Space-illing Curves in Generating Equidistrubuted Sequences and Their Properties in Sampling of Images

Intensive streams of video sequences arise more and more frequently in monitoring the quality of production processes. Such streams not only have to be processed on-line, but also stored in order to document production quality and to investigate possible causes of insufficient quality. Direct storage of a video stream, coming with the intensity 10-30 frames per secondwith a resolution of 1-8megapixels, from one productionmonthwould require 100-500 terra bytes of a disk (or tape) space. A common remedy is to apply compression algorithms (like MPEG or H264), but compression algorithms usually introduce changes in gray-levels or colors, which is undesirable from the point of view of identifying defects and their causes. For these reasons we return to the traditional idea of sampling images, followed by loss-less compression. However, classical sampling on a rectangular grid is insufficient for our purposes, since it is still too demanding from the point of view of storage capacity. Our experience of using equidistributed (or quasirandom) sequences as experimental sites in nonparametric regression function estimation Rafajłowicz and Schwabe (2003); Rafajłowicz and Schwabe (2006); Rafajłowicz and Skubalska-Rafajłowicz (2003) suggests that such sequences can be good candidates for sampling sites. Roughly speaking, the reason is in that the projection of a 100× 100 rectangular grid on the axes has 100 points, while a typical equidistributed sequence of the length 104 provides again 104 points when projected onto the same axes. The idea of using equidistributed (EQD) sequences in sampling images was firstly described in Thevenaz (2008), where it was used for image registration. Our goals are different and we need more specialized sampling schemes than a "general purpose" Halton’s sequence, which was used in Thevenaz (2008). Our aim is to propose a new method of generating equidistributed sequences, which is based on space-filling curves. Due to the remarkable properties of space-filling curves (SFC), which preserve volumes and (to some extent) neighborhoods, the proposed sequences are wellsuited for sampling of images in such a way that samples can be processed similarly as an original image. We concentrate mainly on 2D images here, but 3D images are also covered by the theoretical properties. Simple reconstruction schemes, which are well-suited for industrial images, are also briefly discussed. We also indicate ways of generating sampling sequences 7