Structured linear algebra problems in adaptive optics imaging

Johnathan M. Bardsley; Sarah Knepper; James Nagy

doi:10.1007/s10444-011-9172-9

Structured linear algebra problems in adaptive optics imaging

Johnathan M. Bardsley, Sarah Knepper, James Nagy

Mathematics

Emory University

Research output: Contribution to journal › Article › peer-review

19 Scopus citations

Abstract

A main problem in adaptive optics is to reconstruct the phase spectrum given noisy phase differences. We present an efficient approach to solve the least-squares minimization problem resulting from this reconstruction, using either a truncated singular value decomposition (TSVD)-type or a Tikhonov-type regularization. Both of these approaches make use of Kronecker products and the generalized singular value decomposition. The TSVD-type regularization operates as a direct method whereas the Tikhonov-type regularization uses a preconditioned conjugate gradient type iterative algorithm to achieve fast convergence.

Original language	English
Pages (from-to)	103-117
Number of pages	15
Journal	Advances in Computational Mathematics
Volume	35
Issue number	2
DOIs	https://doi.org/10.1007/s10444-011-9172-9
State	Published - Nov 2011

Keywords

Adaptive optics
Generalized singular value decomposition
Image deblurring
Kronecker product
LSQR
Preconditioning
Tikhonov regularization
Truncated singular value decomposition
Wavefront reconstruction

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10.1007/s10444-011-9172-9

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title = "Structured linear algebra problems in adaptive optics imaging",

abstract = "A main problem in adaptive optics is to reconstruct the phase spectrum given noisy phase differences. We present an efficient approach to solve the least-squares minimization problem resulting from this reconstruction, using either a truncated singular value decomposition (TSVD)-type or a Tikhonov-type regularization. Both of these approaches make use of Kronecker products and the generalized singular value decomposition. The TSVD-type regularization operates as a direct method whereas the Tikhonov-type regularization uses a preconditioned conjugate gradient type iterative algorithm to achieve fast convergence.",

keywords = "Adaptive optics, Generalized singular value decomposition, Image deblurring, Kronecker product, LSQR, Preconditioning, Tikhonov regularization, Truncated singular value decomposition, Wavefront reconstruction",

author = "Bardsley, {Johnathan M.} and Sarah Knepper and James Nagy",

note = "Funding Information: Research of first author was supported by the National Science Foundation under grant DMS-0915107. Research of third author was supported by the National Science Foundation under grant DMS-0811031, and the Air Force Office of Scientific Research under grant FA9550-09-1-0487.",

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AU - Bardsley, Johnathan M.

AU - Knepper, Sarah

AU - Nagy, James

N1 - Funding Information: Research of first author was supported by the National Science Foundation under grant DMS-0915107. Research of third author was supported by the National Science Foundation under grant DMS-0811031, and the Air Force Office of Scientific Research under grant FA9550-09-1-0487.

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N2 - A main problem in adaptive optics is to reconstruct the phase spectrum given noisy phase differences. We present an efficient approach to solve the least-squares minimization problem resulting from this reconstruction, using either a truncated singular value decomposition (TSVD)-type or a Tikhonov-type regularization. Both of these approaches make use of Kronecker products and the generalized singular value decomposition. The TSVD-type regularization operates as a direct method whereas the Tikhonov-type regularization uses a preconditioned conjugate gradient type iterative algorithm to achieve fast convergence.

AB - A main problem in adaptive optics is to reconstruct the phase spectrum given noisy phase differences. We present an efficient approach to solve the least-squares minimization problem resulting from this reconstruction, using either a truncated singular value decomposition (TSVD)-type or a Tikhonov-type regularization. Both of these approaches make use of Kronecker products and the generalized singular value decomposition. The TSVD-type regularization operates as a direct method whereas the Tikhonov-type regularization uses a preconditioned conjugate gradient type iterative algorithm to achieve fast convergence.

KW - Adaptive optics

KW - Generalized singular value decomposition

KW - Image deblurring

KW - Kronecker product

KW - LSQR

KW - Preconditioning

KW - Tikhonov regularization

KW - Truncated singular value decomposition

KW - Wavefront reconstruction

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JO - Advances in Computational Mathematics

JF - Advances in Computational Mathematics

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Structured linear algebra problems in adaptive optics imaging

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