Restoration of a Feature Closed Class of Two-Dimensional Images

A restoration principle is proposed and demonstrated for a feature closed class of two-dimensional images. Any member of this class of images can be represented as an intersection of unions of half planes in R
²
. Images with corners as well as images with continuous curvature can be generated under this algebraic representation. All resulting images have a well-defined interior and a nonintersecting boundary. Knowledge that the image to be restored comes from this closed class of images allows for an image restoration algorithm that is superior to a point-thickening algorithm. The proposed restoration algorithm produces a restored image that lies within the image to be restored. For a random sample of n points known to lie within the image to be restored, the expected value of the residual area for the image with corners is of order (In n/n) and is proportional to the number of outside corners. For images with continuous curvature, the residual area is of order 1/√n and is proportional to a geometric constant associated with the curvature of the boundary. The restoration algorithm offers insight into the consideration of the shape of the boundary in image restoration.