最近我面对有关使用FFTW一些问题和它的C2C转换（参见： 3D C2C与FFT FFTW库）。由于我所在的使用FFTW LIB我的问题，我为了铁饼这种情况在更具体的方式创造了一个新的课题。 由于我做一个复杂的复杂的变换和真实数据的傅立叶空间我变换的数据应该是对称的：F [N] = CON（F [NN]）

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])

现在我做了测试数据的小块一些转换检查转换后的数据为这种对称。对于一维变换的每一个事物都按预期运行，但对于高维我得到了真正的意外的结果。

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.

我使用 fftwf_plan_dft_2d 来改造一个8x8的灰度图像转换傅立叶空间和复杂的结果为：

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

抱歉数据这个长长的名单，但它显示我的问题。

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

例如对于 F [3] = - 182.721 + 15.8486i 我预计 F [64-3] = F [61] = - 182.721 - 15.8486i ，但你可以看到它是 -84.7868 + 9.10052i 。取而代之的是共轭 F [3] 位于指数5同样的事情，其他对。

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.

如果有一个系统，我无法找到它。

If there is a system I cannot find it.

下面是完整的code：

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];
    }
}

我会AP preciate一些帮助。 - D

I would appreciate some Help. :-D

问候

狼

推荐答案

对于2D傅里叶变换，它仍然是真实的，当x是实数FFT（x）是共轭对称。但是，这在两个方面。因此，在索引（X，Y）元件16 * X + Y应为（16-X，16-y）的元件的缀合物在指数16 *（16-X的mod 16）+（16-γ模16） ，其中当y不为0是272-16 * XY模256。

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.

但我觉得，虽然你说的16×16，你到底指的8x8。所以（X，Y）在8 * x + y为共轭至（8-X，8- y）处8 *（8-X的mod 8）+（8-γ的mod 8）

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).

在具体地，例如，当x = 0的共轭元素是y和8-γ - 包括，例如，3和5，当你发现

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

（当x = 0或y = 0，这样的事情8-Y MOD 8以上的均值为0。）

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