Restoration of manifold-valued images by half-quadratic minimization

Ronny BERGMANN; Raymond H. CHAN; Ralf HIELSCHER; Johannes PERSCH; Gabriele STEIDL

doi:10.3934/ipi.2016001

Restoration of manifold-valued images by half-quadratic minimization

Ronny BERGMANN^*, Raymond H. CHAN, Ralf HIELSCHER, Johannes PERSCH, Gabriele STEIDL

^*Corresponding author for this work

Research output: Journal Publications › Journal Article (refereed) › peer-review

51 Citations (Scopus)

Abstract

The paper addresses the generalization of the half-quadratic minimization method for the restoration of images having values in a complete, connected Riemannian manifold. We recall the half-quadratic minimization method using the notation of the c-transform and adapt the algorithm to our special variational setting. We prove the convergence of the method for Hadamard spaces. Extensive numerical examples for images with values on spheres, in the rotation group SO(3), and in the manifold of positive definite matrices demonstrate the excellent performance of the algorithm. In particular, the method with SO(3)-valued data shows promising results for the restoration of images obtained from Electron Backscattered Diffraction which are of interest in material science.

Original language	English
Pages (from-to)	281-304
Number of pages	24
Journal	Inverse Problems and Imaging
Volume	10
Issue number	2
DOIs	https://doi.org/10.3934/ipi.2016001
Publication status	Published - May 2016
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2016 American Institute of Mathematical Sciences.

Funding

Funding by the DFG within the RTG GrK 1932 Stochastic Models for Innovations in the Engineering Sciences, project area P3, and in the project STE 571/13-1 & BE 5888/2-1 is gratefully acknowledged. Furthermore RC gratefully acknowledges support by the HKRGC Grants No. CUHK300614, CUHK2/CRF/11G, AoE/M-05/12; CUHK DAG No. 4053007, and FIS Grant No. 1907303.

Keywords

DT-MRI
EBSD
Half-uadratic minimization
Manifold-valued data
Quasi-Newton method
Variational restoration methods

Access to Document

10.3934/ipi.2016001

Cite this

@article{553dcaab203d4701b6fa221047f46f50,

title = "Restoration of manifold-valued images by half-quadratic minimization",

abstract = "The paper addresses the generalization of the half-quadratic minimization method for the restoration of images having values in a complete, connected Riemannian manifold. We recall the half-quadratic minimization method using the notation of the c-transform and adapt the algorithm to our special variational setting. We prove the convergence of the method for Hadamard spaces. Extensive numerical examples for images with values on spheres, in the rotation group SO(3), and in the manifold of positive definite matrices demonstrate the excellent performance of the algorithm. In particular, the method with SO(3)-valued data shows promising results for the restoration of images obtained from Electron Backscattered Diffraction which are of interest in material science.",

keywords = "DT-MRI, EBSD, Half-uadratic minimization, Manifold-valued data, Quasi-Newton method, Variational restoration methods",

author = "Ronny BERGMANN and CHAN, {Raymond H.} and Ralf HIELSCHER and Johannes PERSCH and Gabriele STEIDL",

note = "Publisher Copyright: {\textcopyright} 2016 American Institute of Mathematical Sciences.",

year = "2016",

month = may,

doi = "10.3934/ipi.2016001",

language = "English",

volume = "10",

pages = "281--304",

journal = "Inverse Problems and Imaging",

issn = "1930-8337",

publisher = "American Institute of Mathematical Sciences",

number = "2",

}

TY - JOUR

T1 - Restoration of manifold-valued images by half-quadratic minimization

AU - BERGMANN, Ronny

AU - CHAN, Raymond H.

AU - HIELSCHER, Ralf

AU - PERSCH, Johannes

AU - STEIDL, Gabriele

PY - 2016/5

Y1 - 2016/5

N2 - The paper addresses the generalization of the half-quadratic minimization method for the restoration of images having values in a complete, connected Riemannian manifold. We recall the half-quadratic minimization method using the notation of the c-transform and adapt the algorithm to our special variational setting. We prove the convergence of the method for Hadamard spaces. Extensive numerical examples for images with values on spheres, in the rotation group SO(3), and in the manifold of positive definite matrices demonstrate the excellent performance of the algorithm. In particular, the method with SO(3)-valued data shows promising results for the restoration of images obtained from Electron Backscattered Diffraction which are of interest in material science.

AB - The paper addresses the generalization of the half-quadratic minimization method for the restoration of images having values in a complete, connected Riemannian manifold. We recall the half-quadratic minimization method using the notation of the c-transform and adapt the algorithm to our special variational setting. We prove the convergence of the method for Hadamard spaces. Extensive numerical examples for images with values on spheres, in the rotation group SO(3), and in the manifold of positive definite matrices demonstrate the excellent performance of the algorithm. In particular, the method with SO(3)-valued data shows promising results for the restoration of images obtained from Electron Backscattered Diffraction which are of interest in material science.

KW - DT-MRI

KW - EBSD

KW - Half-uadratic minimization

KW - Manifold-valued data

KW - Quasi-Newton method

KW - Variational restoration methods

UR - http://www.scopus.com/inward/record.url?scp=84970984326&partnerID=8YFLogxK

U2 - 10.3934/ipi.2016001

DO - 10.3934/ipi.2016001

M3 - Journal Article (refereed)

AN - SCOPUS:84970984326

SN - 1930-8337

VL - 10

SP - 281

EP - 304

JO - Inverse Problems and Imaging

JF - Inverse Problems and Imaging

IS - 2

ER -

Restoration of manifold-valued images by half-quadratic minimization

Abstract

Bibliographical note

Funding

Keywords

Access to Document

Other files and links

Fingerprint

Cite this