There is a certain family of 'sweeping' operators / functions $S(y; f,k,g)$, where:
- $f$ is a function $f : x \mapsto \mathbb{R}^N$
- $k$ is a 'kernel' function $k : x \mapsto \mathbb{R}^N$
- $y$ represents an 'offset' (independent) variable
- $g$ is a higher-order function $g : f, k; \tilde f, \tilde k, y \mapsto \mathbb{R}^N$, where
- $\tilde f[f]$ is a higher order function modifying $f(x)$, and
- $\tilde k[k,y]$ is a higher order function modifying $k(x)$, in a manner that depends on 'offset' $y$ $-$ i.e., $\tilde k$ results in a 'shifted' (or more generally, an 'adjusted') version of $k$ by $y$ (plus any other modifications, e.g. reflection).
In other words the kernel $k$ 'sweeps' over $f$, and for each position $y$ in the sweep, a particular function $g$ is applied, the result of which becomes the output of $S$ at that position $y$.
The most prominent example of such an operator is of course convolution, where function $g$ takes the specific form of an integral transform:
- $\tilde f[f(x)] = f(x)$
- $\tilde k[k(x), y] = k(y-x)$ $~~~~~~~~~~~~~~~~$ i.e. $k$ is reflected and offset by $y$
- $ \displaystyle g(f,k; \tilde f, \tilde k, y) = \int_{x \in \mathbb{R}^N} ~\tilde f[f(x)] ~~ \tilde k[k(x),y] ~~\mathrm{d}x $
By contrast, an example which follows the same form, but where the kernel 'adjustment' does not represent an 'offset' in the 'space sweeping' sense but is a 'multiplier' instead, is the Fourier transform, where: $\tilde f[f(x)] = f(x)$, $k(x) = e^{-2\pi ix}$, $~~\tilde k[k(x), y] = k(xy)$, and $g$ is the same integral transform $ \displaystyle g(f,k; \tilde f, \tilde k, y) = \int_{x \in \mathbb{R}^N} ~\tilde f[f(x)] ~~ \tilde k[k(x),y] ~~\mathrm{d}x $
Examples of other 'sweeping' functions are morphological dilation and erosion, where $g$ takes a different form involving set operations (i.e. it's not an integral transform). Yet another is in image registration, where $g$ is of the form of a similarity function plus a regularizer, etc.
Is there a general name for this kind of 'sweeping' operation / function $S(y; ~f, k, g)$ that 'sweeps' a kernel $k$ over a function $f$, and for each position $y$ in the sweep it applies a function $g$?
Closest I could come up with was "kernel-based transform" in general, though this does not seem to be a particularly accepted or used term, nor does it capture exactly what I'm referring to (in particular the 'space sweeping' aspect of such a transform). Conversely, I thought maybe it could be referred to as a "generalised morphological operation" but again this does not seem like an accepted or used term, nor does it capture applications in non-morphological contexts (e.g. feature detection). Any ideas?
