Low-complexity Scaling Methods for DCT-II Approximations

TL;DR

The paper tackles efficient construction of 2N-point DCT-II approximations from N-point low-complexity transforms using Hou's recursive factorization, framing JAM scaling as a special case within a broader scalable framework. It develops a family of scaling methods based on parameter matrices $\hat{\mathbf{B}}_N$ and $\hat{\mathbf{G}}_N$ and establishes sufficient orthogonality conditions, supported by extensive error and performance analyses. Empirical results show that several proposed methods (notably Methods VI and VII) outperform JAM in total error energy and coding gain across multiple 16- and 8-point baselines, with multiplierless implementations and favorable arithmetic complexity. A hardware demonstration on an FPGA using an ABDCT core confirms the practicality of the scaling approaches, highlighting favorable timing and power characteristics and showcasing viable paths for energy-efficient DCT-II computation in image/video codecs.

Abstract

This paper introduces a collection of scaling methods for generating $2N$-point DCT-II approximations based on $N$-point low-complexity transformations. Such scaling is based on the Hou recursive matrix factorization of the exact $2N$-point DCT-II matrix. Encompassing the widely employed Jridi-Alfalou-Meher scaling method, the proposed techniques are shown to produce DCT-II approximations that outperform the transforms resulting from the JAM scaling method according to total error energy and mean squared error. Orthogonality conditions are derived and an extensive error analysis based on statistical simulation demonstrates the good performance of the introduced scaling methods. A hardware implementation is also provided demonstrating the competitiveness of the proposed methods when compared to the JAM scaling method.

Figures (3)

• Figure 1: $\|\hat{\mathbf{C}}^\text{II}_{16}-\mathbf{C}^\text{II}_{16}\|_{\mathsf{F}}$ as a function of $\|\hat{\mathbf{C}}^\text{II}_{8}-\mathbf{C}^\text{II}_{8}\|_{\mathsf{F}}$ for the methods outlined in Table \ref{['tab:scaling_families']}.
• Figure 2: Block diagram for the proposed scaling methods for DCT approximation. For the hardware implementation using the ABDCT, the $\mathbf{T}_{N}$ sub-blocks implement the ABDCT.
• Figure 3: Testbed architecture for testing the implemented designs.

Theorems & Definitions (5)

• Proposition 1
• proof
• Lemma 1
• Proposition 2
• proof