Why Do Most of The Papers Use the Frobenius Norm for Denoising? I have an noisy image and I want to remove noise from it; suppose $y$ is noisy image and $A$ is linear mask which makes my image noisy and $x$ is original image, so we have
$$
Ax + \eta = y
$$
and $\eta$ is additive noise. I have read in many papers that the equation below is a good cost function for removing noise:
$$
x^* = \arg\min_{x} \lambda\|A(x) - y \|^2_\mathcal{F} + \|x\|_{TV}
$$
First term aims at minimizing additive noise but why do most of the papers use the Frobenius norm? And second term wants preserve edges and features of images.
The Frobenius Norm has multiple equivalent definitions – the useful for error measure is probably this one:

$$\left\|M\right\|_\mathrm F = \sqrt{\sum_{p\in M}\left\lvert p\right\rvert^2}$$

That's a root square over all pixels.

Root mean squares are very useful cost functions, as they describe the power of a signal.