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