Convolution basic example

Definition

y(t) = h(t)*x(t) = \int_{-\inf}^{\inf} x(\tau)h(t-\tau)d\tau = \int_{-\inf}^{\inf} h(\tau)x(t-\tau)d\tau = x(t)*h(t)

Discreate form

y(n) = \sum_{m=-\inf}^{\inf}x(n-m)h(m) = \sum_{m=-\inf}^{\inf}h(n-m)x(m)

Properties
1. h(t)*x(t) = x(t)*h(t)
2. h*(g*x) = (h*g)*x

In image processing : two dimensions convolution
h is called convolution kernel or mask

y(m, n) = \sum_{i=-k}^{k}\sum_{j=-k}^{k}x(m+i, n+j)h(i,j)

Sliding kernel throughout the image. If image is like in kernel, we will have peak value of y(m, n). –> roughly use to detect same image ??
The pictures below is a good visualization from Wiki.

ref : hmc

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