Optimal Data Structures for Spherical Multiresolution

Transcription

1 Optimal Data Structures for Spherical Multiresolution Analysis and Synthesis J. A. Rod Blais Dept. of Geomatics Engineering Pacific Institute for the Mathematical Sciences University of Calgary, Calgary, AB ca/

2 Introduction Geopotential and related fields are global with continuous spectra Multiresolution analysis and synthesis require regular array data Spherical Quad Tree (SQT) data structures are most appropriate Spherical Harmonic Wavelets (SHWs) using Transforms (SHTs) Equiangular discretizations are most common in applications Equitriangular (near equiareal) are often desirable in practice Other equidistribution strategies are much less appropriate Simulations with Geopotential Models (GMs) are discussed Concluding Remarks

3 Multiresolution Analysis on Sphere (S 2 ) A multiresolution analysis of L 2 (S 2 ) is a nested sequence of subspaces {V n : n = 0, ± 1, ± 2, } such that 2 2 L (S )... V1 V0 V 1... f(t) V n f(st) V n-1, s > 1 (dilations) f(t) V n f(t-k) V n (translations) a scaling or smoothing function (t) s g (st k) such hthat t{ {(t-k)} is i an orthonormal l basis b i of f V 0 a detail or wavelet function (t) s h k(st k) that is a quadratic conjugate of the scaling function (t) k k k

4 Multiresolution Analysis of GMs GM GM GM GM N N/2 0 M M M N N/2 0 using the lowpass filter kernel with decimation (binary scaling) K : GM GM and k :u (, ) u (, ) L N N/2 L N N/2 and highpass filter kernel with decimation (binary scaling) K : GM M and k :u (, ) v (, ) H N N/2 H N N/2 satisfying Bezout s theorem: K K K K I L L H H for reconstruction purposes: KLGMN/2 KHMN/2 GMN

5 Example: SHT of Order N 2B CS c s s s s s c c s s s s c c c s s s c c c c s s cb1,0 cb1,1 cb1,b1 cb c s cn1,0 cn1,1 cn1,b1 c c c B1,1 B1,1 B1,1 N1, B 1,2 B 1,2 B 1,2 N 1,2 B 1,0 B11 1,1 B 1,B 1 B,B B1 B 1,B 1 N 1,B 1 B,0 B,1 B,B 1 B,B B 1,B N 1,B 1,B B1,B1 N1,B1 N1,B N1,B1 N1,N1

6 Multiresolution Synthesis of GMs For a potential function u(, ) in GM u(, ) u (, ) k v (, ) k v (, ) 0 H 0 H N u N(, ) and with more observational information, u(, ) u (, ) k v (, ) k v (, ) N H N H 2N Recall that with the usual (geodetic) conventions, N 1 m N n,m n n0 m n u (, ) u Y (, ) N1 n c cosm s sinm P (cos ) n0 m0 nm nm nm

7 Estimation of Spectral Coefficients Chebychev Quadrature (CQ) Methods: Equiangular grids of e.g. 2Nx4N points for Δθ = Δλ With 2N equispaced parallels and at least N equispaced meridians Usually excluding the poles for numerical stability» Providing least degree N per data (N 2 coefts, O(N 3 ) oper ns) Least-Squares (LS) Methods: Equiangular grids of e.g. Nx2N points for Δθ = Δλ With at least N parallels and at least N equispaced meridians Usually excluding the poles for numerical stability Multiple least-squares estimation per order» Requiring least data per degree N (N 2 coefts, O(N 4 ) oper ns)

8 SHT Using Chebychev Quadrature SHT: with and z11 z z1k c00 s s 1 z z... z c c... s K z z... z c c... c J1 J2 JK 0 1 c cosm qz P (cos ) J K nm k jk j nm j s nm j1 k1 sinm k n z (c cosm s sinm ) P (cos ) jk nm k nm k nm j n0 m0 in which q j denote the Chebychev quadrature weights [Blais, 2011].

9 SHT Using Least Squares From the previous synthesis formulation z "const " jk const. IDFT (cnm is nm)p nm (cos j ) k1,k nm one has for unknown coefficients and c nm s nm DFT[z jk ] "const." (cnm i s nm )P nm(cos j) k1,k nm implying a least-squares problem per order m, with θ j = j π/n, j = 0, 1,, N-1 λ k = k π/n, k = 0, 1,, 2N-1 c for and with m n, n = 0, 1,, N-1 [Blais, 2011]. nm s nm

10 SHT Computations for Degree N For gridded ddata with at tleast t2n equispaced disolatitude data DFT IDFT z u iv zˆ jk per row jh jh per row jk For gridded data with 2N data per column with constant Δθ Chebychev c is u iv cˆ isˆ nm nm jh jh Quadrature nm nm For gridded data with N data per column with variable Δθ Least c i s u i v cˆ i sˆ nm nm jh jh Squares nm nm Note that only DFTs are generally invertible above.

11 Equidistribution on the Sphere In general for an arbitrary partition of the sphere S 2, N C( n)/n n 1 Discrepancy sup N 2 all CS lim C( n) / N N in which C( k) denotes the characteristic function for the cell C. n 1 With pseudo-random numbers, Disc. O((log log N) 1/2 /N 1/2 ) With quasi-random numbers, Disc. O((log N) s /N ) for dim. s Hence quasi-random (deterministic) sequences are advantageous in higher dimensional simulations [Morokoff & Caflisch, 1994]

12 Quad Tree Data Structures Planar Quad Trees Two-dimensional pyramidal binary data structures Planar domains can be square, rectangular or triangular Most often used for multiresolution analysis/synthesis Spherical Quad Trees Spherical pyramidal binary data structures Spherical partitions can be equiangular or equitriangular Octahedrons and icosahedrons have equitriangular faces Densification can only be near equiareal in general

13 Near Equiareal Strategies Platonic Solids (for exact regularity) Tetrahedron ( 4 vertices, 4 edges, 4 faces) Cube ( 8 vertices, 12 edges, 6 faces) Octahedron ( 6 vertices, 12 edges, 8 faces) Dodecahedron (20 vertices, 30 edges, 12 faces) Icosahedron (12 vertices, 30 edges, 20 faces) Dualities (by interchanging faces and vertices) Tetrahedron Tetrahedron Cube Octahedron Dd Dodecahedron hd Icosahedron

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15 Cube and Octahedron For the octahedron, radially projected face centres are equispaced on parallels enabling the use of FFTs in SHTs.

16 Othd Octahedron Based dexamples Level One: x x x x x x x x Level Two: x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x Level Three: x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x Data matrix: 2 L x 4(2 L -1) for level L=1, 2, 3,

17 Octahedron Based EGM 2008 Level: Grid CQ SHT without MASK CQ SHT with MASK Spectral RMS Spatial RMS Spectral RMS Spatial RMS 1: E E E E+00 2: E E E E E E E E+00 3: E E E E-07 4: E E E E-07 5: E E E E-08 6: E E E E-08 7: E E E E-08 8: E E E E-09 9: E E E E-09 10: E E E E-09 11: E E E E-10 12: E E E E-11

18 Octahedron Based EGM 2008 Level: Grid LS SHT without MASK LS SHT with MASK Spectral RMS Spatial RMS Spectral RMS Spatial RMS 1: E E E E+00 2: E E E E-06 3: E E E E-06 4: E E E E-06 5: E E E E E-06 6: E E E E-06 7: E E E E-06 8: E E E E-06 9: E E E E-06 10: E E E E-06 11: E E E E-06

19 Dodecahedron and Icosahedron For the icosahedron, radially projected face centres are equispaced on parallels enabling the use of FFTs in SHTs.

20 Icosahedron Based Examples Level One: x x x x x x x x x x x x x x x x x x x x Level Two : x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x Level Three: x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x Data matrix: 3 2 L-1 x 10 2 L-1 for level L=1, 2, 3,

21 Icosahedron Based EGM 2008 Level: Grid CQ SHT without MASK CQ SHT with MASK Spectral RMS Spatial RMS Spectral RMS Spatial RMS 2: E E E E-08 3: E E E E-07 4: E E E E-08 5: E E E E-08 6: E E E E-08 7: E E E E-09 8: E E E E-09 9: E E E E-09 10: E E E E-10 11: E E E E-10 10

22 Icosahedron Based EGM 2008 Level: Grid LS SHT without MASK LS SHT with MASK Spectral RMS Spatial RMS Spectral RMS Spatial RMS 1: E E E E-22 2: E E E E-06 3: E E E E-06 4: E E E E-06 5: E E E E-06 6: E E E E-06 7: E E E E-06 8: E E E E-06 9: E E E E-06 10: E E E E-06 06

23 Reuter Grids Homogeneous point distributions ib ti include e.g. Reuter grids Fast computations can be performed using panel clustering New grids are required for each level of decomposition Not a pyramidal data structure for analysis and synthesis Latitude (de eg) 0 eg) Latitude (d Longitude (deg) Longitude (deg)

24 Concluding Remarks Multiresolution analysis/synthesis can be done on 2 or S 2 Equiangular grids are common but not always desirable Equitriangular (near equiareal) are generally convenient Other equidistributed ib t d point sets often have drawbacks SHTs (CQ & LS) analysis/synthesis give comparable RMS With EGM and like data, equitriangular grids work well With noise like data, some analysis problems may arise