[Paper Review] Compressed Sensing and Reconstruction of Unstructured Mesh Datasets

Maher Salloum, Nathan Fabian+2 more|arXiv (Cornell University)|Aug 25, 2015

Sparse and Compressive Sensing Techniques13 references3 citations

TL;DR

This paper proposes compressive sensing (CS) using second-generation wavelets (SGW) and Stagewise Orthogonal Matching Pursuit (StOMP) for lossy compression of unstructured mesh datasets from finite element simulations. It achieves compression ratios of 10–30 with minimal visual degradation, enabling efficient in-situ storage and interactive reconstruction without altering simulation code.

ABSTRACT

Exascale computing promises quantities of data too large to efficiently store and transfer across networks in order to be able to analyze and visualize the results. We investigate Compressive Sensing (CS) as a way to reduce the size of the data as it is being stored. CS works by sampling the data on the computational cluster within an alternative function space such as wavelet bases, and then reconstructing back to the original space on visualization platforms. While much work has gone into exploring CS on structured data sets, such as image data, we investigate its usefulness for point clouds such as unstructured mesh datasets found in many finite element simulations. We sample using second generation wavelets (SGW) and reconstruct using the Stagewise Orthogonal Matching Pursuit (StOMP) algorithm. We analyze the compression ratios achievable and quality of reconstructed results at each compression rate. We are able to achieve compression ratios between 10 and 30 on moderate size datasets with minimal visual deterioration as a result of the lossy compression.

Motivation & Objective

• Address the challenge of exascale simulation data growth, where I/O and storage bottlenecks hinder analysis and visualization.
• Explore compressive sensing (CS) as a non-intrusive, low-overhead method for compressing unstructured mesh datasets, which are common in finite element simulations.
• Extend CS theory from structured lattices (e.g., images) to unstructured point-cloud data such as finite element meshes.
• Enable interactive, high-quality reconstruction of compressed data by selecting wavelet bases and sampling matrices during post-processing.
• Achieve high compression ratios while maintaining visual fidelity and minimizing reconstruction error for transient temperature fields on irregular meshes.

Proposed method

• Apply compressive sensing (CS) to unstructured mesh datasets by sampling in a transform domain using second-generation wavelets (SGW), which are well-suited for irregular geometries and meshes.
• Use random sampling matrices (Φ) incoherent with SGW bases to ensure restricted isometry property (RIP), enabling stable and unique reconstruction from compressed samples.
• Formulate the reconstruction as a sparse recovery problem: solve y = ΦΨs for wavelet coefficients s using the Stagewise Orthogonal Matching Pursuit (StOMP) algorithm.
• Perform compression locally on each processor in a parallel simulation, avoiding inter-processor communication and enabling efficient in-situ compression.
• Reconstruct the full dataset by assembling the individually reconstructed portions on a visualization workstation, allowing for level-of-detail control via wavelet order selection.
• Use the wavelet basis matrix Ψ and sampling matrix Φ to define the system A = ΦΨ, where y = As is solved to recover sparse coefficients s from compressed measurements y.

Experimental results

Research questions

• RQ1 Can compressive sensing be effectively applied to unstructured mesh datasets, which lack regular grid structure, to enable high compression with minimal quality loss?
• RQ2 What compression ratios and reconstruction errors can be achieved using second-generation wavelets (SGW) and StOMP on finite element simulation data with complex, heterogeneous geometries?
• RQ3 How does the presence of local gradients and discontinuities in transient temperature fields affect reconstruction accuracy and error dynamics?
• RQ4 To what extent can in-situ compression be performed without modifying the simulation code, and how does local compression per processor impact reconstruction fidelity?
• RQ5 Can interactive, multi-resolution reconstruction be achieved by selecting wavelet basis and reconstruction parameters post-compression based on desired quality and performance trade-offs?

Key findings

• The method achieves compression ratios between 10 and 30 on moderate-sized unstructured mesh datasets, with minimal visual deterioration at these rates.
• For a compression ratio of 10, reconstructed temperature fields visually match the original data across all processor partitions, indicating strong perceptual fidelity.
• Reconstruction error decreases over time in transient simulations as temperature fields smooth out, with the lowest errors observed at later time steps due to reduced gradients.
• Higher wavelet orders (w=5) reduce reconstruction error in smoother regions but can increase error at early times due to overfitting of sharp gradients.
• A saturation and slight increase in error at very high wavelet orders are observed, indicating a trade-off between resolution and robustness to discontinuities.
• Local gradients and discontinuities significantly impact reconstruction quality, suggesting that compression ratio estimation should account for local data complexity in addition to global smoothness.