Regularizing Ill-Posed Inverse Problems: Deblurring Barcodes

TL;DR

This paper introduces regularization techniques for solving ill-posed inverse problems, specifically focusing on deblurring barcode images by converting the problem into a regularized linear algebra framework.

Contribution

It presents a mathematical modeling approach and demonstrates how to apply regularization to stabilize the solution of ill-posed deblurring problems.

Findings

Regularization improves stability of deblurring solutions

Linear algebra reformulation facilitates computational solutions

Ridge regression effectively addresses ill-posedness

Abstract

This manuscript is designed to introduce students in applied mathematics and data science to the concept of regularization for ill-posed inverse problems. Construct a mathematical model that describes how an image gets blurred. Convert a calculus problem into a linear algebra problem by discretization. Inverting the blurring process should sharpen up an image; this requires the solution of a system of linear algebraic equations. Solving this linear system of equations turns out to be delicate, as deblurring is an example of an ill-posed inverse problem. To address this challenge, recast the system as a regularized least squares problem (also known as ridge regression).