Ramalingam Chellappa

Estimation and choice of neighbors in spatial-interaction models of images

Some aspects of statistical inference for a class of spatial-interaction models for finite images are presented: primarily the simultaneous autoregressive (SAR) models and conditional Markov (CM) models. Each of these models is characterized by a set of neighbors, a set of coefficients, and a noise sequence of specified characteristics. We are concerned with two problems: the estimation of the unknown parameters in both SAR and CM models and the choice of an appropriate model from a class of such competing models. Assuming Gaussian-distributed variables, we discuss maximum likelihood (ML) estimation methods. In general, the ML scheme leads to nonlinear optimization problems. To avoid excessive computation, an iterative scheme is given for SAR models, which gives approximate ML estimates in the Gaussian case and reasonably good estimates in some non-Gaussian situations as well. Likewise, for CM models, an easily computable consistent estimate is given. The asymptotic mean-squared error (mse) of this estimate for a four-neighbor CM model is shown tn be substantially less than the mse of the popular coding estimate. Asymptotically consistent decision rules are given for choosing an appropriate SAR or CM model. The usefulness of the estimation scheme and the decision rule for the choice of neighbors is illustrated by using synthetic patterns. Synthetic patterns obeying known SAR and CM models are generated, and the models corresponding to true and several competing neighbor sets are fitted. The estimation scheme yields estimates close to the parameters of the true models, and the decision rule for the choice of neighbors picks up the true model from the class of competing models.

Stochastic models for closed boundary analysis: Representation and reconstruction

The analysis of closed boundaries of arbitrary shapes on a plane is discussed. Specifically, the problems of representation and reconstruction are considered. A one-to-one correspondence between the given closed boundary and a univariate or multivariate sequence of real numbers is set up. Univariate or multivariate circular autoregressive models are suggested for the representation of the sequence of numbers derived from the closed boundary. The stochastic model representing the closed boundary is invariant to transformations like sealing, translation, choice of starting point, and rotation over angles that are multiples of 2\pi/N , where N is the number of observations. Methods for estimating the unknown parameters of the model are given and a decision rule for choosing the appropriate number of coefficients is included. Constraints on the estimates are derived so that the estimates are invariant to the transformations of the boundaries. The stochastic model enables the reconstruction of a dosed boundary using FFT algorithms. Results of simulations are included and the application to contour coding is discussed.

A model-based approach for estimation of two-dimensional maximum entropy power spectra

A stochastic model-based approach is presented for estimation of the two-dimensional maximum entropy power spectrum (MEPS) from given finite uniform array data. The method consists of fitting an appropriate two-dimensional noncausal Gaussian-Markov random field (GMRF) model to the given data using the maximum likelihood (ML) technique for parameter estimation. The nonlinear criterion function used for ML estimation is similar in structure to the function arising in the deterministic approach of Lang and McClellan. The model-based approach provides new insights into the two-dimensional MEPS estimation problem. For example, using the asymptotic normality of ML estimates, we derive simultaneous confidence bands for the estimated MEPS. It turns out that when the true correlations are generated by a noncausal GMRF model, the two-dimensional MEPS can be obtained by solving linear equations. This approach also suggests techniques for realizing two-dimensional GMRF models from the given correlation data. Several numerical examples are given to illustrate the usefulness of the approach.

Pyramid implementation of optimal-step conjugate-search algorithms for some low-level vision problems

The authors present a parallel pyramid implementation of the line search conjugate gradient algorithm for minimizing the cost function in low-level vision problems. By viewing the global cost function as a Gibbs energy function, it is possible to compute the gradients, inner products, and optimal-step size efficiently using the pyramid. Implementation of this algorithm for shape-from-shading results in a multiresolution conjugate gradient algorithm. The robustness and efficiency of the algorithm are demonstrated for edge detection using the graduated nonconvexity (GNC) algorithm. This formulation is also applied to image estimation based on Markov models. A compound model for the original image is defined that consists of a 2D noncausal Gauss-Markov random field to represent the homogeneous regions and a line process to represent the discontinuities. A deterministic algorithm based on the GNC formulation is derived to obtain a near-optimal maximum a posteriori probability estimate of images corrupted by additive Gaussian noise. >

Relaxation algorithms for MAP estimation of gray-level images with multiplicative noise

The authors present a comparison between stochastic and deterministic relaxation algorithms for maximum a posteriori estimation of gray-level images modeled by noncausal Gauss-Markov random fields (GMRF) and corrupted by film grain noise. The degradation involves nonlinear transformation and multiplicative noise. Parameters for the GMRF model were estimated from the original image using maximum-likelihood techniques. To overcome modeling errors, a constraint minimization approach is suggested for estimating the parameters to ensure the positivity of the power spectral density function. Real image experiments with various noise variances and magnitudes of the nonlinear transformation are presented. >

Graduated nonconvexity algorithm for image estimation using compound Gauss Markov field models

The authors describe the development of a deterministic algorithm for obtaining the global maximum a posteriori probability (MAP) estimate from an image corrupted by additive Gaussian noise. The MAP algorithm requires the probability density function of the original undegraded image and the corrupting noise. It is assumed that the original image is represented by a compound model consisting of a 2-D noncausal Gaussian-Markov random field (GMRF) to represent the homogeneous regions and a line process model to represent the discontinuities. The MAP algorithm is written in terms of the compound GMRF model parameters. The solution to the MAP equations is realized by a deterministic relaxation algorithm that is an extension of the graduated nonconvexity (GNC) algorithm and finds the global MAP estimate in a small number of iterations. As a byproduct, the line process configuration determined by the MAP estimate produces an accurate edge map without any additional cost. Experimental results are given to illustrate the usefulness of the method.<<ETX>>

Noncausal 2-D spectrum estimation for direction finding

A two-dimensional noncausal autoregressive (NCAR) plus additive noise model-based spectrum estimation method is presented for planar array data typical of signals encountered in array processing applications. Since the likelihood function for NCAR plus noise data is nonlinear in the model parameters and is further complicated by the unknown variance of the additive noise, computationally intensive gradient search algorithms are required for computing the estimates. If a doubly periodic lattice is assumed, the complexity of the approximate maximum likelihood (ML) equation is significantly reduced without destroying the theoretical asymptotic properties of the estimates and degrading the observed accuracy of the estimated spectra. Initial conditions for starting the approximate ML computation are suggested. Experimental results that can be used to evaluate the signal-plus-noise approach and compare its performance to those of signal-only methods are presented for Gaussian and simulated planar array data. Statistics of estimated spectrum parameters are given, and estimated spectra for signals with close spatial frequencies are shown. The approximate ML parameter estimates asymptotic properties, such as consistency and normality, are established, and lower bounds for the estimates errors are derived, assuming that the data are Gaussian. >

A mixed-signal VLSI neuroprocessor for image restoration

An analog systolic architecture that uses multiple neuroprocessors for image restoration is presented. For a two-dimensional image, parallel processing is performed for different rows of pixel data and pipelined processing is performed on each row of pixel data. For the image restoration neuroprocessor, local data computation is executed by analog circuitry to achieve full parallelism and to minimize power dissipation. Interprocessor communication is carried out in the digital format to maintain strong signal strength across the chip boundary and to allow multichip operation for high-speed image processing. >

A model based approach for 2-D MEPS analysis

A stochastic model based approach for 2-D MEPS analysis of a given finite uniform array data is presented. The method consists of fitting an appropriate 2-D noncausal Gaussian Markov Random Field (GMRF) model to the given data using maximum likelihood (ML) method for parameter estimation. The nonlinear criterion function used for ML estimation is similar in structure to the function arising in the algebraic approach of Lang and McClellan [1]. An example is given to illustrate the usefulness of our approach.

Correction to 'Estimation and choice of Neighbors in Spatial-Interaction Models of Images'

S +(j/N(i) = (UP,l)) transmit MSG(i, S) to all nodes j such that (i, j) is up For MSG( j, S) if d, =NO then perform START if d, = NO then d, + YES for all k such that k E S, N;‘(k) + (uP,~) transmit MSG( j, S) to all k such that (i, k) is up. For FAIL(I) if di = YES then N,‘(I) + (DOWN,N~(/) . t + 1) transmit MSG(DOWN, i I, N/(l) t) to all k such that (i, k) is up. For WAKE([) if di = YES then N,‘(I) +(YES, N/(I) . t + 1) transmit MSGQJP, i, I, N/([) t) to all k such that k f I and (i, k) is up W + ((e, S, Y, q)/(e, q) = N:(r) for all s, r such that r E S, and N!(r). t > 1) transmit MSGfi, W) to 1. For MSG(DOWN, j, k, t) if di = YES and N!(k). t i t then N/(k) + (DOWN, t) transmit MSG(DOWN, j, k, t) to all I such that (i, I) is up. For MSG(up, j, k, t) if dj = YES and N,‘(k). t i t then N/(k) + (UP, t) transmit MSGQJP, j, k, t) to all I such that (i, I) is up. For MSGUPDATE( j, W) if d, = NO then perform START for all (e, s, r, q) E W if N,S(r) t < q then N:(r) + (e, q), d, + YES else W +W ((e, s, r, q)) if W*O then transmit MSGUPDATE( j, W) to all I such that (i, I) is up.