3-D Image Denoising by Local Smoothing and Nonparametric Regression

Three-dimensional (3-D) images are becoming increasingly popular in image applications, such as magnetic resonance imaging (MRI), functional MRI (fMRI), and other image applications. Observed 3-D images often contain noise that should be removed beforehand for improving the reliability of subsequent image analyses. In the literature, most existing image denoising methods are for 2-D images. Their direct extensions to 3-D cases generally cannot handle 3-D images efficiently, because the structure of 3-D images is often substantially more complicated than that of 2-D images. For instance, edge locations are surfaces in 3-D cases, which are much more challenging to handle, compared to edge curves in 2-D cases. In this article, we propose a novel 3-D image denoising procedure, based on nonparametric estimation of a 3-D jump surface from noisy data. One important feature of this method is its ability to preserve edges and major edge structures, such as intersections of two edge surfaces, pyramids, pointed corners, and so forth. Both theoretical arguments and numerical studies show that it works well in various applications. Software and proofs are available online as supplemental material.