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).[27]

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)