Regularized Image Inpainting

This repository contains an implementation of a convex optimization-based approach to the Regularized Image Inpainting Problem.

Problem Description

The goal is to recover a digital color image $X \in \mathbb{R}^{M \times 3N}$ from a corrupted version $X_{corrupt}$. The image is represented by three channels (Red, Green, Blue) stored side-by-side: $$X = [X_R \quad X_G \quad X_B]$$

A corruption mask $\Omega \in \mathbb{R}^{M \times N}$ identifies damaged pixels (where $\Omega(i,j) = 0$). The corruption operator $\mathcal{A}$ is defined as the entrywise Hadamard product: $$\mathcal{A}(X) = [\Omega \odot X_R \quad \Omega \odot X_G \quad \Omega \odot X_B]$$

Mathematical Definition

The inpainting task is framed as a minimization problem using nuclear norm regularization to promote low-rank structures in the recovered image:

Where:

• $\sigma > 0$ is a regularization parameter.
• $|Z|_*$ denotes the nuclear norm (sum of singular values).
• $\tilde{X} = [X_R^T \quad X_G^T \quad X_B^T]^T$ represents the vertically stacked channels.

Algorithm

The solution is reached by finding a fixed point of the operator $T$:

Iterating $Y_{k+1} = T(Y_k)$ ensures that $\text{prox}_{\gamma g}(Y)$ converges to the optimal solution for appropriate functions $f$ and $g$.

Repository Structure

• algorithm.py: Contains the core logic for the iterative inpainter. It implements the sequence generation $Y_{k+1} = T(Y_k)$ to find the approximate solution for the regularized problem.
• utils.py: A collection of mathematical tools and helper functions, including:
  • Definitions for proximal operators.
  • The linear operator $\mathcal{A}$ for image corruption.
  • Nuclear norm calculation tools.
  • Plotting functions to visualize the original, corrupted, and reconstructed images across various $\sigma$ parameters.

Results

Figure 1: Images obtained for different values of σ with γ = 1 and 50% erased pixels.