cal/cal/spharm.cal

{ RCSid $Id: spharm.cal,v 1.4 2005/02/10 04:53:20 greg Exp $ }
{
        The first few Spherical Harmonics

        Feb 2005        G. Ward
}
                                        { Factorial (n!) }
fact(n) : if(n-1.5, n*fact(n-1), 1);

                                        { Associated Legendre Polynomials 0-8 }
LegendreP2(n,m,x,s) : select(n+1,
        select(m+1, 1),
        select(m+1, x, s),
        select(m+1, .5*(3*x*x - 1), 3*x*s, 3*(1-x*x)),
        select(m+1, .5*x*(5*x*x-3), 1.5*(5*x*x-1)*s, 15*x*(1-x*x), 15*s*s*s),
        select(m+1,
                .125*(3 + x*x*(-30 + x*x*35)),
                2.5*x*(-3 + x*x*7)*s,
                7.5*(7*x*x-1)*(1-x*x),
                105*x*s*s*s,
                105*s*s*s*s),
        select(m+1,
                .125*x*(15 + x*x*(-70 + x*x*63)),
                1.875*s*(1 + x*x*(-14 + x*x*21)),
                52.5*x*(1-x*x)*(3*x*x-1),
                52.5*s*s*s*(9*x*x-1),
                945*x*s*s*s*s,
                945*s*s*s*s*s),
        select(m+1,
                .0625*(-5 + x*x*(105 + x*x*(-315 + x*x*231))),
                2.625*(5 + x*x*(-30 + x*x*33))*s,
                13.125*s*s*(1 + x*x*(-18 + x*x*33)),
                157.5*(11*x*x-3)*x*s*s*s,
                472.5*s*s*s*s*(11*x*x-1),
                10395*x*s*s*s*s*s,
                10395*s*s*s*s*s*s),
        select(m+1,
                .0625*x*(-35 + x*x*(315 + x*x*(-693 + x*x*429))),
                .4375*s*(-5 + x*x*(135 + x*x*(-495 + x*x*429))),
                7.875*x*s*s*(15 + x*x*(-110 + x*x*143)),
                39.375*s*s*(1 + x*x*(-18 + x*x*33)),
                157.5*(11*x*x-3)*x*s*s*s,
                472.5*s*s*s*s*(11*x*x-1),
                10395*x*s*s*s*s*s,
                10395*s*s*s*s*s*s),
        select(m+1,
                .0078125*(35 + x*x*(-1260 + x*x*(6930 + x*x*(-12012 + x*x*6435)))),
                .5625*x*s*(-35 + x*x*(385 + x*x*(-1001 + x*x*715))),
                19.6875*s*s*(-1 + x*x*(33 + x*x*(-143 + x*x*143))),
                433.125*x*s*s*s*(3 + x*x*(-26 + x*x*39)),
                1299.375*s*s*s*s*(1 + x*x*(-26 + x*x*65)),
                67567.5*x*s*s*s*s*s*(5*x*x-1),
                67567.5*s*s*s*s*s*s*(15*x*x-1),
                2027025*x*s*s*s*s*s*s*s,
                2027025*s*s*s*s*s*s*s*s)
        );
                                        { Relation for Legendre with -M }
odd(n) : .5*n - floor(.5*n) - .25;
LegendreP(n,m,x) : if(m+.5,
                LegendreP2(n,m,x,sqrt(1-x*x)),
                fact(n+m)/fact(n-m) * LegendreP2(n,-m,x,sqrt(1-x*x))
                );
                                        { SH normalization factor }
SHnormF(l,m) : sqrt(0.25/PI*(2*l+1)*fact(l-m)/fact(l+m));
                                        { Spherical Harmonics theta function }
SHthetaF(l,m,theta) : SHnormF(l,m)*LegendreP(l,m,cos(theta));

                                        { Spherical Harmonic real portion }
SphericalHarmonicYr(l,m,theta,phi) : SHthetaF(l,m,theta)*cos(m*phi);
                                        { Spherical Harmonic imag. portion }
SphericalHarmonicYi(l,m,theta,phi) : SHthetaF(l,m,theta)*sin(m*phi);

                                        { Ordered, real SH basis functions }
{ Coeff. order based on Basri & Jacobs paper, "Lambertian Reflectance and
        Linear Subspaces," IEEE Trans. on Pattern Analysis & Machine Intel.,
        vol. 25, no. 2, Feb. 2003, pp. 218-33, Eq. (7):

        i       n       m       even/odd
        =       =       =       ========
        1       0       0       x
        2       1       0       x
        3       1       1       e
        4       1       1       o
        5       2       0       x
        6       2       1       e
        7       2       1       o
        8       2       2       e
        9       2       2       o
        10      3       0       x
        11      3       1       e
        ...
}
SH_B4(l,m,o,theta,phi) : if(m-.5, sqrt(2) *
                                if(o, SphericalHarmonicYi(l,m,theta,phi),
                                        SphericalHarmonicYr(l,m,theta,phi)),
                        SHthetaF(l,0,theta) ); 
SH_B3(l,r,theta,phi) : SH_B4(l,floor((r+1.00001)/2),odd(r+1),theta,phi);
SH_B2(l,i,theta,phi) : SH_B3(l,i-l*l-1,theta,phi);
SphericalHarmonicB(i,theta,phi) : SH_B2(ceil(sqrt(i)-1.00001),i,theta,phi);

                                        { Application of SH coeff. f(i) }
SH_F2(n,f,theta,phi) : if(n-.5, f(n)*SphericalHarmonicB(n,theta,phi) +
                                SH_F2(n-1,f,theta,phi), 0);
SphericalHarmonicF(f,theta,phi) : SH_F2(f(0),f,theta,phi);
Revision:	1.5
Committed:	Thu Feb 10 17:07:45 2005 UTC (19 years, 2 months ago) by greg
Branch:	MAIN
CVS Tags:	rad5R4, rad5R2, rad4R2P2, rad5R0, rad5R1, rad3R7P2, rad3R7P1, rad4R2, rad4R1, rad4R0, rad3R8, rad3R9, rad4R2P1, rad5R3, HEAD
Changes since 1.4:	+4 -3 lines
Log Message:	Corrected real series coefficients to include -m factors
#	User	Rev	Content
1	greg	1.5	{ RCSid $Id: spharm.cal,v 1.4 2005/02/10 04:53:20 greg Exp $ }
2	greg	1.1	{
3			The first few Spherical Harmonics
4
5			Feb 2005 G. Ward
6			}
7			{ Factorial (n!) }
8			fact(n) : if(n-1.5, n*fact(n-1), 1);
9
10			{ Associated Legendre Polynomials 0-8 }
11			LegendreP2(n,m,x,s) : select(n+1,
12			select(m+1, 1),
13			select(m+1, x, s),
14			select(m+1, .5(3xx - 1), 3xs, 3(1-x*x)),
15			select(m+1, .5x(5xx-3), 1.5(5xx-1)s, 15x(1-xx), 15sss),
16			select(m+1,
17			.125(3 + xx(-30 + xx*35)),
18			2.5x(-3 + xx7)*s,
19			7.5(7xx-1)(1-x*x),
20			105xsss,
21			105ssss),
22			select(m+1,
23			.125x(15 + xx(-70 + xx63)),
24			1.875s(1 + xx(-14 + xx21)),
25			52.5x(1-xx)(3xx-1),
26			52.5sss(9xx-1),
27			945xsss*s,
28			945ssss*s),
29			select(m+1,
30			.0625(-5 + xx(105 + xx(-315 + xx*231))),
31			2.625(5 + xx(-30 + xx33))s,
32			13.125ss(1 + xx(-18 + xx*33)),
33			157.5(11xx-3)xss*s,
34			472.5ssss(11x*x-1),
35			10395xsssss,
36			10395ssssss),
37			select(m+1,
38			.0625x(-35 + xx(315 + xx(-693 + xx429))),
39			.4375s(-5 + xx(135 + xx(-495 + xx429))),
40			7.875xss(15 + xx(-110 + xx143)),
41			39.375ss(1 + xx(-18 + xx*33)),
42			157.5(11xx-3)xss*s,
43			472.5ssss(11x*x-1),
44			10395xsssss,
45			10395ssssss),
46			select(m+1,
47			.0078125(35 + xx(-1260 + xx(6930 + xx(-12012 + xx*6435)))),
48			.5625xs(-35 + xx(385 + xx(-1001 + xx*715))),
49			19.6875ss(-1 + xx(33 + xx(-143 + xx*143))),
50			433.125xsss(3 + xx(-26 + xx*39)),
51			1299.375ssss(1 + xx(-26 + xx*65)),
52			67567.5xsssss(5x*x-1),
53			67567.5ssssss(15x*x-1),
54			2027025xsssssss,
55			2027025ssssssss)
56			);
57			{ Relation for Legendre with -M }
58			odd(n) : .5n - floor(.5n) - .25;
59			LegendreP(n,m,x) : if(m+.5,
60			LegendreP2(n,m,x,sqrt(1-x*x)),
61	greg	1.4	fact(n+m)/fact(n-m) * LegendreP2(n,-m,x,sqrt(1-x*x))
62	greg	1.1	);
63			{ SH normalization factor }
64			SHnormF(l,m) : sqrt(0.25/PI(2l+1)*fact(l-m)/fact(l+m));
65			{ Spherical Harmonics theta function }
66			SHthetaF(l,m,theta) : SHnormF(l,m)*LegendreP(l,m,cos(theta));
67
68			{ Spherical Harmonic real portion }
69			SphericalHarmonicYr(l,m,theta,phi) : SHthetaF(l,m,theta)cos(mphi);
70			{ Spherical Harmonic imag. portion }
71			SphericalHarmonicYi(l,m,theta,phi) : SHthetaF(l,m,theta)sin(mphi);
72
73			{ Ordered, real SH basis functions }
74	greg	1.3	{ Coeff. order based on Basri & Jacobs paper, "Lambertian Reflectance and
75			Linear Subspaces," IEEE Trans. on Pattern Analysis & Machine Intel.,
76			vol. 25, no. 2, Feb. 2003, pp. 218-33, Eq. (7):
77
78			i n m even/odd
79			= = = ========
80			1 0 0 x
81			2 1 0 x
82			3 1 1 e
83			4 1 1 o
84			5 2 0 x
85			6 2 1 e
86			7 2 1 o
87			8 2 2 e
88			9 2 2 o
89			10 3 0 x
90			11 3 1 e
91			...
92			}
93	greg	1.5	SH_B4(l,m,o,theta,phi) : if(m-.5, sqrt(2) *
94			if(o, SphericalHarmonicYi(l,m,theta,phi),
95	greg	1.1	SphericalHarmonicYr(l,m,theta,phi)),
96	greg	1.5	SHthetaF(l,0,theta) );
97	greg	1.1	SH_B3(l,r,theta,phi) : SH_B4(l,floor((r+1.00001)/2),odd(r+1),theta,phi);
98			SH_B2(l,i,theta,phi) : SH_B3(l,i-l*l-1,theta,phi);
99			SphericalHarmonicB(i,theta,phi) : SH_B2(ceil(sqrt(i)-1.00001),i,theta,phi);
100
101	greg	1.3	{ Application of SH coeff. f(i) }
102	greg	1.1	SH_F2(n,f,theta,phi) : if(n-.5, f(n)*SphericalHarmonicB(n,theta,phi) +
103			SH_F2(n-1,f,theta,phi), 0);
104			SphericalHarmonicF(f,theta,phi) : SH_F2(f(0),f,theta,phi);