MRI Brain Extraction using a Graph Cut based Active Contour Model Noha Youssry El-Zehiry Noha Youssry El-Zehiry and Adel S. Elmaghraby Computer Engineering and Computer Science Department University of Louisville First Annual ORNL Biomedical Science and Engineering Conference March 18th 2009

Motivation and Problem Description

Challenges Inhomogeneities Occlusion Blurred EdgesNoise Cluttered Object Shared Intensities levels

Image segmentation techniques Histogram based methods Model based algorithms Region growing algorithms Graph partitioning methods Neural Network classifiers Clustering Scale Space

OBjective

outline Graph cuts: brief background Active contours: brief background Active contour without edges “Chan-Vese Model” Graph cut optimization for the Chan-Vese energy functional. Brain Extraction Algorithm Results and Conclusion

basic definitions in graph theory v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 Graph G={V,E} Cut S-T Cut Min Cut Min S-T Cut Weighted Graph Cost of the cut C1 C2 1 C1 C2 6 C1 C2

Graph Representable Function A function E of n binary variables is called graph representable if there exists a graph G = (V, E) with terminals s and t and a subset of vertices V 0 = {v 1,..., v n } ⊂ V - {s, t} such that for any configuration x 1,..., x n the value of the energy E(x 1,..., x n ) is equal to a constant plus the cost of a minimum s-t cut among all cuts C = S,T in which v i ∈ S, if x i =0, and v i ∈ T, if x i =1 (1 < i < n). If the constant is equal to zero, we say that E is exactly represented by G and Vo

Class F 2 : Class F 2 is defined as functions that can be written as sum of functions of up to two binary variables at a time, Submodularity of Class F 2 : A class F 2 function is submodular if and only if each term E i,j satisfies the inequality

Theorem 1 The sum of two graph representable functions is graph representable. Graph G1 = G (V, E1) is a representation of F1 = F(x1,..., xn) Graph G2 = G (V, E2) is a representation of F2 = F(x1,...,xn) F1+F2 is graph representable and can be represented by a graph G (V, E) where V is the same set of vertices of G1 and G2 and E is obtained by simply adding the edge weights of E1 and E2

Theorem Let E be a function of n binary variables from the class F 2 Then, E is graph representable if and only if each term E i,j satisfies the submodularity inequality

Deformable Models Parametric Active Contours -Kass, Witkins and Terzoupolos Geometric Active Contours - Osher and Sethian Edge based Region Based

Parametric active contours Kass- snake model (1987) Energy Formulation s=0 s=0.2 s=1 Parametreization: A curve C is represented parametrically C(s)=[x(s) y(s)] s [0,1]

Internal Energy: represents the bending and smoothness of the curve External Energy: controlled mainly by the image gradient and represents how well the contour lies over the boundary of interest and u(x,y) is the image of interest

drawbacks Can not handle topology changes Initialization objective Evolution

Geometric Active Contours Levl Set Methods (Osher-sethian 1989) The curve is not explicitly defined as a function of a certain parameter. The curve is implicitly defined as the zero level of a higher dimensional surface [Level set function=signed distance map]. The higher dimensional surface changes according to the energy formulation. The result of the evolution at any time is obtained by getting the intersection of the surface with the plane z=0

Level Set Methods (Osher-sethian 1989) The zero level set New Zero Level Set The result of the curve evolution

Initialization Evolution result Evolution construct the signed distance mapfind the intersection with z=0

level set function and level set calculus Level set function satisfies the following Heaviside step function Dirac Delta function

active contour without edges chan-vese model (2001) F1 = F2 =0 inf F1(C) + F2(C) F2=0, F1>0 C F1> 0, F2 >0 C C F1=0, F2>0 C C Input Image u o (x,y)

chan-Vese (cont.) Regularization Representation using level sets

chan-Vese (cont.) Initialize the contour Calculate c 1 and c 2 Solve the PDE for the new Phi using gradient descent Update the energy function, c1 and c2 Iterate till the energy is minimized

Gradient Descent Example: minimization of functions of 2 variables High sensitivity to the initialisation, easily stuck in a local minimum (x o,y o ) x y

Graph Cut Optimization of Chan- vese model Discrete formulation of Chan-Vese energy function. Proof of submodularity for the discrete energy function. Correspondence between the energy function and the graph.

Graph Cut Optimization of Chan- vese model s tt s Min Cut Class 1 Class 2

discrete representation for the contour length Cauchy Crofton formula C a set of all lines L a subset of lines L intersecting contour C Euclidean length of C : the number of times line L intersects C courtesy of Boykov and Kolmogorov

cut Metric on grids can approximate Euclidean Metric C Edges of any regular neighborhood system generate families of lines {,,, } Graph nodes are imbedded in R2 in a grid-like fashion graph cut cost for edge weights: the number of edges of family k intersecting C courtesy of Boykov and Kolmogorov wkwk

F(x 1,...,x n ) is submodular and hence graph representable

Algorithm Initialize C Calculate c1 and c2 Construct the graph Find the minimum cut and get the new value for each x p Update c 1 and c 2 and iterate till the energy is minimized

Results Robustness to noise and topology changes

Mammogram Initialization Outside the MAss

Mammogram Initialization inside the MAss

Mammogram Initialization Far from the MAss

illustration of global optimization The convergence of the energy function when optimized using different initializations

Application to the brain extraction problem Brain extraction aims at removing all non brain tissue from the head MRI

Algorithm Apply the curve evolution algorithm to the original MRI slice. The result will group the most homogeneous regions together. Apply connected component analysis to the class of the lower mean intensity value, (alternatively, the one with higher cardinality). Extract the most dominant component ( 2 components), these components represent the brain tissue.

step 1: Curve Evolution

step 2 connected component analysis

step 3: extraction of the dominant component

sagittal view

MRI slice with eye balls

conclusion Brain extraction algorithm using a graph cut based active contour has been presented. Advantages over the existing methods are: Robustness to noise. Robustness to topology changes. Computational Complexity.

references Vladimir Kolmokorov, PhD thesis, Computing geodesics and minimal surfaces via graph cuts, CVPR 2003 Graph cut optimization of the Mumford-Shah functional, VIIP 2007 Active Contour Without Edges, TIP 2001

acknowledgment Vladimir Kolmogorov, University college London. Anre Kezdy, University of Louisville. Pasanna Sahoo, University of Louisville. Luminita Vese, UCLA.

Thank you