Martin Burger Institut für Numerische und Angewandte Mathematik European Institute for Molecular Imaging CeNoS Total Variation and related Methods Segmentation and Level Sets

Martin Burger Total Variation 2 Cetraro, September 2008 Segmentation Basic problem: given a (possibly noisy) image, find the - Edges (discontinuities in grayvalue) - Objects (regions with different key properties) included in the image Distinction into edge-based or region-based segmentation We mainly focus on region-based segmentation, which has various applications in computer vision and in particular medical imaging (segmenting organs, finding tumours, …)

Martin Burger Total Variation 3 Cetraro, September 2008 Segmentation Start with classical (edge-based) segmentation model: Mumford-Shah model Decomposition of image into smooth parts and edges

Martin Burger Total Variation 4 Cetraro, September 2008 Segmentation Match smooth part to image, penalty enforcing smoothness Second penalty needed to avoid segmentation of noisy parts Minimization

Martin Burger Total Variation 5 Cetraro, September 2008 Segmentation Region-based version: 2 regions

Martin Burger Total Variation 6 Cetraro, September 2008 Segmentation Mumford-Shah version

Martin Burger Total Variation 7 Cetraro, September 2008 Segmentation Extreme case: Chan-Vese model

Martin Burger Total Variation 8 Cetraro, September 2008 Segmentation Chan-Vese

Martin Burger Total Variation 9 Cetraro, September 2008 Segmentation How to minimize functionals with respect to shapes ? - Level Set Methods - Total Variation Formulations

Martin Burger Total Variation 10 Cetraro, September 2008 Segmentation Level Set Method

Martin Burger Total Variation 11 Cetraro, September 2008 Segmentation Level Set

Martin Burger Total Variation 12 Cetraro, September 2008 Segmentation Perimeter: Basic Relation by Co-Area Formula

Martin Burger Total Variation 13 Cetraro, September 2008 Segmentation Co-Area Formula

Martin Burger Total Variation 14 Cetraro, September 2008 Segmentation Level Set Formulation: minimize

Martin Burger Total Variation 15 Cetraro, September 2008 Segmentation Approximate Heaviside-Function

Martin Burger Total Variation 16 Cetraro, September 2008 Segmentation Optimality with respect to level set function

Martin Burger Total Variation 17 Cetraro, September 2008 Segmentation Optimality with respect to constants

Martin Burger Total Variation 18 Cetraro, September 2008 Segmentation Gradient descent

Martin Burger Total Variation 19 Cetraro, September 2008 Segmentation Empirical observations - Good segmentation results if objects have reasonably different intensity - Problems with low contrast, some problems with internal intensity variations in objects - Global convergence of level set methods if approximate Dirac delta has global support

Martin Burger Total Variation 20 Cetraro, September 2008 Segmentation Alternative viewpoint: back to functional

Martin Burger Total Variation 21 Cetraro, September 2008 Segmentation TV formulation: minimize

Martin Burger Total Variation 22 Cetraro, September 2008 Segmentation Ignore constants for the moment Basic structure:

Martin Burger Total Variation 23 Cetraro, September 2008 Segmentation Try convex relaxation

Martin Burger Total Variation 24 Cetraro, September 2008 Segmentation Relaxed minimization problem is almost linear Linear minimization problems have minimizers in corners of the simplex defined by constraints Corners of the simplex are defined by functions taking values 0 and 1 only Is the relaxation exact ? If yes, can we compute segmentation from solution of relaxed problem ?

Martin Burger Total Variation 25 Cetraro, September 2008 Segmentation Layer-cake Representation

Martin Burger Total Variation 26 Cetraro, September 2008 Segmentation Co-Area Formula

Martin Burger Total Variation 27 Cetraro, September 2008 Segmentation Exactness of relaxation

Martin Burger Total Variation 28 Cetraro, September 2008 Segmentation Exactness and thresholding

Martin Burger Total Variation 29 Cetraro, September 2008 Segmentation Minimizer of Chan-Vese model can be computed by minimizting relaxed problem and subsequent thresholding

Martin Burger Total Variation 30 Cetraro, September 2008 Segmentation Minimizer of Chan-Vese model can be computed by minimizting relaxed problem and subsequent thresholding

Martin Burger Total Variation 31 Cetraro, September 2008 Segmentation Minimizer of Chan-Vese model can be computed by minimizting relaxed problem and subsequent thresholding

Martin Burger Total Variation 32 Cetraro, September 2008 Segmentation Minimizer of Chan-Vese model can be computed by minimizting relaxed problem and subsequent thresholding

Martin Burger Total Variation 33 Cetraro, September 2008 Segmentation Minimizer of Chan-Vese model can be computed by minimizting relaxed problem and subsequent thresholding

Martin Burger Total Variation 34 Cetraro, September 2008 Segmentation Analysis of Chan-Vese model via relaxed problem

Martin Burger Total Variation 35 Cetraro, September 2008 Segmentation Existence A-priori estimates

Martin Burger Total Variation 36 Cetraro, September 2008 Segmentation A-priori estimates

Martin Burger Total Variation 37 Cetraro, September 2008 Segmentation Compactness

Martin Burger Total Variation 38 Cetraro, September 2008 Segmentation Existence

Martin Burger Total Variation 39 Cetraro, September 2008 Segmentation Region-based Mumford-Shah model

Martin Burger Total Variation 40 Cetraro, September 2008 Segmentation Region-based Mumford-Shah model

Martin Burger Total Variation 41 Cetraro, September 2008 Segmentation Alternative region-based Mumford-Shah model: Restriction of regularization terms to subregions not essential

Martin Burger Total Variation 42 Cetraro, September 2008 Segmentation Alternative region-based Mumford-Shah model: Restriction of regularization terms to subregions not essential