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

TL;DR

This paper reviews recent advances in multiscale, multi-orientation image representations that balance efficiency and complexity, highlighting their geometric properties, invariance, and extensions to non-Euclidean domains.

Contribution

It provides a comprehensive overview of multiscale geometric decompositions, discussing their properties, algorithms, and extensions, offering insights into current research directions.

Findings

Multiscale, multi-orientation dictionaries improve sparsity and invariance in image representations.

Redundant dictionaries require specialized algorithms for efficient sparse coding.

Extensions to non-Euclidean domains expand the applicability of geometric decompositions.

Abstract

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.