Bilateral Filter

\begin{gather}BF[I]_{\bf p} = \frac{1}{W_\bf{p}}  \sum_{ {\bf q} \in S} G_{\sigma_s} (||{\bf p} - {\bf q}||) G_{\sigma_r}(|I_{\bf p} - I_{\bf q} |) I_{\bf q} \\  W_{\bf p} = \sum_{{\bf q}\in S} G_{\sigma_s} (||{\bf p}-{\bf q}||) G_{\sigma_r}(|I_{\bf p} - I_{\bf q} |) \\ G_\sigma ({\bf r}) = e^{ - ||\bf{r}||^2/2\sigma^2 }\end{gather}

 

Bilateral Filter

Gaussian Filter

 

smoothing based on the nonlinear heat eq

// sigmar controls the intensity range that is smoothed out. 
// Higher values will lead to larger regions being smoothed out. 
// The sigmar value should be selected with the dynamic range of the image pixel values in mind.
// sigmas controls smoothing factor. Higher values will lead to more smoothing.
// convolution through using lookup tables.
int BilateralFilter(BYTE *image, int width, int height, 
    double sigmas, double sigmar, int ksize, BYTE* out) {
    //const double sigmas = 1.7;
    //const double sigmar = 50.;
    double sigmas_sq = sigmas * sigmas;
    double sigmar_sq = sigmar * sigmar;
    //const int ksize = 7;
    const int hksz = ksize / 2;
    ksize = hksz * 2 + 1;
    std::vector<double> smooth(width * height, 0);
    // LUT for spatial gaussian;
    std::vector<double> spaceKer(ksize * ksize, 0);
    for (int j = -hksz, pos = 0; j <= hksz; j++) 
        for (int i = -hksz; i <= hksz; i++) 
            spaceKer[pos++] = exp(- 0.5 * double(i * i + j * j)/ sigmas_sq); 
    // LUT for image similarity gaussian;
    double pixelKer[256];
    for (int i = 0; i < 256; i++)
        pixelKer[i] = exp(- 0.5 * double(i * i) / sigmar_sq);

    for (int y = 0, imgpos = 0; y < height; y++) {
        int top = y - hksz;
        int bot = y + hksz;
        for (int x = 0; x < width; x++) {
            int left = x - hksz;
            int right = x + hksz;
            // convolution;
            double wsum = 0;
            double fsum = 0; 	
            int refVal = image[imgpos];
            for (int yy = top, kpos = 0; yy <= bot; yy++) {
                for (int xx = left; xx <= right; xx++) {
                    // check whether the kernel rect is inside the image;
                    if ((yy >= 0) && (yy < height) && (xx >= 0) && (xx < width)) {
                        int curVal = image[yy * width + xx];
                        int idiff = curVal - refVal;
                        double weight = spaceKer[kpos] * pixelKer[abs(idiff)];
                        wsum += weight;
                        fsum += weight * curVal;
                    }
                    kpos++;
                }
            }
            smooth[imgpos++] = fsum / wsum;
        }
    }

    for (int k = smooth.size(); k-- > 0;) {
        int a = int(smooth[k]);
        out[k] = a < 0 ? 0: a > 255 ? 255: a;
    }
    return 1;
}