L1 Splines for Robust, Simple, and Fast Smoothing of Grid Data

TL;DR

This paper introduces an L1 spline smoothing method for grid data that enhances robustness to outliers and maintains computational efficiency through a simple split-Bregman scheme combined with DCT, achieving high-quality results quickly.

Contribution

It proposes replacing the L2 norm with an L1 norm in spline smoothing, enabling robustness to outliers, and introduces an efficient split-Bregman based algorithm for fast computation.

Findings

Robust smoothing results with outlier resistance.

Fast processing times demonstrated in experiments.

High-quality smoothing comparable to traditional methods.

Abstract

Splines are a popular and attractive way of smoothing noisy data. Computing splines involves minimizing a functional which is a linear combination of a fitting term and a regularization term. The former is classically computed using a (weighted) L2 norm while the latter ensures smoothness. Thus, when dealing with grid data, the optimization can be solved very efficiently using the DCT. In this work we propose to replace the L2 norm in the fitting term with an L1 norm, leading to automatic robustness to outliers. To solve the resulting minimization problem we propose an extremely simple and efficient numerical scheme based on split-Bregman iteration combined with DCT. Experimental validation shows the high-quality results obtained in short processing times.