Point Cloud Surface Representations

Transcription

1 Pont Cloud Surface Reresentatons Mark Pauly 2003 see also EG2003 course on Pont-based Comuter Grahcs avalable at: htt://grahcs.stanford.edu/~maauly/pdfs/pontbasedcomutergrahcs_eg03.df

2 Paers Hoe, DeRose, Ducham, McDonald, Stuetzle: Surface Reconstructon from Unorganzed Ponts, SIGGRAPH 92 Carr, Beatson, Cherre, Mtchell, Frght, McCallum, Evans: Reconstructon and Reresentaton of 3D Objects wth Radal Bass Functons, SIGGRAPH 01 Kalaah, Varshney: Statstcal Pont Geometry, Symosum on Geometry Processng,

3 Introducton Many alcatons need a defnton of surface based on ont samles Reducton U-samlng Interrogaton (e.g. ray tracng) Desrable surface roertes Manfold Smooth Local (effcent comutaton) 3

4 Introducton Terms Regular/Irregular, Aroxmaton/Interolaton, Global/Local Standard nterolaton/aroxmaton technques Trangulaton, Least Squares (LS), Radal Bass Functons (RBF) Problems Shar edges, feature sze/nose Functonal -> Manfold 4

5 Terms: Regular/Irregular Regular (on a grd) or rregular (scattered) Neghborhood s unclear for rregular data 5

6 Terms: Aroxmaton/Interolaton Nosy data -> Aroxmaton Perfect data -> Interolaton 6

7 Terms: Global/Local Global aroxmaton Local aroxmaton Localty comes at the exense of smoothness 7

8 Trangulaton Exlot the toology n a trangulaton (e.g. Delaunay) of the data Interolate the data onts on the trangles Pecewse lnear C 0 8

9 Trangulaton: Pecewse lnear Barycentrc nterolaton on smlces (trangles) gven d+1 onts x wth values f and a ont x nsde the smlex defned by x Comute α from x = Σ α x and Σ α = 1 Then f = Σ α f 9

10 Least Squares Fts a rmtve to the data Mnmzes squared dstances between the s and rmtve g g ( x) = a + bx + cx 2 mn g ( g( ) y x 2 10

11 Least Squares - Examle Prmtve s a olynomal g( x) ( 2 ) T 1, x, x c =,... mn 0 = ( ( ) ) 2 T 1,,,... c 2 y j x ( ( ) ) 2 T 1,,,... c y Lnear system of equatons that can be solved usng normal equatons Leads to a system of dm(c) equatons. x x x x 2 11

12 Radal Bass Functons Reresent nterolant as Sum of radal functons r Centered at the data onts f ( x) = w r ( x ) 12

13 13 Radal Bass Functons Solve to comute weghts w Lnear system of equatons ( ) = j j x x y w r () ( ) ( ) ( ) () ( ) ( ) ( ) () = y y y x x x x x x x x x x x x w w w r r r r r r r r r

14 Radal Bass Functons Solvablty deends on radal functon Several choces assure solvablty 2 r d = d log (thn late slne) r ( ) d ( d) = e d 2 / h 2 (Gaussan) h s a data arameter h reflects the feature sze or antcated sacng among onts 14

15 Interolaton Monomal, Lagrange, RBF share the same rncle: Choose bass of a functon sace Fnd weght vector for base elements by solvng lnear system defned by data onts Comute values as lnear combnatons Proertes One costly rerocessng ste Smle evaluaton of functon n any ont 15

16 Interolaton Problems Many onts lead to large lnear systems Evaluaton requres global solutons Solutons RBF wth comact suort Matrx s sarse Stll: soluton deends on every data ont, though dro-off s exonental wth dstance Local aroxmaton aroaches 16

17 Tycal Problems Shar corners/edges Nose vs. feature sze 17

18 Functonal -> Manfold Standard technques are alcable f data reresents a functon Manfolds are more general No arameter doman No knowledge about neghbors 18

19 Imlcts Each orentable n-manfold can be embedded n n+1 sace Idea: Reresent n-manfold as zero-set of a scalar functon n n+1 sace Insde: On the manfold: Outsde: f f f ( x) < 0 ( x) = 0 ( x) > 0 19

20 Imlcts - Illustraton Image courtesy Greg Turk 20

21 Imlcts from ont samles Functon should be zero n data onts f ( ) = 0 Use standard aroxmaton technques to fnd f Trval soluton: Addtonal constrants are needed f =

22 Imlcts from ont samles Constrants defne nsde and outsde Smle aroach (Turk, O Bren) Srnkle addtonal nformaton manually Make addtonal nformaton soft constrants

23 Imlcts from ont samles Use normal nformaton Normals could be comuted from scan Or, normals have to be estmated 23

24 Detour: Local Surface Analyss Estmate local surface roertes from local neghborhoods: No exlct connectvty between samles (as wth trangle meshes) Relace geodesc roxmty wth satal roxmty (requres suffcently hgh samlng densty!) Comute neghborhood accordng to Eucldean dstance 24

25 Neghborhood K-nearest neghbors Can be quckly comuted usng satal datastructures (e.g. kd-tree, octree, bs-tree) Requres sotroc ont dstrbuton 25

26 Neghborhood Imrovement: Angle crteron (Lnsen) Project onts onto tangent lane Sort neghbors accordng to angle Include more onts f angle between subsequent onts s above some threshold 26

27 Neghborhood Local Delaunay trangulaton (Floater) Project onts nto tangent lane Comute local Vorono dagram 27

28 28 Covarance Analyss Covarance matrx of local neghborhood N: wth centrod N j T n n =, 1 1 C = N N 1

29 Covarance Analyss Consder the egenroblem: C v l = λ v l l, l {0,1,2} C s a 3x3, ostve sem-defnte matrx All egenvalues are real-valued The egenvector wth smallest egenvalue defnes the least-squares lane through the onts n the neghborhood,.e. aroxmates the surface normal 29

30 Covarance Analyss Covarance ellsod sanned by the egenvectors scaled wth corresondng egenvalue 30

31 Normal Estmaton Estmate normal drecton by least squares ft Comute consstent orentaton by ncremental roagaton 31

32 Imlcts from ont samles Comute non-zero anchors n the dstance feld Use normal nformaton drectly as constrants f ( n ) =

33 Imlcts from ont samles need to constran dstance to avod selfntersectons f ( + d n ) = 1 33

34 Comutng Imlcts Gven N onts and normals, n and constrants f = 0, f c = Let = + N c An RBF aroxmaton ( ) ( ) d f ( x ) = w r ( x ) leads to a system of lnear equatons 34

35 Comutng Imlcts Practcal roblems: N > Matrx soluton becomes dffcult Dfferent solutons Sarse matrces allow teratve soluton Fast mult-ole methods Smaller number of RBFs 35

36 36 Comutng Imlcts Sarse matrces Needed: Comactly suorted RBFs () ( ) ( ) ( ) () ( ) ( ) ( ) () r r r r r r r r r 0 ) '( 0, ) ( = = > c r d r c d c c

37 Comutng Imlcts Fast mult-ole methods aroxmate soluton usng far- and near-feld exanson herarchcal clusterng of nodes ntroduces fttng error and evaluaton error Storage Solve system Evaluaton Drect Methods O(N^2) O(N^3) O(N) Fast Methods O(N) O(NlogN) O(1) + O(NlogN) setu 37

38 Comutng Imlcts RBF center reducton exlots the redundancy n many ont samled models Greedy aroach (Carr et al.) Start wth random small subset Add RBFs where aroxmaton qualty s not suffcent 38

39 Comutng Imlcts RBF center reducton: Examle 39

40 Imlcts - Conclusons Scalar feld s underconstraned Constrants only defne where the feld s zero, not where t s non-zero Addtonal constrants are needed Sgned felds restrct surfaces to be unbounded All mlct surfaces defne solds 40

41 Paer Hoe, DeRose, Ducham, McDonald, Stuetzle: Surface Reconstructon from Unorganzed Ponts, SIGGRAPH 92 41

42 Summary Goal: Reconstruct olygonal surface from unorganzed set of ont samles Aroach: Aroxmate sgned dstance functon Use contourng method (marchng cubes) to extract trangle mesh 42

43 More Detals Use lnear dstance feld er ont Drecton s defned by normal Normal estmated usng covarance analyss In every ont n sace use the dstance feld of the closest ont (Vorono decomoston) 43

44 More Detals X={x 0,..,.x n } samle of an unknown surface S δ-nosy: x = y + e, y on S, e < δ ρ-dense: Any shere wth radus ρ and center on S contans at least one same x justfcaton for usng k-nearest neghbors Algorthm comlexty: k-nearest neghbors: O(k*logN) normal orentaton: O(NlogN) contourng: O(m), m = #vsted cubes 44

45 Results + shaes of arbtrary toology + smle and effcent comutaton - crude aroxmaton of sgned dstance feld - no toologcal guarantees 45

46 Paer Carr, Beatson, Cherre, Mtchell, Frght, McCallum, Evans: Reconstructon and Reresentaton of 3D Objects wth Radal Bass Functons, SIGGRAPH 01 46

47 Summary Goal: Reconstruct mlct surface from unorganzed ont set Aroach: RBF mlct reresentaton Fast comutaton of matrx soluton usng multole method and RBF center reducton RBF aroxmaton of nosy data 47

48 More Detals RBF nterolaton s(x )=f, =1,...,N addtonal constrants usng normal nformaton smoothest nterolant: s* = argmn s S s accordng to rotaton-nvarant sem-norm. for nosy surface look for least-squares aroxmaton mn ρ s s N N = 1 ( s( x ) f ) 2 48

49 More Detals RBF center reducton 1. Choose subset of nodes and ft RBF s(x) 2. Evaluate resdual e = f s(x ) for all x 3. If {max { e } < fttng accuracy, sto 4. else aend new centers where e s large 5. recomute s(x) and goto 2 49

50 Results + Reconstructon from large ont sets + Irregular samlng dstrbutons + Smooth extraolaton for hole fllng 50

51 Results - Smoothng oeraton does not reserve features - Stll relatvely slow: Fttng tme n order of hours, surface tme n order of mnutes 51

52 Paer Kalaah, Varshney: Statstcal Pont Geometry, Symosum on Geometry Processng,

53 Summary Goal: Effcently reresent ont clouds usng statstcal methods Aroach: Octree subdvson PCA on ostons, normals, and color k-means clusterng and quantzaton 53

54 More Detals Subdvde ont cloud nto clusters usng octree herarchy 54

55 More Detals Aly rncal comonent analyss (PCA) on each cluster (covarance analyss) Treat ostons, normals, colors searately Reresent each cluster by mean + covarance ellsod Collecton of ellsods rovdes statstcal reresentaton of orgnal ont cloud 55

56 More Detals Alcaton: Randomzed Renderng samle PCA ellsods usng trvarate Gaussan PCA ellsod Gaussan random dstrbuton 56

57 More Detals Randomzed renderng 57

58 More Detals Comresson: 58

59 Results + Statstcal aroach well suted for large models + Can handle (some) nose - Decoulng of oston and normals leads to nferor renderng qualty (no coherence) - Comresson (robably) not comettve - Hard to aly nterrogaton or other oerators usng ths reresentaton 59