Harmonic wavelet transform

About: Harmonic wavelet transform is a research topic. Over the lifetime, 9602 publications have been published within this topic receiving 247336 citations.

Fast computation of discrete cosine transform through fast Hartley transform

Abstract: A relationship between the discrete cosine transform (DCT) and the discrete Hartley transform (DHT) is derived. It leads to a new fast and numerically stable algorithm for the DCT.

Digital Ridgelet Transform Based on True Ridge Functions

Abstract: We study a notion of ridgelet transform for arrays of digital data in which the analysis operator uses true ridge functions, as does the synthesis operator. There are fast algorithms for analysis, for synthesis, and for partial reconstruction. Associated with this is a transform which is a digital analog of the orthonormal ridgelet transform (but not orthonormal for finite n). In either approach, we get an overcomplete frame; the result of ridgelet transforming an n × n array is a 2n × 2n array. The analysis operator is invertible on its range; the appropriately preconditioned operator has a tightly controlled spread of singular values. There is a near-parseval relationship. Our construction exploits the recent development by Averbuch et al. (2001) of the Fast Slant Stack, a Radon transform for digital image data; it may be viewed as following a Fast Slant Stack with fast 2-d wavelet transform. A consequence of this construction is that it offers discrete objects (discrete ridgelets, discrete Radon transform, discrete Pseudopolar Fourier domain) which obey inter-relationships paralleling those in the continuum ridgelet theory (between ridgelets, Radon transform, and polar Fourier domain). We make comparisons with other notions of ridgelet transform, and we investigate what we view as the key issue: the summability of the kernel underlying the constructed frame. The sparsity observed in our current implementation is not nearly as good as the sparsity of the underlying continuum theory, so there is room for substantial progress in future implementations.

Fast solution of electromagnetic integral equations using adaptive wavelet packet transform

Abstract: The adaptive wavelet packet transform is applied to sparsify the moment matrices for the fast solution of electromagnetic integral equations. In the algorithm, a cost function is employed to adaptively select the optimal wavelet packet expansion/testing functions to achieve the maximum sparsity possible in the resulting transformed system. The search for the best wavelet packet basis and the moment matrix transformation are implemented by repeated two-channel filtering of the original moment matrix with a pair of quadrature filters. It is found that the sparsified matrix has above-threshold elements that grow only as O(N/sup 1.4/) for typical scatterers. Consequently the operations to solve the transformed moment equation using the conjugate gradient method scales as O(N/sup 1.4/). The additional computational cost for carrying out the adaptive wavelet packet transform is evaluated and discussed.

Discrete Wavelet Transform

Abstract: According to the definition of the continuous wavelet transform (CWT) given in (3.7), Chap. 3, the scale parameter s and translation parameter \(\tau\) can be varied continuously. As a result, performing the CWT on a signal will lead to the generation of redundant information. Although the redundancy is useful in some applications, such as signal denoising and feature extraction where desired performance is achieved at the cost of increased computational time and memory size, other applications may need to emphasize reduced computational time and data size, for example, in image compression and numerical computation. Such requirements illustrate the need for reducing redundancy in the wavelet coefficients among different scales as much as possible, while at the same time, avoiding sacrificing the information contained in the original signal. This can be achieved by parameter discretization, as described in the following section.

The Fourier-Mellin transform and mammalian hearing

Abstract: A combined Fourier–Mellin transform yields a representation of a signal that is independent of delay and scale change. Such a representation should be useful for speech analysis, where delay and scale differences degrade the performance of correlation operations or other similarity measures. At least two different versions of a combined Fourier–Mellin transform can be implemented. The simplest version (the ‖F‖2−‖M‖2 transform) completely eliminates spectral phase information, while a slightly more complicated version (the ?−? transform) preserves some phase information. Both versions can be synthesized with a Fourier transform and an exponential‐sampling algorithm. Exponential sampling produces a frequency scale distortion that is similar to the effect of the cochlea. The ‖F‖2−‖M‖2 transform can also be implemented with a bank of proportional bandwidth filters. If the relative phase between spectral components is preserved, then a Fourier–Mellin transformer can perform compression of linear‐period modulat...