Spatial Smoothness Constraint Research Articles

A Spatial Regularization of LDA for Face Recognition

This paper proposes a new spatial regularization of Fisher linear discriminant analysis (LDA) to reduce the overfitting due to small size sample (SSS) problem in face recognition. Many regularized LDAs have been proposed to alleviate the overfitting by regularizing an estimate of the within-class scatter matrix. Spatial regularization methods have been suggested that make the discriminant vectors spatially smooth, leading to mitigation of the overfitting. As a generalized version of the spatially regularized LDA, the proposed regularized LDA utilizes the non-uniformity of spatial correlation structures in face images in adding a spatial smoothness constraint into an LDA framework. The region-dependent spatial regularization is advantageous for capturing the non-flat spatial correlation structure within face image as well as obtaining a spatially smooth projection of LDA. Experimental results on public face databases such as ORL and CMU PIE show that the proposed regularized LDA performs well especially when the number of training images per individual is quite small, compared with other regularized LDAs.

A Bayesian Framework for Image Segmentation With Spatially Varying Mixtures

A new Bayesian model is proposed for image segmentation based upon Gaussian mixture models (GMM) with spatial smoothness constraints. This model exploits the Dirichlet compound multinomial (DCM) probability density to model the mixing proportions (i.e., the probabilities of class labels) and a Gauss-Markov random field (MRF) on the Dirichlet parameters to impose smoothness. The main advantages of this model are two. First, it explicitly models the mixing proportions as probability vectors and simultaneously imposes spatial smoothness. Second, it results in closed form parameter updates using a maximum a posteriori (MAP) expectation-maximization (EM) algorithm. Previous efforts on this problem used models that did not model the mixing proportions explicitly as probability vectors or could not be solved exactly requiring either time consuming Markov Chain Monte Carlo (MCMC) or inexact variational approximation methods. Numerical experiments are presented that demonstrate the superiority of the proposed model for image segmentation compared to other GMM-based approaches. The model is also successfully compared to state of the art image segmentation methods in clustering both natural images and images degraded by noise.

Detecting and classifying lesions in ultrasound images

A method for determining a candidate lesion region within an ultrasound image. The method includes the steps of: accessing a digital ultrasound image of anatomical tissue; segmenting spatially contiguous pixels in the ultrasound image into a plurality of regions in accordance with substantially similar intensity values and spatial smoothness constraints; and selecting, from the plurality of regions, one or more candidate lesion regions having an intensity value lower than a pre-determined intensity value. In one arrangement, the one or more candidate lesion region is classified into at least one of the following classes: benign, malignant, or unknown.

Use of a Vertical Vorticity Equation in Variational Dual-Doppler Wind Analysis

Abstract The utility of the anelastic vertical vorticity equation in a weak-constraint (least squares error) variational dual-Doppler wind analysis procedure is explored. The analysis winds are obtained by minimizing a cost function accounting for the discrepancies between observed and analyzed radial winds, errors in the mass conservation equation, errors in the anelastic vertical vorticity equation, and spatial smoothness constraints. By using Taylor’s frozen-turbulence hypothesis to shift analysis winds to observation points, discrepancies between radially projected analysis winds and radial wind observations can be calculated at the actual times and locations the data are acquired. The frozen-turbulence hypothesis is also used to evaluate the local derivative term in the vorticity equation. Tests of the analysis procedure are performed with analytical pseudo-observations of an array of translating and temporally decaying counterrotating updrafts and downdrafts generated from a Beltrami flow solution of the Navier–Stokes equations. The experiments explore the value added to the analysis by the vorticity equation constraint in the common scenario of substantial missing low-level data (radial wind observations at heights beneath 1.5 km are withheld from the analysis). Experiments focus on the sensitivity of the most sensitive analysis variable—the vertical velocity component—to values of the weighting coefficients, volume scan period, number of volume scans, and errors in the estimated frozen-turbulence pattern-translation components. Although the vorticity equation constraint is found to add value to many of these analyses, the analysis can become significantly degraded if estimates of the pattern-translation components are largely in error or if the frozen-turbulence hypothesis itself breaks down. However, tests also suggest that these negative impacts can be mitigated if data are available in a rapid-scan mode.

Evaluating the impact of spatio-temporal smoothness constraints on the BOLD hemodynamic response function estimation: an analysis based on Tikhonov regularization

Recently we have proposed the use of Tikhonov regularization with temporal smoothness constraints to estimate the BOLD fMRI hemodynamic response function (HRF). The temporal smoothness constraint was imposed on the estimates by using second derivative information while the regularization parameter was selected based on the generalized cross-validation function (GCV). Using one-dimensional simulations, we previously found this method to produce reliable estimates of the HRF time course, especially its time to peak (TTP), being at the same time fast and robust to over-sampling in the HRF estimation. Here, we extend the method to include simultaneous temporal and spatial smoothness constraints. This method does not need Gaussian smoothing as a pre-processing step as usually done in fMRI data analysis. We carried out two-dimensional simulations to compare the two methods: Tikhonov regularization with temporal (Tik-GCV-T) and spatio-temporal (Tik-GCV-ST) smoothness constraints on the estimated HRF. We focus our attention on quantifying the influence of the Gaussian data smoothing and the presence of edges on the performance of these techniques. Our results suggest that the spatial smoothing introduced by regularization is less severe than that produced by Gaussian smoothing. This allows more accurate estimates of the response amplitudes while producing similar estimates of the TTP. We illustrate these ideas using real data.

Regional three‐dimensional seismic velocity model of the crust and uppermost mantle of northern California

We present a three‐dimensional (3D) tomographic model of the P wave velocity (Vp) structure of northern California. We employed a regional‐scale double‐difference tomography algorithm that incorporates a finite‐difference travel time calculator and spatial smoothing constraints. Arrival times from earthquakes and travel times from controlled‐source explosions, recorded at network and/or temporary stations, were inverted for Vp on a 3D grid with horizontal node spacing of 10 to 20 km and vertical node spacing of 3 to 8 km. Our model provides an unprecedented, comprehensive view of the regional‐scale structure of northern California, putting many previously identified features into a broader regional context and improving the resolution of a number of them and revealing a number of new features, especially in the middle and lower crust, that have never before been reported. Examples of the former include the complex subducting Gorda slab, a steep, deeply penetrating fault beneath the Sacramento River Delta, crustal low‐velocity zones beneath Geysers‐Clear Lake and Long Valley, and the high‐velocity ophiolite body underlying the Great Valley. Examples of the latter include mid‐crustal low‐velocity zones beneath Mount Shasta and north of Lake Tahoe.

A REGRESSION MIXTURE MODEL WITH SPATIAL CONSTRAINTS FOR CLUSTERING SPATIOTEMPORAL DATA

We present a new approach for curve clustering designed for analysis of spatiotemporal data. Such data contains both spatial and temporal patterns that we desire to capture. The proposed methodology is based on regression and Gaussian mixture modeling. The novelty of the herein work is the incorporation of spatial smoothness constraints in the form of a prior for the data labels. This allows to take into account the property of spatiotemporal data according to which spatially adjacent data points have higher probability to belong to the same cluster. The proposed model can be formulated as a Maximum a Posteriori (MAP) problem, where the Expectation Maximization (EM) algorithm is used to estimate the model parameters. Several numerical experiments with both simulated data and real cardiac perfusion MRI data are used for evaluating the methodology. The results are promising and demonstrate the value of the proposed approach.

Introduction of spatial smoothness constraints via linear diffusion for optimization‐based hyperspectral coastal ocean remote‐sensing inversion

An optimization‐based, shallow water remote‐sensing inversion algorithm was recently developed by Lee et al. (1999) that simultaneously derives bottom depth and water column inherent optical properties. Only measured remote‐sensing reflectance is required as input, which is a noted advantage; however, the algorithm is sensitive to noise. Given the observation that bottom depth and optical properties generally change slowly over the spatial domain, we applied a smoothness assumption by modifying the Lee algorithm to accommodate a spatial smoothness constraint. Spatial constraints were introduced through a linear diffusion process. The new spatial constraint model retrievals were compared with those from the original Lee model, as well as from the original model plus smoothing performed as postprocessing. Synthetic and real field data experiments were performed. For the synthetic experiments the Lee method was the most sensitive to noise and posted the largest absolute and standard errors. For the field experiments, Portable Hyperspectral Imager for Low Light Spectroscopy Sensor (PHILLS) imagery was acquired over optically shallow water. An in situ acoustic sensor mounted on a towed Hyperspectral Tethered Spectral Radiometer Buoy (HTSRB) provided measured bottom depth transects, and remote‐sensing reflectance was HTSRB derived. Bathymetry estimates were validated in this research. Lee method bottom depth retrievals were more erratic than those generated via the other methods. Unlike postsmoothing the spatial constraint method corrected large deviations of the Lee method spanning several transect points. Overall, our spatial constraint method yielded the most accurate estimates and represents a significant improvement upon the Lee method. Mean absolute estimation errors for the Lee, postprocessing, and spatial constraint methods were 0.412, 0.389, and 0.335 m, respectively.

Nonparametric Bayesian Image Segmentation

Image segmentation algorithms partition the set of pixels of an image into a specific number of different, spatially homogeneous groups. We propose a nonparametric Bayesian model for histogram clustering which automatically determines the number of segments when spatial smoothness constraints on the class assignments are enforced by a Markov Random Field. A Dirichlet process prior controls the level of resolution which corresponds to the number of clusters in data with a unique cluster structure. The resulting posterior is efficiently sampled by a variant of a conjugate-case sampling algorithm for Dirichlet process mixture models. Experimental results are provided for real-world gray value images, synthetic aperture radar images and magnetic resonance imaging data.

Hierarchical Bayesian estimation for MEG inverse problem

Source current estimation from MEG measurement is an ill-posed problem that requires prior assumptions about brain activity and an efficient estimation algorithm. In this article, we propose a new hierarchical Bayesian method introducing a hierarchical prior that can effectively incorporate both structural and functional MRI data. In our method, the variance of the source current at each source location is considered an unknown parameter and estimated from the observed MEG data and prior information by using the Variational Bayesian method. The fMRI information can be imposed as prior information on the variance distribution rather than the variance itself so that it gives a soft constraint on the variance. A spatial smoothness constraint, that the neural activity within a few millimeter radius tends to be similar due to the neural connections, can also be implemented as a hierarchical prior. The proposed method provides a unified theory to deal with the following three situations: (1) MEG with no other data, (2) MEG with structural MRI data on cortical surfaces, and (3) MEG with both structural MRI and fMRI data. We investigated the performance of our method and conventional linear inverse methods under these three conditions. Simulation results indicate that our method has better accuracy and spatial resolution than the conventional linear inverse methods under all three conditions. It is also shown that accuracy of our method improves as MRI and fMRI information becomes available. Simulation results demonstrate that our method appropriately resolves the inverse problem even if fMRI data convey inaccurate information, while the Wiener filter method is seriously deteriorated by inaccurate fMRI information.

Lateral variation of Lg wave propagation in southern Mexico

In this study we investigated lateral variation of Lg wave propagation in southern Mexico from recordings of 92 crustal earthquakes along 591 travel paths. The efficiency of Lg propagation was measured in terms of Lgto Pn spectral ratio. It was found that Lgpropagation is inefficient for travel paths through the Gulf of Mexico coastal plains and the Gulf of Tehuantepec, areas with thick layers of sediments. An average Lg quality factor, QLg, as a function of frequency for southern Mexico was estimated for the efficient Lg travel paths. The relation obtained for QLg in the frequency range 1.6–8 Hz is QLg(f) = 204 f0.85. The lateral variation of QLg−1was solved as a mixed‐determined inverse tomography problem, separately for each frequency, in which a spatial smoothness constraint was imposed and a priori information was added in poorly covered regions. The spatial resolution obtained was about 200 km. It was found that the Trans‐Mexican Volcanic Belt, the Gulf of Mexico coastal plains, and the area east of 94°W are characterized by lower than average QLgvalues, i.e., higher attenuation. High QLg values were obtained for the Mixteco‐Oaxaca terranes, while for the Guerrero terrane, values similar to the average were obtained. The results show a correlation between QLg and crustal structure and provide valuable information on lateral variation of QLg, which is needed for reliable prediction of ground motion during future earthquakes.

Segmentation of ultrasound B-mode images with intensity inhomogeneity correction.

Displayed ultrasound (US) B-mode images often exhibit tissue intensity inhomogeneities dominated by nonuniform beam attenuation within the body. This is a major problem for intensity-based, automatic segmentation of video-intensity images because conventional threshold-based or intensity-statistic-based approaches do not work well in the presence of such image distortions. Time gain compensation (TGC) is typically used in standard US machines in an attempt to overcome this. However this compensation method is position-dependent which means that different tissues in the same TGC time-range (or corresponding depth range) will be, incorrectly, compensated by the same amount. Compensation should really be tissue-type dependent but automating this step is difficult. The main contribution of this paper is to develop a method for simultaneous estimation of video-intensity inhomogeities and segmentation of US image tissue regions. The method uses a combination of the maximum a posteriori (MAP) and Markov random field (MRF) methods to estimate the US image distortion field assuming it follows a multiplicative model while at the same time labeling image regions based on the corrected intensity statistics. The MAP step is used to estimate the intensity model parameters while the MRF step provides a novel way of incorporating the distributions of image tissue classes as a spatial smoothness constraint. We explain how this multiplicative model can be related to the ultrasonic physics of image formation to justify our approach. Experiments are presented on synthetic images and a gelatin phantom to evaluate quantitatively the accuracy of the method. We also discuss qualitatively the application of the method to clinical breast and cardiac US images. Limitations of the method and potential clinical applications are outlined in the conclusion.

Bayesian 2D Deconvolution: A Model for Diffuse Ultrasound Scattering

Observed medical ultrasound images are degraded representations of the true acoustic tissue reflectance. The degradation is due to blur and speckle, and significantly reduces the diagnostic value of the images. In order to remove both blur and speckle we have developed a new statistical model for diffuse scattering in 2D ultrasound radio-frequency images, incorporating both spatial smoothness constraints and a physical model for diffuse scattering. The modeling approach is Bayesian in nature, and we use Markov chain Monte Carlo methods to obtain the restorations. The results from restorations of some real and simulated radio-frequency ultrasound images are presented, and compared with results produced by Wiener filtering.

Bayesian 2-D deconvolution: a model for diffuse ultrasound scattering.

Observed medical ultrasound images are degraded representations of the true acoustic tissue reflectance. The degradation is due to blur and speckle and significantly reduces the diagnostic value of the images. To remove both blur and speckle, we have developed a new statistical model for diffuse scattering in 2-D ultrasound radio frequency images, incorporating both spatial smoothness constraints and a physical model for diffuse scattering. The modeling approach is Bayesian in nature, and we use Markov chain Monte Carlo methods to obtain the restorations. The results from restorations of some real and simulated radio frequency ultrasound images are presented and compared with results produced by Wiener filtering.

Statistical Errors in Variational Data Assimilation–A Theoretical One-Dimensional Analysis Applied to Doppler Wind Retrieval

Abstract When the velocity field is retrieved by a variational data assimilation method from the tracer (reflectivity) pattern movement observed by a Doppler radar, the accuracy of the retrieved velocity is affected by the observational error and data resolution. The statistical aspect of this problem is examined theoretically with a one-dimensional tracer advection equation. It is found that the retrieved velocity is generally underestimated if the signal-to-noise ratio is less than 1.0 for the gradient of the observed tracer field; so there is a systematic error. This systematic error is caused by the nonlinear response, of the retrieved velocity to the observed data. In addition, there is also a nonsystematic error in the retrieved velocity caused by the observational errors in the tracer field and tracer gradient field. These two types of error can be reduced by using a moving reference frame and by imposing a spatial smoothness constraint on the retrieved field.

Markov models of specular and diffuse scattering in restoration of medical ultrasound images

Observed medical ultrasound images are degraded representations of the true tissue reflectance. The specular reflections at boundaries between regions of different tissue types are blurred, and the diffuse scattering within such regions also contains speckle. This reduces the diagnostic value of such images. In order to remove both blur and speckle, the authors develop a maximum a posteriori deconvolution algorithm for two-dimensional (2-D) ultrasound radio frequency (RF) images based on a new Markov random field image model incorporating spatial smoothness constraints and physical models for specular reflections and diffuse scattering. During stochastic relaxation, the algorithm alternates steps of restoration and segmentation, and includes estimation of reflectance parameters. The smoothness constraints regularize the overall procedure, and the algorithm uses the specular reflection model to locate region boundaries. The resulting restorations of some simulated and real RF images are significantly better than those produced by Wiener filtering.

Penalized Discriminant Analysis

Fisher's linear discriminant analysis (LDA) is a popular data-analytic tool for studying the relationship between a set of predictors and a categorical response. In this paper we describe a penalized version of LDA. It is designed for situations in which there are many highly correlated predictors, such as those obtained by discretizing a function, or the grey-scale values of the pixels in a series of images. In cases such as these it is natural, efficient and sometimes essential to impose a spatial smoothness constraint on the coefficients, both for improved prediction performance and interpretability. We cast the classification problem into a regression framework via optimal scoring. Using this, our proposal facilitates the use of any penalized regression technique in the classification setting. The technique is illustrated with examples in speech recognition and handwritten character recognition.

Recovery of vocal tract geometry fromformants for vowels and fricative consonants using a midsagittal-to-area function conversion model

This study deals with the ill-posed problem of inversion of the articulatory-to-acoustic relationship, i.e., the recovery of vocal tract geometry from formant frequencies. A small database of articulatory-acoustic data has been established for one subject. A midsagittal-to-area function conversion model, which works both for vowels and fricative consonants, has been developed from these data. This model has finally been used as a major constraint for an optimization algorithm based on a gradient descent technique, in order to regularize the ill-posed inversion problem. Other spatial and temporal smoothing constraints have also been used. Single vocal tract configurations, as well as entire [VC] sequences, could be recovered using adequate initial conditions.

Mixture Estimation From Multichannel Image Data

The following problem arises in Computer vision, diagnostic medical imaging, and remote sensing: At each pixel in an image a vector of observations is measured, and the distribution of these measurements is approximated by a mixture model. The goal is to estimate the mixing proportions of the classes by pixel in the image together with any unknown parameters in the latent distributions. In many problems of this type, it is appropriate to incorporate constraints on mixing proportions. This article deals with spatial smoothness constraints, which have been found useful in analyzing sequences of emission tomography images. An estimation methodology using penalized likelihood with multiple smoothing parameters is proposed. Numerical methods for implementing this methodology are developed. This includes an importance sampling technique for approximating the effective degrees of freedom of the solution. The methodology is illustrated with an application to the analysis of a dynamic emission tomography study usi...

Toward a fundamental image preprocessor

A general transformation equation is derived and applied to dynamic and static image differences. It uses translation, rotation, scale, and velocity changes as fundamental object features for image segmentation. A linear minimization algorithm is derived based on a spatial smoothness constraint. The transformation and algorithm constitute an automatic target recognition preprocessor architecture.