# Message Boards

Answer
(Unmark)

GROUPS:

5

# Spherical harmonics interpolation method that minimizes an energy function

Posted 1 year ago

Interpolation on a sphere is an essential problem in engineering, computer graphics, acoustics, and other fields, not to mention its mathematical interest. One approach to spherical interpolation is through spherical harmonics - these functions are smooth, numerically convenient, and easy to analyze. One issue, however, is that interpolation with spherical harmonics generally produces and overdetermined system, as spherical harmonics of degree l 2l+1 l 2 r 1/ 4/3 m m m λ->0 Out[]=
Qualitative introduction. An overdetermined model One problem with using spherical harmonics for interpolation is that they come in “packages”. There are 2 (L+1) l<=L l -l(l+1) ψ[x,y]= ψ 0 v x v y ψ[1,3]=0 ψ[6,15]=26 v y v x x v x v 26 2 (6-1) 2 (15-3) v (1,3) (6,15) m
Mathematical description
Notation and conventions The interpolation can be written as ψ[ϕ,θ]= 2 (L+1) ∑ j=1 x j Y j ( 1 ) where L -π<ϕ<=π -π/2<=θ<=π/2 Y j Y 1 0 Y 0 x j k Y l l<=L 2 (L+1) l 1+3+5+...+(1+2L)= 2 (L+1) Ax=b ( 2 ) where b i ψ i i A ij Y j ϕ i θ i j i A A m n= 2 (L+1) A mn n>=m k Y l k N l
( 3 ) where k N l k P l -l(l+1) 2 ∂ ϕ - 2 k ϕ Y 1 0 Y 0 Y 2 -1 Y 1 Y 3 0 Y 1 Y 4 +1 Y 1 Y 5 -2 Y 2 Y 6 -1 Y 2 Y 1 0 Y 0
Resolving the overdetermined system When a model has more parameters than constraints, one possible solution is to use the extra degrees of freedom to minimize a certain energy function. One way to do that is to express the n x m u x u m u m ψ= x j Y j E= T x W 0 ( 4 ) where W 0 Ax=b x x->x+εδ ε->0 δ Ax=b E A E A W 0 ( 5 ) Given that most physical and real-world functions are defined up to a constant - for example, temperature - the energy function must not punish constant functions. In other words, adding 0 Y 0 ψ W 0 W 0 W 0
( 6 ) The equation above is written in block matrix notation; here, the top-left zero is just the number 0, the off-diagonal zeros denote blocks of n-1 W W 0 ψ E W
( 7 ) We see that the first entry x 1 x x=
( 8 ) where block matrix notation is used: x 1 y n-1
T A ( 9 ) for some m v A A ij Y j ϕ i θ i j A j A i1 Y 1 Y 1 0 Y 0 A i1 i A=(
( 10 ) where c m1 Y 1 B m(n-1)
( 11 ) By looking at the first row of the equation, we get T c v i u i i>=2 v 1 m ∑ i=2 u i u u Wy= T B
( 12 ) where the unnamed matrix is of size mm [unnamedmatrix] ij δ ij δ i1 u x 1 u 1 x=Mu ( 13 ) where M=
( 14 ) In the formula (14), the 1 in the top left corner is the number 1, F (n-1)(m-1) F F= -1 W T B
( 15 ) By components, F ij ( -1 W T B i,j+1 ( -1 W T B i,1 m(m-1) u W 0 Ax=b u u m m x=Mu Ax=b AMu=b ( 16 )AM (
c n1 Y 1 W -1 W T B T (B -1 W C≡B -1 W F F ij C j+1,i C 1,i x u Ax=b T x W 0
Numerical check
Relation to Tikhonov regularization The formula (16) basically minimizes the quantity T (Ax-b) T x W 0 λ-> + 0 |