with respect to simple geometric deformations (translation, rotation). 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 or arbitrary meshed surfaces. The etymology of panorama suggests an overview, based on a choice of partially overlapping "pictures". We hope that this paper will contribute to the appreciation and apprehension of a stream of current research directions in image understanding.Comment: 65 pages, 33 figures, 303 reference

The nonexistence of shearlet scaling functions

AbstractOver the past five years, the directional representation system of shearlets has received much attention and has been shown to exhibit many advantageous properties. Over this time period, there have been a number of attempts to associate shearlet systems with a multiresolution analysis (MRA). However, one can argue that, in each of these attempts, the following statement regarding the resulting shearlet MRA notion is inaccurate: “There exist scaling functions satisfying various desirable properties, such as significant amounts of decay or regularity, nonnegativity, or advantageous refinement or representation conditions. Each such scaling function naturally induces an associated shearlet (either traditional or cone-adapted) that satisfies similar desirable properties. Each such scaling function/associated shearlet pair rationally induces a fast decomposition algorithm for discrete data.” In this article, we attempt to provide explanation for this situation by arguing the great difficulty of associating shearlet systems with such an MRA. We do so by considering two very natural and general notions of shearlet MRA—one which leads to traditional shearlets and one which leads to cone-adapted shearlets—each of which seems to be an excellent candidate to satisfy the above quoted statement. For each of these notions, we prove the nonexistence of associated scaling functions satisfying the above mentioned desirable properties

ShearLab: A Rational Design of a Digital Parabolic Scaling Algorithm

Multivariate problems are typically governed by anisotropic features such as edges in images. A common bracket of most of the various directional representation systems which have been proposed to deliver sparse approximations of such features is the utilization of parabolic scaling. One prominent example is the shearlet system. Our objective in this paper is three-fold: We firstly develop a digital shearlet theory which is rationally designed in the sense that it is the digitization of the existing shearlet theory for continuous data. This implicates that shearlet theory provides a unified treatment of both the continuum and digital realm. Secondly, we analyze the utilization of pseudo-polar grids and the pseudo-polar Fourier transform for digital implementations of parabolic scaling algorithms. We derive an isometric pseudo-polar Fourier transform by careful weighting of the pseudo-polar grid, allowing exploitation of its adjoint for the inverse transform. This leads to a digital implementation of the shearlet transform; an accompanying Matlab toolbox called ShearLab is provided. And, thirdly, we introduce various quantitative measures for digital parabolic scaling algorithms in general, allowing one to tune parameters and objectively improve the implementation as well as compare different directional transform implementations. The usefulness of such measures is exemplarily demonstrated for the digital shearlet transform.Comment: submitted to SIAM J. Multiscale Model. Simu