Interpolation by Optimization

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I have the following interpolation problem:

Given an M by N channel image (1 Channel).
A certain set of pixels are defined as anchors. The rest of the pixels should be interpolated using the original data by minimizing the follwoing cost function:

Where E is the new matrix of interpolated pixels and I is the original image.
Pay attention that the interpolated image equals the original image at the anchor pixels.

The inteprolation weights are given by:

My question is, how can I formalize it in Weighted LS form?
Any optimization could be made to solve it quickly and efficiently?

This interpolation could be used as a minimization for many application in image processing (Edge Preserving Interpolation, Colorization, etc...).

Thank You.