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/*
* fft_utils.i --
*
* Useful routines for FFT operations in Yorick.
*
*-----------------------------------------------------------------------------
*
* Copyright (C) 1995, Eric Thiebaut <thiebaut@obs.univ-lyon1.fr>
*
* This file is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License version 2 as
* published by the Free Software Foundation.
*
* This file is distributed in the hope that it will be useful, but
* WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
*
*-----------------------------------------------------------------------------
*
* History:
* $Id: fft_utils.i,v 1.2 2010-02-10 13:27:12 paumard Exp $
* $Log: fft_utils.i,v $
* Revision 1.2 2010-02-10 13:27:12 paumard
* - Synchronize files with Eric Thiebaut's: fft_utils.i, img.i, plot.i, utils.i.
* - Import emulate_yeti.i
* - Remove basename() and dirname(), standard in pathfun.i.
* - Remove the accents in Erics name to prevent a crash on amd64.
*
* Revision 1.13 2008/07/12 06:38:26 eric
* - Considerable speed-up of fft_gaussian_mtf and fft_gaussian_psf for
* multidimensional array (accounting for the fact that the Gaussian
* is separable). Also FWHM can now be a scalar or a vector with as
* many values as number of dimensions.
* - New function fft_get_ndims.
*
* Revision 1.12 2007/04/24 07:11:43 eric
* - Function grow_dimlist replaced by make_dimlist.
* - Function fft_paste fixed.
*
* Revision 1.11 2005/10/18 18:20:39 eric
* - Oops, the previous bug fix yields a new bug, OK now fft_dist
* should work again for n-D arrays (with n>1).
*
* Revision 1.10 2005/10/18 18:09:19 eric
* - new function: fft_paste;
* - fixed bug in fft_dist for 1-D dimension lists which affected
* functions such as fft_smooth for 1-D arrays (thanks to Christophe
* Pichon for pointing the bug);
*
* Revision 1.1.1.1 2007/12/11 23:55:14 frigaut
* Initial Import - yorick-yutils
*
* Revision 1.9 2004/10/11 11:18:21 eric
* - Fix fft_dist() so that it is as flexible as, e.g., array() for
* the dimension list.
*
* Revision 1.8 2004/08/31 16:20:22 eric
* - New routines for Fourier interpolation: fft_fine_shift,
* fft_unphasor, fft_interp, fft_interp_complex, fft_interp_real.
*
* Revision 1.7 2003/08/23 09:55:57 eric
* - new functions: fft_gaussian_mtf and fft_gaussian_psf;
* - *** POSSIBLE INCOMPATIBILITY *** change order of args in
* fft_recenter and use fft_setup to speed up FFT's;
*
* Revision 1.6 2003/01/31 15:58:32 eric
* - Added new routines: fft_recenter, fft_smooth and reverse_dims.
*
* Revision 1.5 2002/11/20 09:20:56 eric
* - new keywords SQUARE and NYQUIST in fft_dist
* - make use of grow_dimlist in "utils.i"
*
* Revision 1.4 2002/11/14 10:59:38 eric
* - new graphics routines: fft_plh, fft_plc, fft_plfc
* - new routines: abs2, fft_roll_1d, fft_roll_2d, __fft & Co
* - introduction documentation by: help, fft_utils;
*
* Revision 1.3 2001/04/23 14:16:15 eric
* New routine: fft_symmetric_index.
*
* Revision 1.2 2001/04/06 16:42:59 eric
* - new routine: fft_plg
* - fixed routine: fft_recenter_at_max
*
* Revision 1.1 2001/03/23 16:45:54 eric
* Initial revision
*
*-----------------------------------------------------------------------------
*/
require, "utils.i";
local fft_utils;
/* DOCUMENT: FFT utility routines in "fft_utils.i"
This package is mainly written to deal with the particular indexing
rules in FFT transformed arrays. The following routines are provided:
abs2 - squared absolute value.
fft_best_dim - get best dimension to compute FFT.
fft_centroid - get centroid in FFT arrays.
fft_convolve - compute discrete convolution thanks to FFT.
fft_dist - compute length of FFT frequencies/coordinates.
fft_fine_shift, fft_unphasor - shift/roll an array by non-integer
offset by means of Fourier interpolation.
fft_gaussian_mtf - compute Gaussian modulation transfer function.
fft_gaussian_psf - compute Gaussian point spread function.
fft_indgen - generate index of FFT frequencies/coordinates.
fft_interp, fft_interp_complex, fft_interp_real - interpolate array
at non-integer offsets.
fft_plc - plot contours of 2D FFT.
fft_plfc - plot filled contours of 2D FFT.
fft_plg - plot 1D FFT as curve.
fft_plh - plot 1D FFT with stairs.
fft_pli - plot 2D FFT as image.
fft_recenter - recenter array with respect to a template.
fft_recenter_at_max - recenter FFT arrays at their maximum.
fft_roll_1d - roll dimension of 1D arrays.
fft_roll_2d - roll dimension of 2D arrays.
fft_shift_phasor - get complex phasor for arbitrary shift.
fft_smooth - smooth an array by convolution with a gaussian.
fft_symmetric_index - get hermitian-symmetry index for FFT arrays.
reverse_dims - reverse all dimensions of an array.
__fft - expert driver for repeated FFT's with same dimensions.
__fft_init - initialization for __fft.
SEE ALSO: fft, fftw. */
func abs2(x)
/* DOCUMENT abs2(x)
Returns abs(X)^2
SEE ALSO: abs. */
{
if (structof(x) != complex) return x*x;
y = x.im;
x = double(x);
return x*x + y*y;
}
func fft_best_dim(len)
/* DOCUMENT fft_best_dim(len);
Return the smallest integer which is greater or equal LEN and which is
a multiple of powers of 2, 3 and/or 5
SEE ALSO fft_indgen, fft. */
{
best= 2*len;
for (i5=1; i5<=len; i5*=5) {
for (i3= i5; i3<=len; i3*=3) {
i2= i3;
while (i2 < len) i2*=2;
if (i2 == len) return len;
if (i2-len < best-len) best= i2;
}
}
return best;
}
func fft_indgen(dim) { return (u= indgen(0:dim-1)) - dim*(u > dim/2); }
/* DOCUMENT fft_indgen(len)
Return FFT frequencies along a dimension of length LEN.
SEE ALSO: indgen, span, fft_dist, fft_freqlist, fft_symmetric_index. */
func fft_dist(.., nyquist=, square=)
/* DOCUMENT fft_dist(dimlist);
-or- fft_dist(dim1, dim2, ...);
Returns Euclidian lenght of spatial frequencies in frequel units for a
FFT of dimensions DIMLIST.
If keyword NYQUIST is true, the frequel coordinates get rescaled so
that the Nyquist frequency is equal to NYQUIST along every dimension.
This is obtained by using coordinates:
(2.0*NYQUIST/DIM(i))*fft_indgen(DIM(i))
along i-th dimension of lenght DIM(i).
If keyword SQUARE is true, the square of the Euclidian norm is
returned instead.
SEE ALSO: fft_indgen, fft_symmetric_index. */
{
/* Build dimension list. */
local arg, dims;
while (more_args()) {
eq_nocopy, arg, next_arg();
if ((s = structof(arg)) == long || s == int || s == short || s == char) {
/* got an integer array */
if (! (n = dimsof(arg)(1))) {
/* got a scalar */
grow, dims, arg;
} else if (n == 1 && (n = numberof(arg) - 1) == arg(1)) {
/* got a vector which is a valid dimension list */
if (n) grow, dims, arg(2:);
} else {
error, "bad dimension list";
}
} else if (! is_void(arg)) {
error, "unexpected data type in dimension list";
}
}
if (! (n = numberof(dims))) return 0.0; /* scalar array */
if (min(dims) <= 0) error, "negative value in dimension list";
/* Build square radius array one dimension at a time, starting with the
last dimension. */
if (is_void(nyquist)) {
r2 = (u = double(fft_indgen(dims(n))))*u;
while (--n >= 1) {
r2 = r2(-,..) + (u = double(fft_indgen(dims(n))))*u;
}
} else {
s = 2.0*nyquist;
dim = dims(n);
r2 = (u = (s/dim)*fft_indgen(dim))*u;
while (--n >= 1) {
dim = dims(n);
r2 = r2(-,..) + (u = (s/dim)*fft_indgen(dim))*u;
}
}
return (square ? r2 : sqrt(r2));
}
func fft_freqlist(dimlist)
/* DOCUMENT ptr = fft_freqlist(dimlist)
returns a vector of DIMLIST(1) pointers with normalized FFT
frequencies along all dimensions of DIMLIST (must be
dimsof(SOME_ARRAY)) with "adequate" geometry:
*ptr(1) = (2*pi/dimlist(2))*fft_indgen(dimlist(2));
*ptr(2) = [(2*pi/dimlist(3))*fft_indgen(dimlist(3))];
*ptr(3) = [[(2*pi/dimlist(4))*fft_indgen(dimlist(4))]];
...
SEE ALSO: fft_indgen, fft_shift_phasor. */
{
/* Precompute (scaled) Fourier frequencies along every dimensions (the
trick is to build these as "vectors" with adequate dimension list so
as to minimize the number of operations during the search). */
PI = 3.1415926535897932384626433832795029;
ndims = dimlist(1);
ptr = array(pointer, ndims);
for (k=1 ; k<=ndims ; ++k) {
len = dimlist(k + 1);
ws = array(1, k + 1); /* to build dimension list of k-th dimension */
ws(1) = k;
ws(0) = len;
(ws = array(double, ws))(*) = (2.0*PI/len)*fft_indgen(len);
ptr(k) = &ws;
}
return ptr;
}
func fft_smooth(a, fwhm, setup=)
/* DOCUMENT fft_smooth(a, fwhm)
-or- fft_smooth(a, fwhm, setup=workspace)
Returns array A smoothed along all its dimensions by convolution by a
gaussian with full width at half maximum equals to FWHM. See
fft_setup for the meaning of keyword SETUP.
SEE ALSO: fft, fft_setup, fft_gaussian_mtf. */
{
dims = dimsof(a);
if (is_void(setup)) setup=fft_setup(dims);
as_double = (structof(a) != complex);
a = fft((1.0/numberof(a))*fft_gaussian_mtf(dims, fwhm)*
fft(a, +1, setup=setup), -1, setup=setup);
return (as_double ? double(a) : a);
}
local fft_gaussian_psf;
local fft_gaussian_mtf;
/* DOCUMENT fft_gaussian_psf(dimlist, fwhm)
-or- fft_gaussian_mtf(dimlist, fwhm)
Returns normalized Gaussian point spread function (PSF) or
corresponding modulation transfer function (MTF) with dimension list
DIMLIST and full width at half maximum equals to FWHM along each
dimensions (in the PSF space). Up to errors due to limited support,
numerical precision and finite sampling, the PSF and the MTF obey:
sum(PSF) = MTF(1) = 1 (normalization)
MTF = fft(PSF, +1)
PSF = fft(MTF, -1)/numberof(MTF)
where MTF(1) is the 0-th frequency in the MTF. The standard deviation
SIGMA and the FWHM are related by:
FWHM = sqrt(8*log(2))*SIGMA
~ 2.354820045031*SIGMA
Note that, owing to the limited size of the support and/or numerical
precision, these properties may not be perfectly met; for that reason,
_always_ compute directly what you need, e.g. do not take the FFT of
the PSF if what you need is the MTF. Also note that the geometry is
that of the FFT and that for unequal dimension lengths, the PSF has
the same width (in "pixels") along every dimension but not the MTF.
FWHM can be a scalar or a vector with as many values as number of
dimensions.
SEE ALSO: fft_get_ndims, fft_dist, fft_smooth. */
func fft_gaussian_psf(dims, fwhm)
{
ndims = fft_get_ndims(dims);
if (ndims <= 0L) {
if (ndims == 0L) return (1.6651092223153955127063292897904020952612/fwhm);
error, "bad dimension list";
}
//r = (sqrt(log(16.0)/pi)/fwhm) + array(0.0, ndims);
//s = (sqrt(log(16.0))/fwhm) + array(0.0, ndims);
r = (0.9394372786996513337723403284101825868414/fwhm) + array(0.0, ndims);
s = (1.6651092223153955127063292897904020952612/fwhm) + array(0.0, ndims);
j = ndims;
u = s(j)*fft_indgen(dims(j));
p = exp(-u*u);
q = r(j);
while (--j >= 1L) {
u = s(j)*fft_indgen(dims(j));
p = exp(-u*u)*p(-,..);
q *= r(j);
}
return q*p;
}
func fft_gaussian_mtf(dims, fwhm)
{
ndims = fft_get_ndims(dims);
if (ndims <= 0L) {
if (ndims == 0L) return 1.0;
error, "bad dimension list";
}
// s = (pi/sqrt(log(16)))*fwhm/dim
s = 1.8867186677527935983734215966417072034386*fwhm/dims;
j = ndims;
u = s(j)*fft_indgen(dims(j));
p = exp(-u*u);
while (--j >= 1L) {
u = s(j)*fft_indgen(dims(j));
p = exp(-u*u)*p(-,..);
}
return p;
}
func fft_get_ndims(&dims)
/* DOCUMENT fft_get_ndims(dimlist)
Returns the number of dimensions in dimension list DIMLIST and modify
(in-place) DIMLIST to be only the list of dimensions (that is without
the number of dimensions). If DIMLIST is invalid, -1 is returned.
SEE ALSO: dimsof.
*/
{
if (((s = structof(dims)) == long || s == int
|| s == short || s == char) && min(dims) > 0) {
temp = dimsof(dims)(1);
if (temp == 0L) {
dims = long(dims);
return 1L;
} else if (temp == 1L && (ndims = dims(1)) == numberof(dims) - 1L) {
dims = (ndims >= 1L ? long(dims(2:0)) : []);
return ndims;
}
return ndims;
} else if (is_void(dims)) {
return 0L;
}
return -1L;
}
/*
* Notes:
* The FFT of a Gaussian of given FWHM is:
* exp(-pi^2*fwhm^2*(k/dim)^2/log(16))
* where K is the FFT index; hence:
* Nyquist = sqrt(pi^2*fwhm^2*(dim/2/dim)^2/4/log(2))
* = pi*fwhm/4/sqrt(log(2))
* ~ 0.9433593338763967992*fwhm
* The Gaussian is (N is the number of dimensions):
* ((sqrt(log(16)/pi)/fwhm)^n)*exp(-log(16)*(k/fwhm)^2)
* ~ ((0.9394372786996513337/fwhm)^n)*exp(...)
*
* Constants (with 40 significant digits):
* pi/4/sqrt(log(2)) = 0.9433593338763967991867107983208536017193;
* sqrt(log(16)/pi) = 0.9394372786996513337723403284101825868414;
* log(16) = 2.772588722239781237668928485832706272302;
*/
/*---------------------------------------------------------------------------*/
/* SIMPLE FAST FOURIER TRANSFORM */
func __fft_init(dimlist)
/* DOCUMENT __fft_init, dimlist;
Initializes FFT workspace for further calls to __fft (to see).
DIMLIST is the dimension list of the arrays to transform. The routine
defines 2 external symbols:
__fft_setup - used to store the FFT workspace)
__fft_number - used to keep track of the number of calls to __fft
In order to avoid namespace pollution/clash, a routine that uses __fft
should declare these symbols as local before calling __fft_init, e.g.:
local __fft_setup, __fft_number;
__fft_init, dimlist;
SEE ALSO: fft_setup, __fft. */
{
extern __fft_setup, __fft_number;
dims = dimlist(2:);
ndims = numberof(dims);
__fft_setup = array(pointer, ndims);
__fft_number = 0;
for (i=1 ; i<=ndims ; ++i) {
if (! __fft_setup(i)) {
dim = dims(i);
ws = array(double, 6*dim + 15);
fft_init, dim, ws;
__fft_setup(where(dims == dim)) = &ws;
}
}
}
func __fft(x, dir)
/* DOCUMENT __fft(x);
-or- __fft(x, dir);
Replacement for Yorick's fft to speed up fast Fourier transforms (FFT)
in, e.g., iteratives algorithms that necessitate computation of
several FFT's of arrays with same dimension list. The FFT is
performed on all dimensions of X and DIR must be a scalar (default
+1). FFT workspace must be initialized by __fft_init.
SEE ALSO: __fft_init, fft. */
{
extern __fft_setup, __fft_number;
/* Make a private copy of input array even if it is already complex and
get its dimension list. */
if (is_void(dir)) dir = +1;
x = complex(x);
dims = dimsof(x);
ndims = dims(1);
dims = dims(2:0);
if (structof(__fft_setup) != pointer ||
numberof(__fft_setup) != ndims) error, "unitialized FFT workspace";
/* Do the transform along every dimension of X. */
len = 6*dims + 15; // expected length of workspace vectors
std = 1;
top = numberof(x);
for (i=1 ; i<=ndims ; i++) {
dim = dims(i);
ws = __fft_setup(i);
if (numberof(*ws) != len(i) || structof(*ws) != double)
error, swrite(format="bad FFT workspace for dimension length %d", dim);
top /= dim;
fft_raw, dir, x, std, dim, top, ws;
std *= dim;
}
/* increment counter and return result */
if (is_void(__fft_number)) __fft_number= 0;
++__fft_number;
return x;
}
/*---------------------------------------------------------------------------*/
func fft_symmetric_index(..)
/* DOCUMENT fft_symmetric_index(dimlist)
-or- fft_symmetric_index(dim1, dim2, ...);
Returns indices of hermitian-symmetry transform for a FFT with
dimension list DIMLIST. For instance, if A is a N-dimensional array,
then:
AP= A(fft_symmetric_index(dimsof(A)))
is equal to array A with its coordinates negated according to FFT
convention:
AP(X1, X2, ..., XN) = A(-X1, -X2, ..., -XN)
consequently if A is hermitian then:
AP= conj(A).
SEE ALSO: fft_indgen. */
{
/* Build dimension list. */
dimlist = [0];
while (more_args()) make_dimlist, dimlist, next_arg();
/* Compute result starting by last dimension. */
local u;
if ((n = numberof(dimlist)) == 1) return 1; /* scalar array */
for (k=n ; k>1 ; --k) {
dim = dimlist(k);
(q = indgen(dim:1:-1))(1) = 0; // neg. of freq along that dim
u= k<n ? dim*u(-,..) + q : q;
}
return u + 1; /* <== indices start at 1 in Yorick */
}
/*---------------------------------------------------------------------------*/
func _fft_centroid(a1, repeat)
/* DOCUMENT _fft_centroid(a1)
-or- _fft_centroid(a1, repeat)
Working routine for fft_centroid: return position of centroid of 1D array
A1 assuming FFT geometry for the coordinates.
SEE ALSO fft_centroid. */
{
dim= numberof(a1);
u= fft_indgen(dim);
x0= u(a1(mxx));
x= double(u);
do {
w= x0==0 ? a1 : roll(a1, -x0);
if (dim%2 == 0) w(dim/2+1)= 0.0; // remove Nyquist frequency
x1= x0+sum(w*x)/sum(w);
x0p= x0;
x0= long(floor(x1+0.5));
} while (--repeat>0 && abs(x0-x0p)>=1);
return x1;
}
func fft_centroid(a, repeat)
/* DOCUMENT fft_centroid(a)
-or- fft_centroid(a, repeat)
Return the position of centroid of N-dimensional array A assuming
coordinates along dimensions of A are wrapped as in a FFT (see
fft_indgen). The algorithm proceeds by computing the center of gravity
of A around its central element which is the maximum of A for the first
iteration and the closest to the previously computed centroid for
subsequent iterations. The maximum number of iteration is REPEAT
(default: 3; in any cases, at least one iteration is performed). The
Nyquist frequency along each even dimension is omitted to avoid a bias.
SEE ALSO fft_indgen. */
{
if (is_void(repeat)) repeat= 3;
dims= dimsof(a);
if ((ndims= dims(1)) <= 2) {
if (ndims==2) return [_fft_centroid(a(,sum), repeat),
_fft_centroid(a(sum,), repeat)];
if (ndims==1) return _fft_centroid(a, repeat);
return 0.0; // ndims==0
}
result= array(double, ndims);
for (i=1 ; i<=ndims ; ++i) {
dim= dims(i+1);
a1= array(double, numberof(a)/dim, dim);
a1(*)= (i==ndims ? a(*) : transpose(a, [ndims,i])(*));
a1= a1(sum,);
result(i)= _fft_centroid(a1, repeat);
}
return result;
}
/*---------------------------------------------------------------------------*/
/* CENTERING / ROLLING */
func reverse_dims(a)
/* DOCUMENT reverse_dims(a)
Returns array A with all its dimensions reversed.
SEE ALSO: fft_recenter. */
{
n = dimsof(a)(1);
r = ::-1;
if (n == 1) return a(r);
if (n == 2) return a(r, r);
if (n == 3) return a(r, r, r);
if (n == 4) return a(r, r, r, r);
if (n == 5) return a(r, r, r, r, r);
if (n == 6) return a(r, r, r, r, r, r);
if (n == 7) return a(r, r, r, r, r, r, r);
if (n == 8) return a(r, r, r, r, r, r, r, r);
if (n == 9) return a(r, r, r, r, r, r, r, r, r);
if (n == 10) return a(r, r, r, r, r, r, r, r, r, r);
if (n == 11) return a(r, r, r, r, r, r, r, r, r, r, r);
if (n == 12) return a(r, r, r, r, r, r, r, r, r, r, r, r);
if (n == 13) return a(r, r, r, r, r, r, r, r, r, r, r, r, r);
if (n == 14) return a(r, r, r, r, r, r, r, r, r, r, r, r, r, r);
if (n == 15) return a(r, r, r, r, r, r, r, r, r, r, r, r, r, r, r);
if (n == 16) return a(r, r, r, r, r, r, r, r, r, r, r, r, r, r, r, r);
if (n == 17) return a(r, r, r, r, r, r, r, r, r, r, r, r, r, r, r, r, r);
if (n == 18) return a(r, r, r, r, r, r, r, r, r, r, r, r, r, r, r, r, r, r);
error, "too many dimensions";
}
func fft_recenter(x, template, reverse)
/* DOCUMENT fft_recenter(x, template)
-or- fft_recenter(x, template, reverse)
Returns array X rolled so that it matches most closely array TEMPLATE.
X and TEMPLATE may be arrays of real or complex numbers but must have
the same dimension lists. The returned value is roll(X, S) where the
offsets S minimize:
sum(abs(roll(X, S) - TEMPLATE)^2)
If optional argument REVERSE is true, X is also allowed to have all
its dimensions reversed in order to math TEMPLATE.
SEE ALSO: fft, roll, reverse_dims. */
{
ws = fft_setup(dimsof(a));
conj_fft_template = conj(fft(template, +1, setup=ws));
c1 = double(fft(conj_fft_template*fft(x, +1, setup=ws), -1, setup=ws));
max_c1 = max(c1);
if (reverse) {
x2 = reverse_dims(x);
c2 = double(fft(conj_fft_template*fft(x2, +1, setup=ws), -1, setup=ws));
if ((max_c2 = max(c2)) > max_c1) {
eq_nocopy, x, x2;
i = where(c2 == max_c2);
} else {
i = where(c1 == max_c1);
}
c2 = [];
} else {
i = where(c1 == max_c1);
}
c1 = [];
if (numberof(i) != 1) error, swrite("too many maxima (%d)", numberof(i));
index = i(1) - 1; /* 0-based position of the maximum of correlation */
dims = dimsof(x);
ndims = dims(1);
dims = dims(2:);
offset = array(long, ndims);
for (i=1 ; i<=ndims ; ++i) {
dim = dims(i);
offset(i) = index % dim;
index /= dim;
}
//return roll(x, dims - offset);
return (anyof(offset) ? roll(x, dims - offset) : x);
}
func fft_recenter_at_max(z, middle=)
/* DOCUMENT fft_recenter_at_max(z)
Return Z rolled so that its element with maximum value (or maximum
absolute value if Z is complex) is at the origin. If keyword MIDDLE
is true (non-zero and non-nil) the center is at the middle of every
dimension otherwise the center is the first element of the output
array (as assumed by the FFT).
SEE ALSO: roll. */
{
index = (structof(z) == complex ? abs(z) : z)(*)(mxx) - 1;
dims = dimsof(z);
ndims = dims(1);
dims = dims(2:);
offset = array(long, ndims);
for (i=1 ; i<=ndims ; ++i) {
dim = dims(i);
offset(i) = index % dim;
index /= dim;
}
if (middle) dims += (dims+1)/2;
return roll(z, dims - offset);
}
func fft_roll_1d(a, off) {
if ((dimlist = dimsof(a))(1) != 1) error, "expecting 1D array";
n = dimlist(2); k = (n + (off%n))%n; /* wrap offset in the range [0, n-1] */
if (! k) return a;
b = array(structof(a), dimlist);
b(k+1:n) = a(1:n-k); b(1:k) = a(n-k+1:n); return b; }
func fft_roll_2d(a, off1, off2)
/* DOCUMENT fft_roll_1d(v, off)
-or- fft_roll_2d(m, off1, off2)
"rolls" dimensions of the vector V (1D array) or matrix M (2D array)
and return a result with same data type than original array.
SEE ALSO: roll. */
{
if ((dimlist = dimsof(a))(1) != 2) error, "expecting 2D array";
n1 = dimlist(2);
k1 = (n1 + (off1%n1))%n1; /* wrap offset in the range [0, n1-1] */
n2 = dimlist(3);
k2 = (n2 + (off2%n2))%n2; /* wrap offset in the range [0, n2-1] */
case = (! k1) + 2*(! k2);
if (case == 3) return a;
b = array(structof(a), dimlist);
if (case == 0) {
b((r1=k1+1:n1), (r2=k2+1:n2)) = a((s1=1:n1-k1), (s2=1:n2-k2));
b(r1, (t2=1:k2)) = a(s1, (u2=n2-k2+1:n2));
b((t1=1:k1), t2) = a((u1=n1-k1+1:n1), u2);
b(t1, r2) = a(u1, s2);
} else if (case == 1) {
b( , k2+1:n2) = a( , 1:n2-k2);
b( , 1:k2) = a( , n2-k2+1:n2);
} else {
b(k1+1:n1, ) = a(1:n1-k1, );
b(1:k1, ) = a(n1-k1+1:n1, );
}
return b;
}
/*---------------------------------------------------------------------------*/
/* FOURIER INTERPOLATION */
func fft_shift_phasor(off, u)
/* DOCUMENT fft_shift_phasor(off, dimlist)
-or- fft_shift_phasor(off, fft_freqlist(dimlist))
returns complex phasor to apply in FFT space for a shift by OFF cells
in the real space. DIMLIST is a list of dimensions -- the second
calling sequence is to allow for computing the normalized FFT
frequencies only once. The offset OFF must have as many elements as
PTR or as many as dimensions in DIMLIST (i.e. a shift for each
dimension) and may be fractionnal.
This function is intended for Fourier interpolation. For instance,
assuming A is 2-D real array:
z = fft(a, +1);
u = fft_freqlist(dimsof(a));
q = fft_unphasor([0.33, -0.47], u);
Then:
fft_interp(z, q, real=1);
yields the value of A interpolated at coordinate (0.33, -0.47) in FFT
frame, i.e. center of lower left cell is at (0,0). The shfited version
of A by (0.33, -0.47) can be obtained by:
fft_shift(z, q, real=1);
SEE ALSO: fft_freqlist, fft_interp, fft_fine_shift, roll. */
{
if (structof(u) != pointer) u = fft_freqlist(u);
ndims = numberof(u);
for (i=1 ; i<=ndims ; ++i) {
a = (*u(i))*off(i);
if (i == 1) p = cos(a) + 1i*sin(a);
else p *= (cos(a) + 1i*sin(a));
}
return p;
}
func fft_unphasor(z, phasor, setup=, real=) {
z = fft(z*conj(phasor), -1, setup=setup);
return (1.0/numberof(z))*(real ? double(z) : z); }
func fft_fine_shift(a, off, setup=)
/* DOCUMENT fft_fine_shift(a, off, setup=)
-or- fft_unphasor(z, phasor, setup=, real=)
The function fft_fine_shift returns array A shifted by offset OFF
which can be fractionnal. Alternatively, the function fft_unphasor
can be used when the forward FFT of A and/or the complex phasor
corresponding to the shift are already computed:
fft_unphasor(fft(A, +1), fft_shift_phasor(OFF, dimsof(A)),
real=(structof(A) != complex))
yields the same result as fft_fine_shift(A, OFF).
These functions can make use of pre-computed FFT workspace specified
by keyword SETUP (see fft_setup).
TO-DO: Improve code by not Fourier transforming along direction
where OFF is zero (or equal to an integer times the length
of the dimension).
SEE ALSO: fft, fft_setup, fft_shift_phasor, roll. */
{
real = (structof(a) != complex);
dims = dimsof(a);
if (is_void(setup)) setup = fft_setup(dims);
a = fft(fft(a, +1, setup=setup)*fft_shift_phasor(-off, dims), -1);
return (1.0/numberof(a))*(real ? double(a) : a);
}
func fft_interp_real(z, phasor)
{ return (1.0/numberof(z))*double(sum(z*phasor)); }
func fft_interp_complex(z, phasor)
{ return (1.0/numberof(z))*sum(z*phasor); }
func fft_interp(a, off, setup=)
/* DOCUMENT fft_interp(a, off, setup=)
-or- fft_interp_real(z, phasor)
-or- fft_interp_complex(z, phasor)
returns value obtained by Fourier interpolation of A at offset OFF.
The function fft_interp computes the forward FFT of A and can make use
of pre-computed FFT workspace specified by keyword SETUP. The two
other functions (fft_interp_complex, if A is complex; fft_interp_real
otherwise) are usefull when the forward FFT of A and/or the complex
phasor corresponding to the shift are already computed, their
arguments are:
Z = fft(A, +1);
PHASOR = fft_shift_phasor(OFF, dimsof(A));
SEE ALSO: fft, fft_fine_shift, fft_shift_phasor. */
{
real = (structof(a) != complex);
dims = dimsof(a);
if (is_void(setup)) setup = fft_setup(dims);
n = numberof(a);
a = sum(fft(a, +1, setup=setup)*fft_shift_phasor(-off, dims));
return (1.0/n)*(real ? double(a) : a);
}
/*---------------------------------------------------------------------------*/
/* GRAPHICS */
func fft_plh(y, scale=, legend=, hide=, type=, width=, color=, smooth=,
marks=, marker=, mspace=, mphase=)
{ fft_plg, y, scale=scale, legend=legend, hide=hide, type=type, width=width,
color=color, smooth=smooth, marks=marks, marker=marker, mspace=mspace,
mphase=mphase, stair=1; }
func fft_plg(y, scale=, legend=, hide=, type=, width=, color=, smooth=,
marks=, marker=, mspace=, mphase=, stair=)
/* DOCUMENT fft_plg, y;
-or fft_plh, y;
Plot 1-D FFT array Y as a curve, taking care of "rolling" Y and setting
correct coordinates. Keyword SCALE can be used to indicate the
"frequel" scale along X-axis (SCALE is a scalar); by default,
SCALE=1.0.
KEYWORDS legend, hide, type, width, color, closed, smooth
marks, marker, mspace, mphase.
SEE ALSO plh, plg, roll. */
{
if (is_void(scale)) scale= 1.0;
else if (! is_array(scale) || dimsof(scale)(1)!=0)
error, "expecting a scalar for SCALE";
if (! is_array(y) || (dims= dimsof(y))(1)!=1) error, "expecting 1-D array";
dim1= dims(2);
min1= (max1= dim1/2) - dim1 + 1;
if (stair) {
// just=
plh, roll(y, -min1), scale*indgen(min1:max1), legend=legend, hide=hide,
type=type, width=width, color=color,
marks=marks, marker=marker, mspace=mspace, mphase=mphase;
} else {
plg, roll(y, -min1), scale*indgen(min1:max1), legend=legend, hide=hide,
type=type, width=width, color=color, smooth=smooth,
marks=marks, marker=marker, mspace=mspace, mphase=mphase;
}
}
func fft_pli(a, scale=, legend=, hide=, top=, cmin=, cmax=)
/* DOCUMENT fft_pli, a;
Plot 2-D FFT array A as an image, taking care of "rolling" A and setting
correct world boundaries. Keyword SCALE can be used to indicate the
"frequel" scale along both axis (SCALE is a scalar) or along each axis
(SCALE is a 2-element vector: SCALE=[XSCALE,YSCALE]); by default,
SCALE=[1.0, 1.0].
KEYWORDS legend, hide, top, cmin, cmax.
SEE ALSO pli, fft_roll_2d. */
{
local scale1, dim1, min1, max1, scale2, dim2, min2, max2;
__fft_pl2d_limits, a, scale;
pli, fft_roll_2d(bytscl(a, top=top, cmin=cmin, cmax=cmax), -min1, -min2),
scale1*(min1 - 0.5), scale2*(min2 - 0.5),
scale1*(max1 + 0.5), scale2*(max2 + 0.5),
legend=legend, hide=hide;
}
func fft_plc(a, scale=, levs=, type=, width=, color=, smooth=,
legend=, hide=, marks=, marker=, mspace=, mphase=)
/* DOCUMENT fft_plc, a;
Plot contour levels of a 2-D FFT array A, taking care of "rolling" A
and setting correct world boundaries. Keyword SCALE can be used to
indicate the "frequel" scale along both axis (SCALE is a scalar) or
along each axis (SCALE is a 2-element vector: SCALE=[XSCALE,YSCALE]);
by default, SCALE=[1.0, 1.0]. Other keywords have same meaning as in
plc routine.
KEYWORDS scale, levs, type, width, color, smooth,
legend, hide, marks, marker, mspace, mphase.
SEE ALSO plc, roll, fft_plfc. */
{
local scale1, dim1, min1, max1, scale2, dim2, min2, max2;
__fft_pl2d_limits, a, scale;
u1 = (scale1*indgen(min1:max1))(,-:1:dim2);
u2 = (scale2*indgen(min2:max2))(-:1:dim1,);
plc, roll(a, [-min1, -min2]), u2, u1,
levs=levs, type=type, width=width, color=color, smooth=smooth,
legend=legend, hide=hide,
marks=marks, marker=marker, mspace=mspace, mphase=mphase;
}
func fft_plfc(a, scale=, levs=, colors=)
/* DOCUMENT fft_plfc, a;
Plot filled contour levels of a 2-D FFT array A, taking care of
"rolling" A and setting correct world boundaries. Keyword SCALE can
be used to indicate the "frequel" scale along both axis (SCALE is a
scalar) or along each axis (SCALE is a 2-element vector:
SCALE=[XSCALE,YSCALE]); by default, SCALE=[1.0, 1.0]. Other keywords
have same meaning as in plfc routine. As with plfc routine, the
actual level values get saved in external symbol plfc_levs.
KEYWORDS scale, levs, colors.
SEE ALSO plc, roll, fft_plc. */
{
local scale1, dim1, min1, max1, scale2, dim2, min2, max2;
__fft_pl2d_limits, a, scale;
u1 = (scale1*indgen(min1:max1))(,-:1:dim2);
u2 = (scale2*indgen(min2:max2))(-:1:dim1,);
plfc, roll(a, [-min1, -min2]), u2, u1, levs=levs, colors=colors;
}
func __fft_pl2d_limits(z, scale)
/* DOCUMENT __fft_pl2d_limits, z, scale;
Private routine used by fft_pli, fft_plc and fft_plfc.
SEE ALSO fft_pli, fft_plc, fft_plfc. */
{
extern scale1, dim1, min1, max1, scale2, dim2, min2, max2;
if (is_void(scale)) {
scale1 = scale2 = 1.0;
} else if (dimsof(scale)(1) == 0) {
scale1 = scale2 = scale;
} else if (numberof(scale) == 2) {
scale1 = scale(1);
scale2 = scale(2);
} else {
error, "bad number of elements in SCALE";
}
if ((dims = dimsof(z))(1) != 2) error, "expecting 2-D array";
dim1 = dims(2);
min1 = (max1 = dim1/2) - dim1 + 1;
dim2 = dims(3);
min2 = (max2 = dim2/2) - dim2 + 1;
}
/*---------------------------------------------------------------------------*/
func fft_convolve(orig, psf, do_not_roll)
/* DOCUMENT fft_convolve(orig, psf);
-or- fft_convolve(orig, psf, do_not_roll);
Return discrete convolution (computed by FFT) of array ORIG by point
spread function PSF. Unless argument DO_NOT_ROLL is true, PSF is
rolled before. Note: ORIG and PSF must have same dimension list.
SEE ALSO: fft, fft_setup, roll. */
{
real = (structof(orig) != complex && structof(psf) != complex);
dims = dimsof(orig);
ws = fft_setup(dims);
p = fft(orig, -1, setup=ws) * fft((do_not_roll?psf:roll(psf)), -1, setup=ws);
orig = psf = []; // possibly free some memory
fft_inplace, p, +1, setup=ws;
if (real) p = double(p);
return (1.0/numberof(p))*p;
}
#if 0
func fft_of_two_real_arrays(a, b, &ft_a, &ft_b, ljdir, rjdir, setup=)
/* DOCUMENT fft_of_two_real_arrays, a, b, ft_a, ft_b, direction;
-or- fft_of_two_real_arrays, a, b, ft_a, ft_b, ljdir, rjdir;
Computes the FFT of arrays A and B and stores them in TF_A and FT_B
respectively. A and B must have same dimension list. A single FFT is
needed. Agrguments DIRECTION, LJDIR, RJDIR, and keyword SETUP have
the same meaning as for the fft function (which see).
SEE ALSO: fft_setup, fft_inplace. */
{
if (structof(a) == complex || structof(b) == complex)
error, "A and B must be non-complex";
c = a + 1i*b;
fft_inplace, c, ljdir, rjdir, setup=setup;
b = c(fft_symmetric_index(dimsof(c)));
a = c + b;
b = c - b;
ft_a = 0.5*double(a) + 0.5i*(b.im);
ft_b = 0.5*(a.im) - 0.5i*double(b);
}
#endif
/*---------------------------------------------------------------------------*/
/* Notes:
* there are 1 + n/2 "positive" frequencies
* there are (n - 1)/2 "negative" frequencies
*/
func fft_paste(a, b)
/* DOCUMENT fft_paste(a, b)
* -or- fft_paste, a, b;
* Paste array B into array A in the sense of FFT indexing. All
* dimensions of A must be greater or equal the corresponding dimension
* of B. When called as a subroutine, the operation is done in-place.
*
* RESTRICTIONS:
* For even dimensions, the Nyquist frequency from B is not pasted
* into A.
*
* SEE ALSO: fft_indgen.
*/
{
if (! is_array(a) || ! is_array(b))
error, "expecting array argument(s)";
adim = dimsof(a);
bdim = dimsof(b);
if ((n = adim(1) - bdim(1)) != 0) {
if (n > 0) {
grow, bdim, array(1L, n);
bdim(1) = adim(1);
} else {
grow, adim, array(1L, -n);
adim(1) = bdim(1);
}
}
if (anyof(adim < bdim)) error, "destination array is too small";
n = numberof(adim);
ia = ib = 1L; /* indices start at one in Yorick */
sa = numberof(a); /* stride in A */
sb = numberof(b); /* stride in B */
for (k=n ; k>=2 ; --k) {
alen = adim(k);
sa /= alen;
blen = bdim(k);
sb /= blen;
if (blen >= 3) {
j = (blen - 1)/2; /* maximum absolute frequency */
ja = jb = array(long, 2*j + 1);
ja(1:j+1) = indgen(0 : j*sa : sa);
jb(1:j+1) = indgen(0 : j*sb : sb);
ja(j+2:) = indgen((alen - j)*sa : (alen - 1)*sa : sa);
jb(j+2:) = indgen((blen - j)*sb : (blen - 1)*sb : sb);
} else {
ja = jb = 0L;
}
if (k == n) {
ia += ja;
ib += jb;
} else {
ia = ja + ia(-,..);
ib = jb + ib(-,..);
}
}
if (! am_subroutine()) a = a; /* make a copy */
a(ia) = b(ib);
return a;
}
/*---------------------------------------------------------------------------*
* Local Variables: *
* mode: Yorick *
* tab-width: 8 *
* fill-column: 75 *
* c-basic-offset: 2 *
* coding: latin-1 *
* End: *
*---------------------------------------------------------------------------*/
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