Fast Data-independent KLT Approximations Based on Integer Functions

The Karhunen-Lo\`eve 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.