使用FFT进行高斯模糊

Question

I implement a method that blurs an image using a Gaussian like this:

- image I , size = WxH
- kernel K , size = MxM
- padded the kernel PD to the size of the image
  i.e for an image 5x5 and a kernel 3x3 after padding the kernel looks like:
    0 0 0 0 0
    0 x x x 0
    0 x x x 0
    0 x x x 0
    0 0 0 0 0 
where X is the value from the original kernel
- performed 2d fft on the padded kernel PD (FFT_K)
- performed 2d fft on the image I (FFT_I)
- multiplied FFT_I * FFT_K (FFT_RES)
- perfomed fft on FFT_RES
- shifted the FFT_RES (RESULT)

The result contains some aliasing on the edges.

Here is the result:

If you notice in the right image you will see that it is aliased in both dimensions.

Is the above algorithm correct?

The implementation is with C++ and fftw3.

Answer 1

Your above algorithm is correct with a slight caviate.

When you pad your image with 0's, those zeros are actually used in the convolution. In the FFT space, this will add huge high freqency components around the edge of the image. In non-FFT space, this means that, up to 1-the kernel size on the edge, the 0's are rolled in, which will give you odd looking results on the edge. People typically handle this in two ways:

1. Simply throw out 1 kernel size border around the image after convolution.
2. Mirror the edge into the pad instead of filling it with zeros.

For best results, I'll often do both 1 and 2 (to get an actual image and to eliminate the high frequency edge in the Fourier space).