Computer Vision MT25, Convolutions
Flashcards
What notation convention is used for discrete functions $f : \mathbb Z \to \mathbb R$?
Function application is denoted $f[x]$, like you’re indexing an array.
Suppose:
- $f : \mathbb Z \to \mathbb R$
- $g : \mathbb Z \to \mathbb R$
@Define the convolution $f \ast g$, and state which of these is the kernel or filter, and which is the input function.
\[(f \ast g)[x] = \sum^\infty _ {u = -\infty} f[u] g[x - u]\]
- $f$ is the kernel or filter
- $g$ is the input function
Suppose:
- $f : \mathbb Z \to \mathbb R$
- $g : \mathbb Z \to \mathbb R$
- $f$ has finite support in the set $\mathcal M = {0, \ldots, M - 1}$, i.e. for all $m \in \mathbb Z \setminus \mathcal M$, $f[m] = 0$, or equivalently $f : { 0, \ldots, M-1 } \to \mathbb R$
@Define the convolution $f \ast g$.
\[(f \ast g)[x] = \sum^{M-1} _ {u = 0} f[u] g[x - u + \frac{M-1}{2}]\]
(note that in particular, the order in which we look at the filter is right-to-left, this is to make it commutative).
Compute this @example convolution.
Suppose:
- $g : \mathbb Z \times Z \to \mathbb R$
- $f : {0, \ldots, M-1} \times {0, \ldots, N-1} \to \mathbb R$
@Define the convolution $f \ast g$.
\[(f \ast g)[x, y] = \sum^{M-1} _ {u = 0} \sum^{N-1} _ {v = 0} f[u, v] g[x - u + \frac{M-1}{2}, y - v + \frac{M-1}{2}]\]