Quantization-aware Matrix Factorization for Low Bit Rate Image Compression (2408.12691v2)

Abstract: Lossy image compression is essential for efficient transmission and storage. Traditional compression methods mainly rely on discrete cosine transform (DCT) or singular value decomposition (SVD), both of which represent image data in continuous domains and, therefore, necessitate carefully designed quantizers. Notably, these methods consider quantization as a separate step, where quantization errors cannot be incorporated into the compression process. The sensitivity of these methods, especially SVD-based ones, to quantization errors significantly degrades reconstruction quality. To address this issue, we introduce a quantization-aware matrix factorization (QMF) to develop a novel lossy image compression method. QMF provides a low-rank representation of the image data as a product of two smaller factor matrices, with elements constrained to bounded integer values, thereby effectively integrating quantization with low-rank approximation. We propose an efficient, provably convergent iterative algorithm for QMF using a block coordinate descent (BCD) scheme, with subproblems having closed-form solutions. Our experiments on the Kodak and CLIC 2024 datasets demonstrate that our QMF compression method consistently outperforms JPEG at low bit rates below 0.25 bits per pixel (bpp) and remains comparable at higher bit rates. We also assessed our method's capability to preserve visual semantics by evaluating an ImageNet pre-trained classifier on compressed images. Remarkably, our method improved top-1 accuracy by over 5 percentage points compared to JPEG at bit rates under 0.25 bpp. The project is available at https://github.com/pashtari/lrf .

• The paper presents a novel approach that replaces quantization with integer matrix factorization in lossy image compression.
• It employs an iterative Block Coordinate Descent algorithm with closed-form updates to achieve low-rank approximation using integer matrices.
• Experimental results demonstrate higher PSNR, SSIM, and classification accuracy at low bit rates compared to traditional JPEG and SVD techniques.

Quantization-free Lossy Image Compression Using Integer Matrix Factorization

The paper "Quantization-free Lossy Image Compression Using Integer Matrix Factorization" introduces a novel approach to lossy image compression that bypasses the quantization step traditionally required by methods such as JPEG and those based on Singular Value Decomposition (SVD). The authors propose a methodology founded on Integer Matrix Factorization (IMF), which inherently operates without the need for quantization. This is achieved by representing image data as low-rank approximations involving integer values, thus potentially circumventing the quantization errors typically observed in other methods.

Introduction

In the context of image compression, traditional methods, including JPEG and JPEG 2000, use orthogonal linear transformations like Discrete Cosine Transform (DCT) and Discrete Wavelet Transform (DWT) to represent image data in a continuous domain. As a result, these methods require a quantization step prior to byte-level coding, which can introduce significant quantization errors. While SVD-based methods find application due to their energy compaction properties, they again face an acute sensitivity to quantization errors, particularly at low bit rates.

The paper addresses these limitations by proposing IMF as an alternative that maintains integer-only data representations. Through this novel approach, the researchers aim to deliver effective compression performance while minimizing reconstruction errors.

Method

Overall Encoding Framework

The proposed method consists of several key steps:

1. Color Space Transformation: Images are first transformed to the YC\textsubscript{B}C\textsubscript{R} color space.
2. Chroma Downsampling: The chroma components are downsampled.
3. Patchification: The image is split into non-overlapping patches, which are then flattened and stacked to form matrices ready for factorization.
4. Low-rank Approximation Using IMF: These matrices are then low-rank approximated using IMF, which factors them into products of smaller integer matrices.
5. Lossless Compression: The integer matrices are directly encoded using a lossless compression standard.

Each of these steps is carefully designed to leverage integer matrix operations, ensuring that the entire pipeline remains quantization-free.

Integer Matrix Factorization (IMF)

IMF aims to approximate a given matrix $\bm{X} \in \mathbb{R}^{M \times N}$ as a product $\bm{U} \bm{V}^{\mathsf{T}}$ where $\bm{U}$ and $\bm{V}$ are integer matrices within specified bounds. The formulation is crucial because it eliminates the need for quantization, thus significantly reducing errors.

A unique iterative algorithm based on Block Coordinate Descent (BCD) is employed for solving the IMF problem. The authors derive closed-form solutions for updating each factor in this setting, ensuring the convergence and efficiency of the algorithm.

Experiments and Results

The proposed IMF method was extensively evaluated against JPEG and SVD-based compression methods using datasets such as Kodak, CLIC 2024, and the ImageNet validation set. Evaluation metrics included PSNR, SSIM, and classification accuracy for a pre-trained ImageNet classifier on compressed images.

Rate-Distortion Performance

The results indicated that IMF consistently outperformed JPEG and SVD, particularly at low bit rates. For instance, on the Kodak and CLIC 2024 datasets, IMF achieved higher PSNR and SSIM values at bit rates below 0.25 bpp, while remaining competitive at higher bit rates. Additionally, IMF demonstrated superior performance compared to SVD across all evaluated bit rates.

Qualitative Performance

Qualitative analysis further highlighted the advantages of IMF compression. When compared at similar bit rates, IMF provided better preservation of color fidelity and texture, avoiding the typical artifacts seen in JPEG such as blocking and color bleeding.

ImageNet Classification Performance

IMF's ability to maintain semantic integrity in images was also verified. The ResNet-50 classifier achieved over 5% higher top-1 accuracy on IMF-compressed images compared to those compressed using JPEG, particularly noticeable at bit rates below 0.25 bpp.

Discussion

The research reveals that IMF provides a significant improvement over traditional methods by effectively balancing compression efficiency with image quality. The quantization-free nature of IMF addresses the inherent limitations of SVD and JPEG related to quantization errors. Moreover, the methodology's compatibility with existing lossless compression standards adds to its practical applicability.

Auxiliary ablation studies underscored the impact of various parameters such as factor bounds, the number of BCD iterations, and patch size on the performance, suggesting that optimal configuration is crucial for maximizing IMF's benefits.

Conclusion

The proposed IMF method sets a promising direction in the domain of lossy image compression by eliminating the need for quantization, thus enhancing reconstruction accuracy. The efficient iterative algorithm ensures practical feasibility, and empirical results affirm IMF's superior performance in maintaining visual and semantic quality at lower bit rates. Future developments may incorporate entropy-aware regularization to further refine compression efficiency and mitigate any artifacts associated with patchification.