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73
Sampling—50 years after Shannon
 Proceedings of the IEEE
, 2000
"... This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the math ..."
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Cited by 206 (22 self)
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This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the mathematical connections that were made with wavelet theory. To introduce the reader to the modern, Hilbertspace formulation, we reinterpret Shannon’s sampling procedure as an orthogonal projection onto the subspace of bandlimited functions. We then extend the standard sampling paradigm for a representation of functions in the more general class of “shiftinvariant” functions spaces, including splines and wavelets. Practically, this allows for simpler—and possibly more realistic—interpolation models, which can be used in conjunction with a much wider class of (antialiasing) prefilters that are not necessarily ideal lowpass. We summarize and discuss the results available for the determination of the approximation error and of the sampling rate when the input of the system is essentially arbitrary; e.g., nonbandlimited. We also review variations of sampling that can be understood from the same unifying perspective. These include wavelets, multiwavelets, Papoulis generalized sampling, finite elements, and frames. Irregular sampling and radial basis functions are briefly mentioned. Keywords—Bandlimited functions, Hilbert spaces, interpolation, least squares approximation, projection operators, sampling,
Sparse Geometric Image Representations with Bandelets
, 2004
"... This paper introduces a new class of bases, called bandelet bases, which decompose the image along multiscale vectors that are elongated in the direction of a geometric flow. This geometric flow indicates directions in which the image grey levels have regular variations. The image decomposition in ..."
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Cited by 146 (4 self)
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This paper introduces a new class of bases, called bandelet bases, which decompose the image along multiscale vectors that are elongated in the direction of a geometric flow. This geometric flow indicates directions in which the image grey levels have regular variations. The image decomposition in a bandelet basis is implemented with a fast subband filtering algorithm. Bandelet bases lead to optimal approximation rates for geometrically regular images. For image compression and noise removal applications, the geometric flow is optimized with fast algorithms, so that the resulting bandelet basis produces a minimum distortion. Comparisons are made with wavelet image compression and noise removal algorithms.
Visibility of Wavelet Quantization Noise
, 1996
"... The Discrete Wavelet Transform (DWT) decomposes an image into bands that vary in spatial frequency and orientation. It is widely used for image compression. Measures of the visibility of DWT quantization errors are required to achieve optimal compression. Uniform quantization of a single band of coe ..."
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Cited by 109 (1 self)
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The Discrete Wavelet Transform (DWT) decomposes an image into bands that vary in spatial frequency and orientation. It is widely used for image compression. Measures of the visibility of DWT quantization errors are required to achieve optimal compression. Uniform quantization of a single band of coefficients results in an artifact that we call DWT uniform quantization noise; it is the sum of a lattice of random amplitude basis functions of the corresponding DWT synthesis filter. We measured visual detection thresholds for samples of DWT uniform quantization noise in Y, Cb, and Cr color channels. The spatial frequency of a wavelet is r 2 l , where r is display visual resolution in pixels/degree, and l is the wavelet level. Thresholds increase rapidly with wavelet spatial frequency. Thresholds also increase from Y to Cr to Cb, and with orientation from lowpass to horizontal/vertical to diagonal. We construct a mathematical model for DWT noise detection thresholds that is a function of level, orientation, and display visual resolution. This allows calculation of a "perceptually lossless" quantization matrix for which all errors are in theory below the visual threshold. The model may also be used as the basis for adaptive quantization schemes.
Wavelet Families Of Increasing Order In Arbitrary Dimensions
, 1997
"... . We build compactly supported biorthogonal wavelets and perfect reconstruction filter banks for any lattice in any dimension with any number of primal and dual vanishing moments. The resulting scaling functions are interpolating. Our construction relies on the lifting scheme and inherits all of its ..."
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Cited by 46 (0 self)
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. We build compactly supported biorthogonal wavelets and perfect reconstruction filter banks for any lattice in any dimension with any number of primal and dual vanishing moments. The resulting scaling functions are interpolating. Our construction relies on the lifting scheme and inherits all of its advantages: fast transform, inplace calculation, and integerto integer transforms. We show that two lifting steps suffice: predict and update. The predict step can be built using multivariate polynomial interpolation, while update is a multiple of the adjoint of predict. Submitted to IEEE Transactions on Image Processing Over the last decade several constructions of compactly supported wavelets have originated both from signal processing and mathematical analysis. In signal processing, critically sampled wavelet transforms are known as filter banks or subband transforms [32, 43, 54, 56]. In mathematical analysis, wavelets are defined as translates and dilates of one fixed function and ar...
Wavelet theory demystified
 IEEE Trans. Signal Process
, 2003
"... Abstract—In this paper, we revisit wavelet theory starting from the representation of a scaling function as the convolution of a Bspline (the regular part of it) and a distribution (the irregular or residual part). This formulation leads to some new insights on wavelets and makes it possible to red ..."
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Cited by 44 (22 self)
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Abstract—In this paper, we revisit wavelet theory starting from the representation of a scaling function as the convolution of a Bspline (the regular part of it) and a distribution (the irregular or residual part). This formulation leads to some new insights on wavelets and makes it possible to rederive the main results of the classical theory—including some new extensions for fractional orders—in a selfcontained, accessible fashion. In particular, we prove that the Bspline component is entirely responsible for five key wavelet properties: order of approximation, reproduction of polynomials, vanishing moments, multiscale differentiation property, and smoothness (regularity) of the basis functions. We also investigate the interaction of wavelets with differential operators giving explicit time domain formulas for the fractional derivatives of the basis functions. This allows us to specify a corresponding dual wavelet basis and helps us understand why the wavelet transform provides a stable characterization of the derivatives of a signal. Additional results include a new peeling theory of smoothness, leading to the extended notion of wavelet differentiability in thesense and a sharper theorem stating that smoothness implies order. Index Terms—Approximation order, Besov spaces, Hölder smoothness, multiscale differentiation, splines, vanishing moments, wavelets. I.
Embedded Foveation Image Coding
 IEEE TRANSACTIONS ON IMAGE PROCESSING
, 2001
"... The human visual system (HVS) is highly spacevariant in sampling, coding, processing, and understanding. The spatial resolution of the HVS is highest around the point of fixation (foveation point) and decreases rapidly with increasing eccentricity. By taking advantage of this fact, it is possible t ..."
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Cited by 42 (13 self)
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The human visual system (HVS) is highly spacevariant in sampling, coding, processing, and understanding. The spatial resolution of the HVS is highest around the point of fixation (foveation point) and decreases rapidly with increasing eccentricity. By taking advantage of this fact, it is possible to remove considerable highfrequency information redundancy from the peripheral regions and still reconstruct a perceptually good quality image. Great success has been obtained recently by a class of embedded wavelet image coding algorithms, such as the embedded zerotree wavelet (EZW) and the set partitioning in hierarchical trees (SPIHT) algorithms. Embedded wavelet coding not only provides very good compression performance, but also has the property that the bitstream can be truncated at any point and still be decoded to recreate a reasonably good quality image. In this
Wavelet Radiance
 In Fifth Eurographics Workshop on Rendering
, 1994
"... In this paper, we show how wavelet analysis can be used to provide an efficient solution method for global illumination with glossy and diffuse reflections. Wavelets are used to sparsely represent radiance distribution functions and the transport operator. In contrast to previous wavelet methods (fo ..."
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Cited by 36 (3 self)
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In this paper, we show how wavelet analysis can be used to provide an efficient solution method for global illumination with glossy and diffuse reflections. Wavelets are used to sparsely represent radiance distribution functions and the transport operator. In contrast to previous wavelet methods (for radiosity), our algorithm transports light directly among wavelets, and eliminates the pushing and pulling procedures. The framework we describe supports curved surfaces and spatiallyvarying anisotropic BRDFs. We use importance to make the global illumination problem tractable for complex scenes, and we use a final gathering step to improve the visual quality of the solution. 1 Introduction Radiosity algorithms assume that all reflection is ideally diffuse. This assumption, while making the computation of global illumination more tractable, ignores many important effects, such as glossy highlights and mirror reflections. Though more expensive, the simulation of directional reflection is e...
Pairs of Dual Wavelet Frames From Any Two Refinable Functions
, 2000
"... Starting from any two compactly supported refinable functions in L 2 (R) with dilation factor d, we show that it is always possible to construct 2d wavelet functions with compact support such that they generate a pair of dual dwavelet frames in L 2 (R). Moreover, the number of vanishing moments of ..."
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Cited by 29 (18 self)
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Starting from any two compactly supported refinable functions in L 2 (R) with dilation factor d, we show that it is always possible to construct 2d wavelet functions with compact support such that they generate a pair of dual dwavelet frames in L 2 (R). Moreover, the number of vanishing moments of each of these wavelet frames is equal to the approximation order of the dual MRA; this is the highest possible. In particular, when we consider symmetric refinable functions, the constructed dual wavelets are also symmetric or antisymmetric. As a consequence, for any compactly supported refinable function in L 2 (R), it is possible to construct explicitly and easily wavelets that are finite linear combinations of translates (d k), and that generate a wavelet frame with arbitrarily preassigned number of vanishing moments. We illustrate the general theory by examples of such pairs of dual wavelet frames derived from Bspline functions.
Approximation Properties and Construction of Hermite Interpolants and Biorthogonal Multiwavelets
, 2001
"... ..."
Biorthogonal partners and applications
 IEEE Trans. Signal Processing
, 2001
"... Abstract—Two digital filters ( ) and ( ) are said to be biorthogonal partners of each other if their cascade ( ) () satisfies the Nyquist or zerocrossing property. Biorthogonal partners arise in many different contexts such as filterbank theory, exact and least squares digital interpolation, and m ..."
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Cited by 27 (19 self)
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Abstract—Two digital filters ( ) and ( ) are said to be biorthogonal partners of each other if their cascade ( ) () satisfies the Nyquist or zerocrossing property. Biorthogonal partners arise in many different contexts such as filterbank theory, exact and least squares digital interpolation, and multiresolution theory. They also play a central role in the theory of equalization, especially, fractionally spaced equalizers in digital communications. In this paper, we first develop several theoretical properties of biorthogonal partners. We also develop conditions for the existence of biorthogonal partners and FIR biorthogonal pairs and establish the connections to the Riesz basis property. We then explain how these results play a role in many of the abovementioned applications. I.