202 IEEE SIGNAL PROCESSING LETTERS, VOL. 6, NO. 8, AUGUST 1999 Direct Transform to Transform Computation Athanassios N. Skodras, Senior Member, IEEE Abstract— An efficient direct method for the computation of a length- discrete cosine transform (DCT) given two adjacent length- DCT coefficients, is presented in this letter. The computational complexity of the proposed method is lower than the traditional approach for lengths Savings of memory locations and data transfers are also achieved. Index Terms—Compressed-domain processing, discrete cosine transform, downscaling, JPEG, MPEG, transcoding. I. INTRODUCTION N OWADAYS signals, images and video are compressed from their early stages, i.e., just after their acquisition. One such example is a system for processing documents. The images are scanned at high resolution and are immediately compressed by hardware or software, in order to save memory space [1], [2]. Another example comes from the digital video area, where video signals are compressed when transmitted over networks or stored in databases [3], [4]. A similar situation holds for the speech signals [5]. However, in all these cases, it is likely that the signals will have to be processed before being displayed, transmitted, printed, etc. Some of the frequently used processing functions are scaling, filtering, rotation, and translation. Implementing these functions in the compressed domain is advantageous from the computational complexity point of view, as well as the image quality and the memory usage (space and number of accesses). This is because the transition to the time or spatial domain and the recompression of the data are avoided. All existing compres- sion standards—JPEG, MPEG1, MPEG2, H.26x—are based on the discrete cosine transform (DCT), which is applied on blocks of data of certain length. It is this transform that is examined in the present letter. Specifically, the problem that is studied is the direct calculation of the DCT coefficients, when the two adjacent sets of DCT coefficients are given [Fig. 1(a)]. The traditional way of implementing this calculation is depicted in Fig. 1(b). According to this, one has first to recover the data to the time (or spatial) domain by calculating the inverse DCT of the two coefficient sequences and then to retransform them by means of an - point DCT. The proposed approach finds direct application to image browsing, where it may be sufficient to initially deliver an image of lower resolution to the user. Then, based on user’s response, the higher resolution image could be provided by the server. Manuscript received September 25, 1998. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. S. Reeves. The author is with the Electronics Laboratory, University of Patras, GR- 26110 Patras, Greece (e-mail: skodras@physics.upatras.gr). Publisher Item Identifier S 1070-9908(99)05792-2. II. PROPOSED APPROACH Let us assume that the DCT coefficients and of two consecutive data sequences and are given. The problem to be addressed is the efficient computation of given and where are the DCT coefficients of the sequence created by the concatenation of and In other words, and for The normalized forward DCT-II of the length- sequence is given [6] as follows: (1) and the inverse DCT (IDCT or DCT is given as follows: (2) where for and for Notice that and The DCT and the IDCT for the length- sequences are given by similar expressions, the exception being that is replaced by The computation of is performed separately for the even- and the odd-indexed coefficients, as follows. A. Even-Indexed Coefficient Calculation (3) 1070–9908/99$10.00  1999 IEEE