Spherical and Zonal Harmonics

Detailed Description

Classes
struct	cugar::SH_basis< L >

Functions
template<typename Vector3 >
CUGAR_HOST_DEVICE float	cugar::sh (const int32 l, const int32 m, const Vector3 &v)
template<int32 l, typename Vector3 >
CUGAR_HOST_DEVICE float	cugar::sh (const int32 m, const Vector3 &v)
template<int32 l, int32 m, typename Vector3 >
CUGAR_HOST_DEVICE float	cugar::sh (const Vector3 &v)
template<typename ZHVector , typename SHVector , typename Vector3 >
CUGAR_HOST_DEVICE void	cugar::rotate_ZH (const int32 L, const ZHVector &zh_coeff, const Vector3 &d, SHVector &sh_coeff)
template<int32 L, typename ZHVector , typename SHVector , typename Vector3 >
CUGAR_HOST_DEVICE void	cugar::rotate_ZH (const ZHVector &zh_coeff, const Vector3 &d, SHVector &sh_coeff)
template<int32 l, int32 m, typename Vector3 >
CUGAR_HOST_DEVICE float	cugar::rotate_ZH (const float zh_l, const Vector3 &d)
CUGAR_FORCEINLINE CUGAR_HOST_DEVICE float	cugar::sh_legendre_polynomial (const uint32 l, const uint32 m, const float x, const float y)
template<typename OutputIterator >
CUGAR_HOST_DEVICE void	cugar::sh_legendre_integrals (const uint32 n, const float x, OutputIterator r)
CUGAR_HOST_DEVICE float	cugar::sh_polar_integral (const int32 m, const float phi)

Function Documentation

rotate_ZH() [1/3]

template<typename ZHVector , typename SHVector , typename Vector3 >

CUGAR_HOST_DEVICE void cugar::rotate_ZH	(	const int32	L,
		const ZHVector &	zh_coeff,
		const Vector3 &	d,
		SHVector &	sh_coeff
	)

rotate a zonal harmonics to an arbitrary direction vector

Parameters

L	number of bands
zh_coeff	input Zonal Harmonics coefficients
d	input vector
sh_coeff	output Spherical Harmonics coefficients

rotate_ZH() [2/3]

template<int32 L, typename ZHVector , typename SHVector , typename Vector3 >

CUGAR_HOST_DEVICE void cugar::rotate_ZH	(	const ZHVector &	zh_coeff,
		const Vector3 &	d,
		SHVector &	sh_coeff
	)

rotate a zonal harmonics to an arbitrary direction vector, with the number of bands specified at compile-time.

Parameters

zh_coeff	input Zonal Harmonics coefficients
d	input vector
sh_coeff	output Spherical Harmonics coefficients

rotate_ZH() [3/3]

template<int32 l, int32 m, typename Vector3 >

CUGAR_HOST_DEVICE float cugar::rotate_ZH	(	const float	zh_l,
		const Vector3 &	d
	)

return the (l,m) spherical harmonics coefficient of a zonal harmonics function rotated to match a given axis.

Parameters

zh_l	l-band zonal harmonics coefficient
d	input vector

sh() [1/3]

template<typename Vector3 >

CUGAR_HOST_DEVICE float cugar::sh	(	const int32	l,
		const int32	m,
		const Vector3 &	v
	)

evaluate the (l,m)-th basis function on a given vector

Parameters

l	band index
m	subband index
v	input vector

sh() [2/3]

template<int32 l, typename Vector3 >

CUGAR_HOST_DEVICE float cugar::sh	(	const int32	m,
		const Vector3 &	v
	)

evaluate the (l,m)-th basis function on a given vector, where l is determined at compile-time.

Parameters

m	subband index
v	input vector

sh() [3/3]

template<int32 l, int32 m, typename Vector3 >

CUGAR_HOST_DEVICE float cugar::sh

(

const Vector3 &

v

)

evaluate the (l,m)-th basis function on a given vector, where l and m are determined at compile-time.

Parameters

v	input vector

sh_legendre_integrals()

template<typename OutputIterator >

CUGAR_HOST_DEVICE void cugar::sh_legendre_integrals	(	const uint32	n,
		const float	x,
		OutputIterator	r
	)

evaluate the associated Legendre integrals for a basis of order n see Algorithm 1 in:

Importance Sampling Spherical Harmonics, W.Jarosz, N.Carr, H.W.Jensen

Parameters

n	SH order
x	cos(theta), the point of evaluation of the associated (indefinite) Legendre integrals
r	a pointer to the output evaluations for all (l,m) with (0 <= l < n), (0 <= m <= l)

sh_legendre_polynomial()

CUGAR_FORCEINLINE CUGAR_HOST_DEVICE float cugar::sh_legendre_polynomial	(	const uint32	l,
		const uint32	m,
		const float	x,
		const float	y
	)

evaluate the associated Legendre polynomial P_l^m on x = cos(theta), y = sin(theta) = sqrt(1 - x*x)

sh_polar_integral()

CUGAR_HOST_DEVICE float cugar::sh_polar_integral	(	const int32	m,
		const float	phi
	)

evaluate the integral of the polar angle function Phi^m(phi) see equation 14 in:

Importance Sampling Spherical Harmonics, W.Jarosz, N.Carr, H.W.Jensen

Parameters

m	required order
phi	the point of evaluation, in angular coordinates