R-3 Repository :: Browsing by Author "Nowak, Robert David"

Browsing by Author "Nowak, Robert David"

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Adaptive Wavelet Transforms via Lifting
(1998) Claypoole, Roger L.; Baraniuk, Richard G.; Nowak, Robert David; Digital Signal Processing (http://dsp.rice.edu/)
This paper develops new algorithms for adapted multiscale analysis and signal adaptive wavelet transforms. We construct our adaptive transforms with the lifting scheme, which decomposes the wavelet transform into prediction and update stages. We adapt the prediction stage to the signal structure and design the update stage to preserve the desirable properties of the wavelet transform. We incorporate this adaptivity into the redundant and non-redundant transforms; the resulting transforms are scale and spatially adaptive. We study applications to signal estimation; our new transforms show improved denoising performance over existing (non-adaptive) orthogonal transforms.
Adaptive Wavelet Transforms via Lifting
(1999-01-15) Claypoole, Roger L.; Baraniuk, Richard G.; Nowak, Robert David; Digital Signal Processing (http://dsp.rice.edu/)
This paper develops new algorithms for adapted multiscale analysis and signal adaptive wavelet transforms. We construct our adaptive transforms with the lifting scheme, which decomposes the wavelet transform into prediction and update stages. We adapt the prediction stage to the signal structure and design the update stage to preserve the desirable properties of the wavelet transform. We incorporate this adaptivity into the redundant and non-redundant transforms; the resulting transforms are scale and spatially adaptive. We study applications to signal estimation; our new transforms show improved denoising performance over existing (non-adaptive) orthogonal transforms.
Adaptive Wavelet Transforms via Lifting
(1998-05-01) Claypoole, Roger L.; Baraniuk, Richard G.; Nowak, Robert David; Digital Signal Processing (http://dsp.rice.edu/)
This paper develops two new adaptive wavelet transforms based on the lifting scheme. The lifting construction exploits a spatial-domain, prediction-error interpretation of the wavelet transform and provides a powerful framework for designing customized transforms. We use the lifting construction to adaptively tune a wavelet transform to a desired signal by optimizing data-based prediction error criteria. The performances of the new transforms are compared to existing wavelet transforms, and applications to signal denoising are investigated.
Adaptive Weighted Highpass Filters Using Multiscale Analysis
(1998-07-01) Nowak, Robert David; Baraniuk, Richard G.; Digital Signal Processing (http://dsp.rice.edu/)
In this paper, we propose a general framework for studying a class of weighted highpass filters. Our framework, based on a multiscale signal decomposition, allows us to study a wide class of filters and to assess the merits of each. We derive an automatic procedure to tune a filter to the local structure of the image under consideration. The entire algorithm is fully automatic and requires no parameter specification from the user. Several simulations demonstrate the efficacy of the method.
A Bayesian Multiscale Approach to Joint Image Restoration and Edge Detection
(1999-07-20) Wan, Yi; Nowak, Robert David; Center for Multimedia Communications (http://cmc.rice.edu/)
This paper presents a novel wavelet-based method for simultaneous image restoration and edge detection. The Bayesian framework developed here is general enough to treat a wide class of linear inverse problems involving (white or colored) Gaussian observation noises, but we focus on convolution operators. In our new approach, a signal prior is developed by modeling the signal/image wavelet coefficients as independent Gaussian mixture random variables. We specify a uniform (non-informative) distribution on the mixing parameters, which leads to an extremely simple iterative algorithm for joint MAP restoration and edge detection. This algorithm is similar to the popular EM algorithm in that it alternates between a state estimation step and a maximization step, yet it is much simpler in each step and has a very intuitive derivation. Moreover, we show that our algorithm converges monotonically to a local maximum of the posterior distribution. Experimental results show that this new method can perform better than wavelet-vaguelette type methods that are based on linear inverse filtering followed by wavelet coefficient denoising.
Coding Theoretic Approach to Image Segmentation
(2001-10-20) Ndili, Unoma; Nowak, Robert David; Figueiredo, Mario; Digital Signal Processing (http://dsp.rice.edu/)
In this paper, using a coding theoretic approach, we implement Rissanen's concept of minimum description length (MDL) for segmenting an image into piecewise homogeneous regions. Our image model is a Gaussian random field whose mean and variance functions are piecewise constant across the image. The image pixels are (conditionally) independent and Gaussian, given the mean and variance functions. The model is intended to capture variations in both intensity (mean value) and texture (variance). We adopt a multi-scale tree based approach to develop two segmentation algorithms, using MDL to penalize overly complex segmentations. One algorithm is based on an adaptive (greedy) rectangular partitioning scheme. The second algorithm is an optimally-pruned wedgelet decorated dyadic partitioning. We compare the two schemes with an alternative constant variance dyadic CART (classification and regression tree) scheme which accounts only for variations in mean, and demonstrate their performance with SAR image segmentation problems.
CORT: Classification Or Regression Trees
(2003-06-20) Scott, Clayton; Willett, Rebecca; Nowak, Robert David; Digital Signal Processing (http://dsp.rice.edu/)
In this paper we challenge three of the underlying principles of CART, a well know approach to the construction of classification and regression trees. Our primary concern is with the penalization strategy employed to prune back an initial, overgrown tree. We reason, based on both intuitive and theoretical arguments, that the pruning rule for classification should be different from that used for regression (unlike CART). We also argue that growing a treestructured partition that is specifically fitted to the data is unnecessary. Instead, our approach to tree modeling begins with a nonadapted (fixed) dyadic tree structure and partition, much like that underlying multiscale wavelet analysis. We show that dyadic trees provide sufficient flexibility, are easy to construct, and produce near-optimal results when properly pruned. Finally, we advocate the use of a negative log-likelihood measure of empirical risk. This is a more appropriate empirical risk for non-Gaussian regression problems, in contrast to the sum-of-squared errors criterion used in CART regression.
CORT: Classification Or Regression Trees
(2003-04-20) Scott, Clayton; Willett, Rebecca; Nowak, Robert David; Digital Signal Processing (http://dsp.rice.edu/)
In this paper we challenge three of the underlying principles of CART, a well know approach to the construction of classification and regression trees. Our primary concern is with the penalization strategy employed to prune back an initial, overgrown tree. We reason, based on both intuitive and theoretical arguments, that the pruning rule for classification should be different from that used for regression (unlike CART). We also argue that growing a treestructured partition that is specifically fitted to the data is unnecessary. Instead, our approach to tree modeling begins with a nonadapted (fixed) dyadic tree structure and partition, much like that underlying multiscale wavelet analysis. We show that dyadic trees provide sufficient flexibility, are easy to construct, and produce near-optimal results when properly pruned. Finally, we advocate the use of a negative log-likelihood measure of empirical risk. This is a more appropriate empirical risk for non-Gaussian regression problems, in contrast to the sum-of-squared errors criterion used in CART regression.
Distributed Image Compression for Sensor Networks using Correspondence Analysis and Super-Resolution
(2003-09-01) Wagner, Raymond; Nowak, Robert David; Baraniuk, Richard G.; Digital Signal Processing (http://dsp.rice.edu/)
We outline a distributed coding technique for images captured from sensors with overlapping fields of view in a sensor network. First, images from correlated views are roughly registered (relative to a sensor of primary interest) via a low-bandwidth data-sharing method involving image feature points and feature point correspondence. An area of overlap is then identified, and each sensor transmits a low-resolution version of the common image block to the receiver, amortizing the coding cost for that block among the set of sensors. Super-resolution techniques are finally employed at the receiver to reconstruct a high-resolution version of the common block. We discuss the registration and super-resolution techniques used and present examples of each step in the proposed coding process. A numerical analysis illustrating the potential coding benefit follows, and we conclude with a brief discussion of the key issues remaining to be resolved on the path to coder robustness.
Efficient Methods for Identification of Volterra Filters
(1994) Nowak, Robert David; Van Veen, Barry D.; Digital Signal Processing (http://dsp.rice.edu/)
A major drawback of the truncated Volterra series or "Volterra filter" for system identification is the large number of parameters required by the standard filter structure. The corresponding estimation problem requires the solution of a large system of simultaneous linear equations. Two methods for simplifying the estimation problem are discussed in this paper. First, a Kronecker product structure for the Volterra filter is reviewed. In this approach the inverse of the large correlation matrix is expressed as a Kronecker product of small matrices. Second, a parallel decomposition of the Volterra filter based on uncorrelated, symmetric inputs is introduced. Here the Volterra filter is decomposed into a parallel combination of smaller orthogonal "sub-filters." It is shown that each sub-filter is much smaller than the full Volterra filter and hence the parallel decomposition offers many advantages for estimating the Volterra kernels. Simulations illustrate application of the parallel structure with random and pseudorandom excitations. Input conditions that guarantee the existence of a unique estimate are also reviewed.
Generalized likelihood ratio detection for fMRI using complex data
(1999-04-20) Nan, F.Y.; Nowak, Robert David; Digital Signal Processing (http://dsp.rice.edu/)
The majority of fMRI studies obtain functional information using statistical tests based on the magnitude image reconstructions. Recently, a complex correlation (CC) test was proposed based on teh complex image data in order to take advantage of phase information in teh signal. However, the CC test ignores additional phase information in the baseline component of the data. In this paper, a new detector for fMRI based on a Generalized Likelihood Ration Test (GLRT) is proposed. The GLRT exploits the fact that the fMRI response signal as well as the baseline component of the data share a common phase. Theoretical analysis and Monte Carlo simulation are used to explore the performance of the new detector. At relatively low signal intensities, the GLRT outperforms both the standard magnitude data test and the CC test. At high signal intensities, the GLRT performs as well as teh standard magnitude data test and significantly better than the CC test.
Hidden Markov Models for Wavelet-based Signal Processing
(1996-11-01) Crouse, Matthew; Baraniuk, Richard G.; Nowak, Robert David; Digital Signal Processing (http://dsp.rice.edu/)
Current wavelet-based statistical signal and image processing techniques such as shrinkage and filtering treat the wavelet coefficients as though they were statistically independent. This assumption is unrealistic; considering the statistical dependencies between wavelet coefficients can yield substantial performance improvements. In this paper we develop a new framework for wavelet-based signal processing that employs hidden Markov models to characterize the dependencies between wavelet coefficients. To illustrate the power of the new framework, we derive a new signal denoising algorithm that outperforms current scalar shrinkage techniques.
Hierarchical Wavelet-Based Image Model for Pattern Analysis and Synthesis
(2000-07-20) Scott, Clayton; Nowak, Robert David; Center for Multimedia Communications (http://cmc.rice.edu/); Digital Signal Processing (http://dsp.rice.edu/)
Despite their success in other areas of statistical signal processing, current wavelet-based image models are inadequate for modeling patterns in images, due to the presence of unknown transformations (e.g., translation, rotation, scaling) inherent in most pattern observations. In this paper we introduce a hierarchical wavelet-based framework for modeling patterns in digital images. This framework takes advantage of the efficient image representations afforded by wavelets, while accounting for unknown pattern transformations. Given a trained model, we can use this framework to synthesize pattern observations. If the model parameters are unknown, we can infer them from labeled training data using TEMPLAR (Template Learning from Atomic Representations), a novel template learning algorithm with linear complexity. TEMPLAR employs minimum description length (MDL) complexity regularization to learn a template with a sparse representation in the wavelet domain. We illustrate template learning with examples, and discuss how TEMPLAR applies to pattern classification and denoising from multiple, unaligned observations.
Image Restoration Using the EM Algorithm and Wavelet-Based Complexity Regularization
(2002-05-20) Figueiredo, Mario; Nowak, Robert David; Digital Signal Processing (http://dsp.rice.edu/)
This paper introduces an expectation-maximization (EM) algorithm for image restoration (deconvolution) based on a penalized likelihood formulated in the wavelet domain. Regularization is achieved by promoting a reconstruction with low-complexity, expressed in terms of teh wavelet coefficients, taking advantage of the well known sparsity of wavelet representations. Previous works have investigated wavelet-based restoration but, except for certain special cases, teh resulting criteria are solved approximately or requre very demanding optimization methods. The EM algorithm herein proposed combines the efficient image representation offered by the discrete wavelet transform (DWT) with the diagonalization of the convolution operator obtained in teh Fourier domain. The algorithm alternates between an E-step based on teh fast Fourier transform (FFT) and a DWT-based M-step, resulting in an efficient iterative process requiring O(NlogN) operations per iteration. Thus, it is the first image restoration algorithm that optimizes a wavelet-based penalized likelihood criterion and has computational complexity comparable to that of standard wavelet denoising or frequency domain deconvolution methods. The convergence behavior of the algorithm is investigated, and it is shown that under mild conditions the algorithm converges to a globally optimal restoration. Morever, our new approach outperforms several of the best existing methods in benchmark tests, and in some cases is also much less computationally demanding.
Lifting Construction of Non-Linear Wavelet Transforms
(1998-10-01) Claypoole, Roger L.; Baraniuk, Richard G.; Nowak, Robert David; Digital Signal Processing (http://dsp.rice.edu/)
This paper analyzes non-linear wavelet transforms using the lifting construction. The lifting construction exploits a spatial-domain, prediction-error interpretation of the wavelet transform and provides a powerful framework for designing customized transforms. We use the lifting construction to better understand the performance of wavelet transforms that utilize median and Volterra filters.
Low Rank Estimation of Higher Order Statistics
(1995-12-01) Nowak, Robert David; Van Veen, Barry D.; Digital Signal Processing (http://dsp.rice.edu/)
Low rank estimators for higher order statistics are considered in this paper. Rank reduction methods offer a general principle for trading estimator bias for reduced estimator variance. The bias-variance tradeoff is analyzed for low rank estimators of higher order statistics using a tensor product formulation for the moments and cumulants. In general the low rank estimators have a larger bias and smaller variance than the corresponding full rank estimator. Often a tremendous reduction in variance is obtained in exchange for a slight increase in bias. This makes the low rank estimators extremely useful for signal processing algorithms based on sample estimates of the higher order statistics. The low rank estimators also offer considerable reductions in the computational complexity of such algorithms. The design of subspaces to optimize the tradeoffs between bias, variance, and computation is discussed and a noisy input, noisy output system identification problem is used to illustrate the results.
Model-based Inverse Halftoning with Wavelet-Vaguelette Deconvolution
(2000-09-01) Neelamani, Ramesh; Nowak, Robert David; Baraniuk, Richard G.; Center for Multimedia Communications (http://cmc.rice.edu/); Digital Signal Processing (http://dsp.rice.edu/)
In this paper, we demonstrate based on the linear model of Kite that inverse halftoning is equivalent to the well-studied problem of deconvolution in the presence of colored noise. We propose the use of the simple and elegant wavelet-vaguelette deconvolution (WVD) algorithm to perform the inverse halftoning. Unlike previous wavelet-based algorithms, our method is model-based; hence it is adapted to different error diffusion halftoning techniques. Our inverse halftoning algorithm consists of inverting the convolution operator followed by denoising in the wavelet domain. For signals in a Besov space, our algorithm possesses asymptotically (as the number of samples nears infinity near-optimal rates of error decay. Hence for images in a Besov space, it is impossible to improve significantly on the inverse halftoning performance of the WVD algorithm at high resolutions. Using simulations, we verify that our algorithm outperforms or matches the performances of the best published inverse halftoning techniques in the mean square error (MSE) sense and also provides excellent visual performance.
Multiresolution Nonparametric Intensity and Density Estimation
(2002-05-20) Willett, Rebecca; Nowak, Robert David; Digital Signal Processing (http://dsp.rice.edu/)
This paper introduces a new multiscale method for nonparametric piecewise polynomial intensity and density estimation of point processes. Fast, piecewise polynomial, maximum penalized likelihood methods for intensity and density estimation are developed. The recursive partitioning scheme underlying these methods is based on multiscale likelihood factorizations which, unlike conventional wavelet decompositions, are very well suited to applications with point process data. Experimental results demonstrate that multiscale methods can outperform wavelet and kernel based density estimation methods.
A Multiscale Bayesian Framework for Linear Inverse Problems and Its Application to Image Restoration
(2001-01-20) Wan, Yi; Nowak, Robert David; Digital Signal Processing (http://dsp.rice.edu/)
In this paper we develop a wavelet-based statistical method for solving linear inverse problems. The Bayesian framework developed here is general enough to treat a wide class of linear inverse problems involving (white or colored) Gaussian observation noise. In this approach, a signal prior is developed by modeling the signal/imgage wavelet coefficients as independent Gaussian mixture random variabls. We first specify a uniform (non-informative) distribution on the mixing parameters, which leads to a simple and efficient iterative algorithm for MAP estimation. This algorithm is similar to the EM algorithm in that it alternates between a state estimation step and a maximization step. Moreover, we show that our algorithm converges monotonically to a local maximum of the posterior distribution. We next generalize the result to non-uniform priors and develop an efficient integer programming algorithm that enables a similar alternating optimization procedure. Experimental reults show that this new method outperforms recent results, including multiscale Kalman filtering and wavelet-vaguelette type methods based on linear inverse filtering followed by wavelet coefficient denoising.
Multiscale Density Estimation
(2003-08-20) Willett, Rebecca; Nowak, Robert David; Digital Signal Processing (http://dsp.rice.edu/)
The nonparametric density estimation method proposed in this paper is computationally fast, capable of detecting density discontinuities and singularities at a very high resolution, spatially adaptive, and offers near minimax convergence rates for broad classes of densities including Besov spaces. At the heart of this new method lie multiscale signal decompositions based on piecewise-polynomial functions and penalized likelihood estimation. Upper bounds on the estimation error are derived using an information-theoretic risk bound based on squared Hellinger loss. The method and theory share many of the desirable features associated with wavelet-based density estimators, but also offers several advantages including guaranteed non-negativity, bounds on the L1 error, small-sample quantification of the estimation errors, and additional flexibility and adaptability. In particular, the method proposed here can adapt the degrees as well as the locations of the polynomial pieces. For a certain class of densities, the error of the variable degree estimator converges at nearly the parametric rate. Experimental results demonstrate the advantages of the new approach compared to traditional density estimators and wavelet-based estimators.

Browsing by Author "Nowak, Robert David"

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