Performance Analysis of DCT, Hadamard, and PCA in Block-Based Image Compression

TL;DR

This study systematically compares fixed transforms (DCT, Hadamard) with a learned transform (PCA) for block-based image compression across multiple block sizes and rates. By evaluating PSNR, rate–distortion, and energy compaction, the work shows that PCA only outperforms fixed transforms when block dimensionality is large enough to support reliable covariance estimation, while the DCT remains near-optimal for standard $8\times8$ blocks and low bitrates. The Hadamard transform consistently performs worst due to poor energy concentration. The findings explain the persistent dominance of DCT in practical codecs and highlight the limitations of block-wise learned transforms, suggesting future work on training with larger datasets and additional transforms.

Abstract

Block based image compression relies on transform coding to concentrate signal energy into a small number of coefficients. While classical codecs use fixed transforms such as the Discrete Cosine Transform (DCT), data driven methods such as Principal Component Analysis (PCA) are theoretically optimal for decorrelation. This paper presents an experimental comparison of DCT, Hadamard, and PCA across multiple block sizes and compression rates. Using rate distortion and energy compaction analysis, we show that PCA outperforms fixed transforms only when block dimensionality is sufficiently large, while DCT remains near optimal for standard block sizes such as $8\times8$ and at low bit rates. These results explain the robustness of DCT in practical codecs and highlight the limitations of block wise learned transforms.

Figures (6)

• Figure 1: Rate–distortion curves for $4\times4$ blocks. DCT achieves the highest PSNR, while PCA does not provide significant benefit at this scale.
• Figure 2: Rate–distortion curves for $8\times8$ blocks. The DCT achieves the highest PSNR across most rates, while PCA does not provide significant improvement at this scale.
• Figure 3: Rate–distortion curves for $16\times16$ blocks. PCA begins to outperform DCT and Hadamard at medium and high rates as the increased dimensionality enables better covariance estimation.
• Figure 4: Rate–distortion curves for $32\times32$ blocks. PCA dramatically outperforms both DCT and Hadamard, achieving high PSNR even at low rates due to strong energy compaction.
• Figure 5: Rate–distortion behavior of DCT, Hadamard, and PCA transforms across different block sizes, showing the transition from DCT dominance at small blocks to PCA dominance at large blocks.