How to solve a linear system on a very large scale

I have a very large scale of linear system equations
enter the description of the image here where $ N_ {i} $ denotes the neighborhood of a pixel $ i $. $ C_ {i} $ and $ psi_ {i} $ and $ omega_ {i, j} $ are known but $ widetilde {C_ {i}} $ and $ widetilde {C_ {j}} $ are unknown. And $ C_ {i} $,
$ widetilde {C_ {i}} $ and $ widetilde {C_ {j}} $ can be considered as a number. I know it's a linear system on a very large scale, but I do not know how to solve it. It is the treatment of images and the algorithm of image processing