image foresting transform

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Image Foresting Transformfor Image Segmentation

Presented by:Michael FangWeilong Yang

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A Few Things to RecallImage Segmentation

◦Finding homogeneous regionsGraph-based Methods

◦Treating images as graphsImage Foresting Transform

◦Unification◦Efficiency◦Simplicity

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Directed GraphsA directed graph is a pair (I, A), where I is a set of nodes and A is a set of ordered pairs of nodes.

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PathsA path is a sequence t1, t2, …, tk

of distinct nodes in the graph, such that (ti, ti+1) A for 1 i k – 1.

A path is trivial if k = 1;Path denotes the

concatenation of two paths, and , where ends at t and begins at t.

Path = s, t denotes theconcatenation of the longest prefix of and the last arc (s, t).

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Path CostsA path-cost function is a mapping

that assigns to each path a cost (), in some ordered set of cost values.

A function is said to be monotonic-incremental (MI) when

(t) = h(t),( s, t) = () (s, t),

where h(t) is a handicap cost value and satisfies: x’ x x’ (s, t) x (s, t) and x (s, t) x, for x, x’ and (s, t) A.

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Examples of MI Cost Functions

Additive cost function sum(t) = h(t),

sum( s, t) = sum() + w(s, t),where w(s, t) is a fixed non-negative arc weight.

Max-arc cost function max(t) = h(t),

max( s, t) = max{max(), w(s, t)}, where w(s, t) is a fixed arc weight.

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Predecessor Map and Spanning ForestA predecessor map is a function P that

assigns to each node t I either some other node in I, or a distinctive marker nil I – in which case t is the root of the map.

A spanning forest is a predecessor map which takes every node to nil in a finite number of iterations (i.e., it contains no cycles).

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Paths of the Forest PFor any node t I, there is a path

P*(t) which is obtained in backward by following the predecessor nodes along the path.

P*(c) = a, b, c, where P(c) = b, P(b) = a,

P(a) = nil

P*(i) = i, where P(i) = nil

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Optimum-path Forest

An optimum-path forest is a spanning forest P, where (P*(t)) is minimum for all nodes t I. Consider cost function sum in the example below.

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An Image as a Directed GraphA grayscale image I is a pair (I, I),

where I is a finite set of pixels (points in Z2) and I assigns to each pixel t I a value I(t) in some arbitrary value space.

An adjacency relation A is a binary relation between pixels of I, which is usually translation-invariant.

Once A has been fixed, image I can be interpreted as a directed graph, whose nodes are the image pixels in I and whose arcs are defined by A.

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Seed Pixels

In some applications, we would like to use a predefined path-cost function but constrain the search to paths that start in a given set S I of seed pixels. This constraint can be modeled by defining

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IFT Algorithm for Image Segmentation

1. Path Cost

2. Four-Connected Adjacency

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IFT algorithm with FIFO policy(1)

Initialization

It

C(t)

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Growing Process


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Another Example

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Input Image

Gradient Image

Seeds Labeling

IFT

Framework of Image segmentation by IFT

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Experiment Results (1)

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SummaryBasic concept of the Image

Foresting TransformIFT for image segmentationExperiment results

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References (1)1. Martin, D., Fowlkes, C., Tal, D., and Malik, J., A database of human segmented natural

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7. Boykov, Y.Y.[Yuri Y.], Jolly, M.P.[Marie-Pierre], Interactive Graph Cuts for Optimal Boundary and Region Segmentation of Objects in N-D Images, ICCV01(I: 105-112).

8. Andrew, Y.N., Jordan, M. and Weiss, Y., On Spectral Clustering: Analysis and an Algorithm, NIPS(14), 2002, pp. 849–856.

9. Weiss, Y.[Yair], Segmentation using Eigenvectors: A Unifying View, ICCV99(975-982).

10. Felzenszwalb, P.F.[Pedro F.], Huttenlocher, D.P.[Daniel P.], Image Segmentation Using Local Variation, CVPR98(98-104).

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References (2)11. Felzenszwalb, P.F.[Pedro F.], Huttenlocher, D.P.[Daniel P.], Efficient Graph-Based

Image Segmentation, IJCV(59), No. 2, September 2004, pp. 167-181.

12. Haxhimusa, Y.[Yll] and Kropatsch, W.G.[Walter G.], Segmentation Graph Hierarchies, SSPR&SPR(18-20) August 2004, pp. 343–351.

13. Haxhimusa, Y.[Yll], Ion, A.[Adrian], Kropatsch, W.G.[Walter G.], Illetschko, T.[Thomas], Evaluating Minimum Spanning Tree Based Segmentation Algorithms, CAIP05(579).

14. Falcão, A.X.[Alexandre X.], Stolfi, J.[Jorge], de Alencar Lotufo, R.[Roberto], The Image Foresting Transform: Theory, Algorithms, and Applications, PAMI(26), No. 1, January 2004, pp. 19-29.

15. Falcão, A.X.[Alexandre X.], Bergo, F.[Felipe] and Miranda, P.[Paulo], Image Segmentation by Tree Pruning, Proceedings of the Computer Graphics and Image Processing, XVII Brazilian Symposium, 2004, pp. 65–71.

16. Ding, J., Ma, R., Chen, S. and Wang, B., A Fast Directed Tree Based Neighborhood Clustering for Image Segmentation, NIP(4233), 2006, pp. 369–378.

17. Li, K.[Kang], Wu, X.D.[Xiao-Dong], Chen, D.Z.[Danny Z.], Sonka, M.[Milan], Optimal Surface Segmentation in Volumetric Images: A Graph-Theoretic Approach, PAMI(28), No. 1, January 2006, pp. 119-134.

18. Grady, L.[Leo], Random Walks for Image Segmentation, PAMI(28), No. 11, November 2006, pp. 1768-1783.

19. Pednekar, A.S.[Amol S.], Kakadiaris, I.A.[Ioannis A.], Image Segmentation Based on Fuzzy Connectedness Using Dynamic Weights, IP(15), No. 6, June 2006, pp. 1555-1562.

20. Luo, Q., Khoshgoftaar, T.M., Unsupervised Multiscale Color Image Segmentation Based on MDL Principle, IP(15), No. 9, August 2006, pp. 2755-2761.

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References (3)21. Arbeláez, P.A.[Pablo A.], Cohen, L.D.[Laurent D.], A Metric Approach to Vector-

Valued Image Segmentation, IJCV(69), No. 1, August 2006, pp. 119-126.22. Cremers, D.[Daniel], Osher, S.J.[Stanley J.], Soatto, S.[Stefano], Kernel Density

Estimation and Intrinsic Alignment for Shape Priors in Level Set Segmentation, IJCV(69), No. 3, September 2006, pp. 335-351.

23. Bresson, X.[Xavier], Vandergheynst, P.[Pierre], Thiran, J.P.[Jean-Philippe], A Variational Model for Object Segmentation Using Boundary Information and Shape Prior Driven by the Mumford-Shah Functional, IJCV(68), No. 2, June, 2006, pp. 145-162.

24. Boykov, Y.Y.[Yuri Y.], Funka-Lea, G.[Gareth], Graph Cuts and Efficient N-D Image Segmentation, IJCV(70), No. 2, November 2006, pp. 109-131.

25. Bresson, X.[Xavier], Vandergheynst, P.[Pierre], Thiran, J.P.[Jean-Philippe], Multiscale Active Contours, IJCV(70), No. 3, December 2006, pp. 197-211

26. Tu, Z.W.[Zhuo-Wen], Zhu, S.C.[Song-Chun], Parsing Images into Regions, Curves, and Curve Groups, IJCV(69), No. 2, August 2006, pp. 223-249.

27. Seghers, D., Loeckx, D.[Dirk], Maes, F.[Frederik], Vandermeulen, D., Suetens, P.[Paul], Minimal Shape and Intensity Cost Path Segmentation, MedImg(26), No. 8, August 2007, pp. 1115-1129.

28. Papandreou, G., Maragos, P., Multigrid Geometric Active Contour Models, IP(16), No. 1, January 2007, pp. 229-240.

29. Tao, W., Jin, H., Zhang, Y., Color Image Segmentation Based on Mean Shift and Normalized Cuts, SMC-B(37), No. 5, October 2007, pp. 1382-1389.

30. Wu, J., Chung, A.C.S., A Segmentation Model Using Compound Markov Random Fields Based on a Boundary Model, IP(16), No. 1, January 2007, pp. 241-252.

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References (4)31. Pavan, M.[Massimiliano], Pelillo, M.[Marcello], Dominant Sets and Pairwise Clustering,

PAMI(29), No. 1, January 2007, pp. 167-172.32. Tai, Y.W.[Yu-Wing], Jia, J.[Jiaya], Tang, C.K.[Chi-Keung], Soft Color Segmentation and Its

Applications, PAMI(29), No. 9, September 2007, pp. 1520-1537.33. Pyun, K., Lim, J., Won, C.S., Gray, R.M., Image Segmentation Using Hidden Markov

Gauss Mixture Models, IP(16), No. 7, July 2007, pp. 1902-1911.34. Arias, P., Pini, A., Sanguinetti, G., Sprechmann, P., Cancela, P., Fernandez, A., Gomez, A.,

Randall, G., Ultrasound Image Segmentation With Shape Priors: Application to Automatic Cattle Rib-Eye Area Estimation, IP(16), No. 6, June 2007, pp. 1637-1645.

35. Nikou, C.[Christophoros], Galatsanos, N.P.[Nikolaos P.], Likas, A.C.[Aristidis C.], A Class-Adaptive Spatially Variant Mixture Model for Image Segmentation, IP(16), No. 4, April 2007, pp. 1121-1130.

36. Benboudjema, D.[Dalila], Pieczynski, W.[Wojciech], Unsupervised Statistical Segmentation of Nonstationary Images Using Triplet Markov Fields, PAMI(29), No. 8, August 2007, pp. 1367-1378.

37. Michailovich, O.V.[Oleg V.], Rathi, Y.[Yogesh], Tannenbaum, A.[Allen], Image Segmentation Using Active Contours Driven by the Bhattacharyya Gradient Flow, IP(16), No. 11, November 2007, pp. 2787-2801.

38. Zoller, T.[Thomas], Buhmann, J.M.[Joachim M.], Robust Image Segmentation Using Resampling and Shape Constraints, PAMI(29), No. 7, July 2007, pp. 1147-1164.

39. Qiu, H.J.[Huai-Jun], Hancock, E.R.[Edwin R.], Clustering and Embedding Using Commute Times, PAMI(29), No. 11, November 2007, pp. 1873-1890.

40. Rivera, M., Ocegueda, O., Marroquin, J.L., Entropy-Controlled Quadratic Markov Measure Field Models for Efficient Image Segmentation, IP(16), No. 12, December 2007, pp. 3047-3057.

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References (5)41. Chen, S.F.[Shi-Feng], Cao, L.L.[Liang-Liang], Liu, J.Z.[Jian-Zhuang], Tang, X.

[Xiaoou], Iterative MAP and ML Estimations for Image Segmentation, CVPR07(1-6).

42. Allili, M.S.[Mohand Said], Ziou, D.[Djemel], Object of Interest segmentation and Tracking by Using Feature Selection and Active Contours, CVPR07(1-8).

43. Liu, T.[Tie], Sun, J.[Jian], Zheng, N.N.[Nan-Ning], Tang, X.[Xiaoou], Shum, H.Y.[Heung-Yeung], Learning to Detect A Salient Object, CVPR07(1-8).

44. Cremers, D.[Daniel], Rousson, M.[Mikael], Deriche, R.[Rachid], A Review of Statistical Approaches to Level Set Segmentation: Integrating Color, Texture, Motion and Shape, IJCV(72), No. 2, April 2007, pp. 195-215.

45. Tai, X.C.[Xue-Cheng], Christiansen, O.[Oddvar], Lin, P.[Ping], SkjÆlaaen, I.[Inge], Image Segmentation Using Some Piecewise Constant Level Set Methods with MBO Type of Projection, IJCV(73), No. 1, June 2007, pp. 61-76.

46. Riklin-Raviv, T.[Tammy], Kiryati, N.[Nahum], Sochen, N.A.[Nir A.], Prior-based Segmentation and Shape Registration in the Presence of Perspective Distortion, IJCV(72), No. 3, May 2007, pp. 309-328.

47. Sofou, A., Maragos, P.[Petros], Generalized Flooding and Multicue PDE-Based Image Segmentation, IP(17), No. 3, March 2008, pp. 364-376.

48. Ding, J., Ma, R., Chen, S., A Scale-Based Connected Coherence Tree Algorithm for Image Segmentation, IP(17), No. 2, February 2008, pp. 204-216.

49. Du, X., Bui, T.D., A New Model for Image Segmentation, SPLetters(15), 2008, pp. 182-185.

50. Orbanz, P.[Peter], Buhmann, J.M.[Joachim M.], Nonparametric Bayesian Image Segmentation, IJCV(77), No. 1-3, May 2008, pp. 25-45.

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