Question

Why does the convolution of a clear phantom image with a Gaussian function result in a smooth (blurry) image (*use Fourier transform and frequency domain perspective to explain).

Answer 1

First, let us define what is meant by convolving,It is defined as the integral of the product of the two functions f and g after one is reversed and shifted(lets say by an amount ), mathematically

Convolving a image with gaussian destribution is like a two dimensional transformation of the type

which is the convolution of f with the gaussian function of the type ​ . The above equation is sort of a fourier transformation of a function f, called Weierstrass transform. In precise language, the Weierstrass transform is convolution with a Gaussian and hence it is a multiplication of the Fourier transformed signal with a Gaussian, followed by application of the inverse Fourier transform.

We know, the phantom image, will consist of a wide range of frequencies, depending on the color of the image pixel e.g. blue will correspond to a higher frequency and red with a lower frequency.So basically, our image is a overlap of different frequencies.

From the above expression for Weierstrass transform, we can see for ourselves that Weierstrass transform of cos(bx) is e⁻² cos(t) i.e upon convolving the function with a gaussian function, the signal still contains the frequency , but the amplitude has decreased by a factor . This also means that higher is the frequency , higher is the amount of attenuation in its amplitude. In other words, the higher frequency waves will be filtered out such that, the convolution of image with a gaussian function results in a low pass filter (only lower frequncies are passed), which results in smoothening(blurring) the image. Hope you got your answer!