Deflated Preconditioned Conjugate Gradients for Nonlinear Diffusion Image Enhancement

Xiujie Shan; Martin van Gijzen

doi:10.1007/978-3-030-55874-1_45

Deflated Preconditioned Conjugate Gradients for Nonlinear Diffusion Image Enhancement

^*Corresponding author for this work

Numerical Analysis

Research output: Chapter in Book/Conference proceedings/Edited volume › Conference contribution › Scientific › peer-review

3 Citations (Scopus)

44 Downloads (Pure)

Abstract

Nonlinear diffusion equations have been successfully used for image enhancement by reducing the noise in the image while protecting the edges. In discretized form, the denoising requires the solution of a sequence of linear systems. The underlying system matrices stem from a discrete diffusion operator with large jumps in the diffusion coefficients. As a result these matrices can be very ill-conditioned, which leads to slow convergence for iterative methods such as the Conjugate Gradient method. To speed-up the convergence we use deflation and preconditioning. The deflation vectors are defined by a decomposition of the image. The resulting numerical method is easy to implement and matrix-free. We evaluate the performance of the method on a simulated image and on a measured low-field MR image for various types of deflation vectors.

Original language	English
Title of host publication	Numerical Mathematics and Advanced Applications, ENUMATH 2019 - European Conference
Editors	Fred J. Vermolen, Cornelis Vuik
Publisher	Springer
Pages	459-468
Number of pages	10
ISBN (Print)	978-3-0305-5873-4
DOIs	https://doi.org/10.1007/978-3-030-55874-1_45
Publication status	Published - 2021
Event	European Conference on Numerical Mathematics and Advanced Applications, ENUMATH 2019 - Egmond aan Zee, Netherlands Duration: 30 Sept 2019 → 4 Oct 2019

Publication series

Name	Lecture Notes in Computational Science and Engineering
Volume	139
ISSN (Print)	1439-7358
ISSN (Electronic)	2197-7100

Conference

Conference	European Conference on Numerical Mathematics and Advanced Applications, ENUMATH 2019
Country/Territory	Netherlands
City	Egmond aan Zee
Period	30/09/19 → 4/10/19

Bibliographical note

Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care
Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.

Access to Document

10.1007/978-3-030-55874-1_45

Shan_VanGijzen2021_Chapter_DeflatedPreconditionedConjugatFinal published version, 413 KB

Cite this

Shan, X., & van Gijzen, M. (2021). Deflated Preconditioned Conjugate Gradients for Nonlinear Diffusion Image Enhancement. In F. J. Vermolen, & C. Vuik (Eds.), Numerical Mathematics and Advanced Applications, ENUMATH 2019 - European Conference (pp. 459-468). (Lecture Notes in Computational Science and Engineering; Vol. 139). Springer. https://doi.org/10.1007/978-3-030-55874-1_45

@inproceedings{a192b305615e4aa0aaa2a37e94f83e0f,

title = "Deflated Preconditioned Conjugate Gradients for Nonlinear Diffusion Image Enhancement",

abstract = "Nonlinear diffusion equations have been successfully used for image enhancement by reducing the noise in the image while protecting the edges. In discretized form, the denoising requires the solution of a sequence of linear systems. The underlying system matrices stem from a discrete diffusion operator with large jumps in the diffusion coefficients. As a result these matrices can be very ill-conditioned, which leads to slow convergence for iterative methods such as the Conjugate Gradient method. To speed-up the convergence we use deflation and preconditioning. The deflation vectors are defined by a decomposition of the image. The resulting numerical method is easy to implement and matrix-free. We evaluate the performance of the method on a simulated image and on a measured low-field MR image for various types of deflation vectors.",

author = "Xiujie Shan and {van Gijzen}, Martin",

note = "Green Open Access added to TU Delft Institutional Repository {\textquoteleft}You share, we take care!{\textquoteright} – Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public. ; European Conference on Numerical Mathematics and Advanced Applications, ENUMATH 2019 ; Conference date: 30-09-2019 Through 04-10-2019",

year = "2021",

doi = "10.1007/978-3-030-55874-1_45",

language = "English",

isbn = "978-3-0305-5873-4",

series = "Lecture Notes in Computational Science and Engineering",

publisher = "Springer",

pages = "459--468",

editor = "Vermolen, {Fred J.} and Cornelis Vuik",

booktitle = "Numerical Mathematics and Advanced Applications, ENUMATH 2019 - European Conference",

}

Shan, X & van Gijzen, M 2021, Deflated Preconditioned Conjugate Gradients for Nonlinear Diffusion Image Enhancement. in FJ Vermolen & C Vuik (eds), Numerical Mathematics and Advanced Applications, ENUMATH 2019 - European Conference. Lecture Notes in Computational Science and Engineering, vol. 139, Springer, pp. 459-468, European Conference on Numerical Mathematics and Advanced Applications, ENUMATH 2019, Egmond aan Zee, Netherlands, 30/09/19. https://doi.org/10.1007/978-3-030-55874-1_45

Deflated Preconditioned Conjugate Gradients for Nonlinear Diffusion Image Enhancement. / Shan, Xiujie; van Gijzen, Martin.
Numerical Mathematics and Advanced Applications, ENUMATH 2019 - European Conference. ed. / Fred J. Vermolen; Cornelis Vuik. Springer, 2021. p. 459-468 (Lecture Notes in Computational Science and Engineering; Vol. 139).

Research output: Chapter in Book/Conference proceedings/Edited volume › Conference contribution › Scientific › peer-review

TY - GEN

T1 - Deflated Preconditioned Conjugate Gradients for Nonlinear Diffusion Image Enhancement

AU - Shan, Xiujie

AU - van Gijzen, Martin

N1 - Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.

PY - 2021

Y1 - 2021

N2 - Nonlinear diffusion equations have been successfully used for image enhancement by reducing the noise in the image while protecting the edges. In discretized form, the denoising requires the solution of a sequence of linear systems. The underlying system matrices stem from a discrete diffusion operator with large jumps in the diffusion coefficients. As a result these matrices can be very ill-conditioned, which leads to slow convergence for iterative methods such as the Conjugate Gradient method. To speed-up the convergence we use deflation and preconditioning. The deflation vectors are defined by a decomposition of the image. The resulting numerical method is easy to implement and matrix-free. We evaluate the performance of the method on a simulated image and on a measured low-field MR image for various types of deflation vectors.

AB - Nonlinear diffusion equations have been successfully used for image enhancement by reducing the noise in the image while protecting the edges. In discretized form, the denoising requires the solution of a sequence of linear systems. The underlying system matrices stem from a discrete diffusion operator with large jumps in the diffusion coefficients. As a result these matrices can be very ill-conditioned, which leads to slow convergence for iterative methods such as the Conjugate Gradient method. To speed-up the convergence we use deflation and preconditioning. The deflation vectors are defined by a decomposition of the image. The resulting numerical method is easy to implement and matrix-free. We evaluate the performance of the method on a simulated image and on a measured low-field MR image for various types of deflation vectors.

UR - http://www.scopus.com/inward/record.url?scp=85106436807&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-55874-1_45

DO - 10.1007/978-3-030-55874-1_45

M3 - Conference contribution

AN - SCOPUS:85106436807

SN - 978-3-0305-5873-4

T3 - Lecture Notes in Computational Science and Engineering

SP - 459

EP - 468

BT - Numerical Mathematics and Advanced Applications, ENUMATH 2019 - European Conference

A2 - Vermolen, Fred J.

A2 - Vuik, Cornelis

PB - Springer

T2 - European Conference on Numerical Mathematics and Advanced Applications, ENUMATH 2019

Y2 - 30 September 2019 through 4 October 2019

ER -

Deflated Preconditioned Conjugate Gradients for Nonlinear Diffusion Image Enhancement

Abstract

Publication series

Conference

Bibliographical note

Access to Document

Other files and links

Fingerprint

Cite this