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The standard formulations of the Kalman filter (KF) and extended Kalman filter (EKF) require the storage and multiplication of matrices of size n × n, where n is the size of the state space, and the inversion of matrices of size m × m, where m is the size of the observation space. Thus when both m and n are large, implementation issues arise. In this paper,… (More)

The noise contained in data measured by imaging instruments is often primarily of Poisson type. This motivates, in many cases, the use of the Poisson negative-log likelihood function in place of the ubiquitous least squares data fidelity when solving image deblurring problems. We assume that the underlying blurring operator is compact, so that, as in the… (More)

In image processing applications, image intensity is often measured via the counting of incident photons emitted by the object of interest. In such cases, image data-noise is accurately modeled by a Poisson distribution. This motivates the use of Poisson maximum likelihood estimation for image reconstruction. However, when the underlying model equation is… (More)

The connection between Bayesian statistics and the technique of regularization for inverse problems has been given significant attention in recent years. For example, Bayes' Law is frequently used as motivation for variational regularization methods of Tikhonov type. In this setting , the regularization function corresponds to the negative-log of the prior… (More)

SUMMARY The standard formulations of the Kalman filter (KF) and extended Kalman filter (EKF) require storing and multiplication of matrices of size n × n, where n is the size of the state space, and the inversion of matrices of size m × m, where m is the size of the observation space. For large dimensions implementation issues arise. In this paper we… (More)

Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Abstract. High-dimensional inverse problems present a challenge for Markov… (More)

We consider a large-scale convex minimization problem with nonnegativity constraints that arises in astronomical imaging. We develop a cost functional which incorporates the statistics of the noise in the image data and Tikhonov regularization to induce stability. We introduce an efficient hybrid gradient projection-reduced Newton (active set) method. By "… (More)

We consider the problem of solving ill-conditioned linear systems Ax = b subject to the nonnegativity constraint x ≥ 0, and in which the vector b is a realization of a random vectorˆb, i.e. b is noisy. We explore what the statistical literature tells us about solving noisy linear systems; we discuss the effect that a substantial black background in the… (More)

In this paper, we present an algorithm for the restoration of images with an unknown, spatially-varying blur. Existing computational methods for image restoration require the assumption that the blur is known and/or spatially-invariant. Our algorithm uses a combination of techniques. First, we section the image, and then treat the sections as a sequence of… (More)

Image data is often collected by a charge coupled device (CCD) camera. CCD camera noise is known to be well-modeled by a Poisson distribution. If this is taken into account, the negative-log of the Poisson likelihood is the resulting data-fidelity function. We derive, via a Taylor series argument, a weighted least squares approximation of the negative-log… (More)