I have seen lots and lots of questions about convolution here, however I have not seen one addressing my specific doubts.
The wikipedia page on convolution, has a section on visualizing the function:
At the beginning of the document it claims:
In mathematics (in particular, functional analysis) convolution is a mathematical operation on two functions (f and g) that produces a third function expressing how the shape of one is modified by the other
I am especially focused on function expressing how the shape of one is modified by the other.
I understand this wording is meant more as a high level "wishy washy" description than it is about any formalization, however I am trying to understand the properties of convolution as function mixing operation.
To give a clear example of what I mean, linear interpolation is an easy concept. It "mixes" 2 functions $f_1$ and $f_2$ such that the resulting function $g$ is "$1-x$ percent $f_1$ and x percent $f_2$"
In this same spirit, I am trying to understand how convolution "modifies one function through the shape of another".
Naively I first thought that convolution describes the overlapping areas of the functions as one is slided over the other. I got this from the visual explanation. However the claim that the yellow area is the area of overlap of the 2 functions at time $t$ doesn't seem correct. This is only true if one of the 2 functions is a box function with height 1.
Other wise $$\int_a^b f(\tau) g(t - \tau)d\tau$$ discretized as:
$$\sum_{i=0}^n f(\tau_i^*)g(t-\tau_i^*)\Delta\tau$$
Can't possibly be the area, since the overlapping area wouldn't be the product, it would be $min(f(\tau_i^*), g(t-\tau_i^*)) \Delta\tau$ which just happens to be equal to the product when one of the functions is a box function with height 1. Thus the diagram's explanation is somewhat misleading.
Thus I am still not sure about how a convolution merges the shapes of the images. Computationally, what it is doing is fairly straightforward, I am not making a question about what a convolution is, but about how convolutions manipulate their input functions.
Hopefully that makes sense.