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MRI Brain Extraction using a Graph Cut based Active Contour Model

MRI Brain Extraction using a Graph Cut based Active Contour Model. Noha Youssry El-Zehiry and Adel S. Elmaghraby Computer Engineering and Computer Science Department University of Louisville

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MRI Brain Extraction using a Graph Cut based Active Contour Model

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  1. MRI Brain Extraction using a Graph Cut based Active Contour Model • 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

  2. Motivation and Problem Description

  3. Challenges • Inhomogeneities • Occlusion Blurred Edges Noise • Cluttered Object • Shared Intensities levels

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

  5. OBjective

  6. 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

  7. basic definitions in graph theory C1 • Graph G={V,E} C2 C1 v2 • Weighted Graph 6 6 3 v4 v3 • Cut C1 2 v1 7 • Cost of the cut 1 4 v6 v5 1 • S-T Cut v8 v7 1 1 • Min Cut C2 C2 • Min S-T Cut

  8. Class F2: Class F2 is defined as functions that can be written as sum of functions of up to two binary variables at a time, • Submodularity of Class F2: A class F2 function is submodular if and only if each term Ei,j satisfies the inequality

  9. Theorem • Let E be a function of n binary variables from the class F2 • Then, E is graph representable if and only if each term Ei,j satisfies the submodularity inequality

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

  11. C C C C F1> 0, F2 >0 F2=0, F1>0 F1=0, F2>0 active contour without edgeschan-vese model (2001) • F1 = F2 =0 • inf F1(C) + F2(C) Input Image uo(x,y) C

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

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

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

  15. 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.

  16. s s t t Graph Cut Optimization of Chan-vese model Min Cut Class 2 Class 1

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

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

  19. F(x1, ... ,xn) is submodular and hence graph representable

  20. Algorithm • Initialize C • Calculate c1 and c2 • Construct the graph • Find the minimum cut and get the new value for each xp • Update c1 and c2 and iterate till the energy is minimized

  21. Results Robustness to noise and topology changes

  22. Mammogram Initialization Outside the MAss

  23. Mammogram Initialization inside the MAss

  24. Mammogram Initialization Far from the MAss

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

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

  27. 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.

  28. step 1: Curve Evolution

  29. step 2 connected component analysis

  30. step 3: extraction of the dominant component

  31. sagittal view

  32. MRI slice with eye balls

  33. 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.

  34. references • Vladimir Kolmokorov, PhD thesis, 2004. • 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

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

  36. Thank you

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