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”.

This paper describes an approach for decomposing a signal into the sum of an oscillatory component and a transient component. The method uses a newly developed rational-dilation wavelet transform (WT), a selfinverting constant-Q transform with an adjustable Q-factor (quality-factor). We propose that the oscillatory component be modeled as signal that can be sparsely represented using a high Q-factor WT; likewise, we propose that the transient component be modeled as a piecewise smooth signal that can be sparsely represented using a low Q-factor WT. Because the low and high Q-factor wavelet transforms are highly distinct (having low coherence), morphological component analysis (MCA) successfully yields the desired decomposition of a signal into an oscillatory and non-oscillatory component. The method, being non-linear, is not constrained by the limits of conventional LTI filtering. Keywords: wavelets, sparsity, morphological component analysis, constant Q, Q-factor 1.

Numerous signals arising from physiological and physical processes, in addition to being non-stationary, are moreover a mixture of sustained oscillations and non-oscillatory transients that are difficult to disentangle by linear methods. Examples of such signals include speech, biomedical, and geophysical signals. Therefore, this paper describes a new nonlinear signal analysis method based on signal resonance, rather than on frequency or scale, as provided by the Fourier and wavelet transforms. This method expresses a signal as the sum of a ‘high-resonance ’ and a ‘low-resonance ’ component — a high-resonance component being a signal consisting of multiple simultaneous sustained oscillations; a low-resonance component being a signal consisting of non-oscillatory transients of unspecified shape and duration. The resonance-based signal decomposition algorithm presented in this paper utilizes sparse signal representations, morphological component analysis, and constant-Q (wavelet) transforms with adjustable Q-factor. Keywords: sparse signal representation, constant-Q transform, wavelet transform, morphological component analysis 1.