From previous activities we have familiarized ourselves with Fourier Transform, giving us idea on what result would we get if we apply FT on certain patterns. In this activity we would use our knowledge of Fourier Transform to enhance certain images by removing visible repetitive patterns on them using filter masks.

A. Convolution theorem

Before we do the actual filtering, we familiarize ourselves first with generating shapes and getting their FT’s using scilab, the convolution theorem, and working with dirac deltas.


Figure 1. Dots and their Fourier Transform

Figure 2. Circles of incresing radius (left) and their Fourier Transforms (right)

Figure 3. Squares of increasing widths (left) and their Fourier Transforms (right)

Figure 4. Gaussians of increasing variance (left) and their Fourier Transforms (right)

Dirac deltas in random locations

The above dirac deltas were convolved to two patterns:

a= [1 1 1 : -2 -2 -2 : 1 1 1 ];

b= [1 -2 1: 1 -2 1: 1 -2 1];

Convolution. Upper: dirac deltas convoloved with a. Lower: dirac deltas convolved with b

From the result of the convolution it is apparent that the patterns convolved with the dirac deltas were just repeated at the positions of the dirac deltas. This confirms the earlier statement that “The convolution of a dirac delta and a function f(t) results in a replicationof f(t) in the location of the dirac delta” from the manual.

Arranged dirac deltas of increasing frequencies (top to bottom, left images) and their respective FT (right)

— to be continued…