Synthetic Boundary Conditions. As mentioned in Section 2.1, it is unlikely that a true image scene would be modeled well by periodic boundary conditions, and zero boundary conditions only make sense for scenes with a black background. There may be some rare cases when reflective and anti-reflective boundary conditions provide a good model of the true image scene outside the field of view. In this subsection we develop an approach that provides a more realistic extension of pixels across the boundary. For example, texture and edges should be extended sensibly. The motivation for our approach comes from observing that the problem of defining appropriate boundary conditions is similar to the image recovery problem, in which part of the image is damaged and the aim is to recover missing pixels. In our case, the region we wish to recover corresponds to those pixels outside the boundary. Two common approaches for the image recovery problem are image inpainting [5] and texture synthesis [24]. Image inpainting tries to extend the geometric structure of the image, while texture synthesis extends the texture pattern into the unknown region. In this chapter we use the texture synthesis approach. With the image recovery idea in mind, we wish to determine a relation- ship between (unknown) pixel values outside the boundary to those pixel values inside the boundary. Using a basic texture synthesis approach, we can try to find a pixel in the viewable region whose neighborhood (e.g., a rectangular region) is most similar to the corresponding neighborhood of the boundary pixel we wish to fill in. If this idea is applied to a blurred image, it can extend edges across the boundary well, but there is little hope that it can also extend the texture, as texture information is lost in blurring. Hence, in- stead of copying single pixels, we propose to copy small patches that contain the required texture information. This idea is similar to the generalization of texture synthesis to image quilting [23]. To describe more precisely our approach for synthetic boundary condi- tions, we need a bit of notation. Let D r0, n 1s r0, n 1s (domain) B pr m, n m 1s r m, n m 1sq zD (border)
