Applying a 2D Convolution Using 2D FFT

Question

So I was following the article Victor Podlozhnyuk (nVidia) - FFT Based 2D Convolution (Page 7).

I have expanded the kernel to the correct way they have done it. However when it comes to the part on clamping to the edge its very confusing. I would love for someone to explain this as i require the image convolved as the same shape before the padding, but in this article it just shows how to convolve with clamp to edge but not how to recover it.

code as of now

def circExt(k, rows, cols):
    radiusV = np.floor(k.shape[0] / 2)
    radiusH = np.floor(k.shape[1] / 2)
  
    k_c = k

    k_c = np.pad(k_c,((0, int(rows-k.shape[0])), (0, int(cols-k.shape[1]))))

    k_c = np.roll(k_c, int(-radiusV), axis = 0)
    k_c = np.roll(k_c, int(-radiusH), axis = 1)
    
    return k_c
    

def testfft2(image, kernel):
    imageC = image.copy()
    kSize = (kernel.size // 2) + 1
    kernelShape = tuple(ti//2 for ti in kernel.shape)
    
    k_width = kernel.shape[1]
    k_height = kernel.shape[0]
    
    centery ,centerx = kernelShape
    
    imageC = np.pad(imageC, ((0, int(centery)), (0, int(centerx))) , mode = "edge")

    imageC = np.pad(imageC,((0, int(k_height-centery-1)), (0, int(k_width-centerx-1))), mode = "wrap" )

    kernelShift = circExt(kernel, imageC.shape[0], imageC.shape[1])   

    imageC = np.fft.fft2(imageC)
    kernel= np.fft.fft2(kernelShift)
    output = np.real(np.fft.ifft2(np.multiply(imageC, kernel)))

    return output

Answer 1

You may follow the answers to the following questions which implements the paper you linked above:

• Kernel Convolution in Frequency Domain - Cyclic Padding (Exact same paper).
• 2D Frequency Domain Convolution Using FFT (Convolution Theorem).
• Applying 2D Image Convolution in Frequency Domain with Replicate Border Conditions in MATLAB.
• Replicate MATLAB's conv2() in Frequency Domain.
• How to Use Convolution Theorem to Apply a 2D Convolution on an Image.

All the above include code you may use to implement the paper.