Introduction to the Fourier Transform

Convolution

The convolution of two functions f and g are given by:

The mathematical process of convolution in normal space is simply multiplication in the Fourier domain, making it easier to implement in the latter space.

From the previous section, it has been observed  that the FFT process if a simulation of passing an image through an infinitely large converging lens. However, lens have finite dimensions in reality. To simulate the imaging system with finite lenses, convolution is performed.

In this part, circles with different radii were generated. These were the convoluted one-by-one with the following image.

The result is as follows.

Convolution of ‘VIP’ and different sized circles. (Thank you gifcreator.me)

Note that for the smallest radius, the image is not discernible. However, as the radius of the circle increases, the convolution more closely follows the original object. Note also that there is a minimum radii required to capture the entire image. Using a larger radii after this minimum does not greatly enhance visual quality.

The importance of this exercise is that the circle simulates the lens used in the imaging system. A larger lens has more focusing power, thus leading to better image quality. Also, there is an optimum radius that is large enough to capture the image, but small enough so as to save money (larger lens = more space = not efficient).