Jump surface estimation, edge detection, and image restoration

Surface estimation is important in many applications. When conventional smoothing procedures (e.g., running averages, local polynomial kernel smoothing procedures, smoothing spline procedures) are used for estimating jump surfaces from noisy data, jumps are blurred at the same time when noise is removed. In recent years, new smoothing methodologies have been proposed in the statistical literature for detecting jumps in surfaces and for estimating jump surfaces with jumps preserved. We provide a review of these methodologies. Because a monochrome image can be considered a jump surface of the image intensity function, with jumps at the outlines of objects, edge detection and image restoration problems in image processing are closely related to the jump surface estimation problem in statistics. We also review major methodologies on edge detection and image restoration, and discuss connections and differences among these methods and related methods in the statistical literature.