Statistical inference on stationary random fields

Statistical methods are developed to model random processes on multidimensional Euclidean space from observed data. Statistical inference techniques are used to estimate model parameters and test hypotheses concerning stationarity, isotropy, and number of parameters. Algorithms are described for fitting parametric models and testing between alternative model structures. Stochastic partial difference equation models of multidimensional processes are discussed in detail. Computer generated data from a known model are used to directly demonstrate the statistical procedures.     

On stationarizability for nonstationary 2-D random fields usingdiscrete wavelet transforms

The emphasis in this article is on the study of nonstationary two-dimensional (2-D) random fields with wide-sense stationary increments, wide-sense stationary jumps, and 2-D fractional Brownian motion (fBm) fields. The effort made in this work is to develop a realizable method of stationarization provided for nonstationary 2-D random fields. We also present the correlation functions of the discrete wavelet transform relating to 2-D fBm fields that will decay hyperbolically fast.     

Cramer-Rao bound on the estimation accuracy of complex-valuedhomogeneous Gaussian random fields

This paper considers the problem of the achievable accuracy in jointly estimating the parameters of a complex-valued two-dimensional (2-D) Gaussian and homogeneous random field from a single observed realization of it. Based on the 2-D Wold decomposition, the field is modeled as a sum of purely indeterministic, evanescent, and harmonic components. Using this parametric model, we first solve a key problem common to many open problems in parametric estimation of homogeneous random fields: that of expressing the field mean and covariance functions in terms of the model parameters. Employing the parametric representation of the observed field mean and covariance, we derive a closed-form expression for the Fisher information matrix (FIM) of complex-valued homogeneous Gaussian random fields with mixed spectral distribution. Consequently, the Cramer-Rao lower bound on the error variance in jointly estimating the model parameters is evaluated     

Recursive-in-order least-squares parameter estimation algorithm for 2-D noncausal Gaussian Markov random field model

We present in this paper a recursive-in-order least-squares (LS) algorithm to compute efficiently the parameters of a 2-D Gaussian Markov random field (GMRF) model. The algorithm is based on the fact that the least-squares estimation of the parameters of a 2-D noncausal GMRF model is consistent and the coefficient matrix in the normal equation has near-to-block-Toeplitz structure. Hence, it has a Levinson-like form for the updating of model parameters by introducing auxiliary variables. Moreover, this paper proposes the concept ofrecursive path for 2-D recursive-in-order algorithms, and points out that there exists a tradeoff between fast computation of the parameters and accurate choice of model support; a compromise recursive path is then suggested where the orders change alternately in two directions. The computational complexity of the developed algorithm is analyzed, and the results show that the algorithm is more efficient when either the image size or the model support is larger. It is found that the total number of multiplications (mps) involved in the new algorithm is only about 14% of that in the conventional LS method when the image size is 512 × 512 and the neighbor set of the model is a 17 × 17 window. Computer simulation results using the recursive-in-order algorithm developed in this paper and the conventional LS method are given to verify the correctness of the new algorithm.This work was supported by the NSERC under Grants A-4070 and A-7739, and by the FCAR, Grant H-70.     

Simulated annealing in compound Gaussian random fields [imageprocessing]

Recently, a stochastic relaxation technique called simulated annealing has been developed to search for a globally optimal solution in image estimation and restoration problems. The convergence of simulated annealing has been proved only for random fields with a compact range space. Because of this, images were modeled as random fields with bounded discrete or continuous values. However, in most image processing problems, it is more natural to model the image as a random field with values in a noncompact space, e.g. conditional Gaussian models. The proof of convergence of the stochastic relaxation method is extended to a class of compound Gauss-Markov random fields. Simulation results are provided to show the power of these methods     

Electromagnetic scattering interference between two shallow objects buried under 2-D random rough surfaces

A rigorous electromagnetic model has been used to analyze the scattering from two dielectric shallow objects buried under the two-dimensional (2-D) random rough ground (3-D scattering problem) as a means of predicting false alarms. The method of moments (MoM) accelerated by the steepest descent fast multipole method (SDFMM) is used to compute the equivalent electric and magnetic surface currents on all scatterers (i.e., the rough ground and the two buried objects). The roughness parameters influence the scattering interference mechanism of the two objects, however, a large separation distance (e.g., several correlation lengths) showed stronger effect for small ground roughness.     

Automatic detection of gadolinium-enhancing multiple sclerosis lesions in brain MRI using conditional random fields

Gadolinium-enhancing lesions in brain magnetic resonance imaging of multiple sclerosis (MS) patients are of great interest since they are markers of disease activity. Identification of gadolinium-enhancing lesions is particularly challenging because the vast majority of enhancing voxels are associated with normal structures, particularly blood vessels. Furthermore, these lesions are typically small and in close proximity to vessels. In this paper, we present an automatic, probabilistic framework for segmentation of gadolinium-enhancing lesions in MS using conditional random fields. Our approach, through the integration of different components, encodes different information such as correspondence between the intensities and tissue labels, patterns in the labels, or patterns in the intensities. The proposed algorithm is evaluated on 80 multimodal clinical datasets acquired from relapsing-remitting MS patients in the context of multicenter clinical trials. The experimental results exhibit a sensitivity of 98% with a low false positive lesion count. The performance of the proposed algorithm is also compared to a logistic regression classifier, a support vector machine and a Markov random field approach. The results demonstrate superior performance of the proposed algorithm at successfully detecting all of the gadolinium-enhancing lesions while maintaining a low false positive lesion count.     

Spatio-temporal fMRI analysis using Markov random fields

Functional magnetic resonance images (fMRI's) provide high-resolution datasets which allow researchers to obtain accurate delineation and sensitive detection of activation areas involved in cognitive processes. To preserve the resolution of this noninvasive technique, refined methods are required in the analysis of the data. In this paper, we first discuss the widely used methods based on a statistical parameter map (SPM) analysis exposing the different shortcomings of this approach when considering high-resolution data. First, the often used Gaussian filtering results in a blurring effect and in delocalization of the activated area. Secondly, the SPM approach only considers false alarms due to noise but not rejections of activated voxels. We propose to embed the fMRI analysis problem into a Bayesian framework consisting of two steps: i) data restoration and ii) data analysis. We, therefore, propose two Markov random fields (MRF's) to solve these two problems. Results on three protocols (visual, motor and word recognition) are shown for two SPM approaches and compared with the proposed MRF approach.     

Multiple-incidence and multifrequency for profile reconstruction of random rough surfaces using the 3-D electromagnetic fast multipole model

A fast algorithm for reconstructing the profile of random rough surfaces using electromagnetic scattering data is presented. The algorithm is based on merging a fast forward solver and an efficient optimization technique. The steepest descent fast multipole method is used as the three-dimensional fast forward solver. A rapidly convergent descent method employing a "marching-on" strategy for processing multifrequency and multi-incidence angle data is introduced to minimize an underlying cost function. The cost function represents the error between true (synthetic) and simulated scattered field data. Several key issues that impact the accuracy in reconstructing the rough profile are examined in this work, e.g., the location and number of receivers, the incident and scattered directions, the surface roughness, and details regarding the manner in which sensitivity information is computed in the inversion scheme. The results show that using the multiple-incidence (one angle at a time) and the multifrequency (one frequency at a time) strategies lead to improve the profile reconstruction.     

A recursive model-reduction method for approximate inference in Gaussian Markov random fields.

This paper presents recursive cavity modeling--a principled, tractable approach to approximate, near-optimal inference for large Gauss-Markov random fields. The main idea is to subdivide the random field into smaller subfields, constructing cavity models which approximate these subfields. Each cavity model is a concise, yet faithful, model for the surface of one subfield sufficient for near-optimal inference in adjacent subfields. This basic idea leads to a tree-structured algorithm which recursively builds a hierarchy of cavity models during an "upward pass" and then builds a complementary set of blanket models during a reverse "downward pass." The marginal statistics of individual variables can then be approximated using their blanket models. Model thinning plays an important role, allowing us to develop thinned cavity and blanket models thereby providing tractable approximate inference. We develop a maximum-entropy approach that exploits certain tractable representations of Fisher information on thin chordal graphs. Given the resulting set of thinned cavity models, we also develop a fast preconditioner, which provides a simple iterative method to compute optimal estimates. Thus, our overall approach combines recursive inference, variational learning and iterative estimation. We demonstrate the accuracy and scalability of this approach in several challenging, large-scale remote sensing problems.     

A simple information theoretic proof of the maximum entropyproperty of some Gaussian random fields

A well known result of Burg (1967) and Kunsch (1981) identifies a Gaussian Markov random field with autocovariances specified on a finite part L of the n-dimensional integer lattice, as the random field with maximum entropy among all random fields with same autocovariance values on L. A simple information theoretic proof of a version of the maximum entropy theorem for random fields in n dimensions is presented in the special case that the given autocovariances are compatible with a unilateral autoregressive process.     

Registration of 3-D head surfaces using multiple landmarks

A mathematical procedure based on Newton's method is described that enables surface measurements to be registered, or normalized, with respect to spatial position, orientation, and, optionally, scale in three dimensions. An operator is required to identify homologous landmarks on the computer graphics images of surfaces to be registered. In this application, where the method is used to measure changes in facial shape, these landmarks are restricted to parts of the surface that have remained unchanged between the surfaces to be registered. Error in the registration of landmarks is minimized in a least-squares sense; hence multiple landmarks are favored to minimize the effect of individual errors produced by the measuring system and the operator. Examples are presented using measurements of the head taken with an optical surface scanner and a conventional X-ray computed tomography scanner.     

Parameter estimation of 2-D random amplitude polynomial-phasesignals

Phase information has fundamental importance in many two-dimensional (2-D) signal processing problems. In this paper, we consider 2-D signals with random amplitude and a continuous deterministic phase. The signal is represented by a random amplitude polynomial phase model. A computationally efficient estimation algorithm for the signal parameters is presented. The algorithm is based on the properties of the mean phase differencing operator, which is introduced and analyzed. Assuming that the signal is observed in additive white Gaussian noise and that the amplitude field is Gaussian as well, we derive the Cramer-Rao lower bound (CRB) on the error variance in jointly estimating the model parameters. The performance of the algorithm in the presence of additive white Gaussian noise is illustrated by numerical examples and compared with the CRB     

On computation of electromagnetic fields on planar surfaces from fields specified on nearby surfaces

A technique is presented for solution of the inverse problem of calculating the electric field on a planar surface from the electric field specified on a nearby surface. An integral equation is derived that relates two orthogonal components of the electric field on the nearby surface to the respective components of the plane wave spectrum of the planar electric field. The integral equation is solved by an iterative technique, and the planar near field is calculated by an inverse Fourier transform of the plane wave spectrum.     

高斯粗糙面与临空目标复合电磁散身寸的KH-MOM解

提出了一种求解一维粗糙面与二维无限长临空目标复合电磁散射特性的新型混合算法。混合算法只需在粗糙面上进行一次积分运算,即可用基尔霍夫-亥姆霍兹方程(KH)描述电磁波经粗糙面后的散射情况,再用矩量法(MoM)分析目标的散射问题,通过KH与MoM的混合来体现粗糙面与目标之间的耦合作用。经与不同方法的对比,验证了混合方法的正确性,体现了混合方法较数值法在求解效率上的巨大优势。计算了粗糙面与临空目标的统计复合散射特性,分析了粗糙面的起伏参数、临空目标的形状以及粗糙面介质的电参数对复合散射特性的影响。     

Recursive anisotropic 2-D Gaussian filtering based on a triple-axis decomposition.

We describe a recursive algorithm for anisotropic 2-D Gaussian filtering, based on separating the filter into the cascade of three, rather two, 1-D filters. The filters operate along axes obtained by integer horizontal and/or vertical pixel shifts. This eliminates interpolation, which removes spatial inhomogeneity in the filter, and produces more elliptically shaped kernels. It also results in a more regular filter structure, which facilitates implementation in DSP chips. Finally, it improves matching between filters with the same eccentricity and width, but different orientations. Our analysis and experiments indicate that the computational complexity is similar to an algorithm that operates along two axes (<11 ms for a 512 x 512 image using a 3.2-GHz Pentium 4 PC). On the other hand, given a limited set of basis filter axes, there is an orientation dependent lower bound on the achievable aspect ratios.     

Tomographic reconstruction of 3-D vector fields using inner productprobes

Inner product probe measurements are defined for tomographic reconstruction of 3-D vector fields. It is shown that one set of measurements is required to reconstruct an irrotational field, two are required to reconstruct a solenoidal field and special probes are required to reconstruct the components of an arbitrary field.     

Bayesian multichannel image restoration using compound Gauss-Markov random fields

We develop a multichannel image restoration algorithm using compound Gauss-Markov random fields (CGMRF) models. The line process in the CGMRF allows the channels to share important information regarding the objects present in the scene. In order to estimate the underlying multichannel image, two new iterative algorithms are presented and their convergence is established. They can be considered as extensions of the classical simulated annealing and iterative conditional methods. Experimental results with color images demonstrate the effectiveness of the proposed approaches.