Data Representation Using the Weyl Transform

Abstract

The Weyl transform is introduced as a rich framework for data representation. Transform coefficients are connected to the Walsh-Hadamard transform of multiscale autocorrelations, and different forms of dyadic periodicity in a signal are shown to appear as different features in its Weyl coefficients. The Weyl transform has a high degree of symmetry with respect to a large group of multiscale transformations, which allows compact yet discriminative representations to be obtained by pooling coefficients. The effectiveness of the Weyl transform is demonstrated through the example of textured image classification.

Acoustic Seafloor Classification Using the Weyl Transform of Multibeam Echosounder Backscatter Mosaic

“…When dealing with the texture image classification tasks, we are interested in representations that are invariant under a given group of transformations, such as translation, scaling, and rotation. The Weyl transform has recently shown remarkable results in the context of texture classification with standard texture images [40], outperforming some common textural descriptors including Histograms of Oriented Gradients (HOG) [45] and Local Binary Pattern (LBP) [46]. HOG and LBP are very popular methods for texture description in computer vision, such as human detection and biomedical recognition problems.…”

“…, b m−1 ) ∈ Z m 2 are two binary m-tuples. Formally, the binary Heisenberg-Weyl group HW 2 m of order 2 2m+2 is defined as [40], the signed permutation matrices D(a, b) with a T b = 0 form an orthonormal basis of the vector space of real square symmetric matrices with respect to the inner product given by R, S := tr (R T S). In particular, each real symmetric matrix R can be represented as a linear combination of the basis elements as…”

“…Specifically, this investigation focuses on texture-based feature extraction methods. We introduce a novel image-based feature extraction approach for the classification of the seafloor sediment type based on the Weyl Transform, that was recently proved effective in other domains including medical image analysis [40,41]. The Weyl transform, defined as a mapping between a signal and its autocorrelation coefficients, has the desirable property of being scale and orientation invariant.…”

“…The Weyl transform, defined as a mapping between a signal and its autocorrelation coefficients, has the desirable property of being scale and orientation invariant. Consequently, patches sampled from the same texture will share similar Weyl coefficients [40,42]. On this basis, we develop an effective and robust MBES backscatter feature extraction method which, compared to well-established techniques, can significantly enhance thematic cluster separation.…”

A New Integer-to-Integer Transform

“…In [12], 2-D gcd-delta functions that contain only zeroes and ones are used to generate integer 2D DFT pairs. The Weyl transform which has binary valued inner elements is described in [13]. Integer-to-integer approximations of DST are still an active research area, as can be seen in [14].…”