Can Diffusion Equations Restore Damaged Paintings? A Practical Guide

This is a famous painting (image shown) that has spots reducing its aesthetic value and needs restoration.

Traditional restoration relies on skilled conservators, which is time‑consuming and may not perfectly recover the original.

Scientists are exploring mathematical and computer‑based techniques for image restoration.

Diffusion equation method is an effective, theoretically grounded approach for image inpainting. This article introduces how to use diffusion equations for image repair and explains its mathematical principles and applications.

Mathematical Representation of Digital Images

Digital images consist of a rectangular grid of pixels, each with a color or grayscale value. For grayscale images, each pixel’s intensity ranges from 0 (black) to 255 (white). The image can be represented as a function f(x, y) where (x, y) are pixel coordinates and f(x, y) is the intensity.

Applying the Diffusion Equation to Image Inpainting

In the early 19th century, mathematician Joseph Fourier studied heat propagation in solids, such as a metal rod heated at one end. Knowing the temperature distribution at time zero and the heat equation, one can compute temperature at any later time.

In image inpainting, the diffusion equation uses known grayscale values around a missing region to infer the unknown values, analogous to heat diffusion. The process involves:

Identify damaged or missing regions whose pixel values are unknown.

Set initial conditions: known pixels retain their original values; unknown pixels may be initialized to zero or random values.

Solve the diffusion equation numerically (e.g., finite difference or finite element methods) to iteratively update unknown pixel values.

Iterate until convergence criteria are met, such as error below a threshold or a maximum number of iterations.

Although diffusion‑based inpainting yields good results, it can blur edges. Researchers have proposed improvements such as total variation (TV) flow, which preserves edge information during restoration.