Region Segmentation Readings: Chapter 10: 10.1 Additional Materials Provided

Transcription

1 Regon Segmentaton Readngs: hater 10: 10.1 Addtonal Materals Provded K-means lusterng tet EM lusterng aer Grah Parttonng tet Mean-Shft lusterng aer 1

2 Image Segmentaton Image segmentaton s the oeraton of arttonng an mage nto a collecton of connected sets of els. 1. nto regons, whch usually cover the mage 2. nto lnear structures, such as - lne segments - curve segments 3. nto 2D shaes, such as - crcles - ellses - rbbons long, symmetrc regons 2

3 Eamle: Regons 3

4 Man Methods of Regon Segmentaton 1. Regon Growng 2. Slt and Merge 3. lusterng 4

5 lusterng There are K clusters 1,, K wth means m 1,, m K. The least-squares error s defned as K 2 D = - m k. k=1 k Out of all ossble arttons nto K clusters, choose the one that mnmzes D. Why don t we ust do ths? If we could, would we get meanngful obects? 5

6 K-Means lusterng Form K-means clusters from a set of n-dmensonal vectors 1. Set c teraton count to 1 2. hoose randomly a set of K means m 1 1,, m K For each vector comute D, m k c, k=1, K and assgn to the cluster wth nearest mean. 4. Increment c by 1, udate the means to get m 1 c,,m K c. 5. Reeat stes 3 and 4 untl k c = k c+1 for all k. 6

7 K-Means Eamle 1 7

8 K-Means Eamle 2 8

9 K-Means Eamle 3 9

10 K-means Varants Dfferent ways to ntalze the means Dfferent stong crtera Dynamc methods for determnng the rght number of clusters K for a gven mage The EM Algorthm: a robablstc formulaton of K-means 10

11 K-Means Boot Ste: Intalze K clusters: 1,, K Each cluster s reresented by ts mean m Iteraton Ste: Estmate the cluster for each data ont Re-estmate the cluster arameters 11

12 K-Means Eamle 12

13 K-Means Eamle Where do the red onts belong? 13

14 K-means vs. EM K-means EM luster mean mean, varance, Reresentaton and weght luster randomly select ntalze K Intalzaton K means Gaussan dstrbutons Eectaton assgn each ont soft-assgn each ont to closest mean to each dstrbuton Mamzaton comute means comute new arams of current clusters of each dstrbuton 14

15 Notaton Nµ, σ s a 1D normal Gaussan dstrbuton wth mean µ and standard devaton σ so the varance s σ 2. 15

16 Nµ, s a multvarate Gaussan dstrbuton wth mean µ and covarance matr. What s a covarance matr? R G B R σ 2 R σ RG σ RB G σ GR σ 2 G σ GB B σ BR σ BG σ 2 B varancex: σ X 2 = - µ 2 1/N covx,y = - µ y - µ y 1/N 16

17 1. Suose we have a set of clusters: 1, 2,..., K over a set of data onts X = { 1, 2,..., N }. P s the robablty or weght of cluster. P s the robablty of cluster gven ont. P s the robablty of ont belongng to cluster. 2. Suose that a cluster s reresented by a Gaussan dstrbuton Nµ,σ. Then for any ont : 17

18 EM: Eectaton-Mamzaton Boot Ste: Intalze K clusters: 1,, K Iteraton Ste: µ, Σ and P for each cluster. Estmate the cluster of each data ont Re-estmate the cluster arameters, Σ, µ For each cluster Eectaton Mamzaton 18

19 1-D EM wth Gaussan Dstrbutons Each cluster s reresented by a Gaussan dstrbuton Nµ, σ. Intalzaton: For each cluster ntalze ts mean µ, varance σ 2, and weght α. Nµ 1, σ 1 α 1 = P 1 Nµ 2, σ 2 α 2 = P 2 Nµ 3, σ 3 α 3 = P 3 19

20 Eectaton For each ont and each cluster comute P. P = P P / P P = Σ P P Where do we get P and P? 20

21 1. Use the df for a normal dstrbuton: 2. Use α = P from the current arameters of cluster. 21

22 Mamzaton Havng comuted P for each ont and each cluster, use them to comute new mean, varance, and weght for each cluster. = µ = T µ µ N = 22 σ 2 =

23 1 ={r 1, g 1, b 1 } 2 ={r 2, g 2, b 2 } ={r, g, b } luster Parameters µ 1,Σ 1, 1 for 1 µ 2,Σ 2, 2 for 2 µ k,σ k, k for k Mult-Dmensonal Eectaton Ste for olor Image Segmentaton Inut Known Inut Estmaton Outut + lassfcaton Results = = 23

24 1 ={r 1, g 1, b 1 } 2 ={r 2, g 2, b 2 } ={r, g, b } luster Parameters µ 1,Σ 1, 1 for 1 µ 2,Σ 2, 2 for 2 µ k,σ k, k for k Mult-dmensonal Mamzaton Ste for olor Image Segmentaton = Σ T µ µ Inut Known Inut Estmaton Outut + lassfcaton Results = µ N = 24

25 Full EM Algorthm Mult-Dmensonal Boot Ste: Intalze K clusters: 1,, K Iteraton Ste: Eectaton Ste Mamzaton Ste µ, Σ and P for each cluster. = = = Σ T µ µ = µ N = 25

26 Vsualzng EM lusters ellses show one, two, and three standard devatons mean 26

27 EM Demo Demo htt:// Eamle htt://www-2.cs.cmu.edu/~awm/tutorals/gmm13.df 27

28 EM Alcatons Blobworld: Image segmentaton usng Eectaton-Mamzaton and ts alcaton to mage queryng Y s Generatve/Dscrmnatve Learnng of obect classes n color mages 28

29 Blobworld: Samle Results 29

30 Janbo Sh s Grah-Parttonng An mage s reresented by a grah whose nodes are els or small grous of els. The goal s to artton the vertces nto dsont sets so that the smlarty wthn each set s hgh and across dfferent sets s low. 30

31 Mnmal uts Let G = V,E be a grah. Each edge u,v has a weght wu,v that reresents the smlarty between u and v. Grah G can be broken nto 2 dsont grahs wth node sets A and B by removng edges that connect these sets. Let cuta,b = uεa, vεb wu,v. One way to segment G s to fnd the mnmal cut. 31

32 uta,b cuta,b = uεa, vεb wu,v A B w1 w2 32

33 Normalzed ut Mnmal cut favors cuttng off small node grous, so Sh roosed the normalzed cut. cuta, B cuta,b NcutA,B = assoa,v assob,v normalzed cut assoa,v = wu,t u A, t V How much s A connected to the grah as a whole. 33

34 Eamle Normalzed ut A B 3 3 NcutA,B =

35 Sh turned grah cuts nto an egenvector/egenvalue roblem. Set u a weghted grah G=V,E V s the set of N els E s a set of weghted edges weght w gves the smlarty between nodes and Length N vector d: d s the sum of the weghts from node to all other nodes N N matr D: D s a dagonal matr wth d on ts dagonal N N symmetrc matr W: W = w 35

36 Let be a characterstc vector of a set A of nodes = 1 f node s n a set A = -1 otherwse Let y be a contnuous aromaton to Solve the system of equatons D W y = λ D y for the egenvectors y and egenvalues λ Use the egenvector y wth second smallest egenvalue to bartton the grah y => => A If further subdvson s merted, reeat recursvely 36

37 How Sh used the rocedure Sh defned the edge weghts w, by w, = e * - F-F 2 / σi - X-X 2 e / σx f X-X 2 < r 0 otherwse where X s the satal locaton of node F s the feature vector for node I whch can be ntensty, color, teture, moton The formula s set u so that w, s 0 for nodes that are too far aart. 37

38 Eamles of Sh lusterng See Sh s Web Page htt:// 38

39 Problems wth Grah uts Need to know when to sto Very Slooooow Problems wth EM Local mnma Need to know number of segments Need to choose generatve model 39

40 Mean-Shft lusterng Smle, lke K-means But you don t have to select K Statstcal method Guaranteed to converge to a fed number of clusters. 40

41 Fndng Modes n a Hstogram How Many Modes Are There? Easy to see, hard to comute 41

42 Mean Shft [omancu & Meer] Iteratve Mode Search 1. Intalze random seed, and wndow W 2. alculate center of gravty the mean of W: 3. Translate the search wndow to the mean 4. Reeat Ste 2 untl convergence 42

43 Numerc Eamle Must Use Normalzed Hstogram! wndow W centered at 12 mean shft H N 5/15 4/15 3/15 2/15 1/15 N = 105/15+114/15+123/15+132/15+141/15 =

44 Mean Shft Aroach Intalze a wndow around each ont See where t shfts ths determnes whch segment t s n Multle onts wll shft to the same segment 44

45 Segmentaton Algorthm Frst run the mean shft rocedure for each data ont and store ts convergence ont z. Lnk together all the z s that are closer than.5 from each other to form clusters Assgn each ont to ts cluster Elmnate small regons 45

46 Mean-shft for mage segmentaton Useful to take nto account satal nformaton nstead of R, G, B, run n R, G, B,, y sace 46

47 References Sh and Malk, Normalzed uts and Image Segmentaton, Proc. VPR arson, Belonge, Greensan and Malk, Blobworld: Image Segmentaton Usng Eectaton-Mamzaton and ts Alcaton to Image Queryng, IEEE PAMI, Vol 24, No. 8, Aug omancu and Meer, Mean shft analyss and alcatons, Proc. IV