LOGISMOS Cost Function Design

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1 LOGISMOS Cost Function Design MICCAI 2010 Tutorial: Graph Algorithmic Techniques for Biomedical Image Segmentation Mona K. Garvin, Ph.D. The University of Iowa

2 Overview Part I A user s perspective of LOGISMOS cost function design (no graphs!) input cost functions + constraints Part II Representation of LOGISMOS cost functions in graphs optimal surface set

3 Part I: A user s perspective of LOGISMOS cost function design (no graphs!)

9 Two categories of cost functions On-surface costs In-region costs

10 Two categories of cost functions On-surface costs In-region costs On-surface costs re ect the unlikeliness of belonging to particular surfaces.

11 On-surface costs re ect the unlikeliness of belonging to particular surfaces Each voxel has n on-surface costs corresponding to the unlikeliness of belonging to each surface. Cost of a surface f i(x, y): C fi (x,y) = {(x,y,z) z=f i (x,y)} Cost of a surface set {f 1(x, y),..., fn(x, y)}: n C {f1 (x,y),...,f n (x,y)} = i=1 c surfi (x, y, z) C fi (x,y)

12 Example using only on-surface costs Find indicated 7 surfaces in OCT image using only on-surface costs. OCT Image

13 Example using only on-surface costs OCT Image Encourage dark-to-bright transitions Encourage bright-to-dark transitions

14 Example using only on-surface costs Seven cost images: OCT Image

15 Example using only on-surface costs OCT Image D-B 1 Seven cost images:

16 Example using only on-surface costs D-B B-D 1 2 Seven cost images: OCT Image

17 Example using only on-surface costs 1 Seven cost images: D-B B-D B-D OCT Image 1 2 3

18 1 Example using only on-surface costs Seven cost images: D-B B-D B-D D-B OCT Image

19 1 Example using only on-surface costs Seven cost images: D-B B-D B-D D-B B-D OCT Image 5

20 1 Example using only on-surface costs Seven cost images: D-B B-D B-D D-B B-D D-B 3 4 OCT Image 5 6

21 1 Example using only on-surface costs Seven cost images: D-B B-D B-D D-B B-D D-B 3 B-D 4 OCT Image 5 6 7

22 1 Example using only on-surface costs Seven cost images: D-B B-D B-D D-B B-D D-B 3 B-D 4 OCT Image smoothness constraints + thickness constraints

23 Example using only on-surface costs A non-optimal set of surfaces 1 Seven cost images: D-B B-D B-D D-B B-D D-B 3 B-D 4 OCT Image 5 6 7

24 Example using only on-surface costs OCT Image Result The optimal set of surfaces

25 Two categories of cost functions (x,y,z) R 2 c reg2 (x, y, z) On-surface costs In-region costs (x,y,z) R 1 c reg1 (x, y, z) (x,y,z) R 0 c reg0 (x, y, z) In-region costs re ect the unlikeliness of belonging to particular regions.

26 In-region costs re ect the unlikeliness of belonging to particular regions Having n surfaces corresponds to n+1 regions. Each voxel has n+1 in-region costs corresponding to the unlikeliness of belonging to each region. Cost of a region: C Ri = (x,y,z) R i c regi (x, y, z) Cost of a surface set {f 1(x, y),..., fn(x, y)}: n C {f1 (x,y),f 2 (x,y),...,f n (x,y)} = i=0 C Ri

27 Example using only in-region costs OCT Image (labeled surfaces) Find indicated 7 surfaces in OCT image using only in-region costs.

28 Example using only in-region costs OCT Image (labeled regions) Find indicated 7 surfaces in OCT image using only in-region costs.

29 Example using only in-region costs OCT Image (labeled regions) Eight cost images:

30 Example using only in-region costs OCT Image (labeled regions) Eight cost images: D D 0 3 D D 5 7 (variations to emphasize dark regions)

31 Example using only in-region costs OCT Image (labeled regions) Eight cost images: D B D D B D (variations to emphasize bright regions)

32 Example using only in-region costs Eight cost images: D B M D M D B D 6 7 OCT Image (labeled regions) (variations to emphasize medium regions)

33 Example using only in-region costs OCT Image (labeled regions) Eight cost images: D B M D M D B smoothness constraints + thickness constraints 7 D

34 Example using only in-region costs A non-optimal set of surfaces Eight cost images: D B M D M D B D OCT Image (labeled regions)

35 Example using only in-region costs OCT Image Result The optimal set of surfaces

36 Cost function using both on-surface and in-region costs (x,y,z) R 2 c reg2 (x, y, z) {(x,y,z) z=f 2 (x,y)} c surf 2 (x, y, z) (x,y,z) R 1 c reg1 (x, y, z) (x,y,z) R 0 c reg0 (x, y, z) {(x,y,z) z=f 1 (x,y)} c surf 1 (x, y, z)

37 Cost function using both on-surface and in-region costs Surface set cost function: C {f1 (x,y),f 2 (x,y),...,f n (x,y)} = n n C fi (x,y) + i=1 i=0 C Ri (x,y,z) R 2 c reg2 (x, y, z) (x,y,z) R 1 c reg1 (x, y, z) (x,y,z) R 0 c reg0 (x, y, z) {(x,y,z) z=f 2 (x,y)} c surf 2 (x, y, z) {(x,y,z) z=f 1 (x,y)} c surf 1 (x, y, z) On-surface costs: C fi (x,y) = c surfi (x, y, z) {(x,y,z) z=f i (x,y)} In-region costs: C Ri = c regi (x, y, z) (x,y,z) R i

38 Example using both on-surface and in-region costs Find indicated 7 surfaces in OCT image using both on-surface and in-region costs. OCT Image

39 Example using both on-surface and in-region costs Seven on-surface and eight in-region cost images: D-B B-D B-D D-B B-D D-B B-D surf1 surf2 surf3 surf4 surf5 surf6 surf7 D B M D M D B D reg0 reg1 reg2 reg3 reg4 reg5 reg6 reg7

Example using both on-surface and in-region costs 1 2

40 Example using both on-surface and in-region costs OCT Image Result The optimal set of surfaces

41 Part II: Representation of LOGISMOS cost functions in graphs Graph representation

42 Goal: ensure cost of surface set corresponds (within a constant) to cost of corresponding closed set Surface set cost: Closed set cost: n C fi (x,y) + n C Ri n C fi (x,y) + n C Ri + K i=1 i=0 i=1 i=0 Minimum surface set cost Minimum closed set Graph representation

43 Two categories of cost functions On-surface costs In-region costs Graph representation

44 Two categories of cost functions On-surface costs In-region costs Graph representation

45 On-surface cost representation (x,y,z) R 2 c reg2 (x, y, z) {(x,y,z) z=f 2 (x,y)} c surf 2 (x, y, z) (x,y,z) R 1 c reg1 (x, y, z) (x,y,z) R 0 c reg0 (x, y, z) {(x,y,z) z=f 1 (x,y)} c surf 1 (x, y, z) Graph representation

46 On-surface cost representation node weight: w on surfi (x, y, z) = { c surfi (x, y, z) if z =0 c surfi (x, y, z) c surfi (x, y, z 1) otherwise Graph representation

47 Graph representation of on-surface costs (toy example) cost image graph representation Graph representation

48 Graph representation of on-surface costs (toy example) surf. cost= cost image graph representation Graph representation

49 Graph representation of on-surface costs (toy example) surf. cost= CS cost= cost image graph representation Graph representation

50 Graph representation of on-surface costs (toy example) Problem: empty closed set less expensive! surf. cost= CS cost= cost image graph representation Graph representation

51 Graph representation of on-surface costs (toy example) Ensure non-empty closed set will be minimum cost image subtract (sum of last row + 1) = graph representation Graph representation

52 Graph representation of on-surface costs (toy example) cost image graph representation Graph representation

53 Graph representation of on-surface costs (toy example) cost image surf. cost=50 CS cost= K + 50 (K = -181) graph representation Graph representation

54 Two categories of cost functions On-surface costs In-region costs Graph representation

55 In-region cost representation (x,y,z) R 2 c reg2 (x, y, z) {(x,y,z) z=f 2 (x,y)} c surf 2 (x, y, z) (x,y,z) R 1 c reg1 (x, y, z) (x,y,z) R 0 c reg0 (x, y, z) {(x,y,z) z=f 1 (x,y)} c surf 1 (x, y, z) Graph representation

56 In-region cost representation (x,y,z) R2 creg 2 (x,y,z) R1 creg 1 (x, y, z) (x, y, z) node weight (in subgraph associated with surface i) w in regi (x, y, z) =c regi 1 (x, y, z) c regi (x, y, z) (region below) (region above) (x,y,z) R0 creg 0 (x, y, z) + (x,y,z) R 2 c reg2 (x, y, z) + (x,y,z) R 1 c reg2 (x, y, z) + _ (x,y,z) R 1 c reg2 (x, y, z) _ (x,y,z) R 0 c reg2 (x, y, z) (x,y,z) R 0 c reg2 (x, y, z) used Used with surface surface (x,y,z) R 1 c reg1 (x, y, z) + _ + (x,y,z) R 0 c reg1 (x, y, z) (x,y,z) R 0 c reg1 (x, y, z) used with surface 1 Used with surface 1 (x,y,z) R 0 c reg0 (x, y, z) Graph representation

57 In-region cost representation node (x,y,z) R2 creg (x, y, z) 2 (x,y,z) R1 creg 1 (x, y, z) weight (in subgraph associated with surface i) w in regi (x, y, z) =c regi 1 (x, y, z) c regi (x, y, z) (region below) (region above) (x,y,z) R0 creg 0 (x, y, z) + (x,y,z) R 2 c reg2 (x, y, z) + (x,y,z) R 1 c reg2 (x, y, z) + _ (x,y,z) R 1 c reg2 (x, y, z) _ (x,y,z) R 0 c reg2 (x, y, z) (x,y,z) R 0 c reg2 (x, y, z) used Used with surface surface (x,y,z) R 1 c reg1 (x, y, z) + _ + (x,y,z) R 0 c reg1 (x, y, z) closed set (CS) costs (x,y,z) R 0 c reg1 (x, y, z) used with surface 1 Used with surface 1 (x,y,z) R 0 c reg0 (x, y, z) constant Graph representation

58 Graph representation of in-region costs (toy example) Region 2 Region 1 Region 0 Graph 2 Graph 1 Graph representation

59 Graph representation of in-region costs (toy example) Region 2 Region 1 Region 0 Graph 2 Graph 1 Region Cost = 120 CS Cost = -320 Graph representation

60 Graph representation of in-region costs (toy example) Region 2 Region 1 Region 0 Graph 2 Graph 1 Total from 2 = 440 = -K CS Cost = = Graph representation

61 Graph representation of in-region costs (toy example) Region 2 Region 1 Region 0 Graph 2 Graph 1 Total from 2 = 440 = -K K=-440 CS Cost = = K Graph representation

62 Graph representation of in-region costs (toy example) Region 2 Region 1 Region 0 Graph 2 Graph 1 Region Cost = 120 K=-440 CS Cost = = K Graph representation

63 Summary and references Summary

64 Summary Two major cost function types for a set of surfaces: On-surface costs re ect the unlikeliness of belonging to particular surfaces. In-region costs re ect the unlikeliness of belonging to particular regions. The underlying optimality of the LOGISMOS approach allows one to design cost functions without having to think about the graph representation. The cost of a set of surfaces (using on-surface and inregion cost functions) is re ected in the vertex weights of the graph (so that the cost of the closed set corresponds, within a constant, to the cost of the set of surfaces). Summary

65 References K. Li, X. Wu, D. Z. Chen, and M. Sonka, Optimal surface segmentation in volumetric images A graph-theoretic approach, IEEE Trans. Pattern Anal. Mach. Intell., vol. 28, no. 1, pp , Jan M. Haeker (Garvin), X. Wu, M. D. Abràmoff, R. Kardon, and M. Sonka, Incorporation of regional information in optimal 3-D graph search with application for intraretinal layer segmentation of optical coherence tomography images, in IPMI 2007, LNCS, vol. 4584, Springer, 2007, pp M. K. Garvin, M. D. Abràmoff, X. Wu, S. R. Russell, T. L. Burns, and M. Sonka, Automated 3-D intraretinal layer segmentation of macular spectral-domain optical coherence tomography images, IEEE Trans. Med. Imag., 2009, vol. 28, no. 9, pp , Sept Y. Yin, X. Zhang, R. Williams, X. Wu, D. D. Anderson, M. Sonka, LOGISMOS Layered Optimal Graph Image Segmentation of Multiple Objects and Surfaces: Cartilage Segmentation in the Knee Joint, IEEE Trans. Med. Imag., in press. Summary

Graph-Based Image Segmentation: LOGISMOS. Milan Sonka & The IIBI Team

Graph-Based Image Segmentation: LOGISMOS Milan Sonka & The IIBI Team Iowa Institute for Biomedical Imaging The University of Iowa, Iowa City, IA, USA Background Computer-aided segmentation of medical images