Image Compression Using Singular Value Decomposition

TL;DR

This study investigates image compression via singular value decomposition (SVD) by constructing low-rank reconstructions and evaluating them with relative Frobenius error and compression ratio. Using a 1,000-image ImageNet subset, it compares SVD-based compression against JPEG, JPEG2000, and WEBP across a range of tolerance levels for reconstruction error. The results show that while SVD can visually approximate originals, its compression efficiency is consistently worse than industry codecs, especially at low error tolerances, limiting practical viability. The paper concludes that SVD-based methods are mostly academic in this context, with future work suggested around quantization and alternative color spaces to improve performance.

Abstract

Images are a substantial portion of the internet, making efficient compression important for reducing storage and bandwidth demands. This study investigates the use of Singular Value Decomposition and low-rank matrix approximations for image compression, evaluating performance using relative Frobenius error and compression ratio. The approach is applied to both grayscale and multichannel images to assess its generality. Results show that the low-rank approximations often produce images that appear visually similar to the originals, but the compression efficiency remains consistently worse than established formats such as JPEG, JPEG2000, and WEBP at comparable error levels. At low tolerated error levels, the compressed representation produced by Singular Value Decomposition can even exceed the size of the original image, indicating that this method is not competitive with industry-standard codecs for practical image compression.

Figures (11)

• Figure 1: 12 random colored pictures of various dimensions from the ILSVRC2017 dataset
• Figure 2: Grayscale image of dogs using rank $k = 5, 10, 20, 50, 100$ approximation (singular value decomposition) and original grayscale image.
• Figure 3: Visualization of error of approximation using rank $k = 5, 10, 20, 50, 100$ approximation(singular value decomposition) and original grayscale image. Brighter areas signify higher pixel wise error.
• Figure 4: Compression Ratio vs Relative Frobenius Error for grayscale Image
• Figure 5: Colored image of dogs using rank $k = 5, 10, 20, 50, 100$ approximation (singular value decomposition) and original colored image.