Complex Wavelet Bases, Steerability, and the Marr-Like Pyramid

Abstract

Abstract-Our aim in this paper is to tighten the link between wavelets, some classical image-processing operators, and David Marr's theory of early vision. The cornerstone of our approach is a new complex wavelet basis that behaves like a smoothed version of the Gradient-Laplace operator. Starting from first principles, we show that a single-generator wavelet can be defined analytically and that it yields a semi-orthogonal complex basis of , irrespective of the dilation matrix used. We also provide an efficient FFT-based filterbank implementation. We then propose a slightly redundant version of the transform that is nearly translation-invariant and that is optimized for better steerability (Gaussian-like smoothing kernel). We call it the Marr-like wavelet pyramid because it essentially replicates the processing steps in Marr's theory of early vision. We use it to derive a primal wavelet sketch which is a compact description of the image by a multiscale, subsampled edge map. Finally, we provide an efficient iterative algorithm for the reconstruction of an image from its primal wavelet sketch.Index Terms-Feature extraction, primal sketch, steerable filters, wavelet design. MULTISCALE transforms are powerful tools for signal and image processing, computer vision, and for modeling biological vision. A prominent example is the 1-D wavelet transform, which acts as a multiscale version of an th-order derivative operator, where is the number of vanishing moments of the wavelet [1]. Its extension to multiple dimensions and to 2-D, in particular, is typically achieved by forming tensorproduct basis functions. However, such separable wavelets are not well matched to the singularities occuring in images such as lines and edges which can be arbitrarily oriented and even curved. Consequently, there has been a considerable research effort in developing alternative multiscale transforms that are better tuned to the geometry of natural images. Notable examples of these "geometrical x-lets" include biologically-inspired 2-D Gabor transforms [2] [24]. The derivation of the filters is not based on wavelets, but rather obtained through a numerical optimization process. Special constraints are imposed to ensure that the frequency response of the filters is essentially polar-separable and that the decomposition is simple to invert numerically (approximate tight-frame property).In this paper, we present an alternative approach based on an explicit analytical and spline-based formulation of complex wavelet bases of . Special care is given to design basis functions that best match the properties of the visual system, in accordance with Marr's theory of early vision [25]. To motivate our construction, we specify a number of properties that are highly desirable and that are fulfilled, sometimes implicitly, by a number of classical image-processing algorithms.• Invariance. We aim at invariance with respect to elementary geometric operations such as translation, scaling, and rotation. Traditional wavelet transforms only satisfy these propert...

Multiresolution Monogenic Signal Analysis Using the Riesz–Laplace Wavelet Transform

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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.…”

“…5.1.1] that yields a pair of x-and y-derivative wavelets. Edges in the multiscale gradient signal are then detected based on a wavelet-domain version of the Canny edge detector, which includes nonmaximum suppression and hysteresis thesholding [14]. Note that the Canny edge detector requires an estimation of the strength and orientation of the gradient for each point of the image, which is obtained by steering the gradient-like wavelets at every scale and location.…”

“…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.…”

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