Construction of Hilbert Transform Pairs of Wavelet Bases and Gabor-like Transforms

We propose a novel method for constructing Hilbert transform (HT) pairs of wavelet bases based on a fundamental approximation-theoretic characterization of scaling functions--the B-spline factorization theorem. In particular, starting from well-localized scaling functions, we construct HT pairs of biorthogonal wavelet bases of L^2(R) by relating the corresponding wavelet filters via a discrete form of the continuous HT filter. As a concrete application of this methodology, we identify HT pairs of spline wavelets of a specific flavor, which are then combined to realize a family of complex wavelets that resemble the optimally-localized Gabor function for sufficiently large orders. Analytic wavelets, derived from the complexification of HT wavelet pairs, exhibit a one-sided spectrum. Based on the tensor-product of such analytic wavelets, and, in effect, by appropriately combining four separable biorthogonal wavelet bases of L^2(R^2), we then discuss a methodology for constructing 2D directional-selective complex wavelets. In particular, analogous to the HT correspondence between the components of the 1D counterpart, we relate the real and imaginary components of these complex wavelets using a multi-dimensional extension of the HT--the directional HT. Next, we construct a family of complex spline wavelets that resemble the directional Gabor functions proposed by Daugman. Finally, we present an efficient FFT-based filterbank algorithm for implementing the associated complex wavelet transform.Comment: 36 pages, 8 figure

Gabor Shearlets

In this paper, we introduce Gabor shearlets, a variant of shearlet systems, which are based on a different group representation than previous shearlet constructions: they combine elements from Gabor and wavelet frames in their construction. As a consequence, they can be implemented with standard filters from wavelet theory in combination with standard Gabor windows. Unlike the usual shearlets, the new construction can achieve a redundancy as close to one as desired. Our construction follows the general strategy for shearlets. First we define group-based Gabor shearlets and then modify them to a cone-adapted version. In combination with Meyer filters, the cone-adapted Gabor shearlets constitute a tight frame and provide low-redundancy sparse approximations of the common model class of anisotropic features which are cartoon-like functions.Comment: 24 pages, AMS LaTeX, 4 figure

Radon-Gabor Barcodes for Medical Image Retrieval

In recent years, with the explosion of digital images on the Web, content-based retrieval has emerged as a significant research area. Shapes, textures, edges and segments may play a key role in describing the content of an image. Radon and Gabor transforms are both powerful techniques that have been widely studied to extract shape-texture-based information. The combined Radon-Gabor features may be more robust against scale/rotation variations, presence of noise, and illumination changes. The objective of this paper is to harness the potentials of both Gabor and Radon transforms in order to introduce expressive binary features, called barcodes, for image annotation/tagging tasks. We propose two different techniques: Gabor-of-Radon-Image Barcodes (GRIBCs), and Guided-Radon-of-Gabor Barcodes (GRGBCs). For validation, we employ the IRMA x-ray dataset with 193 classes, containing 12,677 training images and 1,733 test images. A total error score as low as 322 and 330 were achieved for GRGBCs and GRIBCs, respectively. This corresponds to $\approx 81\%$ retrieval accuracy for the first hit.Comment: To appear in proceedings of the 23rd International Conference on Pattern Recognition (ICPR 2016), Cancun, Mexico, December 201