fourier.cpp

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/* +------------------------------------------------------------------------+
   |                     Mobile Robot Programming Toolkit (MRPT)            |
   |                          https://www.mrpt.org/                         |
   |                                                                        |
   | Copyright (c) 2005-2019, Individual contributors, see AUTHORS file     |
   | See: https://www.mrpt.org/Authors - All rights reserved.               |
   | Released under BSD License. See: https://www.mrpt.org/License          |
   +------------------------------------------------------------------------+ */

#include "math-precomp.h"  // Precompiled headers

#include <mrpt/core/bits_math.h>
#include <mrpt/math/fourier.h>
#include <Eigen/Dense>
#include <algorithm>
#include <cmath>

using namespace mrpt;
using namespace std;
using namespace mrpt::math;

// Next we declare some auxiliary functions:
namespace mrpt::math
{
// Replaces data[1..2*nn] by its discrete Fourier transform, if isign is input
// as 1; or replaces
// data[1..2*nn] by nn times its inverse discrete Fourier transform, if isign is
// input as -1.
// data is a complex array of length nn or, equivalently, a real array of length
// 2*nn. nn MUST
// be an integer power of 2 (this is not checked for!).
static void four1(float data[], unsigned long nn, int isign)
{
    unsigned long n, mmax, m, j, i;
    double wtemp, wr, wpr, wpi, wi,
        theta;  // Double precision for the trigonometric recurrences.
    float tempr, tempi;

    n = nn << 1;
    j = 1;

    for (i = 1; i < n;
         i += 2)  // This is the bit-reversal section of the routine.
    {
        if (j > i)
        {
            std::swap(data[j], data[i]);  // Exchange the two complex numbers.
            std::swap(data[j + 1], data[i + 1]);
        }
        m = nn;
        while (m >= 2 && j > m)
        {
            j -= m;
            m >>= 1;
        }
        j += m;
    }
    // Here begins the Danielson-Lanczos section of the routine.
    mmax = 2;
    while (n > mmax)  // Outer loop executed log2 nn times.
    {
        unsigned long istep = mmax << 1;
        theta = isign * (6.28318530717959 /
                         mmax);  // Initialize the trigonometric recurrence.
        wtemp = sin(0.5 * theta);
        wpr = -2.0 * wtemp * wtemp;
        wpi = sin(theta);
        wr = 1.0;
        wi = 0.0;
        for (m = 1; m < mmax; m += 2)  // Here are the two nested inner loops.
        {
            for (i = m; i <= n; i += istep)
            {
                j = i + mmax;  // This is the Danielson-Lanczos formula:
                tempr = (float)(wr * data[j] - wi * data[j + 1]);
                tempi = (float)(wr * data[j + 1] + wi * data[j]);
                data[j] = data[i] - tempr;
                data[j + 1] = data[i + 1] - tempi;
                data[i] += tempr;
                data[i + 1] += tempi;
            }
            wr = (wtemp = wr) * wpr - wi * wpi +
                 wr;  // Trigonometric recurrence.
            wi = wi * wpr + wtemp * wpi + wi;
        }
        mmax = istep;
    }
}

// Calculates the Fourier transform of a set of n real-valued data points.
// Replaces this data (which
// is stored in array data[1..n]) by the positive frequency half of its complex
// Fourier transform.
// The real-valued first and last components of the complex transform are
// returned as elements
// data[1] and data[2], respectively. n must be a power of 2. This routine also
// calculates the
// inverse transform of a complex data array if it is the transform of real
// data. (Result in this case
// must be multiplied by 2/n.)
static void realft(float data[], unsigned long n)
{
    unsigned long i, i1, i2, i3, i4, np3;
    float c1 = 0.5, c2, h1r, h1i, h2r, h2i;
    double wr, wi, wpr, wpi, wtemp,
        theta;  // Double precision for the trigonometric recurrences.
    theta = 3.141592653589793 / (double)(n >> 1);  // Initialize the recurrence.

    c2 = -0.5;
    four1(data, n >> 1, 1);  // The forward transform is here.

    wtemp = sin(0.5 * theta);
    wpr = -2.0 * wtemp * wtemp;
    wpi = sin(theta);
    wr = 1.0 + wpr;
    wi = wpi;
    np3 = n + 3;
    for (i = 2; i <= (n >> 2); i++)  // Case i=1 done separately below.
    {
        i4 = 1 + (i3 = np3 - (i2 = 1 + (i1 = i + i - 1)));
        h1r = c1 * (data[i1] + data[i3]);  // The two separate transforms are
        // separated out of data.
        h1i = c1 * (data[i2] - data[i4]);
        h2r = -c2 * (data[i2] + data[i4]);
        h2i = c2 * (data[i1] - data[i3]);
        data[i1] =
            (float)(h1r + wr * h2r - wi * h2i);  // Here they are recombined to
        // form the true transform of
        // the original real data.
        data[i2] = (float)(h1i + wr * h2i + wi * h2r);
        data[i3] = (float)(h1r - wr * h2r + wi * h2i);
        data[i4] = (float)(-h1i + wr * h2i + wi * h2r);
        wr = (wtemp = wr) * wpr - wi * wpi + wr;  // The recurrence.
        wi = wi * wpr + wtemp * wpi + wi;
    }

    data[1] = (h1r = data[1]) + data[2];
    // Squeeze the first and last data together to get them all within the
    // original array.
    data[2] = h1r - data[2];
}

/**
    Copyright(C) 1997 Takuya OOURA (email: ooura@mmm.t.u-tokyo.ac.jp).
    You may use, copy, modify this code for any purpose and
    without fee. You may distribute this ORIGINAL package.
  */
using FFT_TYPE = float;

static void makewt(int nw, int* ip, FFT_TYPE* w);
static void bitrv2(int n, int* ip, FFT_TYPE* a);
static void cftbsub(int n, FFT_TYPE* a, FFT_TYPE* w);
static void cftfsub(int n, FFT_TYPE* a, FFT_TYPE* w);
static void rftfsub(int n, FFT_TYPE* a, int nc, FFT_TYPE* c);
static void rftbsub(int n, FFT_TYPE* a, int nc, FFT_TYPE* c);

static void cftbsub(int n, FFT_TYPE* a, FFT_TYPE* w)
{
    int j, j1, j2, j3, k, k1, ks, l, m;
    FFT_TYPE wk1r, wk1i, wk2r, wk2i, wk3r, wk3i;
    FFT_TYPE x0r, x0i, x1r, x1i, x2r, x2i, x3r, x3i;

    l = 2;
    while ((l << 1) < n)
    {
        m = l << 2;
        for (j = 0; j <= l - 2; j += 2)
        {
            j1 = j + l;
            j2 = j1 + l;
            j3 = j2 + l;
            x0r = a[j] + a[j1];
            x0i = a[j + 1] + a[j1 + 1];
            x1r = a[j] - a[j1];
            x1i = a[j + 1] - a[j1 + 1];
            x2r = a[j2] + a[j3];
            x2i = a[j2 + 1] + a[j3 + 1];
            x3r = a[j2] - a[j3];
            x3i = a[j2 + 1] - a[j3 + 1];
            a[j] = x0r + x2r;
            a[j + 1] = x0i + x2i;
            a[j2] = x0r - x2r;
            a[j2 + 1] = x0i - x2i;
            a[j1] = x1r - x3i;
            a[j1 + 1] = x1i + x3r;
            a[j3] = x1r + x3i;
            a[j3 + 1] = x1i - x3r;
        }
        if (m < n)
        {
            wk1r = w[2];
            for (j = m; j <= l + m - 2; j += 2)
            {
                j1 = j + l;
                j2 = j1 + l;
                j3 = j2 + l;
                x0r = a[j] + a[j1];
                x0i = a[j + 1] + a[j1 + 1];
                x1r = a[j] - a[j1];
                x1i = a[j + 1] - a[j1 + 1];
                x2r = a[j2] + a[j3];
                x2i = a[j2 + 1] + a[j3 + 1];
                x3r = a[j2] - a[j3];
                x3i = a[j2 + 1] - a[j3 + 1];
                a[j] = x0r + x2r;
                a[j + 1] = x0i + x2i;
                a[j2] = x2i - x0i;
                a[j2 + 1] = x0r - x2r;
                x0r = x1r - x3i;
                x0i = x1i + x3r;
                a[j1] = wk1r * (x0r - x0i);
                a[j1 + 1] = wk1r * (x0r + x0i);
                x0r = x3i + x1r;
                x0i = x3r - x1i;
                a[j3] = wk1r * (x0i - x0r);
                a[j3 + 1] = wk1r * (x0i + x0r);
            }
            k1 = 1;
            ks = -1;
            for (k = (m << 1); k <= n - m; k += m)
            {
                k1++;
                ks = -ks;
                wk1r = w[k1 << 1];
                wk1i = w[(k1 << 1) + 1];
                wk2r = ks * w[k1];
                wk2i = w[k1 + ks];
                wk3r = wk1r - 2 * wk2i * wk1i;
                wk3i = 2 * wk2i * wk1r - wk1i;
                for (j = k; j <= l + k - 2; j += 2)
                {
                    j1 = j + l;
                    j2 = j1 + l;
                    j3 = j2 + l;
                    x0r = a[j] + a[j1];
                    x0i = a[j + 1] + a[j1 + 1];
                    x1r = a[j] - a[j1];
                    x1i = a[j + 1] - a[j1 + 1];
                    x2r = a[j2] + a[j3];
                    x2i = a[j2 + 1] + a[j3 + 1];
                    x3r = a[j2] - a[j3];
                    x3i = a[j2 + 1] - a[j3 + 1];
                    a[j] = x0r + x2r;
                    a[j + 1] = x0i + x2i;
                    x0r -= x2r;
                    x0i -= x2i;
                    a[j2] = wk2r * x0r - wk2i * x0i;
                    a[j2 + 1] = wk2r * x0i + wk2i * x0r;
                    x0r = x1r - x3i;
                    x0i = x1i + x3r;
                    a[j1] = wk1r * x0r - wk1i * x0i;
                    a[j1 + 1] = wk1r * x0i + wk1i * x0r;
                    x0r = x1r + x3i;
                    x0i = x1i - x3r;
                    a[j3] = wk3r * x0r - wk3i * x0i;
                    a[j3 + 1] = wk3r * x0i + wk3i * x0r;
                }
            }
        }
        l = m;
    }
    if (l < n)
    {
        for (j = 0; j <= l - 2; j += 2)
        {
            j1 = j + l;
            x0r = a[j] - a[j1];
            x0i = a[j + 1] - a[j1 + 1];
            a[j] += a[j1];
            a[j + 1] += a[j1 + 1];
            a[j1] = x0r;
            a[j1 + 1] = x0i;
        }
    }
}

static void cftfsub(int n, FFT_TYPE* a, FFT_TYPE* w)
{
    int j, j1, j2, j3, k, k1, ks, l, m;
    FFT_TYPE wk1r, wk1i, wk2r, wk2i, wk3r, wk3i;
    FFT_TYPE x0r, x0i, x1r, x1i, x2r, x2i, x3r, x3i;

    l = 2;
    while ((l << 1) < n)
    {
        m = l << 2;
        for (j = 0; j <= l - 2; j += 2)
        {
            j1 = j + l;
            j2 = j1 + l;
            j3 = j2 + l;
            x0r = a[j] + a[j1];
            x0i = a[j + 1] + a[j1 + 1];
            x1r = a[j] - a[j1];
            x1i = a[j + 1] - a[j1 + 1];
            x2r = a[j2] + a[j3];
            x2i = a[j2 + 1] + a[j3 + 1];
            x3r = a[j2] - a[j3];
            x3i = a[j2 + 1] - a[j3 + 1];
            a[j] = x0r + x2r;
            a[j + 1] = x0i + x2i;
            a[j2] = x0r - x2r;
            a[j2 + 1] = x0i - x2i;
            a[j1] = x1r + x3i;
            a[j1 + 1] = x1i - x3r;
            a[j3] = x1r - x3i;
            a[j3 + 1] = x1i + x3r;
        }
        if (m < n)
        {
            wk1r = w[2];
            for (j = m; j <= l + m - 2; j += 2)
            {
                j1 = j + l;
                j2 = j1 + l;
                j3 = j2 + l;
                x0r = a[j] + a[j1];
                x0i = a[j + 1] + a[j1 + 1];
                x1r = a[j] - a[j1];
                x1i = a[j + 1] - a[j1 + 1];
                x2r = a[j2] + a[j3];
                x2i = a[j2 + 1] + a[j3 + 1];
                x3r = a[j2] - a[j3];
                x3i = a[j2 + 1] - a[j3 + 1];
                a[j] = x0r + x2r;
                a[j + 1] = x0i + x2i;
                a[j2] = x0i - x2i;
                a[j2 + 1] = x2r - x0r;
                x0r = x1r + x3i;
                x0i = x1i - x3r;
                a[j1] = wk1r * (x0i + x0r);
                a[j1 + 1] = wk1r * (x0i - x0r);
                x0r = x3i - x1r;
                x0i = x3r + x1i;
                a[j3] = wk1r * (x0r + x0i);
                a[j3 + 1] = wk1r * (x0r - x0i);
            }
            k1 = 1;
            ks = -1;
            for (k = (m << 1); k <= n - m; k += m)
            {
                k1++;
                ks = -ks;
                wk1r = w[k1 << 1];
                wk1i = w[(k1 << 1) + 1];
                wk2r = ks * w[k1];
                wk2i = w[k1 + ks];
                wk3r = wk1r - 2 * wk2i * wk1i;
                wk3i = 2 * wk2i * wk1r - wk1i;
                for (j = k; j <= l + k - 2; j += 2)
                {
                    j1 = j + l;
                    j2 = j1 + l;
                    j3 = j2 + l;
                    x0r = a[j] + a[j1];
                    x0i = a[j + 1] + a[j1 + 1];
                    x1r = a[j] - a[j1];
                    x1i = a[j + 1] - a[j1 + 1];
                    x2r = a[j2] + a[j3];
                    x2i = a[j2 + 1] + a[j3 + 1];
                    x3r = a[j2] - a[j3];
                    x3i = a[j2 + 1] - a[j3 + 1];
                    a[j] = x0r + x2r;
                    a[j + 1] = x0i + x2i;
                    x0r -= x2r;
                    x0i -= x2i;
                    a[j2] = wk2r * x0r + wk2i * x0i;
                    a[j2 + 1] = wk2r * x0i - wk2i * x0r;
                    x0r = x1r + x3i;
                    x0i = x1i - x3r;
                    a[j1] = wk1r * x0r + wk1i * x0i;
                    a[j1 + 1] = wk1r * x0i - wk1i * x0r;
                    x0r = x1r - x3i;
                    x0i = x1i + x3r;
                    a[j3] = wk3r * x0r + wk3i * x0i;
                    a[j3 + 1] = wk3r * x0i - wk3i * x0r;
                }
            }
        }
        l = m;
    }
    if (l < n)
    {
        for (j = 0; j <= l - 2; j += 2)
        {
            j1 = j + l;
            x0r = a[j] - a[j1];
            x0i = a[j + 1] - a[j1 + 1];
            a[j] += a[j1];
            a[j + 1] += a[j1 + 1];
            a[j1] = x0r;
            a[j1 + 1] = x0i;
        }
    }
}

static void makewt(int nw, int* ip, FFT_TYPE* w)
{
    void bitrv2(int n, int* ip, FFT_TYPE* a);
    int nwh, j;
    FFT_TYPE delta, x, y;

    ip[0] = nw;
    ip[1] = 1;
    if (nw > 2)
    {
        nwh = nw >> 1;
        delta = atan(1.0f) / nwh;
        w[0] = 1;
        w[1] = 0;
        w[nwh] = cos(delta * nwh);
        w[nwh + 1] = w[nwh];
        for (j = 2; j <= nwh - 2; j += 2)
        {
            x = cos(delta * j);
            y = sin(delta * j);
            w[j] = x;
            w[j + 1] = y;
            w[nw - j] = y;
            w[nw - j + 1] = x;
        }
        bitrv2(nw, ip + 2, w);
    }
}

/**
    Copyright(C) 1997 Takuya OOURA (email: ooura@mmm.t.u-tokyo.ac.jp).
    You may use, copy, modify this code for any purpose and
    without fee. You may distribute this ORIGINAL package.
  */
static void makect(int nc, int* ip, FFT_TYPE* c)
{
    int nch, j;
    FFT_TYPE delta;

    ip[1] = nc;
    if (nc > 1)
    {
        nch = nc >> 1;
        delta = atan(1.0f) / nch;
        c[0] = 0.5f;
        c[nch] = 0.5f * cos(delta * nch);
        for (j = 1; j <= nch - 1; j++)
        {
            c[j] = 0.5f * cos(delta * j);
            c[nc - j] = 0.5f * sin(delta * j);
        }
    }
}

/**
    Copyright(C) 1997 Takuya OOURA (email: ooura@mmm.t.u-tokyo.ac.jp).
    You may use, copy, modify this code for any purpose and
    without fee. You may distribute this ORIGINAL package.
  */
static void bitrv2(int n, int* ip, FFT_TYPE* a)
{
    int j, j1, k, k1, l, m, m2;
    FFT_TYPE xr, xi;

    ip[0] = 0;
    l = n;
    m = 1;
    while ((m << 2) < l)
    {
        l >>= 1;
        for (j = 0; j <= m - 1; j++)
        {
            ip[m + j] = ip[j] + l;
        }
        m <<= 1;
    }
    if ((m << 2) > l)
    {
        for (k = 1; k <= m - 1; k++)
        {
            for (j = 0; j <= k - 1; j++)
            {
                j1 = (j << 1) + ip[k];
                k1 = (k << 1) + ip[j];
                xr = a[j1];
                xi = a[j1 + 1];
                a[j1] = a[k1];
                a[j1 + 1] = a[k1 + 1];
                a[k1] = xr;
                a[k1 + 1] = xi;
            }
        }
    }
    else
    {
        m2 = m << 1;
        for (k = 1; k <= m - 1; k++)
        {
            for (j = 0; j <= k - 1; j++)
            {
                j1 = (j << 1) + ip[k];
                k1 = (k << 1) + ip[j];
                xr = a[j1];
                xi = a[j1 + 1];
                a[j1] = a[k1];
                a[j1 + 1] = a[k1 + 1];
                a[k1] = xr;
                a[k1 + 1] = xi;
                j1 += m2;
                k1 += m2;
                xr = a[j1];
                xi = a[j1 + 1];
                a[j1] = a[k1];
                a[j1 + 1] = a[k1 + 1];
                a[k1] = xr;
                a[k1 + 1] = xi;
            }
        }
    }
}

/**
    Copyright(C) 1997 Takuya OOURA (email: ooura@mmm.t.u-tokyo.ac.jp).
    You may use, copy, modify this code for any purpose and
    without fee. You may distribute this ORIGINAL package.
  */
static void cdft(int n, int isgn, FFT_TYPE* a, int* ip, FFT_TYPE* w)
{
    if (n > (ip[0] << 2))
    {
        makewt(n >> 2, ip, w);
    }
    if (n > 4)
    {
        bitrv2(n, ip + 2, a);
    }
    if (isgn < 0)
    {
        cftfsub(n, a, w);
    }
    else
    {
        cftbsub(n, a, w);
    }
}

static void rftfsub(int n, FFT_TYPE* a, int nc, FFT_TYPE* c)
{
    int j, k, kk, ks;
    FFT_TYPE wkr, wki, xr, xi, yr, yi;

    ks = (nc << 2) / n;
    kk = 0;
    for (k = (n >> 1) - 2; k >= 2; k -= 2)
    {
        j = n - k;
        kk += ks;
        wkr = 0.5f - c[kk];
        wki = c[nc - kk];
        xr = a[k] - a[j];
        xi = a[k + 1] + a[j + 1];
        yr = wkr * xr + wki * xi;
        yi = wkr * xi - wki * xr;
        a[k] -= yr;
        a[k + 1] -= yi;
        a[j] += yr;
        a[j + 1] -= yi;
    }
}

static void rdft(int n, int isgn, FFT_TYPE* a, int* ip, FFT_TYPE* w)
{
    int nw, nc;
    FFT_TYPE xi;

    nw = ip[0];
    if (n > (nw << 2))
    {
        nw = n >> 2;
        makewt(nw, ip, w);
    }
    nc = ip[1];
    if (n > (nc << 2))
    {
        nc = n >> 2;
        makect(nc, ip, w + nw);
    }
    if (isgn < 0)
    {
        a[1] = 0.5f * (a[0] - a[1]);
        a[0] -= a[1];
        if (n > 4)
        {
            rftfsub(n, a, nc, w + nw);
            bitrv2(n, ip + 2, a);
        }
        cftfsub(n, a, w);
    }
    else
    {
        if (n > 4)
        {
            bitrv2(n, ip + 2, a);
        }
        cftbsub(n, a, w);
        if (n > 4)
        {
            rftbsub(n, a, nc, w + nw);
        }
        xi = a[0] - a[1];
        a[0] += a[1];
        a[1] = xi;
    }
}

static void rftbsub(int n, FFT_TYPE* a, int nc, FFT_TYPE* c)
{
    int j, k, kk, ks;
    FFT_TYPE wkr, wki, xr, xi, yr, yi;

    ks = (nc << 2) / n;
    kk = 0;
    for (k = (n >> 1) - 2; k >= 2; k -= 2)
    {
        j = n - k;
        kk += ks;
        wkr = 0.5f - c[kk];
        wki = c[nc - kk];
        xr = a[k] - a[j];
        xi = a[k + 1] + a[j + 1];
        yr = wkr * xr - wki * xi;
        yi = wkr * xi + wki * xr;
        a[k] -= yr;
        a[k + 1] -= yi;
        a[j] += yr;
        a[j + 1] -= yi;
    }
}

/**
    Copyright(C) 1997 Takuya OOURA (email: ooura@mmm.t.u-tokyo.ac.jp).
    You may use, copy, modify this code for any purpose and
    without fee. You may distribute this ORIGINAL package.

-------- Real DFT / Inverse of Real DFT --------
    [definition]
        <case1> RDFT
            R[k1][k2] = sum_j1=0^n1-1 sum_j2=0^n2-1 a[j1][j2] *
                            cos(2*pi*j1*k1/n1 + 2*pi*j2*k2/n2),
                            0<=k1<n1, 0<=k2<n2
            I[k1][k2] = sum_j1=0^n1-1 sum_j2=0^n2-1 a[j1][j2] *
                            sin(2*pi*j1*k1/n1 + 2*pi*j2*k2/n2),
                            0<=k1<n1, 0<=k2<n2
        <case2> IRDFT (excluding scale)
            a[k1][k2] = (1/2) * sum_j1=0^n1-1 sum_j2=0^n2-1
                            (R[j1][j2] *
                            cos(2*pi*j1*k1/n1 + 2*pi*j2*k2/n2) +
                            I[j1][j2] *
                            sin(2*pi*j1*k1/n1 + 2*pi*j2*k2/n2)),
                            0<=k1<n1, 0<=k2<n2
        (notes: R[n1-k1][n2-k2] = R[k1][k2],
                I[n1-k1][n2-k2] = -I[k1][k2],
                R[n1-k1][0] = R[k1][0],
                I[n1-k1][0] = -I[k1][0],
                R[0][n2-k2] = R[0][k2],
                I[0][n2-k2] = -I[0][k2],
                0<k1<n1, 0<k2<n2)
    [usage]
        <case1>
            ip[0] = 0; // first time only
            rdft2d(n1, n2, 1, a, t, ip, w);
        <case2>
            ip[0] = 0; // first time only
            rdft2d(n1, n2, -1, a, t, ip, w);
    [parameters]
        n1     :data length (int)
                n1 >= 2, n1 = power of 2
        n2     :data length (int)
                n2 >= 2, n2 = power of 2
        a[0...n1-1][0...n2-1]
               :input/output data (FFT_TYPE **)
                <case1>
                    output data
                        a[k1][2*k2] = R[k1][k2] = R[n1-k1][n2-k2],
                        a[k1][2*k2+1] = I[k1][k2] = -I[n1-k1][n2-k2],
                            0<k1<n1, 0<k2<n2/2,
                        a[0][2*k2] = R[0][k2] = R[0][n2-k2],
                        a[0][2*k2+1] = I[0][k2] = -I[0][n2-k2],
                            0<k2<n2/2,
                        a[k1][0] = R[k1][0] = R[n1-k1][0],
                        a[k1][1] = I[k1][0] = -I[n1-k1][0],
                        a[n1-k1][1] = R[k1][n2/2] = R[n1-k1][n2/2],
                        a[n1-k1][0] = -I[k1][n2/2] = I[n1-k1][n2/2],
                            0<k1<n1/2,
                        a[0][0] = R[0][0],
                        a[0][1] = R[0][n2/2],
                        a[n1/2][0] = R[n1/2][0],
                        a[n1/2][1] = R[n1/2][n2/2]
                <case2>
                    input data
                        a[j1][2*j2] = R[j1][j2] = R[n1-j1][n2-j2],
                        a[j1][2*j2+1] = I[j1][j2] = -I[n1-j1][n2-j2],
                            0<j1<n1, 0<j2<n2/2,
                        a[0][2*j2] = R[0][j2] = R[0][n2-j2],
                        a[0][2*j2+1] = I[0][j2] = -I[0][n2-j2],
                            0<j2<n2/2,
                        a[j1][0] = R[j1][0] = R[n1-j1][0],
                        a[j1][1] = I[j1][0] = -I[n1-j1][0],
                        a[n1-j1][1] = R[j1][n2/2] = R[n1-j1][n2/2],
                        a[n1-j1][0] = -I[j1][n2/2] = I[n1-j1][n2/2],
                            0<j1<n1/2,
                        a[0][0] = R[0][0],
                        a[0][1] = R[0][n2/2],
                        a[n1/2][0] = R[n1/2][0],
                        a[n1/2][1] = R[n1/2][n2/2]
        t[0...2*n1-1]
               :work area (FFT_TYPE *)
        ip[0...*]
               :work area for bit reversal (int *)
                length of ip >= 2+sqrt(n)  ; if n % 4 == 0
                                2+sqrt(n/2); otherwise
                (n = max(n1, n2/2))
                ip[0],ip[1] are pointers of the cos/sin table.
        w[0...*]
               :cos/sin table (FFT_TYPE *)
                length of w >= max(n1/2, n2/4) + n2/4
                w[],ip[] are initialized if ip[0] == 0.
    [remark]
        Inverse of
            rdft2d(n1, n2, 1, a, t, ip, w);
        is
            rdft2d(n1, n2, -1, a, t, ip, w);
            for (j1 = 0; j1 <= n1 - 1; j1++) {
                for (j2 = 0; j2 <= n2 - 1; j2++) {
                    a[j1][j2] *= 2.0 / (n1 * n2);
                }
            }
  */
static void rdft2d(
    int n1, int n2, int isgn, FFT_TYPE** a, FFT_TYPE* t, int* ip, FFT_TYPE* w)
{
    int n, nw, nc, n1h, i, j, i2;
    FFT_TYPE xi;

    n = n1 << 1;
    if (n < n2)
    {
        n = n2;
    }
    nw = ip[0];
    if (n > (nw << 2))
    {
        nw = n >> 2;
        makewt(nw, ip, w);
    }
    nc = ip[1];
    if (n2 > (nc << 2))
    {
        nc = n2 >> 2;
        makect(nc, ip, w + nw);
    }
    n1h = n1 >> 1;
    if (isgn < 0)
    {
        for (i = 1; i <= n1h - 1; i++)
        {
            j = n1 - i;
            xi = a[i][0] - a[j][0];
            a[i][0] += a[j][0];
            a[j][0] = xi;
            xi = a[j][1] - a[i][1];
            a[i][1] += a[j][1];
            a[j][1] = xi;
        }
        for (j = 0; j <= n2 - 2; j += 2)
        {
            for (i = 0; i <= n1 - 1; i++)
            {
                i2 = i << 1;
                t[i2] = a[i][j];
                t[i2 + 1] = a[i][j + 1];
            }
            cdft(n1 << 1, isgn, t, ip, w);
            for (i = 0; i <= n1 - 1; i++)
            {
                i2 = i << 1;
                a[i][j] = t[i2];
                a[i][j + 1] = t[i2 + 1];
            }
        }
        for (i = 0; i <= n1 - 1; i++)
        {
            rdft(n2, isgn, a[i], ip, w);
        }
    }
    else
    {
        for (i = 0; i <= n1 - 1; i++)
        {
            rdft(n2, isgn, a[i], ip, w);
        }
        for (j = 0; j <= n2 - 2; j += 2)
        {
            for (i = 0; i <= n1 - 1; i++)
            {
                i2 = i << 1;
                t[i2] = a[i][j];
                t[i2 + 1] = a[i][j + 1];
            }
            cdft(n1 << 1, isgn, t, ip, w);
            for (i = 0; i <= n1 - 1; i++)
            {
                i2 = i << 1;
                a[i][j] = t[i2];
                a[i][j + 1] = t[i2 + 1];
            }
        }
        for (i = 1; i <= n1h - 1; i++)
        {
            j = n1 - i;
            a[j][0] = 0.5f * (a[i][0] - a[j][0]);
            a[i][0] -= a[j][0];
            a[j][1] = 0.5f * (a[i][1] + a[j][1]);
            a[i][1] -= a[j][1];
        }
    }
}

/**
    Copyright(C) 1997 Takuya OOURA (email: ooura@mmm.t.u-tokyo.ac.jp).
    You may use, copy, modify this code for any purpose and
    without fee. You may distribute this ORIGINAL package.

-------- Complex DFT (Discrete Fourier Transform) --------
    [definition]
        <case1>
            X[k1][k2] = sum_j1=0^n1-1 sum_j2=0^n2-1 x[j1][j2] *
                            exp(2*pi*i*j1*k1/n1) *
                            exp(2*pi*i*j2*k2/n2), 0<=k1<n1, 0<=k2<n2
        <case2>
            X[k1][k2] = sum_j1=0^n1-1 sum_j2=0^n2-1 x[j1][j2] *
                            exp(-2*pi*i*j1*k1/n1) *
                            exp(-2*pi*i*j2*k2/n2), 0<=k1<n1, 0<=k2<n2
        (notes: sum_j=0^n-1 is a summation from j=0 to n-1)
    [usage]
        <case1>
            ip[0] = 0; // first time only
            cdft2d(n1, 2*n2, 1, a, t, ip, w);
        <case2>
            ip[0] = 0; // first time only
            cdft2d(n1, 2*n2, -1, a, t, ip, w);
    [parameters]
        n1     :data length (int)
                n1 >= 1, n1 = power of 2
        2*n2   :data length (int)
                n2 >= 1, n2 = power of 2
        a[0...n1-1][0...2*n2-1]
               :input/output data (double **)
                input data
                    a[j1][2*j2] = Re(x[j1][j2]),
                    a[j1][2*j2+1] = Im(x[j1][j2]),
                    0<=j1<n1, 0<=j2<n2
                output data
                    a[k1][2*k2] = Re(X[k1][k2]),
                    a[k1][2*k2+1] = Im(X[k1][k2]),
                    0<=k1<n1, 0<=k2<n2
        t[0...2*n1-1]
               :work area (double *)
        ip[0...*]
               :work area for bit reversal (int *)
                length of ip >= 2+sqrt(n)  ; if n % 4 == 0
                                2+sqrt(n/2); otherwise
                (n = max(n1, n2))
                ip[0],ip[1] are pointers of the cos/sin table.
        w[0...*]
               :cos/sin table (double *)
                length of w >= max(n1/2, n2/2)
                w[],ip[] are initialized if ip[0] == 0.
    [remark]
        Inverse of
            cdft2d(n1, 2*n2, -1, a, t, ip, w);
        is
            cdft2d(n1, 2*n2, 1, a, t, ip, w);
            for (j1 = 0; j1 <= n1 - 1; j1++) {
                for (j2 = 0; j2 <= 2 * n2 - 1; j2++) {
                    a[j1][j2] *= 1.0 / (n1 * n2);
                }
            }

*/
static void cdft2d(
    int n1, int n2, int isgn, FFT_TYPE** a, FFT_TYPE* t, int* ip, FFT_TYPE* w)
{
    void makewt(int nw, int* ip, FFT_TYPE* w);
    void cdft(int n, int isgn, FFT_TYPE* a, int* ip, FFT_TYPE* w);
    int n, i, j, i2;

    n = n1 << 1;
    if (n < n2)
    {
        n = n2;
    }
    if (n > (ip[0] << 2))
    {
        makewt(n >> 2, ip, w);
    }
    for (i = 0; i <= n1 - 1; i++)
    {
        cdft(n2, isgn, a[i], ip, w);
    }
    for (j = 0; j <= n2 - 2; j += 2)
    {
        for (i = 0; i <= n1 - 1; i++)
        {
            i2 = i << 1;
            t[i2] = a[i][j];
            t[i2 + 1] = a[i][j + 1];
        }
        cdft(n1 << 1, isgn, t, ip, w);
        for (i = 0; i <= n1 - 1; i++)
        {
            i2 = i << 1;
            a[i][j] = t[i2];
            a[i][j + 1] = t[i2 + 1];
        }
    }
}

}  // namespace mrpt::math

void mrpt::math::fft_real(
    CVectorFloat& in_realData, CVectorFloat& out_FFT_Re,
    CVectorFloat& out_FFT_Im, CVectorFloat& out_FFT_Mag)
{
    MRPT_START

    auto n = (unsigned long)in_realData.size();

    // TODO: Test data lenght is 2^N...

    CVectorFloat auxVect(n + 1);

    memcpy(&auxVect[1], &in_realData[0], n * sizeof(auxVect[0]));

    realft(&auxVect[0], n);

    unsigned int n_2 = 1 + (n / 2);

    out_FFT_Re.resize(n_2);
    out_FFT_Im.resize(n_2);
    out_FFT_Mag.resize(n_2);

    for (unsigned int i = 0; i < n_2; i++)
    {
        if (i == (n_2 - 1))
            out_FFT_Re[i] = auxVect[2];
        else
            out_FFT_Re[i] = auxVect[1 + i * 2];

        if (i == 0 || i == (n_2 - 1))
            out_FFT_Im[i] = 0;
        else
            out_FFT_Im[i] = auxVect[1 + i * 2 + 1];

        out_FFT_Mag[i] =
            std::sqrt(square(out_FFT_Re[i]) + square(out_FFT_Im[i]));
    }

    MRPT_END
}

void math::dft2_real(
    const CMatrixFloat& in_data, CMatrixFloat& out_real, CMatrixFloat& out_imag)
{
    MRPT_START

    size_t i, j;
    using float_ptr = FFT_TYPE*;

    // The dimensions:
    size_t dim1 = in_data.rows();
    size_t dim2 = in_data.cols();

    // Transform to format compatible with C routines:
    // ------------------------------------------------------------
    FFT_TYPE** a;
    FFT_TYPE* t;
    int* ip;
    FFT_TYPE* w;

    // Reserve memory and copy data:
    // --------------------------------------
    a = new float_ptr[dim1];
    for (i = 0; i < dim1; i++)
    {
        a[i] = new FFT_TYPE[dim2];
        for (j = 0; j < dim2; j++) a[i][j] = in_data(i, j);
    }

    t = new FFT_TYPE[2 * dim1 + 20];
    ip = new int[(int)ceil(20 + 2 + sqrt((FFT_TYPE)max(dim1, dim2 / 2)))];
    ip[0] = 0;
    w = new FFT_TYPE[max(dim1 / 2, dim2 / 4) + dim2 / 4 + 20];

    // Do the job!
    // --------------------------------------
    rdft2d((int)dim1, (int)dim2, 1, a, t, ip, w);

    // Transform back to MRPT matrix format:
    // --------------------------------------
    out_real.setSize(dim1, dim2);
    out_imag.setSize(dim1, dim2);

    // a[k1][2*k2] = R[k1][k2] = R[n1-k1][n2-k2],
    // a[k1][2*k2+1] = I[k1][k2] = -I[n1-k1][n2-k2],
    //       0<k1<n1, 0<k2<n2/2,
    for (i = 1; i < dim1; i++)
        for (j = 1; j < dim2 / 2; j++)
        {
            out_real(i, j) = (float)a[i][j * 2];
            out_real(dim1 - i, dim2 - j) = (float)a[i][j * 2];
            out_imag(i, j) = (float)-a[i][j * 2 + 1];
            out_imag(dim1 - i, dim2 - j) = (float)a[i][j * 2 + 1];
        }
    // a[0][2*k2] = R[0][k2] = R[0][n2-k2],
    // a[0][2*k2+1] = I[0][k2] = -I[0][n2-k2],
    //     0<k2<n2/2,
    for (j = 1; j < dim2 / 2; j++)
    {
        out_real(0, j) = (float)a[0][j * 2];
        out_real(0, dim2 - j) = (float)a[0][j * 2];
        out_imag(0, j) = (float)-a[0][j * 2 + 1];
        out_imag(0, dim2 - j) = (float)a[0][j * 2 + 1];
    }

    // a[k1][0] = R[k1][0] = R[n1-k1][0],
    // a[k1][1] = I[k1][0] = -I[n1-k1][0],
    // a[n1-k1][1] = R[k1][n2/2] = R[n1-k1][n2/2],
    // a[n1-k1][0] = -I[k1][n2/2] = I[n1-k1][n2/2],
    //    0<k1<n1/2,
    for (i = 1; i < dim1 / 2; i++)
    {
        out_real(i, 0) = (float)a[i][0];
        out_real(dim1 - i, 0) = (float)a[i][0];
        out_imag(i, 0) = (float)-a[i][1];
        out_imag(dim1 - i, 0) = (float)a[i][1];
        out_real(i, dim2 / 2) = (float)a[dim1 - i][1];
        out_real(dim1 - i, dim2 / 2) = (float)a[dim1 - i][1];
        out_imag(i, dim2 / 2) = (float)a[dim1 - i][0];
        out_imag(dim1 - i, dim2 / 2) = (float)-a[dim1 - i][0];
    }

    // a[0][0] = R[0][0],
    // a[0][1] = R[0][n2/2],
    // a[n1/2][0] = R[n1/2][0],
    // a[n1/2][1] = R[n1/2][n2/2]
    out_real(0, 0) = (float)a[0][0];
    out_real(0, dim2 / 2) = (float)a[0][1];
    out_real(dim1 / 2, 0) = (float)a[dim1 / 2][0];
    out_real(dim1 / 2, dim2 / 2) = (float)a[dim1 / 2][1];

    // Free temporary memory:
    for (i = 0; i < dim1; i++) delete[] a[i];
    delete[] a;
    delete[] t;
    delete[] ip;
    delete[] w;

    MRPT_END
}

void math::idft2_real(
    const CMatrixFloat& in_real, const CMatrixFloat& in_imag,
    CMatrixFloat& out_data)
{
    MRPT_START

    size_t i, j;
    using float_ptr = FFT_TYPE*;

    ASSERT_(in_real.rows() == in_imag.rows());
    ASSERT_(in_real.cols() == in_imag.cols());

    // The dimensions:
    size_t dim1 = in_real.rows();
    size_t dim2 = in_real.cols();

    if (mrpt::round2up(dim1) != dim1 || mrpt::round2up(dim2) != dim2)
        THROW_EXCEPTION("Matrix sizes are not a power of two!");

    // Transform to format compatible with C routines:
    // ------------------------------------------------------------
    FFT_TYPE** a;
    FFT_TYPE* t;
    int* ip;
    FFT_TYPE* w;

    // Reserve memory and copy data:
    // --------------------------------------
    a = new float_ptr[dim1];
    for (i = 0; i < dim1; i++) a[i] = new FFT_TYPE[dim2];

    // a[j1][2*j2] = R[j1][j2] = R[n1-j1][n2-j2],
    // a[j1][2*j2+1] = I[j1][j2] = -I[n1-j1][n2-j2],
    //    0<j1<n1, 0<j2<n2/2,
    for (i = 1; i < dim1; i++)
        for (j = 1; j < dim2 / 2; j++)
        {
            a[i][2 * j] = in_real(i, j);
            a[i][2 * j + 1] = -in_imag(i, j);
        }

    // a[0][2*j2] = R[0][j2] = R[0][n2-j2],
    // a[0][2*j2+1] = I[0][j2] = -I[0][n2-j2],
    //    0<j2<n2/2,
    for (j = 1; j < dim2 / 2; j++)
    {
        a[0][2 * j] = in_real(0, j);
        a[0][2 * j + 1] = -in_imag(0, j);
    }

    // a[j1][0] = R[j1][0] = R[n1-j1][0],
    // a[j1][1] = I[j1][0] = -I[n1-j1][0],
    // a[n1-j1][1] = R[j1][n2/2] = R[n1-j1][n2/2],
    // a[n1-j1][0] = -I[j1][n2/2] = I[n1-j1][n2/2],
    //    0<j1<n1/2,
    for (i = 1; i < dim1 / 2; i++)
    {
        a[i][0] = in_real(i, 0);
        a[i][1] = -in_imag(i, 0);
        a[dim1 - i][1] = in_real(i, dim2 / 2);
        a[dim1 - i][0] = in_imag(i, dim2 / 2);
    }

    // a[0][0] = R[0][0],
    // a[0][1] = R[0][n2/2],
    // a[n1/2][0] = R[n1/2][0],
    // a[n1/2][1] = R[n1/2][n2/2]
    a[0][0] = in_real(0, 0);
    a[0][1] = in_real(0, dim2 / 2);
    a[dim1 / 2][0] = in_real(dim1 / 2, 0);
    a[dim1 / 2][1] = in_real(dim1 / 2, dim2 / 2);

    t = new FFT_TYPE[2 * dim1 + 20];
    ip = new int[(int)ceil(20 + 2 + sqrt((FFT_TYPE)max(dim1, dim2 / 2)))];
    ip[0] = 0;
    w = new FFT_TYPE[max(dim1 / 2, dim2 / 4) + dim2 / 4 + 20];

    // Do the job!
    // --------------------------------------
    rdft2d((int)dim1, (int)dim2, -1, a, t, ip, w);

    // Transform back to MRPT matrix format:
    // --------------------------------------
    out_data.setSize(dim1, dim2);

    FFT_TYPE scale = 2.0f / (dim1 * dim2);

    for (i = 0; i < dim1; i++)
        for (j = 0; j < dim2; j++) out_data(i, j) = (float)(a[i][j] * scale);

    // Free temporary memory:
    for (i = 0; i < dim1; i++) delete[] a[i];
    delete[] a;
    delete[] t;
    delete[] ip;
    delete[] w;

    MRPT_END
}

/*---------------------------------------------------------------
                        myGeneralDFT

    sign ->  -1: DFT, 1:IDFT
 ---------------------------------------------------------------*/
static void myGeneralDFT(
    int sign, const CMatrixFloat& in_real, const CMatrixFloat& in_imag,
    CMatrixFloat& out_real, CMatrixFloat& out_imag)
{
    ASSERT_(in_real.rows() == in_imag.rows());
    ASSERT_(in_real.cols() == in_imag.cols());

    // The dimensions:
    size_t dim1 = in_real.rows();
    size_t dim2 = in_real.cols();

    size_t k1, k2, n1, n2;
    float w_r, w_i;
    auto ang1 = (float)(sign * M_2PI / dim1);
    auto ang2 = (float)(sign * M_2PI / dim2);
    float phase;
    float R, I;
    float scale = sign == 1 ? (1.0f / (dim1 * dim2)) : 1;

    out_real.setSize(dim1, dim2);
    out_imag.setSize(dim1, dim2);

    for (k1 = 0; k1 < dim1; k1++)
    {
        for (k2 = 0; k2 < dim2; k2++)
        {
            R = I = 0;  // Accum:

            for (n1 = 0; n1 < dim1; n1++)
            {
                phase = ang1 * n1 * k1;
                for (n2 = 0; n2 < dim2; n2++)
                {
                    w_r = cos(phase);
                    w_i = sin(phase);

                    R += w_r * in_real(n1, n2) - w_i * in_imag(n1, n2);
                    I += w_i * in_real(n1, n2) + w_r * in_imag(n1, n2);

                    phase += ang2 * k2;
                }  // end for k2
            }  // end for k1

            // Save result:
            out_real(k1, k2) = R * scale;
            out_imag(k1, k2) = I * scale;

        }  // end for k2
    }  // end for k1
}

/*---------------------------------------------------------------
                        dft2_complex
 ---------------------------------------------------------------*/
void math::dft2_complex(
    const CMatrixFloat& in_real, const CMatrixFloat& in_imag,
    CMatrixFloat& out_real, CMatrixFloat& out_imag)
{
    MRPT_START

    ASSERT_(in_real.rows() == in_imag.rows());
    ASSERT_(in_real.cols() == in_imag.cols());

    // The dimensions:
    size_t dim1 = in_real.rows();
    size_t dim2 = in_real.cols();
    size_t i, j;

    bool dim1IsPower2 = (mrpt::round2up(dim1) == dim1);
    bool dim2IsPower2 = (mrpt::round2up(dim2) == dim2);

    // FFT or DFT??
    if (dim1IsPower2 && dim2IsPower2)
    {
        // ----------------------------------------
        //          Optimized FFT:
        // ----------------------------------------
        using float_ptr = FFT_TYPE*;

        // Transform to format compatible with C routines:
        // ------------------------------------------------------------
        static FFT_TYPE** a = nullptr;
        static FFT_TYPE* t = nullptr;
        static int* ip = nullptr;
        static FFT_TYPE* w = nullptr;

        // Reserve memory
        // --------------------------------------
        static int alreadyInitSize1 = -1, alreadyInitSize2 = -1;

        if (alreadyInitSize1 != (int)dim1 || alreadyInitSize2 != (int)dim2)
        {
            // Create/realloc buffers:
            if (a)
            {
                for (i = 0; i < dim1; i++) delete[] a[i];
                delete[] a;
            }
            if (ip) delete[] ip;
            if (t) delete[] t;
            if (w) delete[] w;

            alreadyInitSize1 = (int)dim1;
            alreadyInitSize2 = (int)dim2;

            a = new float_ptr[dim1];
            for (i = 0; i < dim1; i++) a[i] = new FFT_TYPE[2 * dim2];

            t = new FFT_TYPE[2 * dim1 + 20];
            ip = new int[(int)ceil(
                20 + 2 + sqrt((FFT_TYPE)max(dim1, dim2 / 2)))];
            ip[0] = 0;
            w = new FFT_TYPE[max(dim1 / 2, dim2 / 4) + dim2 / 4 + 20];
        }

        // and copy data:
        // --------------------------------------
        for (i = 0; i < dim1; i++)
            for (j = 0; j < dim2; j++)
            {
                a[i][2 * j + 0] = in_real(i, j);
                a[i][2 * j + 1] = in_imag(i, j);
            }

        // Do the job!
        // --------------------------------------
        cdft2d((int)dim1, (int)(2 * dim2), 1, a, t, ip, w);

        // Transform back to MRPT matrix format:
        // --------------------------------------
        out_real.setSize(dim1, dim2);
        out_imag.setSize(dim1, dim2);

        // a[k1][2*k2] = Re(X[k1][k2]),
        // a[k1][2*k2+1] = Im(X[k1][k2]),
        // 0<=k1<n1, 0<=k2<n2
        for (i = 0; i < dim1; i++)
            for (j = 0; j < dim2; j++)
            {
                out_real(i, j) = (float)a[i][j * 2 + 0];
                out_imag(i, j) = (float)a[i][j * 2 + 1];
            }

    }  // end FFT
    else
    {
        // ----------------------------------------
        //          General DFT:
        // ----------------------------------------
        printf("Using general DFT...\n");
        myGeneralDFT(-1, in_real, in_imag, out_real, out_imag);
    }

    MRPT_END
}

/*---------------------------------------------------------------
                        idft2_complex
 ---------------------------------------------------------------*/
void math::idft2_complex(
    const CMatrixFloat& in_real, const CMatrixFloat& in_imag,
    CMatrixFloat& out_real, CMatrixFloat& out_imag)
{
    MRPT_START

    ASSERT_(in_real.rows() == in_imag.rows());
    ASSERT_(in_real.cols() == in_imag.cols());

    // The dimensions:
    size_t dim1 = in_real.rows();
    size_t dim2 = in_real.cols();
    size_t i, j;

    bool dim1IsPower2 = (mrpt::round2up(dim1) == dim1);
    bool dim2IsPower2 = (mrpt::round2up(dim2) == dim2);

    // FFT or DFT??
    if (dim1IsPower2 && dim2IsPower2)
    {
        using float_ptr = FFT_TYPE*;
        // ----------------------------------------
        //          Optimized FFT:
        // ----------------------------------------

        // Transform to format compatible with C routines:
        // ------------------------------------------------------------
        static FFT_TYPE** a = nullptr;
        static FFT_TYPE* t = nullptr;
        static int* ip = nullptr;
        static FFT_TYPE* w = nullptr;

        // Reserve memory
        // --------------------------------------

        // and copy data:
        // --------------------------------------
        static int alreadyInitSize1 = -1, alreadyInitSize2 = -1;

        if (alreadyInitSize1 != (int)dim1 || alreadyInitSize2 != (int)dim2)
        {
            // Create/realloc buffers:
            if (a)
            {
                for (i = 0; i < dim1; i++) delete[] a[i];
                delete[] a;
            }
            if (ip) delete[] ip;
            if (t) delete[] t;
            if (w) delete[] w;

            alreadyInitSize1 = (int)dim1;
            alreadyInitSize2 = (int)dim2;

            a = new float_ptr[dim1];
            for (i = 0; i < dim1; i++) a[i] = new FFT_TYPE[2 * dim2];
            t = new FFT_TYPE[2 * dim1 + 20];
            ip = new int[(int)ceil(
                20 + 2 + sqrt((FFT_TYPE)max(dim1, dim2 / 2)))];
            ip[0] = 0;
            w = new FFT_TYPE[max(dim1 / 2, dim2 / 4) + dim2 / 4 + 20];
        }

        // and copy data:
        // --------------------------------------
        for (i = 0; i < dim1; i++)
            for (j = 0; j < dim2; j++)
            {
                a[i][2 * j + 0] = in_real(i, j);
                a[i][2 * j + 1] = in_imag(i, j);
            }

        // Do the job!
        // --------------------------------------
        cdft2d((int)dim1, (int)(2 * dim2), -1, a, t, ip, w);

        // Transform back to MRPT matrix format:
        // --------------------------------------
        out_real.setSize(dim1, dim2);
        out_imag.setSize(dim1, dim2);

        FFT_TYPE scale = 1.0f / (dim1 * dim2);

        // a[k1][2*k2] = Re(X[k1][k2]),
        // a[k1][2*k2+1] = Im(X[k1][k2]),
        // 0<=k1<n1, 0<=k2<n2
        for (i = 0; i < dim1; i++)
            for (j = 0; j < dim2; j++)
            {
                //              out_real(i,j)=(float)(a[i][j*2+0]*scale);
                //              out_imag(i,j)=(float)(a[i][j*2+1]*scale);
                out_real(i, j) = (float)(a[i][j * 2 + 0]);
                out_imag(i, j) = (float)(a[i][j * 2 + 1]);
            }

        out_real.asEigen() *= scale;
        out_imag.asEigen() *= scale;

        // The element (0,0) is purely real!
        out_imag(0, 0) = 0;

    }  // end FFT
    else
    {
        // ----------------------------------------
        //          General DFT:
        // ----------------------------------------
        printf("Using general DFT...\n");
        myGeneralDFT(1, in_real, in_imag, out_real, out_imag);
    }

    MRPT_END
}

void mrpt::math::cross_correlation_FFT(
    const CMatrixFloat& A, const CMatrixFloat& B, CMatrixFloat& out_corr)
{
    MRPT_START

    ASSERT_(A.cols() == B.cols() && A.rows() == B.rows());
    if (mrpt::round2up(A.rows()) != A.rows() ||
        mrpt::round2up(A.cols()) != A.cols())
        THROW_EXCEPTION("Size of input matrices must be powers of two.");

    // Find smallest valid size:
    size_t x, y;
    const size_t lx = A.cols();
    const size_t ly = A.rows();

    const CMatrixFloat& i1 = A;
    const CMatrixFloat& i2 = B;

    // FFT:
    CMatrixFloat I1_R, I1_I, I2_R, I2_I, ZEROS(ly, lx);
    math::dft2_complex(i1, ZEROS, I1_R, I1_I);
    math::dft2_complex(i2, ZEROS, I2_R, I2_I);

    // Compute the COMPLEX division of I2 by I1:
    for (y = 0; y < ly; y++)
        for (x = 0; x < lx; x++)
        {
            float r1 = I1_R(y, x);
            float r2 = I2_R(y, x);

            float ii1 = I1_I(y, x);
            float ii2 = I2_I(y, x);

            float den = square(r1) + square(ii1);
            I2_R(y, x) = (r1 * r2 + ii1 * ii2) / den;
            I2_I(y, x) = (ii2 * r1 - r2 * ii1) / den;
        }

    // IFFT:
    CMatrixFloat res_R, res_I;
    math::idft2_complex(I2_R, I2_I, res_R, res_I);

    out_corr.setSize(ly, lx);
    for (y = 0; y < ly; y++)
        for (x = 0; x < lx; x++)
            out_corr(y, x) = sqrt(square(res_R(y, x)) + square(res_I(y, x)));

    MRPT_END
}