Adjoint method in PDE-based image compression

We consider a shape optimization based method for finding the best interpolation data in the compression of images with noise. The aim is to reconstruct missing regions by means of minimizing a data fitting term in an <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> -norm, for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mn>1</mml:mn> <mml:mo>⩽</mml:mo> <mml:mi>p</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mo>+</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:math> , between original images and their reconstructed counterparts using linear diffusion PDE-based inpainting. Reformulating the problem as a constrained optimization over sets (shapes), we derive the topological asymptotic expansion of the considered shape functionals with respect to the insertion of small ball (a single pixel) using the adjoint method. Based on the achieved distributed topological shape derivatives, we propose a numerical approach to determine the optimal set and present numerical experiments showing the efficiency of our method. Numerical computations are presented that confirm the usefulness of our theoretical findings for PDE-based image compression.