MEDICAL IMAGE COMPUTING (CAP 5937) LECTURE 10: Medical Image Segmentation as an Energy Minimization Problem

Transcription

1 SPRING 06 MEDICAL IMAGE COMPUTING (CAP 97) LECTURE 0: Medical Image Segmentation as an Energy Minimization Problem Dr. Ulas Bagci HEC, Center for Research in Computer Vision (CRCV), University of Central Florida (UCF), Orlando, FL 84. or

2 Energy functional Outline Data and Smoothness terms Graph Cut Min cut Max Flow Applications in Radiology Images

3 Labeling & Segmentation Labeling is a common way for modeling various computer vision problems (e.g. optical flow, image segmentation, stereo matching, etc)

4 4 Labeling & Segmentation Labeling is a common way for modeling various computer vision problems (e.g. optical flow, image segmentation, stereo matching, etc) The set of labels can be discrete (as in image segmentation) L = {l,...,l m } with L = m

5 Labeling & Segmentation Labeling is a common way for modeling various computer vision problems (e.g. optical flow, image segmentation, stereo matching, etc) The set of labels can be discrete (as in image segmentation) L = {l,...,l m } with L = m Or continuous (tracking, etc.) L R n for n

6 6 Labeling is a function Labels are assigned to sites (pixel locations)

7 7 Labeling is a function Labels are assigned to sites (pixel locations) For a given image, we have = N cols.n rows

8 8 Labeling is a function Labels are assigned to sites (pixel locations) For a given image, we have = N cols.n rows Identifying a labeling function (with segmentation) is f :! L

9 9 Labeling is a function Labels are assigned to sites (pixel locations) For a given image, we have Identifying a labeling function (with segmentation) is f :! L = N cols.n rows We aim at calculating a labeling function that minimizes a given (total) error or energy

10 0 Labeling is a function Labels are assigned to sites (pixel locations) For a given image, we have Identifying a labeling function (with segmentation) is f :! L = N cols.n rows We aim at calculating a labeling function that minimizes a given (total) error or energy E(f) = X p [E data (p, f p )+ X qa(p) E smooth (f p,f q )] *A is an adjacency relation between pixel locations

11 Example of Energy Minimization Methods Credit: Zhao et al., IJNMBE

12 Energy Function? Penalizing results which are not compatible with the observed images/volumes

13 Energy Function? Penalizing results which are not compatible with the observed images/volumes Unary (data) cost (inverted)

14 4 Energy Function? Penalizing results which are not compatible with the observed images/volumes Enforcing spatial coherence. Pairwise (boundary) cost (inverted)

15 Energy Function? Penalizing results which are not compatible with the observed images/volumes Enforcing spatial coherence. Pairwise (boundary) cost (inverted) p q

16 6 Segmentation as an Energy Minimization Problem E data assigns non-negative penalties to a pixel location p when assigning a label to this location. 0//

17 7 Segmentation as an Energy Minimization Problem E data assigns non-negative penalties to a pixel location p when assigning a label to this location. E smooth assigns non-negative penalties by comparing the assigned labels f p and f q at adjacent positions p and q.

18 8 Segmentation as an Energy Minimization Problem E data assigns non-negative penalties to a pixel location p when assigning a label to this location. E smooth assigns non-negative penalties by comparing the assigned labels f p and f q at adjacent positions p and q. pixel p Edge q Node/vertex Image as a graph representation

19 9 Segmentation as an Energy Minimization Problem E data assigns non-negative penalties to a pixel location p when assigning a label to this location. E smooth assigns non-negative penalties by comparing the assigned labels f p and f q at adjacent positions p and q. This optimization model is characterized by local interactions along edges between adjacent pixels, and often called MRF (Markov Random Field) model.

20 0 E(f) = X p Energy Function-Details [E data (p, f p )+ X qa(p) E smooth (f p,f q )]

21 E(f) = X p Example Data Term: Energy Function-Details [E data (p, f p )+ X qa(p) E smooth (f p,f q )] E data (p, f p )= (p) (p = 0) = logp(p BG) (p = ) = logp(p FG)

22 E(f) = X p Example Data Term: Energy Function-Details Example Smoothness Term: [E data (p, f p )+ X qa(p) E smooth (f p,f q )] E data (p, f p )= (p) (p = 0) = logp(p BG) (p = ) = logp(p FG) (p, q) =K pq (p 6= q) where K pq = exp( (I p I q ) /( )) p, q

23 E(f) = X p Energy Function-Details [E data (p, f p )+ X qa(p) E smooth (f p,f q )] E(p) = X p p(p)+ X p X pq(p, q) qa(p) p = argmin pl E(p)

24 4 E(f) = X p Energy Function-Details [E data (p, f p )+ X qa(p) E smooth (f p,f q )] E(p) = X p p(p)+ X p X pq(p, q) qa(p) p = argmin pl E(p) To solve this problem, transform the energy functional into mincut/max-flow problem and solve it!

25 Graph Cuts for Optimal Boundary Detection (Boykov ICCV 00) Binary label: foreground vs. background User labels some pixels Exploit Statistics of known Fg & Bg Smoothness of label Turn into discrete graph optimization Graph cut (min cut / max flow) F F F F F F B B B B B

26 6 Graph Cut Each pixel is connected to its neighbors in an undirected graph Goal: split nodes into two sets A and B based on pixel values, and try to classify Neighbors in the same way too!

27 7 Graph Cut Each pixel is connected to its neighbors in an undirected graph Goal: split nodes into two sets A and B based on pixel values, and try to classify Neighbors in the same way too!

28 8 Each pixel = node Add two nodes F & B Graph-Cut Labeling: link each pixel to either F or B F Desired result F F F F F B B B B B

29 9 Cost Function: Data term Put one edge between each pixel and both F & G Weight of edge = minus data term F B

30 0 Cost Function: Smoothness term Add an edge between each neighbor pair Weight = smoothness term F B

31 Min-Cut Energy optimization equivalent to graph min cut Cut: remove edges to disconnect F from B Minimum: minimize sum of cut edge weight F cut B

32 Min-Cut Source 9 v v 4 Graph (V, E, C) Vertices V = {v, v... v n } Edges E = {(v, v )...} Costs C = {c (, )...} Sink

33 Min-Cut What is a st-cut? Source An st-cut (S,T) divides the nodes between source and sink. 9 v v 4 What is the cost of a st-cut? Sum of cost of all edges going from S to T Sink = 6

34 4 Min-Cut What is a st-cut? Source v v 9 Sink = 7 An st-cut (S,T) divides the nodes between source and sink. What is the cost of a st-cut? Sum of cost of all edges going from S to T What is the st-mincut? st-cut with the minimum cost

35 How to compute min-cut? Source 9 v v Sink 4 Solve the dual maximum flow problem Compute the maximum flow between Source and Sink Constraints Edges: Flow < Capacity Nodes: Flow in & Flow out Min-cut\Max-flow Theorem In every network, the maximum flow equals the cost of the st-mincut

36 6 Flow = 0 Source Max-Flow Algorithms Augmenting Path Based Algorithms v v 9 Sink 4. Find path from source to sink with positive capacity. Push maximum possible flow through this path. Repeat until no path can be found Algorithms assume non-negative capacity

37 7 Flow = 0 Source Max-Flow Algorithms Augmenting Path Based Algorithms v v 9 Sink 4. Find path from source to sink with positive capacity. Push maximum possible flow through this path. Repeat until no path can be found Algorithms assume non-negative capacity

38 8 Flow = 0 + Source Max-Flow Algorithms Augmenting Path Based Algorithms - 9 v v - Sink 4. Find path from source to sink with positive capacity. Push maximum possible flow through this path. Repeat until no path can be found Algorithms assume non-negative capacity

39 9 Flow = Source Max-Flow Algorithms Augmenting Path Based Algorithms v v 0 9 Sink 4. Find path from source to sink with positive capacity. Push maximum possible flow through this path. Repeat until no path can be found Algorithms assume non-negative capacity

40 40 Flow = Source Max-Flow Algorithms Augmenting Path Based Algorithms v v 0 9 Sink 4. Find path from source to sink with positive capacity. Push maximum possible flow through this path. Repeat until no path can be found Algorithms assume non-negative capacity

41 4 Flow = Source Max-Flow Algorithms Augmenting Path Based Algorithms v v 0 9 Sink 4. Find path from source to sink with positive capacity. Push maximum possible flow through this path. Repeat until no path can be found Algorithms assume non-negative capacity

42 4 Flow = + 4 Source Max-Flow Algorithms Augmenting Path Based Algorithms v v 0 Sink 0. Find path from source to sink with positive capacity. Push maximum possible flow through this path. Repeat until no path can be found Algorithms assume non-negative capacity

43 4 Flow = 6 Source Max-Flow Algorithms Augmenting Path Based Algorithms v v 0 Sink 0. Find path from source to sink with positive capacity. Push maximum possible flow through this path. Repeat until no path can be found Algorithms assume non-negative capacity

44 44 Flow = 6 Source Max-Flow Algorithms Augmenting Path Based Algorithms v v 0 Sink 0. Find path from source to sink with positive capacity. Push maximum possible flow through this path. Repeat until no path can be found Algorithms assume non-negative capacity

45 4 Flow = 6 + Source Max-Flow Algorithms Augmenting Path Based Algorithms v v Sink 0. Find path from source to sink with positive capacity. Push maximum possible flow through this path. Repeat until no path can be found Algorithms assume non-negative capacity

46 46 Flow = 7 Source Max-Flow Algorithms Augmenting Path Based Algorithms v v Sink 0. Find path from source to sink with positive capacity. Push maximum possible flow through this path. Repeat until no path can be found Algorithms assume non-negative capacity

47 47 Flow = 7 Source Max-Flow Algorithms Augmenting Path Based Algorithms v v Sink 0. Find path from source to sink with positive capacity. Push maximum possible flow through this path. Repeat until no path can be found Algorithms assume non-negative capacity

48 48 Another Example-Max Flow source 9 Ford & Fulkerson algorithm (96) 6 4 Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) 6 8 Find the path in the residual graph sink End

49 49 Another Example-Max Flow source 9 Ford & Fulkerson algorithm (96) 6 4 Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) 6 8 Find the path in the residual graph sink End

50 0 Another Example-Max Flow source 9 Ford & Fulkerson algorithm (96) 6 4 Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) 6 8 Find the path in the residual graph sink End flow =

51 Another Example-Max Flow source sink 9 + Ford & Fulkerson algorithm (96) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) End Find the path in the residual graph flow =

52 Another Example-Max Flow source 9 Ford & Fulkerson algorithm (96) 4 Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) 6 Find the path in the residual graph sink End flow =

53 Another Example-Max Flow source 9 Ford & Fulkerson algorithm (96) 4 Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) 6 Find the path in the residual graph sink End flow =

54 4 Another Example-Max Flow source 9 Ford & Fulkerson algorithm (96) 4 Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) 6 Find the path in the residual graph sink End flow = 6

55 6 Another Example-Max Flow source sink Ford & Fulkerson algorithm (96) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) End Find the path in the residual graph flow = 6

56 6 Another Example-Max Flow source 6 Ford & Fulkerson algorithm (96) 4 Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) 6 Find the path in the residual graph sink End flow = 6

57 7 Another Example-Max Flow source 6 Ford & Fulkerson algorithm (96) 4 Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) 6 Find the path in the residual graph sink End flow = 6

58 8 Another Example-Max Flow source 6 Ford & Fulkerson algorithm (96) 4 Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) 6 Find the path in the residual graph sink End flow =

59 9 6 Another Example-Max Flow source sink Ford & Fulkerson algorithm (96) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) End Find the path in the residual graph flow =

60 60 6 Another Example-Max Flow source 4 sink 7 6 Ford & Fulkerson algorithm (96) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) End Find the path in the residual graph flow =

61 6 6 Another Example-Max Flow source 4 sink 7 6 Ford & Fulkerson algorithm (96) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) End Find the path in the residual graph flow =

62 6 6 Another Example-Max Flow source 4 sink 7 6 Ford & Fulkerson algorithm (96) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) End Find the path in the residual graph flow =

63 6 Another Example-Max Flow - source Ford & Fulkerson algorithm (96) 4 7 Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) 6 - Find the path in the residual graph sink End flow =

64 64 Another Example-Max Flow source Ford & Fulkerson algorithm (96) 4 7 Find the path from source to sink 6 6 While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) Find the path in the residual graph sink End flow =

65 6 Another Example-Max Flow source Ford & Fulkerson algorithm (96) 4 7 Find the path from source to sink 6 6 While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) Find the path in the residual graph sink End flow =

66 66 Another Example-Max Flow source Ford & Fulkerson algorithm (96) 4 7 Find the path from source to sink 6 6 While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) Find the path in the residual graph sink End flow =

67 67 Another Example-Max Flow source Ford & Fulkerson algorithm (96) 4-7 Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) Find the path in the residual graph sink End flow =

68 68 Another Example-Max Flow source Ford & Fulkerson algorithm (96) 7 Find the path from source to sink 4 6 While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) Find the path in the residual graph sink End flow =

69 69 Another Example-Max Flow source Ford & Fulkerson algorithm (96) 7 Find the path from source to sink 4 6 While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) Find the path in the residual graph sink End flow =

70 70 Another Example-Max Flow source Ford & Fulkerson algorithm (96) 7 Find the path from source to sink 4 6 While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) Find the path in the residual graph sink End flow =

71 7 Another Example-Max Flow source Ford & Fulkerson algorithm (96) 7 Why is the solution globally optimal? 6 4 sink flow =

72 7 S Another Example-Max Flow source Ford & Fulkerson algorithm (96) 7 Why is the solution globally optimal? 6. Let S be the set of reachable nodes in the residual graph 4 sink flow =

73 7 Another Example-Max Flow S source 9 Ford & Fulkerson algorithm (96) 6 4 Why is the solution globally optimal?. Let S be the set of reachable nodes in the 6 8 residual graph. The flow from S to V - S equals to the sum of capacities from S to V S sink

74 74 / / / 0/ /6 0/ 0/ Another Example-Max Flow source /6 8/8 sink /4 0/ flow = 8/9 / 0/ / / / / Individual flows obtained by summing up all paths Ford & Fulkerson algorithm (96) Why is the solution globally optimal?. Let S be the set of reachable nodes in the residual graph. The flow from S to V - S equals to the sum of capacities from S to V S. The flow from any A to V - A is upper bounded by the sum of capacities from A to V A 4. The solution is globally optimal

75 Another Example-Max Flow 7 source sink S T cost = 8 source sink S T

76 76 Another Example-Max Flow S source cost = 6 8 sink T

77 77 Another Example-Max Flow source 9 x 4 x x x 4 6 x 6 x C(x) = x + 9x + 4x + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x 4 (-x ) + x (-x ) + 6x (-x ) + x 6 (-x ) + x (-x 6 ) + x 6 (-x ) + x 4 (-x ) + x 6 (-x ) + 6(-x 4 ) + 8(-x ) + (-x 6 ) 6 8 sink

78 78 Another Example-Max Flow source 9 x 4 x x x 4 6 x 6 x C(x) = x + 9x + 4x + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x 4 (-x ) + x (-x ) + x (-x ) + x 6 (-x ) + x (-x 6 ) + x 6 (-x ) + x 4 (-x ) + x 6 (-x ) + 6(-x 4 ) + (-x ) + (-x 6 ) + x + x (-x ) + x (-x ) + (-x ) 6 8 sink

79 79 Another Example-Max Flow source 9 x 4 x x x 4 6 x 6 x 6 8 sink C(x) = x + 9x + 4x + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x 4 (-x ) + x (-x ) + x (-x ) + x 6 (-x ) + x (-x 6 ) + x 6 (-x ) + x 4 (-x ) + x 6 (-x ) + 6(-x 4 ) + (-x ) + (-x 6 ) + x + x (-x ) + x (-x ) + (-x ) x + x (-x ) + x (-x ) + (-x ) = + x (-x ) + x (-x )

80 80 Another Example-Max Flow source 9 x 4 x x x 4 x 6 x 6 sink C(x) = + x + 6x + 4x + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x 4 (-x ) + x (-x ) + x (-x ) + x 6 (-x ) + x (-x 6 ) + x (-x 4 ) + 6(-x 4 ) + (-x ) + (-x 6 ) + x (-x ) + x + x 6 (-x ) + x (-x 6 ) + (-x ) x + x 6 (-x ) + x (-x 6 ) + (-x ) = + x (-x 6 ) + x 6 (-x )

81 8 Another Example-Max Flow source 6 x 4 x x x 4 x 6 x C(x) = 6 + x + 6x + 4x + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x 4 (-x ) + x (-x ) + x (-x ) + x 6 (-x ) + x (-x 6 ) + x (-x 4 ) + 6(-x 4 ) + x (-x 6 ) + x 6 (-x ) + (-x ) + (-x 6 ) + x (-x ) 6 sink

82 Another Example-Max Flow 8 S source x x 7 x 4 x x 4 sink min cut C(x) = + x + 4x + x (-x ) + x (-x ) + 7x (-x ) + x (-x 4 ) + x (-x ) + 6x (-x 6 ) + 6x (-x ) + x (-x 4 ) + 4(-x 4 ) + x (-x 6 ) + x 6 (-x ) 6 cost = 0 x 6 cost = 0 T S set of reachable nodes from s All coefficients positive Must be global minimum

83 8 History of Max-Flow Algorithms Augmenting Path and Push-Relabel n: #nodes m: #edges U: maximum edge weight Algorithms assume nonnegative edge weights

84 Software Packages for Optimization 84

85 Applications Used in Energy Minimization Based Segmentation Methods 8

86 Applications 86

87 87 Interactive Organ Segmentation (Boykov and Jolly, MICCAI 000) Segmentation of multiple objects. (a-c): Cardiac MRI. (d): Kidney CE- MR angiography

88 88 Interactive Organ Segmentation (Boykov and Jolly, MICCAI 000) Segmentation of the right lung in CT. (a): representative D slice of original D data. (b): segmentation results on the slice in (a). (c-d) D visualization of segmentation results.

89 89 AUTOMATIC HEART ISOLATION FOR CT CORONARY VISUALIZATION USING GRAPH-CUTS (Funka-Lea et al, ISBI 006) Isolating the entire heart allows the coronary vessels on the surface of the heart to be easily visualized despite the proximity of surrounding organs such as the ribs and pulmonary blood vessels.

90 90 AUTOMATIC HEART ISOLATION FOR CT CORONARY VISUALIZATION USING GRAPH-CUTS (Funka-Lea et al, ISBI 006) Isolating the entire heart allows the coronary vessels on the surface of the heart to be easily visualized despite the proximity of surrounding organs such as the ribs and pulmonary blood vessels. Numerous techniques have been described for segmenting the left ventricle of the heart in images from various types of medical scanners but rarely has the entire heart been segmented.

91 9 AUTOMATIC HEART ISOLATION FOR CT CORONARY VISUALIZATION USING GRAPH-CUTS (Funka-Lea et al, ISBI 006) Isolating the entire heart allows the coronary vessels on the surface of the heart to be easily visualized despite the proximity of surrounding organs such as the ribs and pulmonary blood vessels. Numerous techniques have been described for segmenting the left ventricle of the heart in images from various types of medical scanners but rarely has the entire heart been segmented. Graph-cut formation:

92 9 AUTOMATIC HEART ISOLATION FOR CT CORONARY VISUALIZATION USING GRAPH-CUTS (Funka-Lea et al, ISBI 006) Isolating the entire heart allows the coronary vessels on the surface of the heart to be easily visualized despite the proximity of surrounding organs such as the ribs and pulmonary blood vessels. Numerous techniques have been described for segmenting the left ventricle of the heart in images from various types of medical scanners but rarely has the entire heart been segmented. Graph-cut formation:

93 9 AUTOMATIC HEART ISOLATION FOR CT CORONARY VISUALIZATION USING GRAPH-CUTS (Funka-Lea et al, ISBI 006) Isolating the entire heart allows the coronary vessels on the surface of the heart to be easily visualized despite the proximity of surrounding organs such as the ribs and pulmonary blood vessels. Numerous techniques have been described for segmenting the left ventricle of the heart in images from various types of medical scanners but rarely has the entire heart been segmented. Graph-cut formation: (C is center)

94 94 AUTOMATIC HEART ISOLATION FOR CT CORONARY VISUALIZATION USING GRAPH-CUTS (Funka-Lea et al, ISBI 006) Isolating the entire heart allows the coronary vessels on the surface of the heart to be easily visualized despite the proximity of surrounding organs such as the ribs and pulmonary blood vessels. Numerous techniques have been described for segmenting the left ventricle of the heart in images from various types of medical scanners but rarely has the entire heart been segmented. Graph-cut formation: (C is center) If cos(.) <0, (0 otherwise.)

95 9 AUTOMATIC HEART ISOLATION FOR CT CORONARY VISUALIZATION USING GRAPH-CUTS (Funka-Lea et al, ISBI 006) Top Left: A balloon is expanded within the heart. The heart wall pushes the balloon toward the heart center as the balloon grows. Top right: volume rendering of original heart volume. Bottom left: heart cropped based on segmentation mask. Bottom right: volume rendering after automatic heart isolation algorithm.

96 96 Measurement of hippocampal atrophy using 4D graph-cut segmentation: Application to ADNI (Wolz, et al. NeuroImage 0) Proposed a method for simultaneously segmenting longitudinal magnetic resonance (MR) images

97 97 Measurement of hippocampal atrophy using 4D graph-cut segmentation: Application to ADNI (Wolz, et al. NeuroImage 0) Proposed a method for simultaneously segmenting longitudinal magnetic resonance (MR) images D MRI + time component (longitudinal)

98 98 Measurement of hippocampal atrophy using 4D graph-cut segmentation: Application to ADNI (Wolz, et al. NeuroImage 0) Proposed a method for simultaneously segmenting longitudinal magnetic resonance (MR) images D MRI + time component (longitudinal) A 4D graph is used to represent the longitudinal data: edges are weighted based on spatial and intensity priors and connect spatially and temporally neighboring voxels represented by vertices in the graph.

99 99 Measurement of hippocampal atrophy using 4D graph-cut segmentation: Application to ADNI (Wolz, et al. NeuroImage 0) Proposed a method for simultaneously segmenting longitudinal magnetic resonance (MR) images D MRI + time component (longitudinal) A 4D graph is used to represent the longitudinal data: edges are weighted based on spatial and intensity priors and connect spatially and temporally neighboring voxels represented by vertices in the graph. Solving the min-cut/max-flow problem on this graph yields the segmentation for all time-points in a single step

100 00 Measurement of hippocampal atrophy using 4D graph-cut segmentation: Application to ADNI (Wolz, et al. NeuroImage 0) Proposed a method for simultaneously segmenting longitudinal magnetic resonance (MR) images D MRI + time component (longitudinal) A 4D graph is used to represent the longitudinal data: edges are weighted based on spatial and intensity priors and connect spatially and temporally neighboring voxels represented by vertices in the graph. Solving the min-cut/max-flow problem on this graph yields the segmentation for all time-points in a single step Time-series image can be considered as a single (4D) image, with the following dimensions: x,y,z and t

101 0 Measurement of hippocampal atrophy using 4D graph-cut segmentation: Application to ADNI (Wolz, et al. NeuroImage 0) Proposed a method for simultaneously segmenting longitudinal magnetic resonance (MR) images D MRI + time component (longitudinal) A 4D graph is used to represent the longitudinal data: edges are weighted based on spatial and intensity priors and connect spatially and temporally neighboring voxels represented by vertices in the graph. Solving the min-cut/max-flow problem on this graph yields the segmentation for all time-points in a single step Time-series image can be considered as a single (4D) image, with the following dimensions: x,y,z and t 6 spatial neighbors in D And temporal neighbors and

102 0 Measurement of hippocampal atrophy using 4D graph-cut segmentation: Application to ADNI (Wolz, et al. NeuroImage 0) a. Right hippocampus segmentation (baseline), b. follow up segmentation ( months)

103 0 Integrated Graph Cuts for Brain Image Segmentation (Song et al, MICCAI 006) In addition to image intensity, tissue priors and local boundary information are integrated into the edge weight metrics in the graph.

104 04 Integrated Graph Cuts for Brain Image Segmentation (Song et al, MICCAI 006) In addition to image intensity, tissue priors and local boundary information are integrated into the edge weight metrics in the graph. Example of the graph with three terminals for brain MRI tissue segmentation of gray matter (GM), white matter (WM), and cerebrospinal fluid (CSF). The set of nodes V includes all voxels and terminals. The set of edges E includes all n-links and t-links. n-links: voxel-to-voxel edges t-links: voxel-to-terminal edges

105 Integrated Graph Cuts for Brain Image Segmentation (Song et al, MICCAI 006) 0

106 06 Integrated Graph Cuts for Brain Image Segmentation (Song et al, MICCAI 006) Regularization term Data term Atlas term Pairwise term

107 07 Integrated Graph Cuts for Brain Image Segmentation (Song et al, MICCAI 006) Regularization term Data term Atlas term (T, segmentation results)

108 08 Summary Data and Smoothness Terms -> Graph based segmentation methods Additional terms can(should) be added into segmentation formulation based observation/need and problem definition Problems formulated as a MRF task can be solved by max-flow/mincut

109 09 Slide Credits and References Fredo Durand M. Tappen R. Szelisky J.Malcolm, Graph Cut in Tensor Scale Interactive Graph Cuts for Optimal Boundary & Region Segmentation of Objects in N-D images. Yuri Boykov and Marie-Pierre Jolly. In International Conference on Computer Vision, (ICCV), vol. I, A Comparative Study of Energy Minimization Methods for Markov Random Fields. Rick Szeliski, Ramin Zabih, Daniel Scharstein, Olga Veksler, Vladimir Kolmogorov, Aseem Agarwala, Marshall Tappen, Carsten Rother. ECCV P. Kumar, Oxford University.