9,511 research outputs found
Fast Mojette Transform for Discrete Tomography
A new algorithm for reconstructing a two dimensional object from a set of one
dimensional projected views is presented that is both computationally exact and
experimentally practical. The algorithm has a computational complexity of O(n
log2 n) with n = N^2 for an NxN image, is robust in the presence of noise and
produces no artefacts in the reconstruction process, as is the case with
conventional tomographic methods. The reconstruction process is approximation
free because the object is assumed to be discrete and utilizes fully discrete
Radon transforms. Noise in the projection data can be suppressed further by
introducing redundancy in the reconstruction. The number of projections
required for exact reconstruction and the response to noise can be controlled
without comprising the digital nature of the algorithm. The digital projections
are those of the Mojette Transform, a form of discrete linogram. A simple
analytical mapping is developed that compacts these projections exactly into
symmetric periodic slices within the Discrete Fourier Transform. A new digital
angle set is constructed that allows the periodic slices to completely fill all
of the objects Discrete Fourier space. Techniques are proposed to acquire these
digital projections experimentally to enable fast and robust two dimensional
reconstructions.Comment: 22 pages, 13 figures, Submitted to Elsevier Signal Processin
Projections Onto Convex Sets (POCS) Based Optimization by Lifting
Two new optimization techniques based on projections onto convex space (POCS)
framework for solving convex and some non-convex optimization problems are
presented. The dimension of the minimization problem is lifted by one and sets
corresponding to the cost function are defined. If the cost function is a
convex function in R^N the corresponding set is a convex set in R^(N+1). The
iterative optimization approach starts with an arbitrary initial estimate in
R^(N+1) and an orthogonal projection is performed onto one of the sets in a
sequential manner at each step of the optimization problem. The method provides
globally optimal solutions in total-variation, filtered variation, l1, and
entropic cost functions. It is also experimentally observed that cost functions
based on lp, p<1 can be handled by using the supporting hyperplane concept
OPED reconstruction algorithm for limited angle problem
The structure of the reconstruction algorithm OPED permits a natural way to
generate additional data, while still preserving the essential feature of the
algorithm. This provides a method for image reconstruction for limited angel
problems. In stead of completing the set of data, the set of discrete sine
transforms of the data is completed. This is achieved by solving systems of
linear equations that have, upon choosing appropriate parameters, positive
definite coefficient matrices. Numerical examples are presented.Comment: 17 page
Phase and TV Based Convex Sets for Blind Deconvolution of Microscopic Images
In this article, two closed and convex sets for blind deconvolution problem
are proposed. Most blurring functions in microscopy are symmetric with respect
to the origin. Therefore, they do not modify the phase of the Fourier transform
(FT) of the original image. As a result blurred image and the original image
have the same FT phase. Therefore, the set of images with a prescribed FT phase
can be used as a constraint set in blind deconvolution problems. Another convex
set that can be used during the image reconstruction process is the epigraph
set of Total Variation (TV) function. This set does not need a prescribed upper
bound on the total variation of the image. The upper bound is automatically
adjusted according to the current image of the restoration process. Both of
these two closed and convex sets can be used as a part of any blind
deconvolution algorithm. Simulation examples are presented.Comment: Submitted to IEEE Selected Topics in Signal Processin
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