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 R2. 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.