Difference in the interpretation of cross-correlation and convolution? I'm confused on the interpretation between convolution and cross-correlation integral transforms. The convolution integral is defined as
$f (t) \otimes g (t) = \int f(\tau) g(t-\tau) d\tau$
and the cross-correlation is defined as
$f (t) * g(t) = \int f^{*}(\tau)g(t+\tau)$
However, when looking up the final interpretation of this result, convolution is cited as saying it is "how one function is shaped by another one" while cross-correlation is "measure of similarity", yet I am confused as to how you can extract two seemingly different interpretations from the formulas above since the difference seems quite small (complex conjugate and no reflection of function $g$).
I've taken a look at it and all I see is that convolution seems to be totally symmetric, since $f\otimes g = g\otimes f$ where as this is not necessarily the case for cross-correlation. Is convolution just a measure of the similarity of the shape only while cross-correlation encompasses both the correlation of the shape and the order at which they come by? I see on wikipedia that cross correlation is the convolution of an adjoint $f$ function but this doesn't help me.
 A: I don't know where you got these interpretations from and the context within which they are discussed. On their own, they make little sense indeed. However, they are motivated by the applications that these integrals are used or encountered.
The convolution integral naturally appears in signals and systems theory, when a signal $x(t), t \in \mathbb{R},$ passes through a "linear time invariant filter" (or "channel") with, so called, impulse response $g(t), t \in \mathbb{R}$. The convolution integral provides the output for all $t\in \mathbb{R}$. Many operations in nature such as propagation of electromagnetic signals over a medium have this mathematical representation.
The cross-correlation integral is essentially the inner product of the "vector" $x(\tau), \tau \in \mathbb{R}$ and the "vector" $g(\tau), \tau \in \mathbb{R}$, shifted by $t$. Recall from elementary linear algebra that the inner product of two vectors is a measure of their similarity, thus it makes sense to interpret cross-correlation as such.
Clearly, the two operations are equivalent in the sense that you can always treat cross-correlation as a convolution with an adjoint version of the second function, and vice versa. However, the applications these operations correspond to are so common that it turns out it is more convenient to use both terms to clearly identify what we are talking about.
