Fast & Robust Image Interpolation using Gradient Graph Laplacian Regularizer

TL;DR

This paper introduces a gradient graph Laplacian regularizer (GGLR) for image interpolation that effectively promotes piecewise planar signals, reducing artifacts like the staircase effect, and offers fast, high-quality reconstruction with lower complexity.

Contribution

The paper proposes a novel GGLR method that generalizes traditional GLR to better handle planar image patches, improving interpolation quality and computational efficiency.

Findings

GGLR outperforms existing methods in image interpolation quality.

The approach effectively reduces the staircase effect in reconstructed images.

GGLR achieves faster convergence due to its quadratic programming formulation.

Abstract

In the graph signal processing (GSP) literature, it has been shown that signal-dependent graph Laplacian regularizer (GLR) can efficiently promote piecewise constant (PWC) signal reconstruction for various image restoration tasks. However, for planar image patches, like total variation (TV), GLR may suffer from the well-known "staircase" effect. To remedy this problem, we generalize GLR to gradient graph Laplacian regularizer (GGLR) that provably promotes piecewise planar (PWP) signal reconstruction for the image interpolation problem -- a 2D grid with random missing pixels that requires completion. Specifically, we first construct two higher-order gradient graphs to connect local horizontal and vertical gradients. Each local gradient is estimated using structure tensor, which is robust using known pixels in a small neighborhood, mitigating the problem of larger noise variance when computing gradient of gradients. Moreover, unlike total generalized variation (TGV), GGLR retains the quadratic form of GLR, leading to an unconstrained quadratic programming (QP) problem per iteration that can be solved quickly using conjugate gradient (CG). We derive the means-square-error minimizing weight parameter for GGLR, trading off bias and variance of the signal estimate. Experiments show that GGLR outperformed competing schemes in interpolation quality for severely damaged images at a reduced complexity.