myntfalskj4

2022-07-16

Suppose that I have an image with $n$ pixels and the following equation,
$\sum _{r\in P}\left(U\left(r\right)-\sum _{s\in N\left(r\right)}{w}_{rs}U\left(s\right){\right)}^{2}=0$
$U=\left(U\left(1\right),U\left(2\right),...,U\left(n\right)\right)$ are the variables;
$P$ is the set of all the $n$ pixels;
$N\left(r\right)$ is the set of all neighboring pixels of $r$. For example, for an image with $3×3$ pixels, $N\left(1\right)=\text{{2, 4, 5}}$
${w}_{rs}$ is a weighting function where $\sum _{s\in N\left(r\right)}{w}_{rs}=1$
Note: I just mentioned that it is an image to better explain the role of the set $N$ and the function ${w}_{rs}$.
Well, it's easy to show that "one" solution for that equation are the scalar multiples of the all-ones vector. I mean, $U=k\cdot \left(1,1,...,1\right)$ is clearly a solution.
My question is: there's a way to proof that this is the "only" solution for the above equation?

Charlize Manning

Expert