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Martin Rumpf
lecture course at the summerschool
july 3rd – 7th, Evolène
Partial Differential Equationsin Image and Surface Processing
motivation
imaging is geometry processing
geometry processing can benefit from imaging
ReferencesSapiro, Geometric partial differential equations and image analysisCambridge University Press, 2001
Aubert, Kornprobst, Mathematical Problems in Image ProcessingSpringer, 2002
Osher, Fedkiw, Level Set Methods and dynamic implicit surfacesSpringer, 2003
Sethian, Level set methods and fast marching methods,Cambridge University Press, 1999
Alvarez, Guichard, Lions, Morel, Coll,Axioms and Fundamental Equations of Image ProcessingArch. Ration. Mech. Anal. 123, 199-257, 1993
geometry images
[Gu, Gortler, Hoppe ´02]
parametrization
surface fairing
isotropic denoising
anisotropic denoising
3d ultra sound
denoising images
noisy initial data denoised image
anisotropicdenoising
cartoon extraction based on anisotropic functionals
aerial imagesisotropic anisotropic
orientation
surface / image restoration based on anisotropic area functionals
noise corruptedMR angiography
surface restoration
image restoration (inpainting)
surface matching
?
image matching
a variety of image modalities:
CT, MRI (T1,T2), FLAIR, PET, ....
CTMRT
Matching image morphologies
= Matching of image contours(edge surfaces and regular contours)
matchedinitial mismatchwith edge set
PETCT
matching CT and PET images
numerical relaxation
matching brains and cortical surfaces of two different patients
explicit surfaces (notation)
implicit surfaces (notation)
Finite elements in
Finite elements on explicit surfaces
Finite elements on ensemble of level sets
An axiomatic approach to scale space
[Alvarez, Guichard, Lions, Morel, Coll ´92]
[Rec][Trans][Comp][Loc]
[Reg]
[GS][G]
[Iso]
surface fairing
anisotropic denoising
noisy data timestep 1 timestep 2 timestep 7
denoising 3D images
original image
anisotropic diffusion(Cf. [Weickert ´98])
anisotropic geometric diffusion
Perona Malikmodel
MCM
denoising 3D images
3d ultra sound
denoising 3D images
noisy initial data denoised image
anisotropicdenoising
morphological image denoising in 4D
gradient descent1D
>1D
infinite dimensional problems
restoration of surfacesgradient descent:
Willmore flow
cf. [Ballester et al. 01],[Masnou, Morel ´98],[Mumford, Nitzberg ´96]
image restoration (inpainting)
anisotropic energy functionals and gradient descent
[Belletini, Paolini´96]
classification:
cf. [Esedoglu, Osher ´03]
curve smoothing
anisotropic isotropic
classifikation:
surface fairing
surface / image restoration based on anisotropic area functionals
noise corruptedMR angiography
cartoon extraction based on anisotropic functionals
aerial imagesisotropic anisotropic
orientation
matching surfaces
physical interpretation
(tangential distortion) (normal bending)
(matching of feature sets)
deformation on the parameter domain and on the surface
matching regular contour surfaces
level sets
[Droske, R.´03]
mismatch
FLAIR
drawback
MR
matching image morphologies
matching singular edge surfacesand regular contour surfaces
in explicit:
matching edge sets via a level set approach
[Droske, Ring ´05]
[Mumford, Shah ´86 ]
recall (free discontinuity problem):
phase field approximation [Ambrosio, Tortorelli ´91]
singular morphology matching via a phase field approach
[Droske´05], [Droske, Ring, R.´05]
matchedinitial mismatchwith edge set
PETCT
matching shapes of CT and PET images
[Berkels, Droske, Han, Hornegger, R. ´06]
matching cortical surfaces
MRI
numerical relaxation
matching brains and cortical surfaces of two different patients
matching edge sets and regular contour sets
= matching singular and regular morphology
[Droske, R. ´05]
initial mismatch
MRI T1 FLAIR initial mismatch
initial final
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