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Person# Kunal Narayan Chaudhury

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Wavelet

A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy

Algorithm

In mathematics and computer science, an algorithm (ˈælɡərɪðəm) is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algo

Fourier transform

In physics and mathematics, the Fourier transform (FT) is a transform that converts a function into a form that describes the frequencies present in the original function. The output of the transfo

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Kunal Narayan Chaudhury, Michaël Unser, John Paul Ward

The Riesz transform is a natural multidimensional extension of the Hilbert transform, and it has been the object of study for many years due to its nice mathematical properties. More recently, the Riesz transform and its variants have been used to construct complex wavelets and steerable wavelet frames in higher dimensions. The flip side of this approach, however, is that the Riesz transform of a wavelet often has slow decay. One can nevertheless overcome this problem by requiring the original wavelet to have sufficient smoothness, decay, and vanishing moments. In this paper, we derive necessary conditions in terms of these three properties that guarantee the decay of the Riesz transform and its variants, and, as an application, we show how the decay of the popular Simoncelli wavelets can be improved by appropriately modifying their Fourier transforms. By applying the Riesz transform to these new wavelets, we obtain steerable frames with rapid decay.

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It is well known that spatial averaging can be realized (in space or frequency domain) using algorithms whose complexity does not scale with the size or shape of the filter. These fast algorithms are generally referred to as constant-time or O(1) algorithms in the image-processing literature. Along with the spatial filter, the edge-preserving bilateral filter involves an additional range kernel. This is used to restrict the averaging to those neighborhood pixels whose intensity are similar or close to that of the pixel of interest. The range kernel operates by acting on the pixel intensities. This makes the averaging process nonlinear and computationally intensive, particularly when the spatial filter is large. In this paper, we show how the O(1) averaging algorithms can be leveraged for realizing the bilateral filter in constant time, by using trigonometric range kernels. This is done by generalizing the idea presented by Porikli, i.e., using polynomial kernels. The class of trigonometric kernels turns out to be sufficiently rich, allowing for the approximation of the standard Gaussian bilateral filter. The attractive feature of our approach is that, for a fixed number of terms, the quality of approximation achieved using trigonometric kernels is much superior to that obtained by Porikli using polynomials.

Kunal Narayan Chaudhury, Sebanti Sanyal

It is well-known that box filters can be efficiently computed using pre-integration and local finite-differences. By generalizing this idea and by combining it with a nonstandard variant of the central limit theorem, we had earlier proposed a constant-time or O(1) algorithm that allowed one to perform space-variant filtering using Gaussian-like kernels. The algorithm was based on the observation that both isotropic and anisotropic Gaussians could be approximated using certain bivariate splines called box splines. The attractive feature of the algorithm was that it allowed one to continuously control the shape and size (covariance) of the filter, and that it had a fixed computational cost per pixel, irrespective of the size of the filter. The algorithm, however, offered a limited control on the covariance and accuracy of the Gaussian approximation. In this paper, we propose some improvements of our previous algorithm.