# A “convolution”-like operator for “moving difference” of functions?

What is the following operator called?

$$(f\star g)(t) = \int_{-\infty}^\infty \|f(\tau)-g(\tau-t)\|dt$$

I am thinking it can peehaps be built as convolution of exponential and/or logarithmic mapping of the functions $f,g$ somehow, but what name would the operator have?

For context, solving the problem

$$t_o=\min_{t}\{(f\star g)(t)\}$$ for known $f,g$ is something that can be done in signal processing and pattern recognition.

For example extracting symbols from an image (f) given a template (g):

Where are looking for "d", but apparently for our font "h" looks quite similar.

• I don't know a name for this particular operation (and it might have one), but this is just the $L^1$ norm of $f-g_t$ where $g_t$ is the translation of $g$ by $t$ - i.e. $g_t(\tau)=g(\tau-t)$. – Milo Brandt Jul 29 '18 at 23:08