Fast Data-independent KLT Approximations Based on Integer Functions

TL;DR

This work addresses the high computational cost and data-dependency of the Karhunen-Loève transform (KLT) by introducing data-independent, low-complexity approximations derived from integer-rounding functions. By constraining transform matrices to lightweight, multiplierless entries and optimizing across a range of correlation coefficients $\rho$, the authors identify several 8-point transforms that rival or exceed the exact KLT in image compression scenarios, sometimes outperforming the DCT as well. A two-stage optimization with $k$-means clustering yields robust transform candidates for different $\rho$ regimes, and fast algorithms are developed by factorizing the transform matrices into sparse components, significantly reducing arithmetic complexity. Hardware validation on an FPGA demonstrates practical feasibility, with one transform ($\textbf{T}_1$) offering the best resource efficiency and others (e.g., $\textbf{T}_{16}$, $\textbf{T}_{18}$) showing strong performance at higher resource budgets. Overall, the paper extends the utility of KLT-like transforms to low-power, real-time image encoding by delivering data-independent, efficient, and hardware-friendly approximations across a broad range of signal correlations.

Abstract

The Karhunen-Loève transform (KLT) stands as a well-established discrete transform, demonstrating optimal characteristics in data decorrelation and dimensionality reduction. Its ability to condense energy compression into a select few main components has rendered it instrumental in various applications within image compression frameworks. However, computing the KLT depends on the covariance matrix of the input data, which makes it difficult to develop fast algorithms for its implementation. Approximations for the KLT, utilizing specific rounding functions, have been introduced to reduce its computational complexity. Therefore, our paper introduces a category of low-complexity, data-independent KLT approximations, employing a range of round-off functions. The design methodology of the approximate transform is defined for any block-length $N$, but emphasis is given to transforms of $N = 8$ due to its wide use in image and video compression. The proposed transforms perform well when compared to the exact KLT and approximations considering classical performance measures. For particular scenarios, our proposed transforms demonstrated superior performance when compared to KLT approximations documented in the literature. We also developed fast algorithms for the proposed transforms, further reducing the arithmetic cost associated with their implementation. Evaluation of field programmable gate array (FPGA) hardware implementation metrics was conducted. Practical applications in image encoding showed the relevance of the proposed transforms. In fact, we showed that one of the proposed transforms outperformed the exact KLT given certain compression ratios.

Figures (10)

• Figure 1: Example of $\rho$ intervals in groups.
• Figure 2: Signal flow graph for $\mathbf{T}_1$, $\mathbf{T}_3$, and $\mathbf{T}_{13}$.
• Figure 3: Signal flow graph for $\mathbf{T}_{16}$ and $\mathbf{T}_{17}$.
• Figure 4: Signal flow graph for $\mathbf{T}_{18}$.
• Figure 5: Blocks $\mathbf{M}_1$ and $\mathbf{M}_2$ from the signal flow graphs.