13.3.2 The Convolution Operation

The convolution operation is a mathematical operation which takes two functions $f(x)$ and $g(x)$ and produces a third function $h(x)$. Mathematically, convolution is defined as:

$$h(x) = f(x) * g(x) = \int^{+\infty}_{-\infty} f(\tau)g(x - \tau)d\tau$$ (13)

$g(x)$ is referred to as the filter. The integral only needs to be evaluated over the range where $g(x - \tau)$ is nonzero (called the support of the filter).[30]

In spatial domain image processing, you discretize the operation. $f(x)$ becomes an array of pixels $F[x]$. The kernel $g(x)$ is an array of values $G[0...(width-1)]$ (assume finite support). Equation 13 becomes:

$$H[x] = \sum^{width-1}_{i = 0} F[x+i]G[i]$$ (14)