# Image reconstruction using low-light components

Let's say we have a regular photo and three low-light photos illuminated in different colors. Each pixel is a three-component vector $$q=(R,G,B)$$. Then $$q_k^{A}$$ is the $$k$$-th pixel of the regular photo and $$q_k^{B}$$ $$q_k^{C}$$ $$q_k^{D}$$ be the $$k$$-th pixel of the three low-light versions.

The task is to reconstruct the regular photo from the three low-light photos where:

$$q_k^{A} = F^{A}q_k^{B} + F^{C}q_k^{C} + F^{D}q_k^{D} + q_{const}$$. Clearly, $$F^A, F^{B}, F^{C}$$ are $$3 \times 3$$ matrices and $$q_{const}$$ is a vector.

The task is to perform least-squares fit for the 30 components (three matrices and the vector). Specifically, we should minimize:

$$S = \frac{1}{MN}\sum\limits_{k=0}^{MN-1}{||-q_k^{A} + F^{A}q_k^{B} + F^{C}q_k^{C} + F^{D}q_k^{D} + q_{const}||^2}$$

What is the proper way to approach this problem? I have implemented linear regression of different types before, but in this case I am not sure how to proceed.

• I'd go for auto encoders. For each well-lit photo make ill-lit pair. Then you can feed ill-lit ones and try to reconstruct them as well-lit. Two questions though: how big is your data set? how bad ill-lit photos are? If there's none detail captured it's more like generating new data from learnt distribution than just reconstruction. – Piotr Rarus - Reinstate Monica Jan 2 at 7:34