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

TL;DR

This work addresses the computational burden of the DCT in image/video codecs by deriving low‑complexity, orthogonal DCT approximations through an angle‑based greedy optimization that treats each DCT row individually. By exploring compact search spaces and enforcing orthogonality, the authors obtain new 8‑point transforms, notably \widehat{\mathbf{C}}_1, which achieves superior MSE, coding gain, and transform efficiency relative to state‑of‑the‑art approximations, and even outperforms the exact DCT at some compression levels. A fast butterfly‑like implementation reduces arithmetic cost to 24 additions and 6 bit shifts, and FPGA/hardware realizations verify practical viability. The approximations are evaluated on still images and in HEVC video coding, showing substantial gains in many regimes while maintaining close or superior perceptual quality, underscoring the potential of angle‑based DCT approximations for real‑time, resource‑constrained applications.

Abstract

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.

Figures (12)

• Figure 1: Algorithm for the proposed method.
• Figure 2: Curves for the coding gain error of $\widehat{\mathbf{C}}_1$, $\widehat{\mathbf{C}}_\text{LO}$, and $\widehat{\mathbf{C}}_6$, relative to the exact DCT, for $0<\rho<1$.
• Figure 3: Signal flow graph of the proposed transform, relating the input data $x_n$, $n = 0, 1, \ldots , 7$, to its correspondent coefficients $\tilde{X}_k$, $k = 0, 1, \ldots , 7$, where $\mathbf{\tilde{X}} = \mathbf{x} \cdot \mathbf{T}_1$. of $\mathbf{T}_1$. Dashed arrows representing multiplication by $-1$.
• Figure 4: Architectures for (a) $\mathbf{T}_1$, (b) $\mathbf{T}_\text{LO}$, and (c) $\mathbf{T}_6$.
• Figure 5: Curves for the average of (a) MSE; (b) PSNR; and (c) SSIM corresponding to 44 images.