A preconditioning technique for fast iterative image restoration

Chalidabhongse, Junavit

doi:10.1109/iscas.1992.230519

Cited by 2 publications

(2 citation statements)

References 7 publications

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“…The construction of the preconditioners for the ill-conditioned image restoration problems that we consider use the underlying approximations of Section 2, together with the fact that any BCCB matrix has the spectral factorization given in (3). (5) where the scalar a is called a regularization parameter and the matrix L is called a regularization operator. L is usually chosen to be the identity matrix or a discretization of a 2-D differentiation operator, chosen to enforce a smoothing criteria on the restored image.…”

Section: Regularization and Fft-based Preconditioningmentioning

confidence: 99%

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<title>Image restoration using fast Fourier and wavelet transforms</title>

Harrod¹,

Nagy

Plemmons

1994

Substance Identification Analytics

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Image restoration can be modeled as a discrete, ill-posed, 2-D inverse problem where, if the blurring function (PSF) h is spatially invariant, then the blurring operator will have a displacement stracture.We consider solving this image restoration problem by a preconditioned conjugate gradient least squares algorithm. Because the problem is ill-posed, it is necessary to use regularization in order to compute a reasonable approrimation to the trae image. We describe preconditioning techniques for two different regularization procedures; Tikhonov regularization, and iterative regularization with the conjugate gradient algorithm. In particular, we show how one can incorporate preconditioning, using a block circulant operator, into the conjugate gradient algorithm for these regularization schemes. We thus obtain new fast and stable 2-D FFT based iterative methods in which preconditioning takes place in the Fourier domain, while the iteration takes place in the spatial domain. If the least squares restoration problem is M-by-N, then the overall numerical complexity of our algorithm is O(M log N). Moreover, the preconditioning does not affect the resolution of the restored image. Numerical tests comparing several underlying circulant preconditioners, as well as tests with and without our preconditioned regularization procedure, are reported for a model problem. Also discussed are implementations of the algorithms on the new Cray massively parallel processor (MPP) system. In other more general (i. e. non spatially invariant PSF) situations, it is often still the case that the blurring matrix is generated by a "smooth" PSF h on a regular grid. For general non-periodic situations, the FFT cannot be effectively used. Here, our purpose is to explore the possible use of new wavelet transform based preconditioned conjugate gradient iterative methods of solution for these problems. The decompositions and subsequent iterations may be done in O(M log N) or even 0(M) operations for M-by-N operators H, by preconditioning with wavelets.

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Section: Regularization and Fft-based Preconditioningmentioning

confidence: 99%

“…Recently, the conjugate gradient (CC) method has been used to solve the image restoration problem [3,4,5,26]. The classical CG method is an iterative method of fundamental importance for solving Hermitian positive definite systems of linear equations Hf = g, cf., Golub and Van Loan [15].…”

Section: Introductionmentioning

confidence: 99%

<title>Image restoration using fast Fourier and wavelet transforms</title>

Harrod¹,

Nagy

Plemmons

1994

Substance Identification Analytics

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Iterative image restoration using approximate inverse preconditioning

Nagy

Plemmons

Torgersen

1996

IEEE Trans. on Image Process.

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Removing a linear shift-invariant blur from a signal or image can be accomplished by inverse or Wiener filtering, or by an iterative least-squares deblurring procedure. Because of the ill-posed characteristics of the deconvolution problem, in the presence of noise, filtering methods often yield poor results. On the other hand, iterative methods often suffer from slow convergence at high spatial frequencies. This paper concerns solving deconvolution problems for atmospherically blurred images by the preconditioned conjugate gradient algorithm, where a new approximate inverse preconditioner is used to increase the rate of convergence. Theoretical results are established to show that fast convergence can be expected, and test results are reported for a ground-based astronomical imaging problem.

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A preconditioning technique for fast iterative image restoration

Cited by 2 publications

References 7 publications

<title>Image restoration using fast Fourier and wavelet transforms</title>

<title>Image restoration using fast Fourier and wavelet transforms</title>

Iterative image restoration using approximate inverse preconditioning

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