In the paper
A circle detection approach based on Radon Transform by Erman Okman and Gozde B. Akar.
I have a few questions on some basics.
first of all what does
$$ds^2 = dx^2 + dy^2$$
mean, It says it is the infinitisimal part but what does it mean specificly. I guessed it could be a substitution.
an other part is the given the following integral.
$$ g_1(x)=\int_{-\infty}^{\infty} f(x,y)dy$$
what exactly is the directional integral
$$g_1[m] = \sum _n f[m,n] $$
more specifically it says " which are evaluated by taking integral of $g_1(x)$ and over $[m-0.5, m+0.5]$ and $[n-0.5, n+0.5]$, respectively."
does this mean we do a several integration of the form $$ g_1 [m]= \sum_n \int_{n-0.5} ^{n+0.5} f(m,y)dy$$
and if yes is there a simple way to implement it in matlab. Well I have a discrete Image which means the image is only defined on a grid $[1,\dots,m ]\times [1,\dots n]$. How would the integral look like and how can I integrate over one variable on a function $f(x,y)$ with two variables.
I am happy for any hint. talk to you later
The relation $dx^2=dx^2+dy^2$ was called the characteristic triangle by Leibniz. Here $ds$ is an infinitesimal element of length (i.e. $s$ is the arclength parameter in modern terminology) along a curve.