3,722 research outputs found
Fixing Nonconvergence of Algebraic Iterative Reconstruction with an Unmatched Backprojector
We consider algebraic iterative reconstruction methods with applications in
image reconstruction. In particular, we are concerned with methods based on an
unmatched projector/backprojector pair; i.e., the backprojector is not the
exact adjoint or transpose of the forward projector. Such situations are common
in large-scale computed tomography, and we consider the common situation where
the method does not converge due to the nonsymmetry of the iteration matrix. We
propose a modified algorithm that incorporates a small shift parameter, and we
give the conditions that guarantee convergence of this method to a fixed point
of a slightly perturbed problem. We also give perturbation bounds for this
fixed point. Moreover, we discuss how to use Krylov subspace methods to
efficiently estimate the leftmost eigenvalue of a certain matrix to select a
proper shift parameter. The modified algorithm is illustrated with test
problems from computed tomography
The Optimal Projection Equations for Static and Dynamic Output Feedback: The Singular Case
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/57882/1/OptProjSingularTAC1987.pd
The Optimal Projection Equations for Reduced-Order State Estimation: The Singular Measurement Noise Case
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/57879/1/OptProjSingStateEstTAC1987.pd
Geometry of oblique projections
Let A be a unital C*-algebra. Denote by P the space of selfadjoint
projections of A. We study the relationship between P and the spaces of
projections P_a determined by the different involutions #_a induced by positive
invertible elements a in A. The maps f_p: P \to P_a sending p to the unique q
in P_a with the same range as p and \Omega_a: P_a \to P sending q to the
unitary part of the polar decomposition of the symmetry 2q-1 are shown to be
diffeomorphisms. We characterize the pairs of idempotents q, r in A with
|q-r|<1 such that there exists a positive element a in A verifying that q, r
are in P_a. In this case q and r can be joined by an unique short geodesic
along the space of idempotents Q of A.Comment: 25 pages, Latex, to appear in Studia Mathematic
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