Download - 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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IFT algorithm with FIFO policy(2)
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Growing Process
IFT algorithm with FIFO policy(3)
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IFT algorithm with FIFO policy(4)
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IFT algorithm with FIFO policy(4)
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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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Experiment Results (2)
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Experiment Results (3)
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Experiment Results (4)
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SummaryBasic concept of the Image
Foresting TransformIFT for image segmentationExperiment results
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References (2)11. Felzenszwalb, P.F.[Pedro F.], Huttenlocher, D.P.[Daniel P.], Efficient Graph-Based
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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.
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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.
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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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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.
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Thank You!Questions?