We address the problem of restoring an image from its noisy convolutions with two or more unknown finite impulse response (FIR) filters. We develop theoretical results about the existence and uniqueness of solutions, and show that under some generically true assumptions, both the filters and the image can be determined exactly in the absence of noise, and stably estimated in its presence. We present efficient algorithms to estimate the blur functions and their sizes. These algorithms are of two types, subspace-based and likelihood-based, and are extensions of techniques proposed for the solution of the multichannel blind deconvolution problem in one dimension. We present memory and computation-efficient techniques to handle the very large matrices arising in the two-dimensional (2-D) case. Once the blur functions are determined, they are used in a multichannel deconvolution step to reconstruct the unknown image. The theoretical and practical implications of edge effects, and "weakly exciting" images are examined. Finally, the algorithms are demonstrated on synthetic and real data.

We address the problem of restoring an image from its noisy convolutions with two or more blur functions (channels). Deconvolution from multiple blurs is, in general, better conditioned than from a single blur, and can be performed without regularization for moderate noise levels. We characterize the problem of missing data at the image boundaries, and show that perfect reconstruction is impossible (even in the no-noise case) almost surely unless there are at least three channels. Conversely, when there are at least three channels, we show that perfect reconstruction is not only possible almost surely in the absence of noise, but also that it can be accomplished by finite impulse response (FIR) filtering. Such FIR reconstruction is vastly more efficient computationally than the least-squares solution, and is suitable for low noise levels. Even in the high-noise case, the estimates obtained by FIR filtering provide useful starting points for iterative least-squares algorithms. We present results on the minimum possible sizes of such deconvolver filters. We derive expressions for the mean-square errors in the FIR reconstructions, and show that performance comparable to that of the least-squares reconstruction may be obtained with relatively small deconvolver filters. Finally, we demonstrate the FIR reconstruction on synthetic and real data.

We present two "fast" approaches to the NP-hard problem of computing a maximally sparse approximate solution to linear inverse problems, also known as the best subset selection. The first approach, a heuristic, is an iterative algorithm globally convergent to sparse elements of any given convex, compact S/spl sub/R/sup mx/. We demonstrate its effectiveness in bandlimited extrapolation and in sparse filter design. The second approach is a polynomial-time greedy sequential backward elimination algorithm. We show that if A has full column rank and /spl epsiv/ is small enough, then the algorithm will find the sparsest x satisfying /spl par/Ax-b/spl par//spl les//spl epsiv/, if such exists.

We present an iterative algorithm for computing sparse solutions (or sparse approximate solutions) to linear inverse problems. The algorithm is intended to supplement the existing arsenal of techniques. It is shown to converge to the local minima of a function of the form used for picking out sparse solutions, and its connection with existing techniques explained. Finally, it is demonstrated on subset selection and deconvolution examples. The fact that the proposed algorithm is sometimes successful when existing greedy algorithms fail is also demonstrated.

We address the problem of restoring an image from its noisy convolutions with two or more unknown blur functions. We extend some of the results developed for the multichannel blind deconvolution problem in one dimension. When the unknown blur functions have no common factors, we present algorithms to estimate them quickly and accurately. Once the blur functions are available, they can be used to reconstruct the unknown image. We show that, under certain conditions, the last step can be achieved by FIR filtering with a perfect reconstruction filter bank. Some results on the effects of noise on these algorithms are also presented.

This paper addresses the exploratory visualization of multispectral image data. In such data, each component of the vector pixel corresponds to a different imaging modality or a different combination of imaging parameters, and may provide different levels of contrast sensitivity between different regions of the underlying image. We address the problem of presenting this multidimensional data to human observers by synthesizing a display matched to their visual capabilities. Specifically, we seek to determine a data-adaptive linear projection of the vector data to one dimension that produces a grayscale image providing maximum discrimination between the different regions of the underlying object. The approach is equivalent to the extraction of the best linear feature of the vector field. Several new feature-extraction criteria that take into account both the spatial and multivariate structures of the data are proposed and illustrated by simulations on test images.

This paper addresses the problem of the visualization of vector-valued images. Attempting to synthesize a display matched to the capabilities of a human observer, we have reduced the problem to the extraction of the best linear feature of the vector field. In previous work, we have proposed and demonstrated several new nonparametric feature extraction criteria (projection indices) that make use of both the spatial and multivariate structures of the data. We present a theoretical analysis of these projection indices and a Monte-Carlo study of their effectiveness. The study uses performance measures derived from a decision-theoretic model of the human observer.>

This paper addresses the problem of the visualization of vector-valued images. It is attempted to synthesize a display matched to the capabilities of a human observer. The problem is reduced to the extraction of the best linear feature of the vector field. Several new nonparametric feature extraction criteria (projection indices) that make use of both the spatial and multivariate structures of the data have been recently proposed and demonstrated. A theoretical analysis of these projection indices and a Monte-Carlo study of their effectiveness is presented here. The study uses performance measures derived from a decision-theoretic model of the human observer.