# What are some techniques to recover an original image from three different compressed versions?

This question is theoretical. Let's say you have three different highly compressed images of a same source image, each compressed with a different algorithm and thus "lossy" in different ways.

What would be the possible techniques to recover the best possible image, using the three different compressed versions ?

Commonly, people use the "mean square error (MSE)" (or its log version, known as the peak signal noise ratio). Assume the ground-truth image is $X$, and the recovered image is $\hat{X}$. Then MSE is defined as $${\rm{MSE}}(X,\hat{X}) = \sum_{i,j}(X[i,j]-\hat{X}[i,j])^2/N$$ where $N$ is the number of pixels in $X$, and $X[i,j]$ denotes a pixels located at $i$th row and $j$th column in $X$.
Without loss of generality, say we have $n$ compressed counterparts are $C_1,C_2,\cdots,C_n$, and we reconstruct $\hat{X}$ as $$\hat{X}=f(C_1,C_2,\cdots,C_n)$$ where $f(\cdot)$ is a method to recover the original image from its compressed counterparts. And your objective is to find an appropriate function $f(\cdot)$ and minimizes the MSE.
To simplify the problem, people often restrict the function $f(\cdot)$ to some fixed type but with parameters to be learned from training data. For example, one simplest function can be the linear combination i.e. $$f(C_1,C_2,\cdots,C_n) = \sum_{k}w_kC_k$$ where $w_k$ is the weight of using the $k$th compressed image in the recovered image. This is a classic regression problem and can be solved easily (go and check the normal equation, which gives the closed form solution to find all $w_k$s). However, you should note the following things:
• 4). my guess says the linear regression method is not the best way, but it is a very good start point. Later your may try other $f(\cdot)$, like kernel regressions, or adding more terms, e.g. the total variation term.