Low-complexity 8-point DCT approximations based on integer functions

The discrete cosine transform (DCT) is widely regarded as a key operation in digital signal processing [15,51]. In fact, the Karhunen-Loève transform (KLT) is the asymptotic equivalent of the DCT, being the former an optimal transform in terms of decorrelation and energy compaction properties [1,15,18,23,37,51]. When high correlated first-order Markov signals are considered [15, 51]—such as natural images [37]— the DCT can closely emulate the KLT [1]. The DCT has been considered and effectively adopted in a number of methods for image and video coding [6]. In fact, the DCT is the central mathematical operation for the following standards: JPEG [46,58], MPEG-1 [52], MPEG-2 [28], H.261 [30], H.263 [31], H.264 [33,40,61,61], and the recent HEVC [8,49,54]. In all above standards, the particular 8-point DCT is considered. Thus, developing fast algorithms for the efficient evaluation of the 8-point DCT is a main task in the circuits, systems, and signal processing communities. Archived literature contains a multitude of fast algorithms for this particular blocklength [26, 57]. Remarkably extensive reports have been generated amalgamating scattered results for the 8-point DCT [15,51]. Among the most popular techniques, we mention the following algorithms: Wang factorization [59], Lee DCT for power-of-two blocklengths [35], Arai DCT scheme [2], Loeffler algorithm [39], Vetterli-Nussbaumer algorithm [57], Hou algorithm [26], and Feig-Winograd factorization [20]. All these methods are classical results in the field and have been considered for practical applications [38, 52, 56]. For instance, the Arai DCT scheme was employed in various recent hardware implementations of the DCT [19,42, 50]. Naturally, DCT fast algorithms that result in major computational savings compared to direct computation of the DCT were already developed decades ago. In fact, the intense research in the field has led R. J. Cintra is with the Signal Processing Group, Departamento de Estat́ıstica, Universidade Federal de Pernambuco, Recife, Brazil. E-mail: [email protected] F. M. Bayer is with the Departamento de Estat́ıstica and Laboratório de Ciências Espaciais de Santa Maria (LACESM), Universidade Federal de Santa Maria, Santa Maria, RS, Brazil. Email: [email protected]. C. J. Tablada is with the Signal Processing Group and the Graduate Program in Statistics, Universidade Federal de Pernambuco, Recife, PE, Brazil.