Convolution explained

1D convolution

input = [$w$] , filter=[$k$] $\rightarrow$ output=[$w$]

For example:

input =[1111111] and filter=[0.25,0.5,0.25], hence output=[1111111]

2D Convolution operation as a mtarix multiplication

input = [$w, h$] , filter=[$k, k$] $\rightarrow$ output=[$w, h$]

$$\begin{align} (f * g)(t) = \int _0 ^t f(t - \tau) g(\tau) d \tau \end{align}$$

Convolution consists in blending 2 functions together.

The example below shows the convolution of a step function and a Gaussian function.

Each point $t$ in a convolve signal ($f*g$) has signal:

$$\begin{align} (f*g)(t) = \int_{-\inf} ^{+\inf} f(\tau) g(t - \tau) d \tau \end{align}$$

It is the integral of a point wise multiplication between all elements in the signal $f(\tau)$ and all elements in the signal $g(t - \tau)$ shifted by $t$.

2D Convolution: step by step

We will convolve those 2 signals: $x(\tau)$ and $h(\tau)$.

$$\begin{align} y(t) = \int_{-\inf} ^{+\inf} x(\tau) h(t - \tau) d \tau \end{align}$$

$h(\tau)$ is the function that we shift. Typically, we shift the shortest or simplest signal. We will maintain and shift $h(t - \tau)$.