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Sr. Research Scientist, Photoshop Group, Adobe Systems
The Art Restoration / Inpainting ProblemIn the early stages of our research that led to the Photoshop Healing Brush we explored the problem of repairing paintings by removing cracks and seams, as shown in these (unpublished) December 1999 results below. Soon afterwards, this same "area reconstruction" problem and a different solution gained wide public acclaim as Image Inpainting thanks to a very well received paper at SIGGRAPH 2000 by M. Bertalmío, G. Sapiro, V. Caselles and C. Ballester. To briefly compare the two methods:
(1) Our 1999 algorithm (which is part of the Healing Brush code in Photoshop) was based on the bi-Laplace equation, imposing both Dirichlet and Neumann boundary conditions at the same time. In other words, the image inside the area of the crack is replaced with a bi-harmonic function, which matches the outside image at the boundary, and also matches the outside normal image derivative at the boundary. This achieves smooth reconstruction and continuation of outside features into the reconstructed area, which would be impossible with a second order equation. We presented more detail in our invited talk at the ECCV workshop on Applications of Computer Vision. See the related paper.
(2) Image Inpainting solves a fluid dynamics-related equation for the inpainted area, identifying the image with the stream function, as clarified in a very insightful paper by M. Bertalmío, A. Bertozzi and G. Sapiro, which everyone interested in those problems should definitely read.
Despite these different mathematical approaches, both methods give visually pleasing results perhaps because both methods make an effort to "bring in" structures from outside based on higher order PDE, and not simply match boundary pixel values.