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

• A method for deriving low-complexity DCT approximations based on integer functions is proposed.
• Obtained approximations are multiplierless and require only additions and bit-shifting operations.
• Derived approximations generalize several existing methods, including the SDCT, and offer state-of-the-art performance as an image compression tool.
• A parametric fast algorithm is introduced.

The discrete cosine transform (DCT) is a central mathematical operation in several digital signal processing methods and image/video standards. In this paper, we propose a collection of twelve approximations for the 8-point DCT based on integer functions. Considered functions include: the floor, ceiling, truncation, and rounding-off functions. Sought approximations are required to meet the following specific criteria: (i) very low arithmetic complexity, (ii) orthogonality or quasi-orthogonality, and (iii) low-complexity inversion. By varying a scaling parameter, approximations could be systematically obtained and several existing approximations were identified as particular cases of the proposed methodology. Particular cases include the signed DCT and the rounded DCT. Four new quasi-orthogonal approximations were introduced and their practical relevance was demonstrated. All approximations were given fast algorithms based on matrix factorization methods. Proposed approximations are multiplierless; their computation requires only additions and bit-shifting operations. Additive complexity ranged from 18 to 24 additions. Obtained approximations were compared with the exact DCT and assessed in the context of JPEG-like image compression. As quality assessment measures, we considered the peak signal-to-noise ratio and the structural similarity index. Because its low-complexity and good performance properties, the proposed approximations are suitable for hardware implementation in dedicated architectures.