3D Steerable Wavelets in Practice

Abstract

Abstract-We introduce a systematic and practical design for steerable wavelet frames in 3D. Our steerable wavelets are obtained by applying a 3D version of the generalized Riesz transform to a primary isotropic wavelet frame. The novel transform is self-reversible (tight frame) and its elementary constituents (Riesz wavelets) can be efficiently rotated in any 3D direction by forming appropriate linear combinations. Moreover, the basis functions at a given location can be linearly combined to design custom (and adaptive) steerable wavelets. The features of the proposed method are illustrated with the processing and analysis of 3D biomedical data. In particular, we show how those wavelets can be used to characterize directional patterns and to detect edges by means of a 3D monogenic analysis. We also propose a new inverse-problem formalism along with an optimization algorithm for reconstructing 3D images from a sparse set of wavelet-domain edges. The scheme results in high-quality image reconstructions which demonstrate the feature-reduction ability of the steerable wavelets as well as their potential for solving inverse problems.

Maximally Localized Radial Profiles for Tight Steerable Wavelet Frames

“…First, a multiscale primal sketch [14], or edge map [15], is extracted from the set of wavelet coefficients of the image. An approximation of the original image is then recovered from this small subset of coefficients relying on constrained optimization.…”

“…Steerability is a crucial aspect in many other image-processing applications such as finding the dominant orientation at each image location, detecting contours [12], or identifying features in a rotationinvariant fashion [13]. More recently, algorithms for image reconstruction from the small subset of wavelet coefficients called the "primal sketch" have been proposed relying on the steerable pyramid [14], [15]. In this work, we study the design of wavelet profiles for use in applications relying on steerable tight frames.…”

Optimal Isotropic Wavelets for Localized Tight Frame Representations

“…For a given , there is a constant , depending on , and such that the wavelet coefficients of satisfy (7) for sufficiently large. Proof: The coefficients are (8) Changing variables, we write (9) Finally, we require and use the support condition along with the Cauchy-Schwartz inequality to obtain the bound in (7).…”

“…One can view such an as an infinite sum of compactly supported functions , centered at the lattice points . The function contains the local information of near , and the wavelet coefficients of satisfy the estimate (10) in analogy with (7).…”

Wavelet domain compression analysis for differential interferometric synthetic aperture radar (DInSAR) images

“…Recent research includes: the wavelet Bayesian network image de-noising (Ho & Hwang 2013), the improved bounds for subband-adaptive iterative shrinkage/ thresholding algorithms (Zhang & Kingsbury 2013), the wavelet domain multifractal analysis for static and dynamic texture classification (Hui Ji et al, 2013), the multiscale image fusion using the undecimated wavelet transform with spectral factorization and nonorthogonal filter banks (Ellmauthaler et al, 2013), the 3D steerable wavelets in practice (Chenouard & Unser 2013) and selecting the best wavelet for texture discrimination (Chaudhry et al, 2007).…”