Fast spherical hormonic transform b 92 b 02053

Fast spherical hormonic transform 歐永俊 資 三 b 92 b 02053 1

Background Direct computation: O(N^4) BUT Separable!! Can be reduced to O(N^3) 2

Background Outer IDFT: O(N^3) O(n^2 lgn) FFT Inner SHM: O(N^3) Fast SHM O(n^2 lgn) 3

Background The problem of a fast spherical harmonic transform is now reduced to the efficient calculation of these discrete Legendre transforms. for an arbitrary input vector s with kth component [s]k. 4

STEP 1: using the Legendre recurrence The recurrence relation satisfied by degree: L Stands at level L, iterating the recurrence formula forward r steps degree: r, r-1 associated legendre functions, satisfy follows: 5

STEP 1: using the Legendre recurrence In matrix form( recurrence relation): Purpose : rewrite projections onto high degree Legendre polynomials as sums of projections onto (shifted) legendre Polynomials of lower degree. 6

STEP 1: using the Legendre recurrence stored at early time degree at most r Higher degree inner product can be compute as inner products of stored data and (precomputed) sampled values of the polynomials A, B( remember it satisfies recurrence relation of asso. legendre) “A hint to divide and conquer but not enough!” 7

STEP 2: smoothing and subsampling(working in the “cosine transform” domain) What is not enough? Projections of a data vector onto lower degree trigonometric polynomials should be computed more efficiently. How? Can be done by first lowpass filtering(smoothing) and subsampling the data vector(in cosine transform domain). Then the complexity of computing these inner products grows linearly with n. 8

STEP 2: smoothing and subsampling(working in the “cosine transform” domain) denote the n-dinmensional orthogonal DCT matrix comprised of normalized sampled cosines DCT 9

STEP 2: smoothing and subsampling(working in the “cosine transform” domain) fact 1: orthogonality of C implies fact 2: for any trigonometric polynomial Q of degree at most N-1 the cosine coefficients vanish for n>deg(Q) Computation of the low degree Legendre projections accomplished with very few operations 10

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STEP 2: smoothing and subsampling(working in the “cosine transform” domain) define: critically sampled lowpass operator (bandwidth p) define: truncation operator (keeps first p coordinates of given input vector) is to first to compute cosine representation of a vector the effect of of length N, then remove all frequency components beyond p from s (smoothing), and finally, to keep only those samples necessary to represent this smoothed version(subsampling). 13

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STEP 2: smoothing and subsampling(working in the “cosine transform” domain) Q: degree p Note: the effect of applying to is to simply sample Q on coarser grid O(nlgn) by fast DCT/IDCT 15

STEP 3: divide and conquer low degree transform coefficients: high degree transform coefficients: lower part: (a discrete transform of size N/2) 16

STEP 3: divide and conquer higher part: (Page 7) ( with degree at most N/2) (a discrete transform of size N/2) 17

Summary O(Nlg. N) page 15 two problems of size N/2 Page 16, 17 Overall is 18

Numerical result 19

reference 1. A fast spherical harmonics transform algorithm R Suda, M Takami - Math. Comp, 2002 2. D. Healy, S. Moore, and D. Rockmore, “An FFT for the 2 -sphere improvements and variations” Proc. of ICASSP-96 volume 3. pp. 1323 -1326 20