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87
Quantized Frame Expansions with Erasures
 Applied and Computational Harmonic Analysis
, 2001
"... This paper places frames in a new setting, where some of the elements are deleted. Since proper subsets of fi'ames are sometimes them selves frames, a quantized frame expansion can be a useful representation even when some transform coefficients are lost in transmission. This yields robustness ..."
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Cited by 137 (18 self)
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This paper places frames in a new setting, where some of the elements are deleted. Since proper subsets of fi'ames are sometimes them selves frames, a quantized frame expansion can be a useful representation even when some transform coefficients are lost in transmission. This yields robustness to losses in packet networks such as the Internet
Bivariate Shrinkage Functions for WaveletBased Denoising Exploiting Interscale Dependency
, 2002
"... Most simple nonlinear thresholding rules for waveletbased denoising assume that the wavelet coefficients are independent. However, wavelet coefficients of natural images have significant dependencies. In this paper, we will only consider the dependencies between the coefficients and their parents i ..."
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Cited by 135 (4 self)
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Most simple nonlinear thresholding rules for waveletbased denoising assume that the wavelet coefficients are independent. However, wavelet coefficients of natural images have significant dependencies. In this paper, we will only consider the dependencies between the coefficients and their parents in detail. For this purpose, new nonGaussian bivariate distributions are proposed, and corresponding nonlinear threshold functions (shrinkage functions) are derived from the models using Bayesian estimation theory. The new shrinkage functions do not assume the independence of wavelet coefficients. We will show three image denoising examples in order to show the performance of these new bivariate shrinkage rules. In the second example, a simple subbanddependent datadriven image denoising system is described and compared with effective datadriven techniques in the literature, namely VisuShrink, SureShrink, BayesShrink, and hidden Markov models. In the third example, the same idea is applied to the dualtree complex wavelet coefficients.
Multiresolution markov models for signal and image processing
 Proceedings of the IEEE
, 2002
"... This paper reviews a significant component of the rich field of statistical multiresolution (MR) modeling and processing. These MR methods have found application and permeated the literature of a widely scattered set of disciplines, and one of our principal objectives is to present a single, coheren ..."
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Cited by 122 (18 self)
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This paper reviews a significant component of the rich field of statistical multiresolution (MR) modeling and processing. These MR methods have found application and permeated the literature of a widely scattered set of disciplines, and one of our principal objectives is to present a single, coherent picture of this framework. A second goal is to describe how this topic fits into the even larger field of MR methods and concepts–in particular making ties to topics such as wavelets and multigrid methods. A third is to provide several alternate viewpoints for this body of work, as the methods and concepts we describe intersect with a number of other fields. The principle focus of our presentation is the class of MR Markov processes defined on pyramidally organized trees. The attractiveness of these models stems from both the very efficient algorithms they admit and their expressive power and broad applicability. We show how a variety of methods and models relate to this framework including models for selfsimilar and 1/f processes. We also illustrate how these methods have been used in practice. We discuss the construction of MR models on trees and show how questions that arise in this context make contact with wavelets, state space modeling of time series, system and parameter identification, and hidden
Platelets: A Multiscale Approach for Recovering Edges and Surfaces in PhotonLimited Medical Imaging
 IEEE TRANSACTIONS ON MEDICAL IMAGING
, 2003
"... The nonparametric multiscale platelet algorithms presented in this paper, unlike traditional waveletbased methods, are both well suited to photonlimited medical imaging applications involving Poisson data and capable of better approximating edge contours. This paper introduces platelets, localized ..."
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Cited by 77 (19 self)
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The nonparametric multiscale platelet algorithms presented in this paper, unlike traditional waveletbased methods, are both well suited to photonlimited medical imaging applications involving Poisson data and capable of better approximating edge contours. This paper introduces platelets, localized functions at various scales, locations, and orientations that produce piecewise linear image approximations, and a new multiscale image decomposition based on these functions. Platelets are well suited for approximating images consisting of smooth regions separated by smooth boundaries. For smoothness measured in certain H older classes, it is shown that the error of mterm platelet approximations can decay significantly faster than that of mterm approximations in terms of sinusoids, wavelets, or wedgelets. This suggests that platelets may outperform existing techniques for image denoising and reconstruction. Fast, plateletbased, maximum penalized likelihood methods for photonlimited image denoising, deblurring and tomographic reconstruction problems are developed. Because platelet decompositions of Poisson distributed images are tractable and computationally efficient, existing image reconstruction methods based on expectationmaximization type algorithms can be easily enhanced with platelet techniques. Experimental results suggest that plateletbased methods can outperform standard reconstruction methods currently in use in confocal microscopy, image restoration, and emission tomography.
Bivariate Shrinkage with Local Variance Estimation
, 2002
"... The performance of imagedenoising algorithms using wavelet transforms can be improved significantly by taking into account the statistical dependencies among wavelet coefficients as demonstrated by several algorithms presented in the literature. In two earlier papers by the authors, a simple bivari ..."
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Cited by 74 (5 self)
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The performance of imagedenoising algorithms using wavelet transforms can be improved significantly by taking into account the statistical dependencies among wavelet coefficients as demonstrated by several algorithms presented in the literature. In two earlier papers by the authors, a simple bivariate shrinkage rule is described using a coefficient and its parent. The performance can also be improved using simple models by estimating model parameters in a local neighborhood. This letter presents a locally adaptive denoising algorithm using the bivariate shrinkage function. The algorithm is illustrated using both the orthogonal and dual tree complex wavelet transforms. Some comparisons with the best available results will be given in order to illustrate the effectiveness of the proposed algorithm.
Video denoising using 2D and 3D dualtree complex wavelet transforms
 Wavelet Appl Signal Image Proc. X (Proc. SPIE 5207
, 2003
"... The denoising of video data should take into account both temporal and spatial dimensions, however, true 3D transforms are rarely used for video denoising. Separable 3D transforms have artifacts that degrade their performance in applications. This paper describes the design and application of the n ..."
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Cited by 51 (5 self)
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The denoising of video data should take into account both temporal and spatial dimensions, however, true 3D transforms are rarely used for video denoising. Separable 3D transforms have artifacts that degrade their performance in applications. This paper describes the design and application of the nonseparable oriented 3D dualtree wavelet transform for video denoising. This transform gives a motionbased multiscale decomposition for video — it isolates in its subbands motion along different directions. In addition, we investigate the denoising of video using the 2D and 3D dualtree oriented wavelet transforms, where the 2D transform is applied to each frame individually.
Hilbert Transform Pairs of Wavelet Bases
, 2001
"... This paper considers the design of pairs of wavelet bases where the wavelets form a Hilbert transform pair. The derivation is based on the limit functions defined by the infinite product formula. It is found that the scaling filters should be offset from one another by a half sample. This gives an a ..."
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Cited by 51 (6 self)
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This paper considers the design of pairs of wavelet bases where the wavelets form a Hilbert transform pair. The derivation is based on the limit functions defined by the infinite product formula. It is found that the scaling filters should be offset from one another by a half sample. This gives an alternative derivation and explanation for the result by Kingsbury, that the dualtree DWT is (nearly) shiftinvariant when the scaling filters satisfy the same offset.
Wave atoms and sparsity of oscillatory patterns
 Appl. Comput. Harmon. Anal
, 2006
"... We introduce “wave atoms ” as a variant of 2D wavelet packets obeying the parabolic scaling wavelength ∼ (diameter) 2. We prove that warped oscillatory functions, a toy model for texture, have a significantly sparser expansion in wave atoms than in other fixed standard representations like wavelets, ..."
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Cited by 45 (5 self)
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We introduce “wave atoms ” as a variant of 2D wavelet packets obeying the parabolic scaling wavelength ∼ (diameter) 2. We prove that warped oscillatory functions, a toy model for texture, have a significantly sparser expansion in wave atoms than in other fixed standard representations like wavelets, Gabor atoms, or curvelets. We propose a novel algorithm for a tight frame of wave atoms with redundancy two, directly in the frequency plane, by the “wrapping ” technique. We also propose variants of the basic transform for applications in image processing, including an orthonormal basis, and a shiftinvariant tight frame with redundancy four. Sparsity and denoising experiments on both seismic and fingerprint images demonstrate the potential of the tool introduced.
A Hidden Markov Tree Model for the Complex Wavelet Transform
 IEEE Transactions on Signal Processing
, 2001
"... Multiresolution models such as the hidden Markov tree (HMT) aim to capture the statistical structure of signals and images by leveraging two key wavelet transform properties: wavelet coefficients representing smooth/singular regions in a signal have small/large magnitude, and small/large magnitudes ..."
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Cited by 31 (7 self)
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Multiresolution models such as the hidden Markov tree (HMT) aim to capture the statistical structure of signals and images by leveraging two key wavelet transform properties: wavelet coefficients representing smooth/singular regions in a signal have small/large magnitude, and small/large magnitudes persist through scale. Unfortunately, the HMT based on the conventional (fully decimated) wavelet transform suffers from shiftvariance, making it less accurate and realistic. In this paper, we extend the HMT modeling framework to the complex wavelet transform, which features near shiftinvariance and improved directionality compared to the standard wavelet transform. The complex HMT model is computationally efficient (with lineartime computation and processing algorithms) and applicable to general Bayesian inference problems as a prior density for images. We demonstrate the effectiveness of the model with two applications.
Improving appearance model matching using local image structure
 In Information Processing in Medical Imaging, 18 th International Conference
, 2003
"... Abstract. We show how nonlinear representations of local image structure can be used to improve the performance of model matching algorithms in medical image analysis tasks. Rather than represent the image structure using intensity values or gradients, we use measures that indicate the reliability ..."
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Cited by 23 (6 self)
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Abstract. We show how nonlinear representations of local image structure can be used to improve the performance of model matching algorithms in medical image analysis tasks. Rather than represent the image structure using intensity values or gradients, we use measures that indicate the reliability of a set of local image feature detector outputs. These features are image edges, corners, and gradients. Feature detector outputs in flat, noisy regions tend to be ignored whereas those near strong structure are favoured. We demonstrate that combinations of these features give more accurate and reliable matching between models and new images than modelling image intensity alone. We also show that the approach is robust to nonlinear changes in contrast, such as those found in multimodal imaging. 1