Regularizing Ill-Posed Inverse Problems: Deblurring Barcodes

TL;DR

The paper analyzes regularization for ill-posed inverse problems in image deblurring, modeling blur with kernels and discretizing to a forward operator Af ≈ b. It shows that direct inversion is unstable due to rapidly decaying singular values and noise amplification, and introduces Tikhonov regularization with f_lambda = (A^T A + lambda^2 I)^{-1} A^T b, interpreted via an augmented least-squares problem. Through UPC barcode experiments, it demonstrates how appropriate regularization reliably recovers structure from blurred, noisy data, while naive inversion fails. The SVD provides a principled lens to understand the method: small singular values drive instability, and regularization attenuates their influence, with λ selection guided by the L-curve and application-specific trade-offs, yielding robust, image-deblurred results.

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).

Figures (21)

• Figure 1: An example of a one-dimensional "image" $f(t)$. The vertical lines indicate jump discontinuities in $f$.
• Figure 2: A blurred version of the image from Figure \ref{['ME:fig:f1']}, using the simple blurring function (\ref{['ME:avgker']}) with parameter $z=0.025$ (top) and $z=0.05$ (bottom): the larger the value of $z$, the stronger the blurring effect.
• Figure 3: Three choices of the kernel function $h(s,t)$, centered at $s=0.35$. The top plots use the blurring parameter $z=0.025$; the bottom plots use $z=0.05$. As $z$ gets increases, the kernel spreads more around the point $s$, resulting in more blurring.
• Figure 4: The integral (\ref{['ME:dconv']}) is the area under the blue curve $h(s_j,t)f(t)$ in the left plot. Approximate this quantity with the area of the rectangles of width $1/n$ that touch $h(s_j,t)f(t)$ at $t=t_1,\ldots t_n$, as shown on the left and described in equation (\ref{['ME:midpt']}). This move allows us to replace a calculus problem with a linear algebra problem.
• Figure 5: Python code to construct the three blurring functions (\ref{['ME:avgker']}), (\ref{['ME:hatker']}), and (\ref{['ME:gaussker']}), and to build the matrix ${\bf A}$ defined in (\ref{['ME:blurmat']}).