Why convolution in spatial domain equal to multiplication in frequency domain?

Question

Why it is said that "convolution of an image in spatial domain is equal to multiplication in frequency domain" ? Could anyone please explain it briefly?

Answer 1

StackOverflow, unfortunately, doesn't support MathJaX hence it is hard to show the math here.

One way to explain is that Convolution is Linear Invariant Operator.
As you know, Linear Time / Spatially Invariant Systems basically do one thing - Delay and Scaling.

The Eigen Functions of Delay and Scaling are the Harmonic Functions.
Which means that give a signal described by harmonic signals (Practically its Fourier Transform) Linear Time / Spatially Invariant Operator only scales it by complex number (Scaling and shifting by phase) which is what you do in the Fourier Domain.

It is similar to Diagonalization in Linear Algebra.
For instance let's thing of the Filter we apply on the image as an operator - A.
So the output of the system is y = A x.
If A is diagonalizable as A = P^T D P where D is diagonal matrix and P P^T = I, namely Unitary Matrix.
So y = A x = P^T D P x hence by defining z = P x and t = P y we get t = D z namely we only need to multiply each element in t and not the whole matrix multiplication.

If you think about P as the Fourier Transom operator then instead of doing Matrix Multiplication you can have element wise multiplication in other domain - Fourier Domain.