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Superposed Atomic Representation for Robust High-Dimensional Data Recovery of Multiple Low-Dimensional Structures
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Proposes Superposed Atomic Representation (SAR), a unified atomic-norm framework recovering two low-dimensional components from their sum (subsuming RSR, RPCA, OP, TRPCA), plus a Weighted SAR (WSAR) with an ADMM solver and convergence proof, evaluated on face restoration, color image recovery, and background modeling.
Problem
Recovering multiple components with different low-dimensional structures (sparse, low-rank, subspace) from their superposition in high-dimensional data, which existing methods tackle case by case.
Approach
Unify RSR, RPCA, OP, and TRPCA as special cases of recovering two signals from their sum via atomic norm regularization, then weight each atom's penalty inversely to its coefficient magnitude (WSAR) so significant atoms are penalized less, solved with ADMM.
Key results
- On BSD color image recovery with 10% corrupted pixels, WSAR TRPCA reaches 43.75 PSNR on image 1 versus 40.50 for TRPCA and 41.15 for ETRPCA (Table 2).
- WSAR TRPCA achieves the highest PSNR on all 5 sample images in Table 2, e.g., 36.31 vs 31.08 for TRPCA on image 2.
- WSAR RSR produces restored faces with better visual quality and fewer artifacts than RSR and EGSNR on Extended Yale B occlusion, reported qualitatively only.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use WSAR TRPCA instead of ETRPCA | Tensor robust image recovery · BSD image 1 | PSNR: 41.15 -> 43.75 | Helped |
| Use WSAR TRPCA instead of ETRPCA | Tensor robust image recovery · BSD image 2 | PSNR: 31.75 -> 36.31 | Helped |
| Use WSAR TRPCA instead of ETRPCA | Tensor robust image recovery · BSD image 3 | PSNR: 26.26 -> 32.23 | Helped |
| Use WSAR TRPCA instead of ETRPCA | Tensor robust image recovery · BSD image 4 | PSNR: 27.76 -> 30.02 | Helped |
| Use WSAR TRPCA instead of ETRPCA | Tensor robust image recovery · BSD image 5 | PSNR: 28.27 -> 33.29 | Helped |
| Use WSAR RSR instead of RSR | Robust face restoration · EYaleB | visual restoration quality | Helped |
| Use WSAR TRPCA instead of competing recovery methods | Background modeling · CAVIAR video, frames 60, 70 and 80 | background and foreground-mask quality | Helped |
Limitations
- Theory covers only two atomic norm terms; extension to more structures is stated without analysis.
- Weight-setting strategy relies on a constant tau said to be set adaptively without detailing the procedure.
- Face restoration results are shown qualitatively only in the main text.
Experimental setup
Datasets: Extended Yale B Berkeley Segmentation Dataset (BSD) Scene Background Initialization (SBI) CAVIAR video
Benchmarks: Extended Yale B BSD SBI CAVIAR
Metrics: PSNR visual quality
Baselines: RSR EGSNR RPCA SNN TRPCA ETRPCA
Builds on: RPCA TRPCA robust sparse representation atomic norm ADMM
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