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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.

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  • $\begingroup$ Are $ x $ and $ y $ vectors? $\endgroup$ – Royi Mar 15 '18 at 22:20
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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.

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There are many types of matrix norms. Three are quite standard:

  1. element-wise norms: unfolding the matrix into a long vector, and compute a norm for that vector.
  2. Schatten norms: (power) vector noms over singular values of the matrix.
  3. induced norm: maxima over vector norms with uni-norm vectors.

The Frobenius norm is the most simple: the square root of the sum of squared magnitude of all entries, corresponding to the Euclidean vector norm. It was also called Schur or Hilbert–Schmidt norm. It is also an instance of the Schatten norms, with power two. One of its main property is that it is invariant under rotations or orthogonal transformations (like Fourier or orthogonal wavelets, often used on images), which can be attractive. It can be written as $\sqrt{\mathrm{Trace}(M.M^H)}$. However, other norms are better adapted to additional properties like sparsity or robustness.

When you minimise the square of a Frobenius norm, you "almost" perform the classical least-square" minimization. If you add a penalty term like TV, you can constrain the optimization to satisfy what you'd expect from images, and the square of a Frobenius norm puts you in the framework of proximity operators (a generalization of projections), for which efficient algorithms exist.

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