无法使用 FFT 和 FFTW 检索原始图像

Question

我在我的 C 代码中使用 FFTW，但遇到了一些问题。首先，我可以将原始图像转换为两个图像（mag+phase）并通过逆变换取回原始图像。但是，如果我想得到一个以频率为中心的 mag 文件，它就不再起作用了：最终图像有一些问题。

这是我的一些代码。有人可以帮我找出代码中的错误吗？

编辑：我已修复代码以遵循@francis 的建议，但我的问题始终存在。


enum {
    TYPE_CENTERED,
    TYPE_REGULAR
};

static void fft_to_spectra(fits* fit, fftw_complex *frequency_repr, double *as,
        double *ps, int nbdata) {
    unsigned int i;

    for (i = 0; i < nbdata; i++) {
        double r = creal(frequency_repr[i]);
        double im = cimag(frequency_repr[i]);
        as[i] = hypot(r, im);
        ps[i] = atan2(im, r);
    }
}

static void fft_to_freq(fits* fit, fftw_complex *frequency_repr, double *as, double *ps, int nbdata) {
    unsigned int i;

    for (i = 0; i < nbdata; i++) {
        frequency_repr[i] = as[i] * (cos(ps[i]) + I * sin(ps[i]));
    }
}

void change_symmetry(unsigned int width, unsigned int height, unsigned int i, unsigned int j, unsigned int *x,
        unsigned int *y) {

    if (i < width / 2 && j < height / 2) {
        *x = i + width / 2;
        *y = j + height / 2;
    }
    if (i >= width / 2 && j < height / 2) {
        *x = i - width / 2;
        *y = j + height / 2;
    }
    if (i < width / 2 && j >= height / 2) {
        *x = i + width / 2;
        *y = j - height / 2;
    }
    if (i >= width / 2 && j >= height / 2) {
        *x = i - width / 2;
        *y = j - height / 2;
    }
}

static void centered(WORD *buf, unsigned int width,
        unsigned int height) {
    unsigned int i, j;

    WORD *temp = malloc(width * height * sizeof(WORD));
    for (j = 0; j < height; j++) {
        for (i = 0; i < width; i++) {
            unsigned int x = i;
            unsigned int y = j;
            change_symmetry(width, height, i, j, &x, &y);

            temp[j * width + i] = buf[y * width + x];
        }
    }

    memcpy(buf, temp, sizeof(WORD) * width * height);
    free(temp);
}

static void normalisation_spectra(unsigned int w, unsigned int h, double *modulus, double *phase,
        WORD *abuf, WORD *pbuf) {
    unsigned int i;

    for (i = 0; i < h * w; i++) {
        pbuf[i] = round_to_WORD(((phase[i] + M_PI) * USHRT_MAX_DOUBLE / (2 * M_PI)));
        abuf[i] = round_to_WORD((modulus[i] / w / h));
    }
}

static void save_dft_information_in_gfit(fits *fit) {
    strcpy(gfit.dft.ord, fit->dft.type);
    strcpy(gfit.dft.ord, fit->dft.ord);
}

static void FFTD(fits *fit, fits *x, fits *y, int type_order, int layer) {
    WORD *xbuf = x->pdata[layer];
    WORD *ybuf = y->pdata[layer];
    WORD *gbuf = fit->pdata[layer];
    unsigned int i;
    unsigned int width = fit->rx, height = fit->ry;
    int nbdata = width * height;

    fftw_complex *spatial_repr = fftw_malloc(sizeof(fftw_complex) * nbdata);
    if (!spatial_repr) {
        return;
    }
    fftw_complex *frequency_repr = fftw_malloc(sizeof(fftw_complex) * nbdata);
    if (!frequency_repr) {
        fftw_free(spatial_repr);
        return;
    }

    /* copying image selection into the fftw data */
#ifdef _OPENMP
#pragma omp parallel for num_threads(com.max_thread) private(i) schedule(static) if(nbdata > 15000)
#endif
    for (i = 0; i < nbdata; i++) {
        spatial_repr[i] = (double) gbuf[i];
    }

    /* we run the Fourier Transform */
    fftw_plan p = fftw_plan_dft_2d(height, width, spatial_repr, frequency_repr,
            FFTW_FORWARD, FFTW_ESTIMATE);
    fftw_execute(p);

    /* we compute modulus and phase */
    double *modulus = malloc(nbdata * sizeof(double));
    double *phase = malloc(nbdata * sizeof(double));

    fft_to_spectra(fit, frequency_repr, modulus, phase, nbdata);

    //We normalize the modulus and the phase
    normalisation_spectra(width, height, modulus, phase, xbuf, ybuf);
    if (type_order == TYPE_CENTERED) {
        strcpy(x->dft.ord, "CENTERED");
        centered(xbuf, width, height);
        centered(ybuf, width, height);
    }

    free(modulus);
    free(phase);
    fftw_destroy_plan(p);
    fftw_free(spatial_repr);
    fftw_free(frequency_repr);
}

static void FFTI(fits *fit, fits *xfit, fits *yfit, int type_order, int layer) {
    WORD *xbuf = xfit->pdata[layer];
    WORD *ybuf = yfit->pdata[layer];
    WORD *gbuf = fit->pdata[layer];
    unsigned int i;
    unsigned int width = xfit->rx;
    unsigned int height = xfit->ry;
    int nbdata = width * height;

    double *modulus = calloc(1, nbdata * sizeof(double));
    double *phase = calloc(1, nbdata * sizeof(double));

    if (type_order == TYPE_CENTERED) {
        centered(xbuf, width, height);
        centered(ybuf, width, height);
    }

    for (i = 0; i < height * width; i++) {
        modulus[i] = (double) xbuf[i] * (width * height);
        phase[i] = (double) ybuf[i] * (2 * M_PI / USHRT_MAX_DOUBLE);
        phase[i] -= M_PI;
    }

    fftw_complex* spatial_repr = fftw_malloc(sizeof(fftw_complex) * nbdata);
    if (!spatial_repr) {
        return;
    }

    fftw_complex* frequency_repr = fftw_malloc(sizeof(fftw_complex) * nbdata);
    if (!frequency_repr) {
        fftw_free(spatial_repr);
        return;
    }

    fft_to_freq(fit, frequency_repr, modulus, phase, nbdata);

    fftw_plan p = fftw_plan_dft_2d(height, width, frequency_repr, spatial_repr,
            FFTW_BACKWARD, FFTW_ESTIMATE);
    fftw_execute(p);

    for (i = 0; i < nbdata; i++) {
        double pxl = creal(spatial_repr[i]) / nbdata;
        gbuf[i] = round_to_WORD(pxl);
    }

    free(modulus);
    free(phase);
    fftw_destroy_plan(p);
    fftw_free(spatial_repr);
    fftw_free(frequency_repr);
}

这是我的图像，原始图像和 FFTD（居中）->FFTI 结果

Answer 1

计划是使用标志 FFTW_MEASURE 创建的。因此，计算了多个 DFT，并且输入数组可能被覆盖。这是documentation of FFTW:

中规划器标志描述的开始

FFTW_ESTIMATE specifies that, instead of actual measurements of different algorithms, a simple heuristic is used to pick a (probably sub-optimal) plan quickly. With this flag, the input/output arrays are not overwritten during planning.

FFTW_MEASURE tells FFTW to find an optimized plan by actually computing several FFTs and measuring their execution time. Depending on your machine, this can take some time (often a few seconds). FFTW_MEASURE is the default planning option.

切换到 FFTW_ESTIMATE 或在填充输入数组之前创建计划：

/* we run the Fourier Transform */
fftw_plan p = fftw_plan_dft_2d(width, height, spatial_repr, frequency_repr,
        FFTW_FORWARD, FFTW_MEASURE);

/* copying image selection into the fftw data */
#ifdef _OPENMP
#pragma omp parallel for num_threads(com.max_thread) private(i) schedule(static) if(nbdata > 15000)
#endif
for (i = 0; i < nbdata; i++) {
    spatial_repr[i] = (double) gbuf[i];
}

如果您打算使用单个图像，使用 FFTW_ESTIMATE 是可行的方法。相反，如果您考虑处理多个图像，使用 FFTW_MEASURE 创建一次计划并存储它是一个不错的选择。那么你可以在每次执行 FFT 时使用 New-array Execute Functions：

fftw_execute_dft(p, spatial_repr, frequency_repr);

您可以测试 malloc() 或 fftw_malloc() 的 return 值来检查分配是否正确。如果不是，它 returns NULL。 fftw_malloc() 在 fftw-3.3.6-pl2/kernel/kalloc.c 中作为函数 *X(kernel_malloc)(size_t n) 实现。它会调用 memalign() or _aligned_malloc() 等函数。这两个 return NULL 就像 malloc() 以防失败。最后，我没有在您提供的代码段中发现有关释放内存分配的关键问题。

fft_to_spectra() 中的参数 double nbdata 应该是一个整数。 Valgrind 可能认为它很奇怪...

编辑：change_symmetry() 将被修改为奇数尺寸。类似于：

void change_symmetry_forward(unsigned int width, unsigned int height, unsigned int i, unsigned int j, unsigned int *x,
    unsigned int *y) {
     *x = i + width / 2;
     if (*x>=width){
           *x=*x-width;
     }
     *y = j + height / 2;
     if(*y>=height){
          *y =*y-height;
     }
}

和

void change_symmetry_backward(unsigned int width, unsigned int height, unsigned int i, unsigned int j, unsigned int *x,
    unsigned int *y) {
     *x = i +width- width / 2;
     if (*x>=width){
           *x=*x-width;
     }
     *y = j +height- height / 2;
     if(*y>=height){
          *y =*y-height;
     }
}

无法使用 FFT 和 FFTW 检索原始图像

Cannot retrieve original image using FFT with FFTW

c

valgrind

fftw