A Class of DCT Approximations Based on the Feig-Winograd Algorithm

A new class of matrices based on a parametrization of the Feig-Winograd factorization of 8-point DCT is proposed. Such parametrization induces a matrix subspace, which unifies a number of existing methods for DCT approximation. By solving a comprehensive multicriteria optimization problem, we identified several new DCT approximations. Obtained solutions were sought to possess the following properties: (i) low multiplierless computational complexity, (ii) orthogonality or near orthogonality, (iii) low complexity invertibility, and (iv) close proximity and performance to the exact DCT. Proposed approximations were submitted to assessment in terms of proximity to the DCT, coding performance, and suitability for image compression. Considering Pareto efficiency, particular new proposed approximations could outperform various existing methods archived in literature.Comment: 26 pages, 4 figures, 5 tables, fixed arithmetic complexity in Table I

Multiplierless 16-point DCT Approximation for Low-complexity Image and Video Coding

An orthogonal 16-point approximate discrete cosine transform (DCT) is introduced. The proposed transform requires neither multiplications nor bit-shifting operations. A fast algorithm based on matrix factorization is introduced, requiring only 44 additions---the lowest arithmetic cost in literature. To assess the introduced transform, computational complexity, similarity with the exact DCT, and coding performance measures are computed. Classical and state-of-the-art 16-point low-complexity transforms were used in a comparative analysis. In the context of image compression, the proposed approximation was evaluated via PSNR and SSIM measurements, attaining the best cost-benefit ratio among the competitors. For video encoding, the proposed approximation was embedded into a HEVC reference software for direct comparison with the original HEVC standard. Physically realized and tested using FPGA hardware, the proposed transform showed 35% and 37% improvements of area-time and area-time-squared VLSI metrics when compared to the best competing transform in the literature.Comment: 12 pages, 5 figures, 3 table

Low-complexity 8-point DCT Approximation Based on Angle Similarity for Image and Video Coding

The principal component analysis (PCA) is widely used for data decorrelation and dimensionality reduction. However, the use of PCA may be impractical in real-time applications, or in situations were energy and computing constraints are severe. In this context, the discrete cosine transform (DCT) becomes a low-cost alternative to data decorrelation. This paper presents a method to derive computationally efficient approximations to the DCT. The proposed method aims at the minimization of the angle between the rows of the exact DCT matrix and the rows of the approximated transformation matrix. The resulting transformations matrices are orthogonal and have extremely low arithmetic complexity. Considering popular performance measures, one of the proposed transformation matrices outperforms the best competitors in both matrix error and coding capabilities. Practical applications in image and video coding demonstrate the relevance of the proposed transformation. In fact, we show that the proposed approximate DCT can outperform the exact DCT for image encoding under certain compression ratios. The proposed transform and its direct competitors are also physically realized as digital prototype circuits using FPGA technology.Comment: 16 pages, 12 figures, 10 table

Efficient Computation of the 8-point DCT via Summation by Parts

This paper introduces a new fast algorithm for the 8-point discrete cosine transform (DCT) based on the summation-by-parts formula. The proposed method converts the DCT matrix into an alternative transformation matrix that can be decomposed into sparse matrices of low multiplicative complexity. The method is capable of scaled and exact DCT computation and its associated fast algorithm achieves the theoretical minimal multiplicative complexity for the 8-point DCT. Depending on the nature of the input signal simplifications can be introduced and the overall complexity of the proposed algorithm can be further reduced. Several types of input signal are analyzed: arbitrary, null mean, accumulated, and null mean/accumulated signal. The proposed tool has potential application in harmonic detection, image enhancement, and feature extraction, where input signal DC level is discarded and/or the signal is required to be integrated.Comment: Fixed Fig. 1 with the block diagram of the proposed architecture. Manuscript contains 13 pages, 4 figures, 2 table

The Arithmetic Cosine Transform: Exact and Approximate Algorithms

In this paper, we introduce a new class of transform method --- the arithmetic cosine transform (ACT). We provide the central mathematical properties of the ACT, necessary in designing efficient and accurate implementations of the new transform method. The key mathematical tools used in the paper come from analytic number theory, in particular the properties of the Riemann zeta function. Additionally, we demonstrate that an exact signal interpolation is achievable for any block-length. Approximate calculations were also considered. The numerical examples provided show the potential of the ACT for various digital signal processing applications.Comment: 17 pages, 3 figure

A Multiparametric Class of Low-complexity Transforms for Image and Video Coding

Discrete transforms play an important role in many signal processing applications, and low-complexity alternatives for classical transforms became popular in recent years. Particularly, the discrete cosine transform (DCT) has proven to be convenient for data compression, being employed in well-known image and video coding standards such as JPEG, H.264, and the recent high efficiency video coding (HEVC). In this paper, we introduce a new class of low-complexity 8-point DCT approximations based on a series of works published by Bouguezel, Ahmed and Swamy. Also, a multiparametric fast algorithm that encompasses both known and novel transforms is derived. We select the best-performing DCT approximations after solving a multicriteria optimization problem, and submit them to a scaling method for obtaining larger size transforms. We assess these DCT approximations in both JPEG-like image compression and video coding experiments. We show that the optimal DCT approximations present compelling results in terms of coding efficiency and image quality metrics, and require only few addition or bit-shifting operations, being suitable for low-complexity and low-power systems.Comment: Fixed Figure 1 and typos in the reference lis

An Orthogonal 16-point Approximate DCT for Image and Video Compression

A low-complexity orthogonal multiplierless approximation for the 16-point discrete cosine transform (DCT) was introduced. The proposed method was designed to possess a very low computational cost. A fast algorithm based on matrix factorization was proposed requiring only 60~additions. The proposed architecture outperforms classical and state-of-the-art algorithms when assessed as a tool for image and video compression. Digital VLSI hardware implementations were also proposed being physically realized in FPGA technology and implemented in 45 nm up to synthesis and place-route levels. Additionally, the proposed method was embedded into a high efficiency video coding (HEVC) reference software for actual proof-of-concept. Obtained results show negligible video degradation when compared to Chen DCT algorithm in HEVC.Comment: 18 pages, 7 figures, 6 table

An Integer Approximation Method for Discrete Sinusoidal Transforms

Approximate methods have been considered as a means to the evaluation of discrete transforms. In this work, we propose and analyze a class of integer transforms for the discrete Fourier, Hartley, and cosine transforms (DFT, DHT, and DCT), based on simple dyadic rational approximation methods. The introduced method is general, applicable to several block-lengths, whereas existing approaches are usually dedicated to specific transform sizes. The suggested approximate transforms enjoy low multiplicative complexity and the orthogonality property is achievable via matrix polar decomposition. We show that the obtained transforms are competitive with archived methods in literature. New 8-point square wave approximate transforms for the DFT, DHT, and DCT are also introduced as particular cases of the introduced methodology.Comment: 13 pages, 5 figures, 8 table

Improved 8-point Approximate DCT for Image and Video Compression Requiring Only 14 Additions

Video processing systems such as HEVC requiring low energy consumption needed for the multimedia market has lead to extensive development in fast algorithms for the efficient approximation of 2-D DCT transforms. The DCT is employed in a multitude of compression standards due to its remarkable energy compaction properties. Multiplier-free approximate DCT transforms have been proposed that offer superior compression performance at very low circuit complexity. Such approximations can be realized in digital VLSI hardware using additions and subtractions only, leading to significant reductions in chip area and power consumption compared to conventional DCTs and integer transforms. In this paper, we introduce a novel 8-point DCT approximation that requires only 14 addition operations and no multiplications. The proposed transform possesses low computational complexity and is compared to state-of-the-art DCT approximations in terms of both algorithm complexity and peak signal-to-noise ratio. The proposed DCT approximation is a candidate for reconfigurable video standards such as HEVC. The proposed transform and several other DCT approximations are mapped to systolic-array digital architectures and physically realized as digital prototype circuits using FPGA technology and mapped to 45 nm CMOS technology.Comment: 30 pages, 7 figures, 5 table

Low-complexity Architecture for AR(1) Inference

In this Letter, we propose a low-complexity estimator for the correlation coefficient based on the signed $\operatorname{AR}(1)$ process. The introduced approximation is suitable for implementation in low-power hardware architectures. Monte Carlo simulations reveal that the proposed estimator performs comparably to the competing methods in literature with maximum error in order of $10^{-2}$. However, the hardware implementation of the introduced method presents considerable advantages in several relevant metrics, offering more than 95% reduction in dynamic power and doubling the maximum operating frequency when compared to the reference method.Comment: 7 pages, 3 tables, 4 figure