Practical sequential estimation algorithms inherently involve "iterated conditioning"; that is, the estimate at any algorithm step is conditioned on information passed from the previous step. Except in special cases, standard probability theory cannot deal with operations like iterated conditioning, so it is difficult even to formulate algorithms that make optimal use of the available information. The previously developed product space conditional event algebra (PS-CEA) allows for the definition of "conditional events" as regular events in a probability space, and thus for the extension of Bayesian analysis to cases of iterated conditioning. We describe the PS-CEA approach and give an example of its use in deriving iterated image estimators when the information passed from one stage to the next is the current image estimate.

This paper presents an efficient temporal, spatial and frequency decomposition method that is designed for video coding. In this method, the given video sequence is decomposed into small volumes such that the signal in each volume is relatively homogeneous in frequency characteristics in all three dimensions. Volumes thus formed, therefore, adapt to the spatial nonstationarity and the motion in the video signals. For each volume, a best 3D wavelet packet is generated based on a rate-distortion criterion such that the subbands of this 3D wavelet packet are efficiently quantized. The partitioning of the total signal volume into smaller "homogeneous" volumes is realized according to the same rate-distortion criterion.

This paper presents a new progressive image transmission (PIT) design algorithm in which the resolution and resources (rate or distortion and storage size) at each transmission stage are allowed to be prespecified. This algorithm uses the wavelet transform and tree-structured vector quantizer (TSVQ) techniques. The wavelet transform is used to obtain a pyramid structure representation of an image. The vector quantizer technique is used to design a TSVQ for each subimage so that all the subimages that constitute the image at the current stage can be successively refined according to the resources available at that stage. The resources assigned to each subimage for the successive refinement is determined to optimize the performance at the current stage under the resource constraints. Normally, the resource constraints at each stage are determined by the specification of the transmission time or distortion for image data and the storage complexity of the TSVQ. The resolution at each stage is determined/specified according to the application or as part of the design process to optimize the visual effect.

In this paper, we present an algorithm that locally selects the image features with good discriminatory capabilities and uses a multilayer constrained weak-membrane for boundary detection. We use a general non-stationary image model where image features are allowed to vary slowly within regions and boundaries are defined by abrupt changes in these features. Image features usually have varying discriminatory properties in different regions and the purpose of feature selection is to convey the most discriminatory information for edge detection. The variances of image features are calculated on large and small windows and the difference of these variances are used as a measure of the discriminatory weight of the feature. A layer of constrained weak-membrane is used for each feature and these layers collaborate at every iteration to produce the global discontinuities. Local line configurations are used to constrain the weak-membranes to break only along certain configurations that conform to expected shapes of the boundaries.>

The valid parameter spaces of infinite- and finite-lattice (2-D noncausal) Gaussian Markov random fields (GMRFs) are investigated. For the infinite-lattice fields, the valid parameter space is shown to admit an explicit description; a procedure that yields the valid parameter space is presented. This procedure is applied to the second-order (neighborhood) 2-D GMRFs to obtain an explicit description of their valid parameter spaces. For the finite-lattice fields, it is shown that the valid parameter space does not admit a simple description; the conditions that ensure the positivity of the power spectrum are necessary, sufficient, and irreducible. The set of conditions for the infinite-lattice fields, however, serves as a good set of sufficient conditions for the finite-lattice case. The results readily extend to the class of d-D real and complex GMRFs.>

The Markov line process that has been used in some image segmentation and restoration studies is investigated. Realizations from this model are presented for a wide range of parameter values, and the effects of certain parameters are studied. The maximum pseudolikelihood (MPL) estimation procedure is implemented for the Markov line process. The MPL procedure is applied to several images generated from the model as well as to a hand-drawn image and the edge-detector output of a natural image. It is expected that improved segmentation and restoration results can be obtained, if the Markov line process model is fine-tuned to the class of images under consideration, by estimating the parameters of some typical images in that class.>

An adaptive segmentation algorithm is developed which simultaneously estimates the parameters of the underlying Gibbs random field (GRF)and segments the noisy image corrupted by additive independent Gaussian noise. The algorithm, which aims at obtaining the maximum a posteriori (MAP) segmentation is a simulated annealing algorithm that is interrupted at regular intervals for estimating the GRF parameters. Maximum-likelihood (ML) estimates of the parameters based on the current segmentation are used to obtain the next segmentation. It is proven that the parameter estimates and the segmentations converge in distribution to the ML estimate of the parameters and the MAP segmentation with those parameter estimates, respectively. Due to computational difficulties, however, only an approximate version of the algorithm is implemented. The approximate algorithm is applied on several two- and four-region images with different noise levels and with first-order and second-order neighborhoods.>

Discrete-index Markov-type random processes in one and two dimensions are considered, with emphasis on two-dimensional processes (or fields). Important classes of Markov-type models, their properties, and their relationship are described. Although some new results are given, the authors mainly present a systematic study and grouping of processes according to two fundamental Markov-type properties: strict-sense Markov, defined in terms of conditional probabilities, and wide-sense Markov, defined in terms of linear minimum-mean-square error estimates. Classes of models having special cases of the fundamental properties, including many models which are widely used to represent images are obtained by specifying the index set, the conditioning set used to define the Markov property, and the process distribution. The relationships between unilateral and bilateral models in each class are carefully investigated. Particular attention is given to simultaneous autoregressive models which are shown to be both strict-sense and wide-sense Markov. Classification of processes according to their Markov-type properties helps to clarify the consequences of and relationships between different model assumptions.>

A hierarchical random field model and segmentation algorithm is presented for multilook speckled images. Speckle is modeled using Gaussian random fields that have separable autocorrelations. The model incorporates the correlation found in real speckle. The speckle fields are Markov with second-order neighborhoods, a property used in the segmentation. The segmentation algorithm, which is based on a posteriori formulation, estimating the value of the region field given the speckled observation.>

Addresses the problem of fitting Gaussian Markov random field (GMRF) models to natural textures through maximum likelihood (ML) estimation of the model parameters. In particular, the authors are interested in two implementation issues on ML estimation. First, the log-likelihood function to be maximized is not a concave function with respect to the parameters. Thus the conventional gradient maximization method is not computationally attractive. The second concern comes from the fact that the suppose-to-be covariance matrix expressed in terms of the estimated parameters must be non-negative definite. To resolve these difficulties one uses a version of the so-called multi-start algorithm, which is a variation of the deterministic relaxation algorithm. The algorithm aims at determining many of the local maxima of the likelihood function and determining a maximum among them. This algorithm always yields estimates that are in the allowable set of parameter values, yielding a non-negative definite covariance matrix. Some experimental results with natural textures show that often the best fit of natural images to GMRF models via ML estimation occurs at the boundary of the allowable parameter set.>

The statistical properties of speckle are examined. A general joint probability description is derived; special cases of the formulation are then given in detail, including a local statistical description of speckle which goes beyond the marginal statistics. The statistical model developed for speckle is then used in developing a segmentation algorithm for speckled images. The model parameters necessary for the segmentation algorithm are estimated.

This paper presents a segmentation algorithm for noisy textured images. To represent noisy textured images, we propose a hierarchical stochastic model that consists of three levels of random fields: the region process, the texture processes and the noise. The hierarchical model also includes local blurring and nonlinear image transformation as results of the image corrupting effects. Having adopted a statistical model, the maximum a posteriori (MAP) estimation is used to find the segmented regions through the restored(noise-free) textured image data. Since the joint a posteriori distribution at hand is a Gibbs distribution, we use simulated annealing as a maximization technique. The simulated annealing based segmentation algorithm presented in this paper can also be viewed as a two-step iterative algorithm in the spirit of the EM algorithm [10].

A new technique for estimating the component parameters of a mixture of univariate Gaussian distributions using the method of moments is presented. The method of moments basically involves equating the sample moments to the corresponding mixture moments expressed in terms of component parameters and solving these equations for the unknown parameters. These moment equations, however, are nonlinear in the unknown parameters, and heretofore, an analytic solution of these equations has been obtained only for two-component mixtures [2]. Numerical solutions also tend to be unreliable for more than two components, due to the large number of nonlinear equations and parameters to be solved for. In this correspondence, under the condition that the component distributions have equal variances or equal means, the nonlinear moment equations are transformed into a set of linear equations using Prony's method. The solution of these equations for the unknown parameters is analytically feasible and numerically reliable for mixtures with several components. Numerous examples using the proposed technique for two-, three-, and four-component mixtures are presented.

This paper presents a new approach to the use of Gibbs distributions (GD) for modeling and segmentation of noisy and textured images. Specifically, the paper presents random field models for noisy and textured image data based upon a hierarchy of GD. It then presents dynamic programming based segmentation algorithms for noisy and textured images, considering a statistical maximum a posteriori (MAP) criterion. Due to computational concerns, however, sub-optimal versions of the algorithms are devised through simplifying approximations in the model. Since model parameters are needed for the segmentation algorithms, a new parameter estimation technique is developed for estimating the parameters in a GD. Finally, a number of examples are presented which show the usefulness of the Gibbsian model and the effectiveness of the segmentation algorithms and the parameter estimation procedures.

This paper presents a new approach for estimating the parameters of random fields characterized by Gibbs distributions. Unlike the commonly used "coding method", the parameter estimation procedure presented here involves the solution of a set of linear equations. We also present numerical and pictorial results on application of this estimation procedure on several Gibbs distributed images.

A new image segmentation algorithm is presented, based on recursive Bayes smoothing of images modeled by Markov random fields and corrupted by independent additive noise. The Bayes smoothing algorithm presented is an extension of a 1-D algorithm to 2-D and it yields the a posteriori distribution and the optimum Bayes estimate of the scene value at each pixel, using the total noisy image data. Computational concerns in 2-D, however, necessitate certain simplifying assumptions on the model and approximations on the implementation of the algorithm. In particular, the scene (noiseless image) is modeled as a Markov mesh random field and the algorithm is applied on (horizontal/vertical) strips of the image. The Bayes smoothing algorithm is applied to segmentation of two level test images and remotely sensed SAR data obtained from SEASAT, yielding remarkably good segmentation results even for very low signal to noise ratios.

This paper presents a new statistical approach to image segmentation. Making use of Gibbs distribution models of Markov random fields a dynamic programming based segmentation algorithm is developed. The algorithm is described in detail and examples are given.

A new image segmentation algorithm is presented, based on recursive Bayes smoothing of images modeled by Markov random fields and corrupted by independent additive noise. The Bayes smoothing algorithm yields the a posteriori distribution of the scene value at each pixel, given the total noisy image, in a recursive way. The a posteriori distribution together with a criterion of optimality then determine a Bayes estimate of the scene. The algorithm presented is an extension of a 1-D Bayes smoothing algorithm to 2-D and it gives the optimum Bayes estimate for the scene value at each pixel. Computational concerns in 2-D, however, necessitate certain simplifying assumptions on the model and approximations on the implementation of the algorithm. In particular, the scene (noiseless image) is modeled as a Markov mesh random field, a special class of Markov random fields, and the Bayes smoothing algorithm is applied on overlapping strips (horizontal/vertical) of the image consisting of several rows (columns). It is assumed that the signal (scene values) vector sequence along the strip is a vector Markov chain. Since signal correlation in one of the dimensions is not fully used along the edges of the strip, estimates are generated only along the middle sections of the strips. The overlapping strips are chosen such that the union of the middle sections of the strips gives the whole image. The Bayes smoothing algorithm presented here is valid for scene random fields consisting of multilevel (discrete) or continuous random variables.

When a vestigial sideband (VSB) modulated signal, with carrier present, is demodulated with an envelope detector it suffers an unavoidable distortion. The optimum VSB filter which minimizes this distortion is obtained, and it is shown that the filter transfer function should be flat around the carrier frequency ω0and should not be designed with a large slope at ω0as has heretofore been done.