The Regularised Epsilon-derivative for Image Reintegration

Abstract

Many image-editing tasks are carried out in the gradient domain. Suppose that for an image I the gradient ∇I consists of a pair of fields (p;q); then some image “reintegration” scheme is tasked with converting derivative fields (p;q) back to imagespace I; typically, a Poisson equation solver is used for this task. But what if we have altered (p;q) so that this pair (p;q) is no longer integrable? Then we have to project onto integrable gradient data that will indeed reintegrate to an approximation of the original image. For example, we may wish to alter (p;q) so as to emphasize or de-emphasize some aspects of the image, e.g. ameliorating wrinkles in skin images (or indeed enhancing them in the case of ageing a face image). Here, we propose a new gradient kernel that retains part of the original image, regularising the reintegration back into the image domain. We compare our approach with the Screened-Poisson method which includes a term λ times a “screen” term that moves the solution image back closer to the input image. Effectively, we are doing a similar adjustment, but we show that the results are a good deal better than using Screened-Poisson, which tends to overly blur the output. Moreover, in Screened-Poisson one must choose a value for λ, which may be different for every image – here we determine that our new kernel method does not need to adapt to each image yet delivers comparable or better results.