fftw c2c: missing symmetry in transformed real dat

Question

recently I faced some problems concerning the use of fftw and it's c2c transformation (see: 3d c2c fft with fftw library). As I located my problems in the use of the fftw lib I created a new Question in order to discus this situation in a more concrete way. Since I am doing a complex to complex transform with real data my transformed data in fourier space is supposed to be symmetric: F[n] = con(F[N-n])

Now I did some transformations with small blocks of test-data to check the transformed data for this symmetry. For 1D transform at every things worked as expected, but for higher dimensions I got real unexpected results.

I am using fftwf_plan_dft_2d to transform a 8x8 grayscale image into fourier space and the complex result is given by:

n 
0 real 7971 imag 0 
1 real -437.279 imag -802.151 
2 real -289 imag -566 
3 real -182.721 imag 15.8486 
4 real 31 imag 0 
5 real -182.721 imag -15.8486 
6 real -289 imag 566 
7 real -437.279 imag 802.151 
8 real -1499.79 imag -315.233 
9 real 182.693 imag -74.5563 
10 real 55.9239 imag -12.8234 
11 real -84.7868 imag -9.10052 
12 real -14.4264 imag 211.208 
13 real 289.698 imag 214.723 
14 real 452.659 imag -246.279 
15 real 1136.35 imag -763.85 
16 real 409 imag -134 
17 real -141.865 imag 42.6396 
18 real -33 imag 122 
19 real 129.075 imag -49.7868 
20 real 1 imag -150 
21 real 109.865 imag -84.6396 
22 real 95 imag -142 
23 real -841.075 imag -92.2132 
24 real -108.207 imag -89.2325 
25 real -127.213 imag 28.8995 
26 real -36.6589 imag -8.27922 
27 real -74.6934 imag 43.4437 
28 real 70.4264 imag 29.2082 
29 real -88.3545 imag -81.8499 
30 real -127.924 imag -190.823 
31 real 230.302 imag 8.7229 
32 real -53 imag 0 
33 real -73.1127 imag -22.8578 
34 real -85 imag -82 
35 real -10.8873 imag 51.1421 
36 real -65 imag 0 
37 real -10.8873 imag -51.1421 
38 real -85 imag 82 
39 real -73.1127 imag 22.8578 
40 real -108.207 imag 89.2325 
41 real 230.302 imag -8.7229 
42 real -127.924 imag 190.823 
43 real -88.3545 imag 81.8499 
44 real 70.4264 imag -29.2082 
45 real -74.6934 imag -43.4437 
46 real -36.6589 imag 8.27922 
47 real -127.213 imag -28.8995 
48 real 409 imag 134 
49 real -841.075 imag 92.2132 
50 real 95 imag 142 
51 real 109.865 imag 84.6396 
52 real 1 imag 150 
53 real 129.075 imag 49.7868 
54 real -33 imag -122 
55 real -141.865 imag -42.6396 
56 real -1499.79 imag 315.233 
57 real 1136.35 imag 763.85 
58 real 452.659 imag 246.279 
59 real 289.698 imag -214.723 
60 real -14.4264 imag -211.208 
61 real -84.7868 imag 9.10052 
62 real 55.9239 imag 12.8234 
63 real 182.693 imag 74.5563

Sorry for this long list of data, but it shows my problem.

For example for F[3]=-182.721 + 15.8486i I expected F[64-3] = F[61] = -182.721 - 15.8486i, but as you can see it is -84.7868 + 9.10052i. Instead the conjugate of F[3] is located at index 5. Same thing for other pairs.

If there is a system I cannot find it.

Here is the complete code:

QImage image("/Users/wolle/Desktop/wolf.png");
int w = image.width();
int h = image.height();
int size  = w * h;

cl_float *rawImage = imageToRaw(image); // converts a QImage into an rgb array [0..255]

fftwf_complex *complexImage = (fftwf_complex*) fftwf_malloc(sizeof(fftwf_complex) * size);
fftwf_complex *freqBuffer = (fftwf_complex*) fftwf_malloc(sizeof(fftwf_complex) * size);

// real data to complex data
for (int i = 0; i < size; i++)
{
    complexImage[i][0] = (float)rawImage[i];
    complexImage[i][1] = 0.0f;
}

fftwf_plan forward = fftwf_plan_dft_2d(w, h, complexImage, freqBuffer, FFTW_FORWARD, FFTW_ESTIMATE);

fftwf_execute(forward);

for (int y = 0; y < h; y++)
{
    for (int x = 0; x < w; x++)
    {
        int gid = y * w + x;
        qDebug() << gid  << "real" << freqBuffer[gid][0] << "imag" << freqBuffer[gid][1];
    }
}

I would appreciate some Help. :-D

Greetings

Wolf

Answer 1

For a 2D Fourier transform, it is still true that when x is real FFT(x) is conjugate-symmetric. But that's in two dimensions. So the (x,y) element at index 16*x+y should be the conjugate of the (16-x,16-y) element at index 16*(16-x mod 16)+(16-y mod 16), which when y isn't 0 is 272-16*x-y mod 256.

BUT I think that although you said 16x16 you actually meant 8x8. So (x,y) at 8*x+y is conjugate to (8-x,8-y) at 8*(8-x mod 8) + (8-y mod 8).

In particular, e.g., when x=0 the conjugate elements are y and 8-y -- including, for instance, 3 and 5, as you found.

(When x=0 or y=0, things like "8-y mod 8" above mean 0.)