Unformatted text preview: Image Segmentation Ben Appleton University of Queensland The goal of segmentation General segmentation: Break an image into component objects Object segmentation: Find a particular object of interest Segmentation methodology Identify distinguishing features Colour Texture Shape Use prior knowledge Position, orientation Relation to other objects (eg. a number plate is found toward the bottom of a car) Classical methods Pointsegmentation Thresholding Feature clustering Boundary methods Edge detection Laplacian of Gaussian Region methods Region growing/splitting/merging Watershed Image Simplification Simplifies the segmentation problem by `cartooning' the image
I n +1 ( x, y ) = median I n ( x, y)
( x, y )W ( x , y ) Median filtering
I = div g ( I t
I = div g ( I t + (I 0  I ) { } Nonlinear diffusion ) I I Reactiondiffusion ) I
I Original image Median filtered Nonlinear diffusion Reactiondiffusion Active Contours Snakes Active shapes Level sets Active Contours What are active contours for? `Evolving' surfaces Applications in image analysis Segmentation Multiview reconstruction Object tracking Snakes Represent a surface explicitly as a polytope (polygon or polyhedron) Behaves as a network of point masses connected by springs and thinplate splines External forces derived from image push toward goal Benefits of Snakes Simple idea Widely accepted in medical image analysis Fast Active Shapes (Cootes) Allows snakes to learn object shapes Manually segment several images Determine the vertex covariance matrix Eigenvectors are dominant modes of shape variation Active Shapes
Clenched Splayed Pointing Relaxed First mode (eigenvector) 3 0 3 Problems with Snakes Topology Cannot change topology easily Selfintersection difficult to detect Complexity grows with dimensionality Selfintersection Problems with Snakes Stability Surface points bunch up or spread out Produces instabilities and inaccuracies Bunching Spreading Level Sets  Theory Implicit representation of surfaces Advantages of level sets Simple example Level Sets Represent a surface implicitly as the zerolevel contour of a function (C ) = 0 We are free to choose off the contour Example Representing a Circle y r x ( x, y ) = x + y  r
2 2 Advantages of Level Sets Topology is handled implicitly Extension to higher dimensions is simple Stability is assured Easy to manipulate the implicit contour via the fundamental partial differential equation Manipulate to indirectly move C: The Fundamental PDE of Level Sets (C ) = 0 d (C ) C dt =  F t = + =0 t t where F is the speed function normal to the curve Example Evolving a Circle ( x, y ) = x 2 + y 2  r Level Set representation of a circle Setting F = 1 causes the circle to expand uniformly  = 1 by choice of representation, so we obtain the level set evolution equation: nconditional stability = 1 t Explicit solution: ( x, y , t ) = x 2 + y 2  r  t which means that the circle has radius r + t at time t, as expected Level Sets  Evolution Speed functions Two speed functions for segmentation Speed Functions Surface evolution solely dependent on F Design F directly or derive by variational calculus F a function of geometric variables such as normal direction, curvature and so on F = 1  (C ) + G I N
Regularisation Edgeattraction force
Viscosity = 0.5 Viscosity = 2 Viscosity = 5 Segmentation via MalladiSethianVemuri Function Inflation /deflation Lung xray Segmentation via Geodesic Active Contours
Gradient metric: Minimise energy: E (C ) = g ds C Speed function: F = g N  g (  ) Implicit scheme: t = g + ( g )
g= 1 + 1 + G I MRI of the left ventricle in a heart New Directions  Optimality Snakes and level sets are iterative optimisers Not guaranteed to extract the best answer Graph theory and its continuous extensions allow optimal solutions to the same problems Surfaces and partitionings A partitioning divides a space into disjoint components Boundaries form simple closed surfaces Many Image Analysis (IA) problems seek surfaces or partitionings of a space: Segmentation Object tracking 3D reconstruction Geodesic Active Contours and Surfaces
( ) Proposed for segmentation by Caselles et. al. (PAMI 1997) E C = g ds Weighted minimal surface model C 1 g= Variational level set approach Gradient metric: 1 + G I
Metric Geodesic Active Contours Globally Optimal (Level set evolution) Geodesic Active Contours Chest xray Lung Healthy lung Continuous maximal flows Simulate the flow F of an incompressible fluid Pressure P is conserved P =  div F under flow F t F is driven by fluctuations in P Boundary conditions:
Ps = 1 Pt = 0 F = P t F g Example: Lung Segmentation
Evolving potential field (P) Results
2D segmentation
Globally minimal surfaces (5.1s) Minimum cuts (12.9s) Microscope cell image (231 221) Metric image GOGAC (1.48s) Chest CT (200 160 90) Globally minimal surface (8.5 min) Classical Geodesic Active Surface (22 min) Minimum cut (14 min) Results 3D reconstruction
Minimum cut (230 sec) Minimal surface (140 sec) Summary Active contours have emerged as a powerful tool in segmentation Great improvements have been made as viewpoints change Snakes Level Sets (topology) Level Sets Minimal surfaces (optimality) ...
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 One '09
 MehrtashHarandi
 Image processing, Differential geometry of surfaces, level sets

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