2D Convolution Theorem

Verifying Convolution Theorem in 2D using Images

( All the details are in my blog post here: http://heraqi.blogspot.com.eg/2016/12/verifying-convolution-theorem-on-2d.html )
In our verification experiment, we will apply it to three different images with different level of details: high, medium, and low details. The reason will be clear at the end of this post.
Firstly, let's create a black image W of the same size of I with a small white rectangle in the middle.
For the image I, firstly, we will get the result of multiplying the frequency domain of I by W:
1- Apply Fourier transform on I, let the result be F
2- Calculate the point-wise multiplication of F and W, let the result be F×W
3- Get the inverse Fourier transform of F×W, let this result be R1

Now, let's do the inverse process, for the same image I, we will get the result of the convolution of the spacial domain of I and the inverse Fourier transform of W.
1- Apply inverse Fourier transform on W, let the result be M
2- Calculate the convolution of I and M, let the result be R2

As described in the blog post, the convolution theorem establish that the two processes described above to get R1 and R2 are equivalent; so R1 and R2 should be the same images at the end. If we see that, we verify the convolution theorem on 2D images.