A Simple Multi-Scale Gradient Flow-Based Subspace Reconstruction

We propose multi-scale gradient flow-based inverse problem with application to image denoising and reconstruction. We also propose high order continuous <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$TV$</tex> optimization to solve this problem. The proposed algorithm is fast and easy to implement, and it can recover subspaces corresponding to the objects with similar scale and oscillation oscillation pat-terns. We show that the solution is obtained for the Euler-Lagrange equation by applying higher order weak derivative. The proposed algorithm is highly predictable such that subspaces can be estimated almost exactly through topological completion measured over the subsets. Finally, we discuss two main examples to show the feasibility of the proposed approach. First example is recovering subspace of image contaminated with strong salt-and-pepper noise. The second example generates high quality and high contrast <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$RGB$</tex> image from a very low quality initial image. For the both examples, the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(RGB)$</tex> subspaces are estimated through the proposed topological completion indicators.