Design of Steerable Wavelets to Detect Multifold Junctions

Abstract

Abstract-We propose a framework for the detection of junctions in images. Although the detection of edges and key points is a well examined and described area, the multiscale detection of junction centers, especially for odd orders, poses a challenge in pattern analysis. The goal of this paper is to build optimal junction detectors based on 2D steerable wavelets that are polar-separable in the Fourier domain. The approaches we develop are general and can be used for the detection of arbitrary symmetric and asymmetric junctions. The backbone of our construction is a multiscale pyramid with a radial wavelet function where the directional components are represented by circular harmonics and encoded in a shaping matrix. We are able to detect M-fold junctions in different scales and orientations. We provide experimental results on both simulated and real data to demonstrate the effectiveness of the algorithm.

Maximally Localized Radial Profiles for Tight Steerable Wavelet Frames

“…Theorem 1: Let L w be the Hilbert space whose inner product is specified by (13) and let T be the set of functions in L w satisfiying the tight frame property (6). Then, the operator…”

“…Some well-known algorithms for denoising ([6], including the widely-used Bayesian least-squares Gaussianscale-mixture (BLS-GSM) algorithm [7]), texture analysis (or synthesis) [8], [9], and regularization with sparsity constraints for inverse problems [10], [11] rely on the steerable pyramid, although methods that do not exploit steerability are also available for these tasks. 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].…”

A Sinusoidal Image Model Derived from the Circular Harmonic Vector

“…To pick the coefficients of W it is proposed to maximise the energy of the angular response of the sinusoidal wavelets inside of a window h(θ ) by adapting the method described in [42,43,51] for designing prolate spheroidal wavelets. Each sinusoidal wavelet is 2nd-order rotationally symmetric or anti-symmetric and therefore has its angular response concentrated at two points π radians apart.…”

Adaptive-Resolution Multi-Orientation Analysis of Complex Filamentous Network Images

“…A common approach to complete junctions in this case has been to close the gaps using inference methods 3,8 , but these tend to involve heuristics. Additionally, unlike prior attempts to detect junctions explicitly, our approach does not need predefined angles 2 or symmetries 9 . Our approach applies generally to steerable ridge filters, including to a unifying parametric framework for steerable wavelets 10 .…”