Image Inpainting with the Navier-Stokes Equations

Image inpaining “involves filling in part of an image (or video) using information for the surrounding area”([1]). This report summarizes an application of numerical solutions to the Navier-Stokes equations for this well-studied image processing problem. It exploits the remarkable relationship between the steady state solution of the streamfunction in fluid flow and the (grayscale) image intensity level in image processing. This report is a follow-up of an approach first discovered in 2001 by A. Bertozzi, et. al in [1]. 1 Preliminaries and Motivation Image inpainting (from hereon, simply inpainting) is the technique of filling in a region of an image based on the information outside the region. The distinction between inpainting and denoising should be made clear: deblurring generally attemps to modify regions that are individual pixels, while inpainting involves modifying a larger area. Applications for inpainting are generally to remove unwanted patterns in photos, from scratches and vandalization to superimposed letters. Unless otherwise stated, an image will refer to a grayscale image. 1.1 Grayscale Image Basics A digital grayscale image, I, is an m × n matrix, where at each index, Ii,j consists of an integer value from 0 to 255 (we will only consider rectangular images). The (i, j)th index in I is equivalent to the pixel at the corresponding location. This value is referred to as the graylevel at location (i, j), where 0 corresponds to pure black, 255 to pure white, and all intermediate values to different shades of gray. We let D be the set of indicies (i, j) ∈ {1, 2, . . . , n} × {1, 2, . . . ,m} where I is defined. Throughout this report, however, we equivalently treat I as a function from a discrete domain of indicies {1, 2, . . . , n} × {1, 2, . . . ,m} to the integers mod 256.