Clustering-Based Low-Rank Matrix Approximation for Medical Image Compression

TL;DR

This work introduces an adaptive LoRMA, which partitions a medical image into overlapping patches, groups structurally similar patches into clusters using k-means, and performs SVD within each cluster, which consistently outperforms global SVD in PSNR, SSIM, IoU, EPI, and achieving lower MSE.

Abstract

Medical images are inherently high-resolution and contain locally varying structures crucial for diagnosis. Efficient compression must preserve diagnostic fidelity while minimizing redundancy. Low-rank matrix approximation (LoRMA) techniques have shown strong potential for image compression by capturing global correlations; however, they often fail to adapt to local structural variations across regions of interest. To address this, we introduce an adaptive LoRMA, which partitions a medical image into overlapping patches, groups structurally similar patches into clusters using k-means, and performs SVD within each cluster. We derive the overall compression factor accounting for patch overlap and analyze how patch size influences compression efficiency and computational cost. While applicable to any data with high local variation, we focus on medical imaging due to its pronounced local variability. We evaluate and compare our adaptive LoRMA against global SVD across four imaging modalities: MRI, ultrasound, CT scan, and chest X-ray. Results demonstrate that adaptive LoRMA effectively preserves structural integrity, edge details, and diagnostic relevance, measured by PSNR, SSIM, MSE, IoU, and EPI. Adaptive LoRMA minimizes block artifacts and residual errors, particularly in pathological regions, consistently outperforming global SVD in PSNR, SSIM, IoU, EPI, and achieving lower MSE. It prioritizes clinically salient regions while allowing aggressive compression in non-critical regions, optimizing storage efficiency. Although adaptive LoRMA requires higher processing time, its diagnostic fidelity justifies the overhead for high-compression applications.

Figures (23)

• Figure 1: Overview of the cluster-based and global SVD for data compression. (a) Global SVD applies truncated SVD directly to the original medical image $\mathbf{A}$, yielding low-rank components $\mathbf{U}_{r_g}$, $\boldsymbol{\Sigma}_{r_g}$, and $\mathbf{V}_{r_g}^T$, from which the compressed medical image $\mathbf{A}_{r_g}$ is reconstructed. (b) In cluster-based SVD, the original medical image $\mathbf{A}$ is first split into overlapping $p\times p$ patches $\{\mathbf{P}_i\}_{i=1}^N$. Each patch is vectorized into $\mathbb{R}^{p^2}$ and grouped into $K$ clusters $\{\mathcal{C}_k\}_{k=1}^K$ using k‑means, which are used to form cluster matrices $\mathbf{C}_k\in\mathbb R^{N_{\mathbf{C}_k}\times p^2}$. SVD is then applied independently to each $\mathbf{C}_k$ to produce compressed cluster matrices $\widehat{\mathbf{C}}_k$. Finally, the compressed patches are aggregated to reconstruct the compressed medical image $\widehat{\mathbf{A}}$.
• Figure 2: These plots compare how the number of retained singular values (horizontal axis) affects the cumulative energy (red curve, left vertical axis) and the compression factor (blue curve, right vertical axis) for two randomly selected cluster matrices, $\mathbf{C}_{0}$ (left) and $\mathbf{C}_{2}$ (right). The dashed vertical lines indicate the minimal number of singular values, $r_{\mathbf{C}_k}$, needed for 95% energy retention.
• Figure 3: Boxplot showing the intra-cluster variability (Frobenius norm) for different numbers of clusters. As $K$ increases, the intra-cluster variability decreases, showing a more homogeneous clustering of patches. Smaller $K$ values lead to higher variability as different patch features are grouped together, while larger $K$ leads to lower variability as patches with similar features are better captured.
• Figure 4: Heatmaps of normalized singular value decay across clusters for various patch sizes. Each subfigure corresponds to a different patch size: $4^2$, $8^2$, $16^2$, and $32^2$. The main heatmaps illustrate the singular value distribution for all clusters, with zoomed-in views showing the first 16 singular values. Dashed lines connect the highlighted region in the main heatmap to its corresponding zoomed-in view.
• Figure 5: Comparison of global and cluster-based SVD compression for two MRI brain scans with tumors located in different regions. For the first case (a-h), the tumor is situated in a specific brain region. (a) shows the original MRI image, and (b) presents the corresponding binary mask highlighting the tumor area. (c) and (d) display the reconstructed images using global SVD and cluster-based SVD methods, respectively, at a compression factor of approximately 118. The residual maps within the tumor region are shown in (e) and (f) for global and cluster-based approaches, respectively, while (g) and (h) illustrate the residuals outside the tumor area. For the second case (i-p), the tumor appears in a different brain region. (i) presents the original image, and (j) shows the corresponding tumor mask. The compressed reconstructions using global SVD and cluster-based SVD at a compression factor of about 64 are shown in (k) and (l), respectively. (m) and (n) depict the residuals within the tumor area for the global and cluster-based methods, while (o) and (p) show the residuals outside the tumor area.

Theorems & Definitions (6)

• Proposition 1
• proof : Proof
• Proposition 2
• proof : Proof
• Proposition 3
• proof : Proof