空间域图像卷积

我正在尝试复制结果这个链接 https://stackoverflow.com/q/38709810/159072 using linear卷积于空间域.

图像首先转换为二维double数组，然后进行卷积。图像和内核大小相同。图像在卷积之前进行填充，并在卷积之后进行相应的裁剪。

与基于 FFT 的卷积相比，输出很奇怪并且不正确.

我该如何解决这个问题？

请注意，我从 Matlab 获得了以下图像输出，它与我的 C# FFT 输出相匹配：

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^{Update-1: Following @Ben Voigt's comment, I changed the Rescale() function to replace 255.0 with 1 and thus the output is improved substantially. But, still, the output doesn't match the FFT output (which is the correct one).}

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^{Update-2: Following @Cris Luengo's comment, I have padded the image by stitching and then performed spatial convolution. The outcome has been as follows:}

^{So, the output is worse than the previous one. But, this has a similarity with the 2nd output of the linked answer https://stackoverflow.com/a/38724731/159072 which means a circular convolution is not the solution.}

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^{Update-3: I have used the Sum() function proposed by @Cris Luengo's answer. The result is a more improved version of **Update-1**:}

^{But, it is still not 100% similar to the FFT version.}

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^{Update-4: Following @Cris Luengo's comment, I have subtracted the two outcomes to see the difference:}
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<sup>1. spatial minus frequency domain
2. frequency minus spatial domain</sup>

_{Looks like, the difference is substantial which means, spatial convolution is not being done correctly.}

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源代码：

^{(Notify me if you need more source code to see.)}

    public static double[,] LinearConvolutionSpatial(double[,] image, double[,] mask)
    {
        int maskWidth = mask.GetLength(0);
        int maskHeight = mask.GetLength(1);

        double[,] paddedImage = ImagePadder.Pad(image, maskWidth);

        double[,] conv = Convolution.ConvolutionSpatial(paddedImage, mask);

        int cropSize = (maskWidth/2);

        double[,] cropped = ImageCropper.Crop(conv, cropSize);

        return conv;
    } 
    static double[,] ConvolutionSpatial(double[,] paddedImage1, double[,] mask1)
    {
        int imageWidth = paddedImage1.GetLength(0);
        int imageHeight = paddedImage1.GetLength(1);

        int maskWidth = mask1.GetLength(0);
        int maskHeight = mask1.GetLength(1);

        int convWidth = imageWidth - ((maskWidth / 2) * 2);
        int convHeight = imageHeight - ((maskHeight / 2) * 2);

        double[,] convolve = new double[convWidth, convHeight];

        for (int y = 0; y < convHeight; y++)
        {
            for (int x = 0; x < convWidth; x++)
            {
                int startX = x;
                int startY = y;

                convolve[x, y] = Sum(paddedImage1, mask1, startX, startY);
            }
        }

        Rescale(convolve);

        return convolve;
    } 

    static double Sum(double[,] paddedImage1, double[,] mask1, int startX, int startY)
    {
        double sum = 0;

        int maskWidth = mask1.GetLength(0);
        int maskHeight = mask1.GetLength(1);

        for (int y = startY; y < (startY + maskHeight); y++)
        {
            for (int x = startX; x < (startX + maskWidth); x++)
            {
                double img = paddedImage1[x, y];
                double msk = mask1[x - startX, y - startY];
                sum = sum + (img * msk);
            }
        }

        return sum;
    }

    static void Rescale(double[,] convolve)
    {
        int imageWidth = convolve.GetLength(0);
        int imageHeight = convolve.GetLength(1);

        double maxAmp = 0.0;

        for (int j = 0; j < imageHeight; j++)
        {
            for (int i = 0; i < imageWidth; i++)
            {
                maxAmp = Math.Max(maxAmp, convolve[i, j]);
            }
        }

        double scale = 1.0 / maxAmp;

        for (int j = 0; j < imageHeight; j++)
        {
            for (int i = 0; i < imageWidth; i++)
            {
                double d = convolve[i, j] * scale;
                convolve[i, j] = d;
            }
        }
    } 

    public static Bitmap ConvolveInFrequencyDomain(Bitmap image1, Bitmap kernel1)
    {
        Bitmap outcome = null;

        Bitmap image = (Bitmap)image1.Clone();
        Bitmap kernel = (Bitmap)kernel1.Clone();

        //linear convolution: sum. 
        //circular convolution: max
        uint paddedWidth = Tools.ToNextPow2((uint)(image.Width + kernel.Width));
        uint paddedHeight = Tools.ToNextPow2((uint)(image.Height + kernel.Height));

        Bitmap paddedImage = ImagePadder.Pad(image, (int)paddedWidth, (int)paddedHeight);
        Bitmap paddedKernel = ImagePadder.Pad(kernel, (int)paddedWidth, (int)paddedHeight);

        Complex[,] cpxImage = ImageDataConverter.ToComplex(paddedImage);
        Complex[,] cpxKernel = ImageDataConverter.ToComplex(paddedKernel);

        // call the complex function
        Complex[,] convolve = Convolve(cpxImage, cpxKernel);

        outcome = ImageDataConverter.ToBitmap(convolve);

        outcome = ImageCropper.Crop(outcome, (kernel.Width/2)+1);

        return outcome;
    } 

您当前的输出看起来更像是自相关函数，而不是 Lena 与她自己的卷积。我认为问题可能出在你的身上Sum功能。

如果你看一下 的定义卷积和 https://en.wikipedia.org/wiki/Convolution#Discrete_convolution，您将看到内核（或映像，无关紧要）已镜像：

sum_m( f[n-m] g[m] )

对于单一函数，m出现一个加号，而另一个出现一个减号。

您需要修改您的Sum函数来读取mask1图像按正确顺序排列：

static double Sum(double[,] paddedImage1, double[,] mask1, int startX, int startY)
{
    double sum = 0;

    int maskWidth = mask1.GetLength(0);
    int maskHeight = mask1.GetLength(1);

    for (int y = startY; y < (startY + maskHeight); y++)
    {
        for (int x = startX; x < (startX + maskWidth); x++)
        {
            double img = paddedImage1[x, y];
            double msk = mask1[maskWidth - x + startX - 1, maskHeight - y + startY - 1];
            sum = sum + (img * msk);
        }
    }

    return sum;
}

另一种选择是传递镜像版本mask1到这个函数。