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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 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)

Approximating non-Gaussian noise processes with Gaussian models is standard in data analysis. This is due in large part to the fact that Gaussian models yield parameter estimation problems of least squares form, which have been extensively studied both from the theoretical and computational points of view. In image processing applications, for example, data… (More)

In numerous applications of image processing, e.g. astronomical and medical imaging, data-noise is well-modeled by a Poisson distribution. This motivates the use of the negative-log Poisson likelihood function for data fitting. (The fact that application scientists in both astronomical and medical imaging regularly choose this function for data fitting… (More)

High-dimensional inverse problems present a challenge for Markov chain Monte Carlo (MCMC)-type sampling schemes. Typically, they rely on finding an efficient proposal distribution, which can be difficult for large-scale problems, even with adaptive approaches. Moreover, the au-tocorrelations of the samples typically increase with dimension, which leads to… (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)

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)

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)

Image reconstruction gives rise to some challenging large-scale constrained optimization problems. We consider a convex minimization problem with nonnegativity constraints that arises in astronomical imaging. To solve this problem, we use an efficient hybrid gradient projection-reduced Newton (active-set) method. By "reduced Newton," we mean that we take… (More)

The earth's atmosphere is not a perfect media through which to view objects in outer-space; turbulence in the atmospheric temperature distribution results in refractive index variations that interfere with the propagation of light. As a result, wavefronts are non-planar when they reach the ground. The deviation from planarity of a wavefront is known as… (More)