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This paper addresses the problem of both segmenting and reconstructing a noisy signal or image. The work is motivated by large problems arising in certain scientific applications, such as medical imaging. Two objectives for a segmentation and denoising algorithm are laid out: it should be computationally efficient and capable of generating statistics for the errors in the reconstruction and estimates of the boundary locations. The starting point for the development of a suitable algorithm is a variational approach to segmentation (Shah 1992). This paper then develops a precise statistical interpretation of a one dimensional (1-D) version of this variational approach to segmentation. The 1-D algorithm that arises as a result of this analysis is computationally efficient and capable of generating error statistics. A straightforward extension of this algorithm to two dimensions would incorporate recursive procedures for computing estimates of inhomogeneous Gaussian Markov random fields. Such procedures require an unacceptably large number of operations. To meet the objective of developing a computationally efficient algorithm, the use of previously developed multiscale statistical methods is investigated. This results in the development of an algorithm for segmenting and denoising which is not only computationally efficient but also capable of generating error statistics, as desired.

Computing the linear least-squares estimate of a high-dimensional random quantity given noisy data requires solving a large system of linear equations. In many situations, one can solve this system efficiently using the conjugate gradient (CG) algorithm. Computing the estimation error variances is a more intricate task. It is difficult because the error variances are the diagonal elements of a complicated matrix. This paper presents a method for using the conjugate search directions generated by the CG algorithm to obtain a converging approximation to the estimation error variances. The algorithm for computing the error variances falls out naturally from a novel estimation-theoretic interpretation of the CG algorithm. The paper discusses this interpretation and convergence issues and presents numerical examples.

Magnetic resonance imaging (MRI) has become a widely used research and clinical tool in the study of the human brain. The ability to accurately segment the MRI data set into homogeneous regions such as gray matter, white matter, and cerebro spinal fluid aids in morphological quantification of brain features. The large amount of data associated with typical MRI brain scans makes completely manual segmentation prohibitive on a large scale. We develop an estimation-theoretic interpretation of the segmentation problem which leads to a computationally efficient, statistically based recursive technique for its solution. Being statistically based, the method also provides associated measures of uncertainty of the resulting estimates, which are useful both for evaluation of the estimates as well as their combination with other sources of information.

Magnetic resonance imaging (MRI) has become a widely used research and clinical tool in the study of the human brain. The ability to robustly and accurately quantify repeatable morphological measures from such data is aided by the ability to accurately segment the MRI data set into homogeneous regions such as gray matter, white matter, and cerebro spinal fluid. The large amount of data associated with typical MRI scans makes completely manual segmentation prohibitive on a large scale. In this paper an efficient approach to the segmentation of such MR imagery is presented. The approach uses an estimation-theoretic interpretation of the segmentation problem to develop a computationally efficient, statistically-based recursive technique for its solution. Being statistically based, the method also provides associated measures of uncertainty of the resulting estimates, which are extremely important both for evaluation of the estimates as well as their combination with other sources of information.

This work presents a method for segmenting images based on gradients in the intensity function. Past approaches have centered on formulating the problem in the context of variational calculus as the minimization of a functional involving the image intensity and edge functions. Computational methods for finding the minima of such variational problems are prone to two shortfalls: they are often computationally intensive and almost always incapable of computing error statistics associated with the segmentation. Using a particular variational formulation as a starting point, this paper presents a derivation of an associated statistical formulation using multiscale models. The result is an algorithm which is fast and capable of computing error statistics.