10,667 research outputs found
Quantifying admissible undersampling for sparsity-exploiting iterative image reconstruction in X-ray CT
Iterative image reconstruction (IIR) with sparsity-exploiting methods, such
as total variation (TV) minimization, investigated in compressive sensing (CS)
claim potentially large reductions in sampling requirements. Quantifying this
claim for computed tomography (CT) is non-trivial, because both full sampling
in the discrete-to-discrete imaging model and the reduction in sampling
admitted by sparsity-exploiting methods are ill-defined. The present article
proposes definitions of full sampling by introducing four sufficient-sampling
conditions (SSCs). The SSCs are based on the condition number of the system
matrix of a linear imaging model and address invertibility and stability. In
the example application of breast CT, the SSCs are used as reference points of
full sampling for quantifying the undersampling admitted by reconstruction
through TV-minimization. In numerical simulations, factors affecting admissible
undersampling are studied. Differences between few-view and few-detector bin
reconstruction as well as a relation between object sparsity and admitted
undersampling are quantified.Comment: Revised version that was submitted to IEEE Transactions on Medical
Imaging on 8/16/201
Deterministic Versus Randomized Kaczmarz Iterative Projection
Kaczmarz's alternating projection method has been widely used for solving a
consistent (mostly over-determined) linear system of equations Ax=b. Because of
its simple iterative nature with light computation, this method was
successfully applied in computerized tomography. Since tomography generates a
matrix A with highly coherent rows, randomized Kaczmarz algorithm is expected
to provide faster convergence as it picks a row for each iteration at random,
based on a certain probability distribution. It was recently shown that picking
a row at random, proportional with its norm, makes the iteration converge
exponentially in expectation with a decay constant that depends on the scaled
condition number of A and not the number of equations. Since Kaczmarz's method
is a subspace projection method, the convergence rate for simple Kaczmarz
algorithm was developed in terms of subspace angles. This paper provides
analyses of simple and randomized Kaczmarz algorithms and explain the link
between them. It also propose new versions of randomization that may speed up
convergence
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