void Forward_SO3_Naive_fftw( int bw,
fftw_complex *data,
fftw_complex *coeffs,
fftw_complex *workspace_cx,
fftw_complex *workspace_cx2,
double *workspace_re,
double *weights,
fftw_plan *p1,
int flag )
{
int j, n, n3 ;
int m1, m2 ;
int sampHere, coefHere ;
int coefHere2 ;
int tmpInt ;
double *sinPts, *cosPts, *sinPts2, *cosPts2 ;
double *wigners, *scratch ;
double fudge ;
fftw_complex *coeffsPtr ;
fftw_complex *dataPtr ;
double dn ;
n = 2 * bw ;
n3 = n * n * n ;
/* I'll need these for later */
coeffsPtr = coeffs ;
dataPtr = data ;
sinPts = workspace_re ;
cosPts = sinPts + n ;
sinPts2 = cosPts + n ;
cosPts2 = sinPts2 + n ;
wigners = cosPts2 + n ;
scratch = wigners + ( bw * n ) ; /* wigners need at most bw*n space AT
ANY given orders m1, m2 */
/*
before going further, let's precompute all the sines
and cosines I'll need. No matter what order transform
I'm doing, these'll stay the same.
*/
SinEvalPts( n, sinPts );
CosEvalPts( n, cosPts );
SinEvalPts2( n, sinPts2 );
CosEvalPts2( n, cosPts2 );
/*
I also need to copy the contents of data to workspace_cx2,
given that's where the fftw plan expects data to be. I wish
I didn't have to waste so much memory
*/
memcpy( workspace_cx2, data, sizeof(fftw_complex) * n3 );
/*
Stage 1: FFT the "rows". Instead of treating the signal as
3-D object, I can also think of it as an array of size
(n^2) x n. This means all I'm doing in the first stage
is taking n^2-many FFTs, each of length n.
NOTE: Since I'm reusing the FFT code from SpharmonicKit,
even though I'm doing the FORWARD SO(3) transform
here, I need to call grid_invfourier -> the signs
on the complex exponentials are switched (detailed
explanation to be put here eventually, but trust
me)
*/
fftw_execute( *p1 ) ;
/* normalize the Fourier coefficients (sorry, have to do it) */
/* no, I don't! I can wait till the end */
/*
dn = 1. / sqrt( (double) n ) ;
for ( j = 0 ; j < n*n*n; j++ )
{
workspace_cx[ j ][0] *= dn ;
workspace_cx[ j ][1] *= dn ;
}
*/
/*
Stage 2: transpose!
*/
transpose_cx( workspace_cx, workspace_cx2, n*n, n ) ;
/*
Stage 3: FFT again.
*/
fftw_execute( *p1 ) ;
/* normalize the Fourier coefficients (sorry, have to do it) */
/* no, I don't! I can wait till the end */
/*
dn = 1. / ((double) n) ;
for ( j = 0 ; j < n*n*n; j++ )
{
workspace_cx[ j ][0] *= dn ;
workspace_cx[ j ][1] *= dn ;
}
*/
/*
Stage 4: transpose again! And note I'm using the tmp space
of t2r, t2i again.
*/
transpose_cx( workspace_cx, workspace_cx2, n*n, n ) ;
/*
Stage 5: Do the Wigner transforms. This is the tricky bit.
Since I'm working with two order indeces, m1 and m2, the
for-loops will be more intricate than in the case of the
"ordinary" spherical transform on S^2.
Also, I will be taking advantage of the symmetries of the
Wigner-d functions. As long as I keep my signs and flips
right, the Wigner-d's I precompute for an order (m1, m2)
transform can generally be used in seven more transforms:
(m1,-m2), (m2,m1), (m2,-m1), (-m2,m1), (-m2,-m1), (-m1,m2)
and (-m1,-m2).
I say "general" because, of course, I'll be transforming
at orders (m1,m1), (m1,0) and (0,m1), so I won't get such
a huge reduction. Still, this should save time.
If assumptions are made regarding the original input signal,
e.g. it's strictly real, then one may take advantage of
symmetries of the big D wigners (i.e. function of all 3
parameters alpha, beta, gamma) and so simplify the for-loops
some and hence increase the speed of the program. However,
the for-loops to follow will make no such assumptions.
Whether the signal is real or complex, these for-loops will
handle it.
The for-loops will be "designed" as follows. They will be
divided into cases according to the orders:
0) {f_{0,0}}
1) for 0 <= m1 <= bw-1
compute the coefficients
i) {f_{ m1, m1}}
ii) {f_{-m1,-m1}}
iii) {f_{-m1, m1}}
iv) {f_{ m1,-m1}}
2) for 1 <= m1 <= bw-1
compute the coefficients
i) {f_{ m1, 0}}
ii) {f_{-m1, 0}}
iii) {f_{ 0, m1}}
iv) {f_{ 0,-m1}}
3) for 1 <= m1 <= bw-1
for m1+1 <= m2 <= bw-1
compute the coefficients
i) {f_{ m1, m2}}
ii) {f_{-m1,-m2}}
iii) {f_{ m1,-m2}}
iv) {f_{-m1, m2}}
v) {f_{ m2, m1}}
vi) {f_{-m2,-m1}}
vii) {f_{ m2,-m1}}
viii) {f_{-m2, m1}}
Fasten your seatbelt, folks. It's going to be a bumpy ride.
*/
/***************************/
/* */
/* {f_{0,0}} coefficient */
/* */
/***************************/
/* compute the wigners I'll need */
genWig_L2( 0, 0, bw,
sinPts, cosPts,
sinPts2, cosPts2,
wigners, scratch ) ;
/* now, get the locations of where the
samples I have to transform are, and
where the coefficients have to go */
sampHere = sampLoc_so3( 0, 0, bw ) ;
coefHere = coefLoc_so3( 0, 0, bw ) ;
/* ok, reset sample, coef ptrs */
coeffsPtr = coeffs ;
dataPtr = workspace_cx2 ;
/* now advance by the computed amounts */
dataPtr += sampHere ;
coeffsPtr += coefHere ;
/* now transform the real and imaginary parts
of the data */
wigNaiveAnalysis_fftw( 0, 0, bw, dataPtr,
wigners, weights,
coeffsPtr, workspace_cx ) ;
/*** 0 <= m1 <= bw-1 ***/
for ( m1 = 1 ; m1 < bw ; m1 ++ )
{
/* compute the wigners I'll need */
genWig_L2( m1, m1, bw,
sinPts, cosPts,
sinPts2, cosPts2,
wigners, scratch ) ;
/***************************/
/* */
/* {f_{m1,m1}} coefficient */
/* */
/***************************/
/* now, get the locations of where the
samples I have to transform are, and
where the coefficients have to go */
sampHere = sampLoc_so3( m1, m1, bw ) ;
coefHere = coefLoc_so3( m1, m1, bw ) ;
/* ok, reset sample, coef ptrs */
coeffsPtr = coeffs ;
dataPtr = workspace_cx2 ;
/* now advance by the computed amounts */
dataPtr += sampHere ;
coeffsPtr += coefHere ;
/* now transform the real and imaginary parts
of the data */
wigNaiveAnalysis_fftw( m1, m1, bw, dataPtr,
wigners, weights,
coeffsPtr, workspace_cx ) ;
/*****************************/
/* */
/* {f_{-m1,-m1}} coefficient */
/* */
/*****************************/
if ( flag == 0 ) /* if data is complex */
{
/* now, get the locations of where the
samples I have to transform are, and
where the coefficients have to go */
sampHere = sampLoc_so3( -m1, -m1, bw ) ;
coefHere = coefLoc_so3( -m1, -m1, bw ) ;
/* ok, reset sample, coef ptrs */
coeffsPtr = coeffs ;
dataPtr = workspace_cx2 ;
/* now advance by the computed amounts */
dataPtr += sampHere ;
coeffsPtr += coefHere ;
/* now transform the real and imaginary parts
of the data */
wigNaiveAnalysis_fftw( -m1, -m1, bw, dataPtr,
wigners, weights,
coeffsPtr, workspace_cx ) ;
}
else /* data is real, so use symmetry */
{
coefHere = coefLoc_so3( m1, m1, bw ) ;
coefHere2 = coefLoc_so3( -m1, -m1, bw ) ;
for ( j = 0 ; j < bw - m1 ; j ++ )
{
coeffs[coefHere2+j][0] = coeffs[coefHere+j][0];
coeffs[coefHere2+j][1] = -coeffs[coefHere+j][1];
}
}
/*****************************/
/* */
/* {f_{-m1,m1}} coefficient */
/* */
/*****************************/
/* now, get the locations of where the
samples I have to transform are, and
where the coefficients have to go */
sampHere = sampLoc_so3( -m1, m1, bw ) ;
coefHere = coefLoc_so3( -m1, m1, bw ) ;
/* ok, reset sample, coef ptrs */
coeffsPtr = coeffs ;
dataPtr = workspace_cx2 ;
/* now advance by the computed amounts */
dataPtr += sampHere ;
coeffsPtr += coefHere ;
/* now transform the real and imaginary parts
of the data */
wigNaiveAnalysis_fftwY( -m1, m1, bw, dataPtr,
wigners, weights,
coeffsPtr, workspace_cx ) ;
/*****************************/
/* */
/* {f_{m1,-m1}} coefficient */
/* */
/*****************************/
if ( flag == 0 ) /* data is complex */
{
/* now, get the locations of where the
samples I have to transform are, and
where the coefficients have to go */
sampHere = sampLoc_so3( m1, -m1, bw ) ;
coefHere = coefLoc_so3( m1, -m1, bw ) ;
/* ok, reset sample, coef ptrs */
coeffsPtr = coeffs ;
dataPtr = workspace_cx2 ;
/* now advance by the computed amounts */
dataPtr += sampHere ;
coeffsPtr += coefHere ;
/* now transform the real and imaginary parts
of the data */
wigNaiveAnalysis_fftwY( m1, -m1, bw, dataPtr,
wigners, weights,
coeffsPtr, workspace_cx ) ;
}
else /* data is real, so use symmetry */
{
coefHere = coefLoc_so3( -m1, m1, bw );
coefHere2 = coefLoc_so3( m1, -m1, bw );
for ( j = 0 ; j < bw - m1 ; j ++ )
{
coeffs[coefHere2+j][0] = coeffs[coefHere+j][0];
coeffs[coefHere2+j][1] = -coeffs[coefHere+j][1];
}
}
}
/*** for 1 <= m1 <= bw-1 ***/
for ( m1 = 1 ; m1 < bw ; m1 ++ )
{
/* compute the wigners I'll need */
genWig_L2( m1, 0, bw,
sinPts, cosPts,
sinPts2, cosPts2,
wigners, scratch ) ;
/***************************/
/* */
/* {f_{m1,0}} coefficient */
/* */
/***************************/
/* get the locations of where the
samples I have to transform are, and
where the coefficients have to go */
sampHere = sampLoc_so3( m1, 0, bw ) ;
coefHere = coefLoc_so3( m1, 0, bw ) ;
/* ok, reset sample, coef ptrs */
coeffsPtr = coeffs ;
dataPtr = workspace_cx2 ;
/* now advance by the computed amounts */
dataPtr += sampHere ;
coeffsPtr += coefHere ;
/* now transform the real and imaginary parts
of the data */
wigNaiveAnalysis_fftw( m1, 0, bw, dataPtr,
wigners, weights,
coeffsPtr, workspace_cx ) ;
/***************************/
/* */
/* {f_{-m1,0}} coefficient */
/* */
/***************************/
if ( flag == 0 ) /* data is complex */
{
/* get the locations of where the
samples I have to transform are, and
where the coefficients have to go */
sampHere = sampLoc_so3( -m1, 0, bw ) ;
coefHere = coefLoc_so3( -m1, 0, bw ) ;
/* ok, reset sample, coef ptrs */
coeffsPtr = coeffs ;
dataPtr = workspace_cx2 ;
/* now advance by the computed amounts */
dataPtr += sampHere ;
coeffsPtr += coefHere ;
/* now transform the real and imaginary parts
of the data */
wigNaiveAnalysis_fftwX( -m1, 0, bw, dataPtr,
wigners, weights,
coeffsPtr, workspace_cx ) ;
}
else /* data is real, so use symmetry */
{
coefHere = coefLoc_so3( m1, 0, bw );
coefHere2 = coefLoc_so3( -m1, 0, bw );
if ( (m1 % 2) == 0 )
fudge = 1.0 ;
else
fudge = -1.0 ;
for ( j = 0 ; j < bw - m1 ; j ++ )
{
coeffs[coefHere2+j][0] = fudge * coeffs[coefHere+j][0];
coeffs[coefHere2+j][1] = -fudge * coeffs[coefHere+j][1];
}
}
/***************************/
/* */
/* {f_{0,m1}} coefficient */
/* */
/***************************/
/* get the locations of where the
samples I have to transform are, and
where the coefficients have to go */
sampHere = sampLoc_so3( 0, m1, bw ) ;
coefHere = coefLoc_so3( 0, m1, bw ) ;
/* ok, reset sample, coef ptrs */
coeffsPtr = coeffs ;
dataPtr = workspace_cx2 ;
/* now advance by the computed amounts */
dataPtr += sampHere ;
coeffsPtr += coefHere ;
/* now transform the real and imaginary parts
of the data */
wigNaiveAnalysis_fftwX( 0, m1, bw, dataPtr,
wigners, weights,
coeffsPtr, workspace_cx ) ;
/***************************/
/* */
/* {f_{0,-m1}} coefficient */
/* */
/***************************/
if ( flag == 0 ) /* data is complex */
{
/* get the locations of where the
samples I have to transform are, and
where the coefficients have to go */
sampHere = sampLoc_so3( 0, -m1, bw ) ;
coefHere = coefLoc_so3( 0, -m1, bw ) ;
/* ok, reset sample, coef ptrs */
coeffsPtr = coeffs ;
dataPtr = workspace_cx2 ;
/* now advance by the computed amounts */
dataPtr += sampHere ;
coeffsPtr += coefHere ;
/* now transform the real and imaginary parts
of the data */
wigNaiveAnalysis_fftw( 0, -m1, bw, dataPtr,
wigners, weights,
coeffsPtr, workspace_cx ) ;
}
else /* data is real, so use symmetry */
{
coefHere = coefLoc_so3( 0, m1, bw );
coefHere2 = coefLoc_so3( 0, -m1, bw );
if ( (m1 % 2) == 0 )
fudge = 1.0 ;
else
fudge = -1.0 ;
for ( j = 0 ; j < bw - m1 ; j ++ )
{
coeffs[coefHere2+j][0] = fudge * coeffs[coefHere+j][0];
coeffs[coefHere2+j][1] = -fudge * coeffs[coefHere+j][1];
}
}
}
/***
1 <= m1 <= bw-1
m1+1 <= m2 <= bw-1
***/
for ( m1 = 1 ; m1 < bw ; m1 ++ )
{
for ( m2 = m1 + 1 ; m2 < bw ; m2 ++ )
{
/* compute the wigners I'll need */
genWig_L2( m1, m2, bw,
sinPts, cosPts,
sinPts2, cosPts2,
wigners, scratch ) ;
/***************************/
/* */
/* {f_{m1,m2}} coefficient */
/* */
/***************************/
/* get the locations of where the
samples I have to transform are, and
where the coefficients have to go */
sampHere = sampLoc_so3( m1, m2, bw ) ;
coefHere = coefLoc_so3( m1, m2, bw ) ;
/* ok, reset sample, coef ptrs */
coeffsPtr = coeffs ;
dataPtr = workspace_cx2 ;
/* now advance by the computed amounts */
dataPtr += sampHere ;
coeffsPtr += coefHere ;
/* now transform the real and imaginary parts
of the data */
wigNaiveAnalysis_fftw( m1, m2, bw, dataPtr,
wigners, weights,
coeffsPtr, workspace_cx ) ;
/*****************************/
/* */
/* {f_{-m1,-m2}} coefficient */
/* */
/*****************************/
if ( flag == 0 ) /* data is complex */
{
/* get the locations of where the
samples I have to transform are, and
where the coefficients have to go */
sampHere = sampLoc_so3( -m1, -m2, bw ) ;
coefHere = coefLoc_so3( -m1, -m2, bw ) ;
/* ok, reset sample, coef ptrs */
coeffsPtr = coeffs ;
dataPtr = workspace_cx2 ;
/* now advance by the computed amounts */
dataPtr += sampHere ;
coeffsPtr += coefHere ;
/* now transform the real and imaginary parts
of the data */
wigNaiveAnalysis_fftwX( -m1, -m2, bw, dataPtr,
wigners, weights,
coeffsPtr, workspace_cx ) ;
}
else /* data is real, so use symmetry */
{
coefHere = coefLoc_so3( m1, m2, bw );
coefHere2 = coefLoc_so3( -m1, -m2, bw );
if ( ((m2-m1) % 2) == 0 )
fudge = 1.0 ;
else
fudge = -1.0 ;
for ( j = 0 ; j < bw - m2 ; j ++ )
{
coeffs[coefHere2+j][0] = fudge * coeffs[coefHere+j][0];
coeffs[coefHere2+j][1] = -fudge * coeffs[coefHere+j][1];
}
}
/****************************/
/* */
/* {f_{m1,-m2}} coefficient */
/* */
/****************************/
/* get the locations of where the
samples I have to transform are, and
where the coefficients have to go */
sampHere = sampLoc_so3( m1, -m2, bw ) ;
coefHere = coefLoc_so3( m1, -m2, bw ) ;
/* ok, reset sample, coef ptrs */
coeffsPtr = coeffs ;
dataPtr = workspace_cx2 ;
/* now advance by the computed amounts */
dataPtr += sampHere ;
coeffsPtr += coefHere ;
/* now transform the real and imaginary parts
of the data */
wigNaiveAnalysis_fftwY( m1, -m2, bw, dataPtr,
wigners, weights,
coeffsPtr, workspace_cx ) ;
/*****************************/
/* */
/* {f_{-m1,m2}} coefficient */
/* */
/*****************************/
if ( flag == 0 ) /* data is complex */
{
/* get the locations of where the
samples I have to transform are, and
where the coefficients have to go */
sampHere = sampLoc_so3( -m1, m2, bw ) ;
coefHere = coefLoc_so3( -m1, m2, bw ) ;
/* ok, reset sample, coef ptrs */
coeffsPtr = coeffs ;
dataPtr = workspace_cx2 ;
/* now advance by the computed amounts */
dataPtr += sampHere ;
coeffsPtr += coefHere ;
/* now transform the real and imaginary parts
of the data */
wigNaiveAnalysis_fftwY( -m1, m2, bw, dataPtr,
wigners, weights,
coeffsPtr, workspace_cx ) ;
}
else /* data is real, so use symmetry */
{
coefHere = coefLoc_so3( m1, -m2, bw );
coefHere2 = coefLoc_so3( -m1, m2, bw );
if ( ((m2-m1) % 2) == 0 )
fudge = 1.0 ;
else
fudge = -1.0 ;
for ( j = 0 ; j < bw - m2 ; j ++ )
{
coeffs[coefHere2+j][0] = fudge * coeffs[coefHere+j][0];
coeffs[coefHere2+j][1] = -fudge * coeffs[coefHere+j][1];
}
}
/***************************/
/* */
/* {f_{m2,m1}} coefficient */
/* */
/***************************/
/* get the locations of where the
samples I have to transform are, and
where the coefficients have to go */
sampHere = sampLoc_so3( m2, m1, bw ) ;
coefHere = coefLoc_so3( m2, m1, bw ) ;
/* ok, reset sample, coef ptrs */
coeffsPtr = coeffs ;
dataPtr = workspace_cx2 ;
/* now advance by the computed amounts */
dataPtr += sampHere ;
coeffsPtr += coefHere ;
/* now transform the real and imaginary parts
of the data */
wigNaiveAnalysis_fftwX( m2, m1, bw, dataPtr,
wigners, weights,
coeffsPtr, workspace_cx ) ;
/*****************************/
/* */
/* {f_{-m2,-m1}} coefficient */
/* */
/*****************************/
if ( flag == 0 ) /* data is complex */
{
/* get the locations of where the
samples I have to transform are, and
where the coefficients have to go */
sampHere = sampLoc_so3( -m2, -m1, bw ) ;
coefHere = coefLoc_so3( -m2, -m1, bw ) ;
/* ok, reset sample, coef ptrs */
coeffsPtr = coeffs ;
dataPtr = workspace_cx2 ;
/* now advance by the computed amounts */
dataPtr += sampHere ;
coeffsPtr += coefHere ;
/* now transform the real and imaginary parts
of the data */
wigNaiveAnalysis_fftw( -m2, -m1, bw, dataPtr,
wigners, weights,
coeffsPtr, workspace_cx ) ;
}
else /* data is real, so use symmetry */
{
coefHere = coefLoc_so3( m2, m1, bw );
coefHere2 = coefLoc_so3( -m2, -m1, bw );
if ( ((m2-m1) % 2) == 0 )
fudge = 1.0 ;
else
fudge = -1.0 ;
for ( j = 0 ; j < bw - m2 ; j ++ )
{
coeffs[coefHere2+j][0] = fudge * coeffs[coefHere+j][0];
coeffs[coefHere2+j][1] = -fudge * coeffs[coefHere+j][1];
}
}
/****************************/
/* */
/* {f_{m2,-m1}} coefficient */
/* */
/****************************/
/* get the locations of where the
samples I have to transform are, and
where the coefficients have to go */
sampHere = sampLoc_so3( m2, -m1, bw ) ;
coefHere = coefLoc_so3( m2, -m1, bw ) ;
/* ok, reset sample, coef ptrs */
coeffsPtr = coeffs ;
dataPtr = workspace_cx2 ;
/* now advance by the computed amounts */
dataPtr += sampHere ;
coeffsPtr += coefHere ;
/* now transform the real and imaginary parts
of the data */
wigNaiveAnalysis_fftwY( m1, -m2, bw, dataPtr,
wigners, weights,
coeffsPtr, workspace_cx ) ;
/****************************/
/* */
/* {f_{-m2,m1}} coefficient */
/* */
/****************************/
if ( flag == 0 ) /* data is complex */
{
/* get the locations of where the
samples I have to transform are, and
where the coefficients have to go */
sampHere = sampLoc_so3( -m2, m1, bw ) ;
coefHere = coefLoc_so3( -m2, m1, bw ) ;
/* ok, reset sample, coef ptrs */
coeffsPtr = coeffs ;
dataPtr = workspace_cx2 ;
/* now advance by the computed amounts */
dataPtr += sampHere ;
coeffsPtr += coefHere ;
/* now transform the real and imaginary parts
of the data */
wigNaiveAnalysis_fftwY( -m1, m2, bw, dataPtr,
wigners, weights,
coeffsPtr, workspace_cx ) ;
}
else /* data is real, so use symmetry */
{
coefHere = coefLoc_so3( m2, -m1, bw );
coefHere2 = coefLoc_so3( -m2, m1, bw );
if ( ((m2-m1) % 2) == 0 )
fudge = 1.0 ;
else
fudge = -1.0 ;
for ( j = 0 ; j < bw - m2 ; j ++ )
{
coeffs[coefHere2+j][0] = fudge * coeffs[coefHere+j][0];
coeffs[coefHere2+j][1] = -fudge * coeffs[coefHere+j][1];
}
}
}
}
/* reset coef ptrs */
coeffsPtr = coeffs ;
/* need to normalize, one last time */
dn = (M_PI / ( (double) (bw * n )) );
tmpInt = totalCoeffs_so3( bw ) ;
for ( j = 0 ; j < tmpInt ; j ++ )
{
coeffsPtr[ j ][0] *= dn ;
coeffsPtr[ j ][1] *= dn ;
}
/*** and we're done ! ***/
}
void Forward_SO3_Naive_fftw_wo( int bw,
fftw_complex *data,
fftw_complex *coeffs,
fftw_complex *workspace_cx,
REAL *workspace_re,
fftw_plan *p1,
int flag )
{
int j, n ;
int m1, m2 ;
int sampHere, coefHere ;
int coefHere2 ;
double *sinPts, *cosPts, *sinPts2, *cosPts2 ;
double *wigners, *scratch ;
double fudge ;
fftw_complex *coeffsPtr ;
fftw_complex *dataPtr ;
double dn ;
n = 2 * bw ;
sinPts = workspace_re ;
cosPts = sinPts + n ;
sinPts2 = cosPts + n ;
cosPts2 = sinPts2 + n ;
wigners = cosPts2 + n ;
scratch = wigners + ( bw * n ) ; /* wigners need at most bw*n space AT
ANY given orders m1, m2 */
/*
before going further, let's precompute all the sines
and cosines I'll need. No matter what order transform
I'm doing, these'll stay the same.
*/
SinEvalPts( n, sinPts );
CosEvalPts( n, cosPts );
SinEvalPts2( n, sinPts2 );
CosEvalPts2( n, cosPts2 );
/*
Stage 1: FFT
*/
fftw_execute( *p1 ) ;
/*
Stage 2: Do the Wigner transforms. This is the tricky bit.
Since I'm working with two order indeces, m1 and m2, the
for-loops will be more intricate than in the case of the
"ordinary" spherical transform on S^2.
Also, I will be taking advantage of the symmetries of the
Wigner-d functions. As long as I keep my signs and flips
right, the Wigner-d's I precompute for an order (m1, m2)
transform can generally be used in seven more transforms:
(m1,-m2), (m2,m1), (m2,-m1), (-m2,m1), (-m2,-m1), (-m1,m2)
and (-m1,-m2).
I say "general" because, of course, I'll be transforming
at orders (m1,m1), (m1,0) and (0,m1), so I won't get such
a huge reduction. Still, this should save time.
If assumptions are made regarding the original input signal,
e.g. it's strictly real, then one may take advantage of
symmetries of the big D wigners (i.e. function of all 3
parameters alpha, beta, gamma) and so simplify the for-loops
some and hence increase the speed of the program. However,
the for-loops to follow will make no such assumptions.
Whether the signal is real or complex, these for-loops will
handle it.
The for-loops will be "designed" as follows. They will be
divided into cases according to the orders:
0) {f_{0,0}}
1) for 0 <= m1 <= bw-1
compute the coefficients
i) {f_{ m1, m1}}
ii) {f_{-m1,-m1}}
iii) {f_{-m1, m1}}
iv) {f_{ m1,-m1}}
2) for 1 <= m1 <= bw-1
compute the coefficients
i) {f_{ m1, 0}}
ii) {f_{-m1, 0}}
iii) {f_{ 0, m1}}
iv) {f_{ 0,-m1}}
3) for 1 <= m1 <= bw-1
for m1+1 <= m2 <= bw-1
compute the coefficients
i) {f_{ m1, m2}}
ii) {f_{-m1,-m2}}
iii) {f_{ m1,-m2}}
iv) {f_{-m1, m2}}
v) {f_{ m2, m1}}
vi) {f_{-m2,-m1}}
vii) {f_{ m2,-m1}}
viii) {f_{-m2, m1}}
Fasten your seatbelt, folks. It's going to be a bumpy ride.
*/
/***************************/
/* */
/* {f_{0,0}} coefficient */
/* */
/***************************/
/* compute the wigners I'll need */
genWig_L2( 0, 0, bw,
sinPts, cosPts,
sinPts2, cosPts2,
wigners, scratch ) ;
/* now, get the locations of where the
samples I have to transform are, and
where the coefficients have to go */
sampHere = sampLoc_so3( 0, 0, bw ) ;
coefHere = coefLoc_so3( 0, 0, bw ) ;
/* ok, reset sample, coef ptrs */
coeffsPtr = coeffs ;
dataPtr = data ;
/* now advance by the computed amounts */
dataPtr += sampHere ;
coeffsPtr += coefHere ;
/* now transform the real and imaginary parts
of the data */
wigNaiveAnalysis_fftw( 0, 0, bw, dataPtr,
wigners, coeffsPtr,
workspace_cx ) ;
/*** 0 <= m1 <= bw-1 ***/
for ( m1 = 1 ; m1 < bw ; m1 ++ )
{
/* compute the wigners I'll need */
genWig_L2( m1, m1, bw,
sinPts, cosPts,
sinPts2, cosPts2,
wigners, scratch ) ;
/***************************/
/* */
/* {f_{m1,m1}} coefficient */
/* */
/***************************/
/* now, get the locations of where the
samples I have to transform are, and
where the coefficients have to go */
sampHere = sampLoc_so3( m1, m1, bw ) ;
coefHere = coefLoc_so3( m1, m1, bw ) ;
/* ok, reset sample, coef ptrs */
coeffsPtr = coeffs ;
dataPtr = data ;
/* now advance by the computed amounts */
dataPtr += sampHere ;
coeffsPtr += coefHere ;
/* now transform the real and imaginary parts
of the data */
wigNaiveAnalysis_fftw( m1, m1, bw, dataPtr,
wigners, coeffsPtr,
workspace_cx ) ;
/*****************************/
/* */
/* {f_{-m1,-m1}} coefficient */
/* */
/*****************************/
if ( flag == 0 ) /* if data is complex */
{
/* now, get the locations of where the
samples I have to transform are, and
where the coefficients have to go */
sampHere = sampLoc_so3( -m1, -m1, bw ) ;
coefHere = coefLoc_so3( -m1, -m1, bw ) ;
/* ok, reset sample, coef ptrs */
coeffsPtr = coeffs ;
dataPtr = data ;
/* now advance by the computed amounts */
dataPtr += sampHere ;
coeffsPtr += coefHere ;
/* now transform the real and imaginary parts
of the data */
wigNaiveAnalysis_fftw( -m1, -m1, bw, dataPtr,
wigners, coeffsPtr,
workspace_cx ) ;
}
else /* data is real, so use symmetry */
{
coefHere = coefLoc_so3( m1, m1, bw ) ;
coefHere2 = coefLoc_so3( -m1, -m1, bw ) ;
for ( j = 0 ; j < bw - m1 ; j ++ )
{
coeffs[coefHere2+j][0] = coeffs[coefHere+j][0];
coeffs[coefHere2+j][1] = -coeffs[coefHere+j][1];
}
}
/*****************************/
/* */
/* {f_{-m1,m1}} coefficient */
/* */
/*****************************/
/* now, get the locations of where the
samples I have to transform are, and
where the coefficients have to go */
sampHere = sampLoc_so3( -m1, m1, bw ) ;
coefHere = coefLoc_so3( -m1, m1, bw ) ;
/* ok, reset sample, coef ptrs */
coeffsPtr = coeffs ;
dataPtr = data ;
/* now advance by the computed amounts */
dataPtr += sampHere ;
coeffsPtr += coefHere ;
/* now transform the real and imaginary parts
of the data */
wigNaiveAnalysis_fftwY( -m1, m1, bw, dataPtr,
wigners, coeffsPtr,
workspace_cx ) ;
/*****************************/
/* */
/* {f_{m1,-m1}} coefficient */
/* */
/*****************************/
if ( flag == 0 ) /* data is complex */
{
/* now, get the locations of where the
samples I have to transform are, and
where the coefficients have to go */
sampHere = sampLoc_so3( m1, -m1, bw ) ;
coefHere = coefLoc_so3( m1, -m1, bw ) ;
/* ok, reset sample, coef ptrs */
coeffsPtr = coeffs ;
dataPtr = data ;
/* now advance by the computed amounts */
dataPtr += sampHere ;
coeffsPtr += coefHere ;
/* now transform the real and imaginary parts
of the data */
wigNaiveAnalysis_fftwY( m1, -m1, bw, dataPtr,
wigners, coeffsPtr,
workspace_cx ) ;
}
else /* data is real, so use symmetry */
{
coefHere = coefLoc_so3( -m1, m1, bw );
coefHere2 = coefLoc_so3( m1, -m1, bw );
for ( j = 0 ; j < bw - m1 ; j ++ )
{
coeffs[coefHere2+j][0] = coeffs[coefHere+j][0];
coeffs[coefHere2+j][1] = -1.*coeffs[coefHere+j][1];
}
}
}
/*** for 1 <= m1 <= bw-1 ***/
for ( m1 = 1 ; m1 < bw ; m1 ++ )
{
/* compute the wigners I'll need */
genWig_L2( m1, 0, bw,
sinPts, cosPts,
sinPts2, cosPts2,
wigners, scratch ) ;
/***************************/
/* */
/* {f_{m1,0}} coefficient */
/* */
/***************************/
/* get the locations of where the
samples I have to transform are, and
where the coefficients have to go */
sampHere = sampLoc_so3( m1, 0, bw ) ;
coefHere = coefLoc_so3( m1, 0, bw ) ;
/* ok, reset sample, coef ptrs */
coeffsPtr = coeffs ;
dataPtr = data ;
/* now advance by the computed amounts */
dataPtr += sampHere ;
coeffsPtr += coefHere ;
/* now transform the real and imaginary parts
of the data */
wigNaiveAnalysis_fftw( m1, 0, bw, dataPtr,
wigners, coeffsPtr,
workspace_cx ) ;
/***************************/
/* */
/* {f_{-m1,0}} coefficient */
/* */
/***************************/
if ( flag == 0 ) /* data is complex */
{
/* get the locations of where the
samples I have to transform are, and
where the coefficients have to go */
sampHere = sampLoc_so3( -m1, 0, bw ) ;
coefHere = coefLoc_so3( -m1, 0, bw ) ;
/* ok, reset sample, coef ptrs */
coeffsPtr = coeffs ;
dataPtr = data ;
/* now advance by the computed amounts */
dataPtr += sampHere ;
coeffsPtr += coefHere ;
/* now transform the real and imaginary parts
of the data */
wigNaiveAnalysis_fftwX( -m1, 0, bw, dataPtr,
wigners, coeffsPtr,
workspace_cx ) ;
}
else /* data is real, so use symmetry */
{
coefHere = coefLoc_so3( m1, 0, bw );
coefHere2 = coefLoc_so3( -m1, 0, bw );
if ( (m1 % 2) == 0 )
fudge = 1.0 ;
else
fudge = -1.0 ;
for ( j = 0 ; j < bw - m1 ; j ++ )
{
coeffs[coefHere2+j][0] = fudge * coeffs[coefHere+j][0];
coeffs[coefHere2+j][1] = -fudge * coeffs[coefHere+j][1];
}
}
/***************************/
/* */
/* {f_{0,m1}} coefficient */
/* */
/***************************/
/* get the locations of where the
samples I have to transform are, and
where the coefficients have to go */
sampHere = sampLoc_so3( 0, m1, bw ) ;
coefHere = coefLoc_so3( 0, m1, bw ) ;
/* ok, reset sample, coef ptrs */
coeffsPtr = coeffs ;
dataPtr = data ;
/* now advance by the computed amounts */
dataPtr += sampHere ;
coeffsPtr += coefHere ;
/* now transform the real and imaginary parts
of the data */
wigNaiveAnalysis_fftwX( 0, m1, bw, dataPtr,
wigners, coeffsPtr,
workspace_cx ) ;
/***************************/
/* */
/* {f_{0,-m1}} coefficient */
/* */
/***************************/
if ( flag == 0 ) /* data is complex */
{
/* get the locations of where the
samples I have to transform are, and
where the coefficients have to go */
sampHere = sampLoc_so3( 0, -m1, bw ) ;
coefHere = coefLoc_so3( 0, -m1, bw ) ;
/* ok, reset sample, coef ptrs */
coeffsPtr = coeffs ;
dataPtr = data ;
/* now advance by the computed amounts */
dataPtr += sampHere ;
coeffsPtr += coefHere ;
/* now transform the real and imaginary parts
of the data */
wigNaiveAnalysis_fftw( 0, -m1, bw, dataPtr,
wigners, coeffsPtr,
workspace_cx ) ;
}
else /* data is real, so use symmetry */
{
coefHere = coefLoc_so3( 0, m1, bw );
coefHere2 = coefLoc_so3( 0, -m1, bw );
if ( (m1 % 2) == 0 )
fudge = 1.0 ;
else
fudge = -1.0 ;
for ( j = 0 ; j < bw - m1 ; j ++ )
{
coeffs[coefHere2+j][0] = fudge * coeffs[coefHere+j][0];
coeffs[coefHere2+j][1] = -fudge * coeffs[coefHere+j][1];
}
}
}
/***
1 <= m1 <= bw-1
m1+1 <= m2 <= bw-1
***/
for ( m1 = 1 ; m1 < bw ; m1 ++ )
{
for ( m2 = m1 + 1 ; m2 < bw ; m2 ++ )
{
/* compute the wigners I'll need */
genWig_L2( m1, m2, bw,
sinPts, cosPts,
sinPts2, cosPts2,
wigners, scratch ) ;
/***************************/
/* */
/* {f_{m1,m2}} coefficient */
/* */
/***************************/
/* get the locations of where the
samples I have to transform are, and
where the coefficients have to go */
sampHere = sampLoc_so3( m1, m2, bw ) ;
coefHere = coefLoc_so3( m1, m2, bw ) ;
/* ok, reset sample, coef ptrs */
coeffsPtr = coeffs ;
dataPtr = data ;
/* now advance by the computed amounts */
dataPtr += sampHere ;
coeffsPtr += coefHere ;
/* now transform the real and imaginary parts
of the data */
wigNaiveAnalysis_fftw( m1, m2, bw, dataPtr,
wigners, coeffsPtr,
workspace_cx ) ;
/*****************************/
/* */
/* {f_{-m1,-m2}} coefficient */
/* */
/*****************************/
if ( flag == 0 ) /* data is complex */
{
/* get the locations of where the
samples I have to transform are, and
where the coefficients have to go */
sampHere = sampLoc_so3( -m1, -m2, bw ) ;
coefHere = coefLoc_so3( -m1, -m2, bw ) ;
/* ok, reset sample, coef ptrs */
coeffsPtr = coeffs ;
dataPtr = data ;
/* now advance by the computed amounts */
dataPtr += sampHere ;
coeffsPtr += coefHere ;
/* now transform the real and imaginary parts
of the data */
wigNaiveAnalysis_fftwX( -m1, -m2, bw, dataPtr,
wigners, coeffsPtr,
workspace_cx ) ;
}
else /* data is real, so use symmetry */
{
coefHere = coefLoc_so3( m1, m2, bw );
coefHere2 = coefLoc_so3( -m1, -m2, bw );
if ( ((m2-m1) % 2) == 0 )
fudge = 1.0 ;
else
fudge = -1.0 ;
for ( j = 0 ; j < bw - m2 ; j ++ )
{
coeffs[coefHere2+j][0] = fudge * coeffs[coefHere+j][0];
coeffs[coefHere2+j][1] = -fudge * coeffs[coefHere+j][1];
}
}
/****************************/
/* */
/* {f_{m1,-m2}} coefficient */
/* */
/****************************/
/* get the locations of where the
samples I have to transform are, and
where the coefficients have to go */
sampHere = sampLoc_so3( m1, -m2, bw ) ;
coefHere = coefLoc_so3( m1, -m2, bw ) ;
/* ok, reset sample, coef ptrs */
coeffsPtr = coeffs ;
dataPtr = data ;
/* now advance by the computed amounts */
dataPtr += sampHere ;
coeffsPtr += coefHere ;
/* now transform the real and imaginary parts
of the data */
wigNaiveAnalysis_fftwY( m1, -m2, bw, dataPtr,
wigners, coeffsPtr,
workspace_cx ) ;
/*****************************/
/* */
/* {f_{-m1,m2}} coefficient */
/* */
/*****************************/
if ( flag == 0 ) /* data is complex */
{
/* get the locations of where the
samples I have to transform are, and
where the coefficients have to go */
sampHere = sampLoc_so3( -m1, m2, bw ) ;
coefHere = coefLoc_so3( -m1, m2, bw ) ;
/* ok, reset sample, coef ptrs */
coeffsPtr = coeffs ;
dataPtr = data ;
/* now advance by the computed amounts */
dataPtr += sampHere ;
coeffsPtr += coefHere ;
/* now transform the real and imaginary parts
of the data */
wigNaiveAnalysis_fftwY( -m1, m2, bw, dataPtr,
wigners, coeffsPtr,
workspace_cx ) ;
}
else /* data is real, so use symmetry */
{
coefHere = coefLoc_so3( m1, -m2, bw );
coefHere2 = coefLoc_so3( -m1, m2, bw );
if ( ((m2-m1) % 2) == 0 )
fudge = 1.0 ;
else
fudge = -1.0 ;
for ( j = 0 ; j < bw - m2 ; j ++ )
{
coeffs[coefHere2+j][0] = fudge * coeffs[coefHere+j][0];
coeffs[coefHere2+j][1] = -fudge * coeffs[coefHere+j][1];
}
}
/***************************/
/* */
/* {f_{m2,m1}} coefficient */
/* */
/***************************/
/* get the locations of where the
samples I have to transform are, and
where the coefficients have to go */
sampHere = sampLoc_so3( m2, m1, bw ) ;
coefHere = coefLoc_so3( m2, m1, bw ) ;
/* ok, reset sample, coef ptrs */
coeffsPtr = coeffs ;
dataPtr = data ;
/* now advance by the computed amounts */
dataPtr += sampHere ;
coeffsPtr += coefHere ;
/* now transform the real and imaginary parts
of the data */
wigNaiveAnalysis_fftwX( m2, m1, bw, dataPtr,
wigners, coeffsPtr,
workspace_cx ) ;
/*****************************/
/* */
/* {f_{-m2,-m1}} coefficient */
/* */
/*****************************/
if ( flag == 0 ) /* data is complex */
{
/* get the locations of where the
samples I have to transform are, and
where the coefficients have to go */
sampHere = sampLoc_so3( -m2, -m1, bw ) ;
coefHere = coefLoc_so3( -m2, -m1, bw ) ;
/* ok, reset sample, coef ptrs */
coeffsPtr = coeffs ;
dataPtr = data ;
/* now advance by the computed amounts */
dataPtr += sampHere ;
coeffsPtr += coefHere ;
/* now transform the real and imaginary parts
of the data */
wigNaiveAnalysis_fftw( -m2, -m1, bw, dataPtr,
wigners, coeffsPtr,
workspace_cx ) ;
}
else /* data is real, so use symmetry */
{
coefHere = coefLoc_so3( m2, m1, bw );
coefHere2 = coefLoc_so3( -m2, -m1, bw );
if ( ((m2-m1) % 2) == 0 )
fudge = 1.0 ;
else
fudge = -1.0 ;
for ( j = 0 ; j < bw - m2 ; j ++ )
{
coeffs[coefHere2+j][0] = fudge * coeffs[coefHere+j][0];
coeffs[coefHere2+j][1] = -fudge * coeffs[coefHere+j][1];
}
}
/****************************/
/* */
/* {f_{m2,-m1}} coefficient */
/* */
/****************************/
/* get the locations of where the
samples I have to transform are, and
where the coefficients have to go */
sampHere = sampLoc_so3( m2, -m1, bw ) ;
coefHere = coefLoc_so3( m2, -m1, bw ) ;
/* ok, reset sample, coef ptrs */
coeffsPtr = coeffs ;
dataPtr = data ;
/* now advance by the computed amounts */
dataPtr += sampHere ;
coeffsPtr += coefHere ;
/* now transform the real and imaginary parts
of the data */
wigNaiveAnalysis_fftwY( m1, -m2, bw, dataPtr,
wigners, coeffsPtr,
workspace_cx ) ;
/****************************/
/* */
/* {f_{-m2,m1}} coefficient */
/* */
/****************************/
if ( flag == 0 ) /* data is complex */
{
/* get the locations of where the
samples I have to transform are, and
where the coefficients have to go */
sampHere = sampLoc_so3( -m2, m1, bw ) ;
coefHere = coefLoc_so3( -m2, m1, bw ) ;
/* ok, reset sample, coef ptrs */
coeffsPtr = coeffs ;
dataPtr = data ;
/* now advance by the computed amounts */
dataPtr += sampHere ;
coeffsPtr += coefHere ;
/* now transform the real and imaginary parts
of the data */
wigNaiveAnalysis_fftwY( -m1, m2, bw, dataPtr,
wigners, coeffsPtr,
workspace_cx ) ;
}
else /* data is real, so use symmetry */
{
coefHere = coefLoc_so3( m2, -m1, bw );
coefHere2 = coefLoc_so3( -m2, m1, bw );
if ( ((m2-m1) % 2) == 0 )
fudge = 1.0 ;
else
fudge = -1.0 ;
for ( j = 0 ; j < bw - m2 ; j ++ )
{
coeffs[coefHere2+j][0] = fudge * coeffs[coefHere+j][0];
coeffs[coefHere2+j][1] = -fudge * coeffs[coefHere+j][1];
}
}
}
}
/* reset coef ptrs */
coeffsPtr = coeffs ;
/* need to normalize, one last time */
dn = (M_PI / ( (double) (bw * n )) );
for ( j = 0 ; j < totalCoeffs_so3( bw ) ; j ++ )
{
coeffsPtr[ j ][0] *= dn ;
coeffsPtr[ j ][1] *= dn ;
}
/*** and we're done ! ***/
}