Extensions on Low-complexity DCT Approximations for Larger Blocklengths Based on Minimal Angle Similarity
A. P. Radünz, L. Portella, R. S. Oliveira, F. M. Bayer, R. J. Cintra
TL;DR
This work addresses the need for low-complexity, large-block DCT approximations in image/video coding by extending an angular-error minimization approach to $N\in\{16,32,64\}$, yielding new transforms $\widehat{\mathbf{C}}_N$ that closely mimic the exact DCT $\mathbf{C}_N$ while remaining multiplierless. The method builds on minimizing the angle between exact DCT rows $\mathbf{c}_k^T$ and low-complexity rows $\mathbf{t}_k^T$, with $\widehat{\mathbf{C}}_N=\mathbf{S}_N\mathbf{T}_N$ and $\mathbf{S}_N=\sqrt{[\operatorname{diag}(\mathbf{T}_N\mathbf{T}_N^T)]^{-1}}$, and explores a discrete search over $\mathcal{D}$-valued rows; equivalence classes reduce redundancy. The paper introduces JAM scaling to generate $2N$-point transforms from $N$-point ones and derives fast, butterfly-like algorithms; it also validates the approach through JPEG-like image compression, showing improvements over leading 8-point and larger-length approximations across $N=16,32,64$. Overall, the proposed multiplierless DCT approximations offer a favorable trade-off between reconstruction quality and computational cost, with practical impact for hardware-friendly implementations in modern codecs. All results are presented with careful consideration of energy, distortion, and coding-efficiency metrics, underscoring the method’s relevance to real-world image processing pipelines.
Abstract
The discrete cosine transform (DCT) is a central tool for image and video coding because it can be related to the Karhunen-Loève transform (KLT), which is the optimal transform in terms of retained transform coefficients and data decorrelation. In this paper, we introduce 16-, 32-, and 64-point low-complexity DCT approximations by minimizing individually the angle between the rows of the exact DCT matrix and the matrix induced by the approximate transforms. According to some classical figures of merit, the proposed transforms outperformed the approximations for the DCT already known in the literature. Fast algorithms were also developed for the low-complexity transforms, asserting a good balance between the performance and its computational cost. Practical applications in image encoding showed the relevance of the transforms in this context. In fact, the experiments showed that the proposed transforms had better results than the known approximations in the literature for the cases of 16, 32, and 64 blocklength.