Spatially Adaptive Research Articles

Bayesian symmetrical EEG/fMRI fusion with spatially adaptive priors

In this paper, we propose a novel symmetrical EEG/fMRI fusion method which combines EEG and fMRI by means of a common generative model. We use a total variation (TV) prior to model the spatial distribution of the cortical current responses and hemodynamic response functions, and utilize spatially adaptive temporal priors to model their temporal shapes. The spatial adaptivity of the prior model allows for adaptation to the local characteristics of the estimated responses and leads to high estimation performance for the cortical current distribution and the hemodynamic response functions. We utilize a Bayesian formulation with a variational Bayesian framework and obtain a fully automatic fusion algorithm. Simulations with synthetic data and experiments with real data from a multimodal study on face perception demonstrate the performance of the proposed method.

A Patch-Based Structural Masking Model with an Application to Compression

The ability of an image region to hide or mask a given target signal continues to play a key role in the design of numerous image processing and vision systems. However, current state-of-the-art models of visual masking have been optimized for artificial targets placed upon unnatural backgrounds. In this paper, we (1) measure the ability of natural-image patches in masking distortion; (2) analyze the performance of a widely accepted standard masking model in predicting these data; and (3) report optimal model parameters for different patch types (textures, structures, and edges). Our results reveal that the standard model of masking does not generalize across image type; rather, a proper model should be coupled with a classification scheme which can adapt the model parameters based on the type of content contained in local image patches. The utility of this adaptive approach is demonstrated via a spatially adaptive compression algorithm which employs patch-based classification. Despite the addition of extra side information and the high degree of spatial adaptivity, this approach yields an efficient wavelet compression strategy that can be combined with very accurate rate-control procedures.

A high resolution spatially adaptive vortex method for separating flows. Part I: Two-dimensional domains

A grid-free high-resolution spatially-adaptive vortex method for two-dimensional incompressible flow in bounded domains is presented. The computational algorithm is based on operator splitting in which convection and diffusion are handled separately every time step. In the convection step, computational elements are convected with velocities obtained by fast approximations of the Biot–Savart superposition with second-order Runge–Kutta time integration scheme. Diffusion is performed using the smooth redistribution method that employs a Gaussian basis function for vorticity in the interior. Near solid walls, the core functions are modified to conserve circulation. The no-slip boundary condition is enforced by creating of a vortex sheet that is redistributed to neighboring elements using the redistribution method. The proposed method enables accurate and smooth recovery of the vorticity and does not require explicit use of vortex images or occasional re-meshing. Algorithms for reduction in computational cost by accurately removing elements in overcrowded regions and for spatial adaptivity that allows for variable core sizes and variable element spacing are presented. Computations of flow around an impulsively started cylinder for Reynolds number values of 1000, 3000, and 9500 are preformed to investigate various aspects of the proposed method.

A Predictor‐Corrector Algorithm for Reaction‐Diffusion Equations Associated with Neural Activity on Branched Structures

In this paper, we present a numerical method for solving reaction‐diffusion equations on one‐dimensional branched structures. The method effectively decouples the many branches at the nodal junctions so that the equations can be solved as a system of smaller problems that are purely tridiagonal. This strategy enables a number of improvements. In particular, difficulties associated with implicit methods for closed loops in the network are eliminated, and spatial adaptivity can be easily implemented. We apply this method to the electrical activity of a neuron. A neuron can be effectively represented as a one‐dimensional branched structure, and the model equations, based on the Hodgkin–Huxley cable equations, are a set of reaction equations coupled to a single diffusion process. Spatially adaptive methods are particularly effective for neuronal simulations as the region of activity is often very localized in space. Since the algorithm scales with activity instead of network size, it will make a substantial reduction in computational cost over existing methods.

Homomorphic wavelet thresholding technique for denoising medical ultrasound images

A novel homomorphic wavelet thresholding technique for reducing speckle noise in medical ultrasound images is presented. First, we show that the speckle wavelet coefficients in the logarithmically transformed ultrasound images are best described by the Nakagami family of distributions. By exploiting this speckle model and the Laplacian signal prior, a closed form, data-driven, and spatially adaptive threshold is derived in the Bayesian framework. The spatial adaptivity allows the additional information of the image (such as identification of homogeneous or heterogeneous regions) to be incorporated into the algorithm. Further, the threshold has been extended to the redundant wavelet representation, which yields better results than the decimated wavelet transform. Experimental results demonstrate the improved performance of the proposed method over other well-known speckle reduction filters. The application of the proposed method to a realistic US test image shows that the new technique, named HomoGenThresh, outperforms the best wavelet-based denoising method reported in [1] by more than 1.6 dB, Lee filter by 3.6 dB, Kaun filter by 3.1 dB and band-adaptive soft thresholding [2] by 2.1 dB at an input signal-to-noise ratio (SNR) of 13.6 dB.

Multiresolution local polynomial regression: A new approach to pointwise spatial adaptation

In nonparametric local polynomial regression the adaptive selection of the scale parameter (window size/bandwidth) is a key problem. Recently new efficient algorithms, based on Lepski's approach, have been proposed in mathematical statistics for spatially adaptive varying scale denoising. A common feature of these algorithms is that they form test-estimates [email protected]?h different by the scale [email protected]?H and special statistical rules are exploited in order to select the estimate with the best pointwise varying scale. In this paper a novel multiresolution (MR) local polynomial regression is proposed. Instead of selection of the estimate with the best scale h a nonlinear estimate is built using all of the test-estimates [email protected]?h. The adaptive estimation consists of two steps. The first step transforms the data into noisy spectrum coefficients (MR analysis). On the second step, this noisy spectrum is filtered by the thresholding procedure and used for estimation (MR synthesis).

Estimation of Global Temperature Fields from Scattered Observations by a Spherical-Wavelet-Based Spatially Adaptive Method

SummaryThe paper considers the problem of estimating the entire temperature field for every location on the globe from scattered surface air temperatures observed by a network of weather-stations. Classical methods such as spherical harmonics and spherical smoothing splines are not efficient in representing data that have inherent multiscale structures. The paper presents an estimation method that can adapt to the multiscale characteristics of the data. The method is based on a spherical wavelet approach that has recently been developed for a multiscale representation and analysis of scattered data. Spatially adaptive estimators are obtained by coupling the spherical wavelets with different thresholding (selective reconstruction) techniques. These estimators are compared for their spatial adaptability and extrapolation performance by using the surface air temperature data.

A tree-structured Markov random field model for bayesian image segmentation

We present a new image segmentation algorithm based on a tree-structured binary MRF model. The image is recursively segmented in smaller and smaller regions until a stopping condition, local to each region, is met. Each elementary binary segmentation is obtained as the solution of a MAP estimation problem, with the region prior modeled as an MRF. Since only binary fields are used, and thanks to the tree structure, the algorithm is quite fast, and allows one to address the cluster validation problem in a seamless way. In addition, all field parameters are estimated locally, allowing for some spatial adaptivity. To improve segmentation accuracy, a split-and-merge procedure is also developed and a spatially adaptive MRF model is used. Numerical experiments on multispectral images show that the proposed algorithm is much faster than a similar reference algorithm based on "flat" MRF models, and its performance, in terms of segmentation accuracy and map smoothness, is comparable or even superior.

Wavelet Shrinkage: Asymptopia?

SUMMARY Much recent effort has sought asymptotically minimax methods for recovering infinite dimensional objects—curves, densities, spectral densities, images—from noisy data. A now rich and complex body of work develops nearly or exactly minimax estimators for an array of interesting problems. Unfortunately, the results have rarely moved into practice, for a variety of reasons—among them being similarity to known methods, computational intractability and lack of spatial adaptivity. We discuss a method for curve estimation based on n noisy data: translate the empirical wavelet coefficients towards the origin by an amount √(2 log n)σ/√n. The proposal differs from those in current use, is computationally practical and is spatially adaptive; it thus avoids several of the previous objections. Further, the method is nearly minimax both for a wide variety of loss functions—pointwise error, global error measured in Lp-norms, pointwise and global error in estimation of derivatives—and for a wide range of smoothness classes, including standard Holder and Sobolev classes, and bounded variation. This is a much broader near optimality than anything previously proposed: we draw loose parallels with near optimality in robustness and also with the broad near eigenfunction properties of wavelets themselves. Finally, the theory underlying the method is interesting, as it exploits a correspondence between statistical questions and questions of optimal recovery and information-based complexity.

Ideal spatial adaptation by wavelet shrinkage

SUMMARY With ideal spatial adaptation, an oracle furnishes information about how best to adapt a spatially variable estimator, whether piecewise constant, piecewise polynomial, variable knot spline, or variable bandwidth kernel, to the unknown function. Estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels; however, it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone. We describe a new principle for spatially-adaptive estimation: selective wavelet reconstruction. We show that variable-knot spline fits and piecewise-polynomial fits, when equipped with an oracle to select the knots, are not dramatically more powerful than selective wavelet reconstruction with an oracle. We develop a practical spatially adaptive method, RiskShrink, which works by shrinkage of empirical wavelet coefficients. RiskShrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so. A new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n, where n is the sample size. Moreover no estimator can give a better guarantee than this. Within the class of spatially adaptive procedures, RiskShrink is essentially optimal. Relying only on the data, it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle. In contrast, it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone.