2-D transform-domain resolution translation

The extensive use of discrete transforms such as the Discrete Cosine Transform in image and video coding suggests the investigation on ltering before down sampling (FBDS) and ltering after up sampling (FAUS) methods directly acting on the transform domain [1, 2]. On the other hand, Transform Domain Filtering (TDF) was recently introduced as an important tool for implementing linear ltering and other linear operators directly in the compressed domain [3]. In this paper, we introduce the concept of \transform domain resolution translation" as a combined form of the transform domain FBDS and FAUS issues, and then propose the solution with the context of TDF. We, rst, generalize the TDF to include non-uniform and multirate cases. The former is de ned as a TDF problem in which the original transform domain is of di erent size from the target one, while the latter considers the implementation of sampling rate conversion in the transform domain. The implementation architecture is based on pipelined structures that involve matrix-vector product blocks and vector addition, but is not limited to only hardware eventhough the proposed architecture is easily realizable with distributed arithmetic designs and minimizes arithmetic errors due to the small number of processing stages involved. Chang and Messerschmitt showed that TDF is still useful in the form of software architecture since transform data usually take on immensively compressed form [4]. We show step by step examples of how these generalized notions of TDF can provide general solutions to the sub-adjacent block problem, to the 1-D transform domain resolution translation problem and, nally, to the 2-D transform domain resolution translation problem. Such techniques are particularly useful for fast algorithms for processing compressed images and video, where transform coding is extensively used (e.g., in JPEG, H.261, MPEG-1, MPEG-2 and H.263).