Orthogonal and Symmetric Haar Wavelets on the Sphere

Transcription

1 Orthogonal and Symmetric Haar Wavelets on the Sphere

2 2 Motivation Spherically parametrized signals can be found in many fields: Computer graphics Physics Astronomy Climate modeling Medical imaging...

3 3 Motivation An efficient representation of spherical signals is of importance for many applications

4 4 Motivation An efficient representation of spherical signals is of importance for many applications Of particular interest are the ability to approximate a wide range of signals with a small number of basis function coefficients process signals efficiently in the basis representation

5 5 SOHO Wavelets Orthogonality l 2 optimal approximation can be found efficiently Computations are in many cases more efficient Very compact support Minimal costs for performing wavelet transform Processing of signals is efficient Symmetry Avoids artifacts when a signal is approximated in the basis representation

6 6 Representations for Spherical Signals Spherical Harmonics [MacRobert 1948] Physics and chemistry Geoscience [Clarke 2004] Medical imaging [Katsuyuki 2001] Computer graphics [Cabral 1987, Westin 1992, Ramamoorthi 2002, Kautz 2002, Sloan 2002] Spherical Radial Basis Functions Astronomy and geoscience [Fisher 1993; Freeden 1998] Computer graphics [Green 2006; Tsai 2006]

7 7 Wavelets for Spherical Signals Wavelets defined over Euclidean domains [Ng 2003; Wang 2006; Lalonde 1997] Wavelet bases defined over general subdivision surfaces [Lounsbery 1997] Haar wavelets for general measure spaces [Girardi 1997] Discrete spherical wavelets [Schröder 1995] L p Nearly orthogonal spherical Haar wavelets [Nielson 1997; Bonneau 1999; Rosça 2004] Pseudo wavelets defined over the sphere [Ma 2006]

8 8 Background Inner product norm l 2 Orthogonality Wavelet basis f,g = fgdω S 2 f = f,f f i,f i = δ i,i Ψ={ϕ 0,0,ψ j,m j J,m M(j)}

9 9 Background Fast Wavelet Transform

10 10 Background Fast Wavelet Transform Analysis step λ j,k = γ j,m = h j,k,l λ j+1,l l L(j,k) l L(j,m) g j,m,l λ j+1,l

11 11 Background Fast Wavelet Transform Analysis step λ j+1,l = λ j,k = γ j,m = Synthesis step h j,k,l λ j+1,l l L(j,k) l L(j,m) g j,m,l λ j+1,l h j,k,l λ j,k + g j,m,l γ j,m k K(j,l) m M(j,m)

12 12 Partition Basis functions are defined over a partition Hierarchical subdivision scheme of domain

13 13 Partition Basis functions are defined over a partition Hierarchical subdivision scheme of domain Partition is formed by spherical triangles T j,k

14 14 Partition Basis functions are defined over a partition Hierarchical subdivision scheme of domain Partition is formed by spherical triangles T j,k

15 15 Partition Basis functions are defined over a partition Hierarchical subdivision scheme of domain Partition is formed by spherical triangles T j,k

16 16 Partition Basis functions are defined over a partition Hierarchical subdivision scheme of domain Partition is formed by spherical triangles T j,k

17 17 Partition One new vertex on each of the arcs of T j,k b B C a c A

18 18 Partition One new vertex on each of the arcs of T j,k b B C a c A

19 19 Partition One new vertex on each of the arcs of 4-fold subdivision T j,k Every spherical triangle has four child triangles b B C a c A

20 20 Partition One new vertex on each of the arcs of 4-fold subdivision T j,k Every spherical triangle has four child triangles VIDEO

21 21 Scaling Basis Functions For Haar-like bases scaling functions are defined as ϕ j,k = η j,k χ j,k

22 22 Scaling Basis Functions Scaling functions of the SOHO wavelet basis ϕ j,k = τ j,k αj,k

23 23 Scaling Basis Functions Scaling functions of the SOHO wavelet basis ϕ j,k = τ j,k αj,k Scaling function filter coefficients are therefore αj+1,l k h j,k,l = αj,k

24 24 Scaling Basis Functions Refinement relationship ϕ j,k = h j,k,l ϕ j+1,l l L(j,k)

25 25 Scaling Basis Functions Refinement relationship ϕ j,k = l L(j,k) h j,k,l ϕ j+1,l

26 26 Scaling Basis Functions Refinement relationship ϕ j,k = l L(j,k) h j,k,l ϕ j+1,l

27 27 Scaling Basis Functions Refinement relationship ϕ j,k = l L(j,k) h j,k,l ϕ j+1,l =

28 28 Wavelet Basis Functions Span the difference spaces W j where V j W j = V j+1 V j = span k K(j) (ϕ j,k ) W j = span m M(j) (ψ j,m ) Refinement relationship for wavelet basis functions ψ j,m = k K(j) g j,m,l ϕ j+1,l

29 29 Wavelet Basis Functions Wavelet basis functions associated with defined entirely over child triangles T j,k T j+1,l are

30 30 Wavelet Basis Functions Wavelet basis functions associated with defined entirely over child triangles T j,k T j+1,l are =

31 31 Wavelet Basis Functions Wavelet basis functions associated with defined entirely over child triangles T j,k T j+1,l are The disjoint nature of the domains implies that wavelet basis functions on the same level but defined over different partitions are disjoint ψ i j,k,ψ i j,k = δ k,k

32 32 Wavelet Basis Functions Wavelet basis functions associated with defined entirely over child triangles are The disjoint nature of the domains implies that wavelet basis functions on the same level but defined over different partitions are disjoint ψ i j,k,ψ i j,k = δ k,k T j,k T j+1,l A vanishing integral of the wavelet basis functions guarantees that wavelet basis functions on different levels are orthogonal

33 33 Wavelet Basis Functions Wavelet basis functions associated with defined entirely over child triangles T j,k T j+1,l are The disjoint nature of the domains implies that wavelet basis functions on the same level but defined over different partitions are disjoint A vanishing integral of the wavelet basis functions guarantees that wavelet basis functions on different levels are orthogonal => Only one spherical triangle has to be considered for the derivation of the wavelet basis functions

34 34 Local Synthesis Matrix For fixed, the analysis and synthesis steps of the fast wavelet transform can be expressed as compact matrix-vector products λ 0 j+1 λ 1 j+1 λ 2 j+1 T j,k = h 0 g 0 0 g 1 0 g 2 0 h 1 g 0 1 g 1 1 g 2 1 h 2 g 0 2 g 1 2 g 2 2 λ j γ 0 j γ 1 j λ 3 j+1 } h 3 g 0 3 g 1 3 g 2 3 } γ 2 j

35 35 Local Synthesis Matrix T j,k For fixed, the analysis and synthesis steps of the fast wavelet transform can be expressed as compact matrix-vector products λ 0 j+1 h 0 g0 0 g0 1 g0 2 λ j λ 1 j+1 h 1 g1 0 g1 1 g 2 1 γ 0 j = λ 2 j+1 h 2 g2 0 g2 1 g2 2 γj 1 λ 3 j+1 } h 3 g3 0 g3 1 {{ g3 2 } S j,k γ 2 j

36 36 Symmetry A spherical Haar wavelet basis is symmetric is for fixed T j,k the labeling of the three outer child triangles can be altered, without changing the numerical value of the associated basis function coefficients

37 37 Wavelet Basis Functions Semi-orthogonal wavelet basis ψ 0 j,k,ϕ j,k = ψ 1 j,k,ϕ j,k = ψ 2 j,k,ϕ j,k =0

38 38 Wavelet Basis Functions Semi-orthogonal wavelet basis ψ 0 j,k,ϕ j,k = ψ 1 j,k,ϕ j,k = ψ 2 j,k,ϕ j,k =0 Dirac bra-ket notation [ Φ j,k Ψ j,k ]=0

39 39 Wavelet Basis Functions Semi-orthogonal wavelet basis ψ 0 j,k,ϕ j,k = ψ 1 j,k,ϕ j,k = ψ 2 j,k,ϕ j,k =0 Dirac bra-ket notation [ Φ j,k Ψ j,k ]=0 With the refinement relationship this yields [ Φ j,k Φ j+1,k ] G j,k =0

40 40 Wavelet Basis Functions Nullspace gives semi-orthogonal wavelet basis G j,k = α 1 α0 α 2 α0 α 3 α

41 41 Wavelet Basis Functions Synthesis matrix S j,k = α0 αp α 1 α0 α 2 α0 α 3 α1 αp α2 αp α3 αp α0

42 42 Wavelet Basis Functions Parametric synthesis matrix α0 α1 αp a 1,2 α0 Ŝ j,k = a 1,3 α2 α0 a 1,4 α3 α0 α1 αp a 2,2 a 2,3 a 2,4 α2 αp a 3,2 a 3,3 a 3,4 α3 αp a 4,2 a 4,3 a 4,4

43 43 Wavelet Basis Functions Parametric synthesis matrix α0 α1 αp a 1,2 α0 Ŝ j,k = a 1,3 α2 α0 a 1,4 α3 α0 α1 αp a 2,2 a 2,3 a 2,4 α2 αp a 3,2 a 3,3 a 3,4 α3 αp a 4,2 a 4,3 a 4,4 Orthogonality and symmetry can be formulated as linear system

44 44 Wavelet Basis Functions Area-isometry of the three outer child triangles

45 45 Wavelet Basis Functions Area-isometry of the three outer child triangles => Symmetry can be enforced by the parametrization of the synthesis matrix Ŝ j,k = α0 αp c α 1 α0 c α 1 α0 c α 1 α0 α1 αp b a a α1 αp a b a α1 αp a a b

46 46 Wavelet Basis Functions With Ŝ j,k orthogonality can be enforced with a simple linear system 0 = c α1 αp +2a α1 αp + b α1 αp 0 = c 2 α 1 α 0 + a 2 +2ab 0 = c α 1 +2a α 1 + b α 1

47 47 Wavelet Basis Functions Wavelet basis functions for the SOHO wavelet basis ψ 0 j,k = ψ 1 j,k = ψ 3 j,k = α1 τ (( 2a +1)τ 1 + aτ 2 + aτ 3 ) α 0 α1 α1 τ ( aτ 1 +( 2a +1)τ 2 + aτ 3 ) α 0 α1 α1 τ ( aτ 1 + aτ 2 +( 2a +1)τ 3 ) α 0 α1 with a = α 0 ± α α 0 α 1 3α 0

48 48 Wavelet Basis Functions Wavelet basis functions for the SOHO wavelet basis

49 49 Subdivision Scheme Partition defines only topology

50 50 Subdivision Scheme Partition defines only topology => Positions of new vertices can be chosen so that areaisometry of the outer child partitions is satisfied

51 51 Subdivision Scheme Partition defines only topology => Positions of new vertices can be chosen so that areaisometry of the outer child partitions is satisfied b-β 1 c-β 2 β 1 γ γ β 2

52 52 Subdivision Scheme Our subdivision Geodesic Bisector

53 53 Experiments Comparison of SOHO wavelet basis with six previously proposed spherical Haar wavelet bases Bio-Haar wavelets [Schröder 1995] Pseudo Haar wavelets [Ma 2006] Four nearly orthogonal spherical Haar wavelet bases [Nielson 1997; Bonneau 1999]

54 54 Experiments Comparison of SOHO wavelet basis with six previously proposed spherical Haar wavelet bases Bio-Haar wavelets [Schröder 1995] Pseudo Haar wavelets [Ma 2006] Four nearly orthogonal spherical Haar wavelet bases [Nielson 1997; Bonneau 1999] Three test signals

55 55 Experiments Nonlinear approximation: Reconstruction of the test signals with a subset of all basis function coefficients Increasing number of nonzero basis function coefficients l 2 optimal approximation strategies Octahedron as base polyhedron Signals defined over partitions on level eight of the partition trees: 131,072 basis function coefficients

56 56 Experiments Reconstruction with 64 basis function coefficients

57 57 Experiments Reconstruction with 256 basis function coefficients

58 58 Experiments Reconstruction with 512 basis function coefficients

59 59 Experiments Reconstruction with 2056 basis function coefficients

60 60 Experiments Reconstruction with 8192 basis function coefficients

61 61 Experiments Reconstruction with basis function coefficients

62 62 Experiments Reconstruction with basis function coefficients

63 63 Experiments Reconstruction with all basis function coefficients

64 64 Experiments Comparison of l 2 optimal and k-largest approximation strategy for the Bio-Haar basis L2 Error Texture Map, k largest Texture Map, optimal BRDF, k largest BRDF, optimal Visibility Map, k largest Visibility, optimal Coefficients Retained x 10 4

65 65 Experiments Approximation of texture map L2 Error Texture Map SOHO Bio Haar Pseudo Haar Nielson1 Nielson2 Bonneau1 Bonneau Coefficients Retained

66 66 Experiments Approximation of BRDF L2 Error BRDF SOHO Bio Haar Pseudo Haar Nielson1 Nielson2 Bonneau1 Bonneau Coefficients Retained

67 67 Experiments Approximation of visibility map L2 Error Visibility Map SOHO Bio Haar Pseudo Haar Nielson1 Nielson2 Bonneau1 Bonneau Coefficients Retained

68 68 Experiments Approximation of texture map SOHO Bio-Haar Pseudo Haar

69 69 Experiments Approximation of texture map SOHO Bio-Haar Pseudo Haar

70 70 Experiments Approximation of BRDF SOHO Bio-Haar Pseudo Haar

71 71 Experiments Approximation of visibility map SOHO Bio-Haar Pseudo Haar

72 72 Rotation of Signals Rotation and alignment of signals necessary in many applications

73 73 Rotation of Signals Rotation and alignment of signals necessary in many applications

74 74 Rotation of Signals Rotation and alignment of signals necessary in many applications

75 75 Rotation of Signals Rotation and alignment of signals necessary in many applications

76 76 Rotation of Signals Basis transformation matrix allows to project from the rotated source basis into the unrotated target basis BTM Source Basis Target Basis

77 77 Rotation of Signals Previous work: Planar representations for spherical signals [Wang 2006]

78 78 Rotation of Signals Previous work: Planar representations for spherical signals [Wang 2006] => Numerical computation of basis transformation matrix elements

79 79 Rotation of Signals Previous work: Planar representations for spherical signals [Wang 2006] => Numerical computation of basis transformation matrix elements Spherical Haar wavelet bases: Rotation amounts to translation of basis functions on the sphere

80 80 Rotation of Signals Previous work: Planar representations for spherical signals [Wang 2006] => Numerical computation of basis transformation matrix elements Spherical Haar wavelet bases: Rotation amounts to translation of basis functions on the sphere => Analytic computation of matrix elements

81 81 Rotation of Signals Rotation matrices for spherical Haar wavelet bases have quasi block symmetric structure nz = nz =

82 82 Rotation of Signals Experiments Source Basis Target Basis

83 83 Conclusion SOHO wavelet basis: orthogonal and symmetric spherical Haar wavelet basis Well suited for the efficient approximation and processing of spherical signals Competitive or lower error rates than previously proposed spherical Haar wavelet bases Signals represented in the SOHO wavelet basis can be rotated with basis transformation matrices: analytic computation of the matrix elements possible

84 84 Outlook What are sufficient conditions for a symmetric and orthogonal spherical Haar wavelet basis? Any constraint necessary? What alternative constraints next to the areaisometry of the three outer child triangles exist? Does a smooth, orthogonal and symmetric spherical wavelet basis exist? Advantages in practice?

85 85 Outlook Many fields might benefit from the use of the SOHO wavelet basis Computer graphics Medical imaging Climate modeling Astronomy...

86 86 Outlook Operation-aware representations Design of bases for specific applications and operations such as rotation Basis transformation matrices are an interesting starting point Overcomplete representations Are they useful for applications beyond data analysis?

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88 88 References [Nielson 1997] Gregory M. Nielson, Il-Hong Jung, and Junwon Sung. Haar Wavelets over Triangular Domains with Applications to Multiresolution Models for Flow over a Sphere. In VIS 97: Proceedings of the 8th Conference on Visualization 97, pages 143 ff., Los Alamitos, CA, USA, IEEE Computer Society Press. [Bonneau 1999] Georges-Pierre Bonneau. Optimal Triangular Haar Bases for Spherical Data. In VIS 99: Proceedings of the Conference on Visualization 99, pages , Los Alamitos, CA, USA, IEEE Computer Society Press. [Rosça 2004] Daniela Rosça. Optimal Haar Wavelets on Spherical Triangulations. Pure Mathematics and Applications, 15(2), [Ma 2006] Wan-Chun Ma, Chun-Tse Hsiao, Ken-Yi Lee, Yung-Yu Chuang, and Bing-Yu Chen. Real-Time Triple Product Relighting Using Spherical Local-Frame Parameterization. The Visual Computer, (9-11): , Pacific Graphics 2006 Conference Proceedings. [Wang 2006] Rui Wang, Ren Ng, David Luebke, and Greg Humphreys. Efficient Wavelet Rotation for Environment Map Rendering. In Proceedings of the 2006 Eurographics Symposium on Rendering. Springer-Verlag, Vienna, Published as Rendering Techniques [Stollnitz 1996] Eric J. Stollnitz, Tony D. Derose, and David H. Salesin. Wavelets for Computer Graphics: Theory and Applications. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 1996.