Robust Principal Component Completion

Yinjian Wang; Wei Li; Yuanyuan Gui; James E. Fowler; Gemine Vivone

cs.CVMar 26, 2026

Robust Principal Component Completion

Authors: Yinjian Wang, Wei Li, Yuanyuan Gui, James E. Fowler, Gemine Vivone

Organizations: School of Information and Electronics, Beijing Institute of Technology, and the National Key Laboratory of Science and Technology on Space-Born Intelligent Information Processing, 100081 Beijing, China · Department of Electrical and Computer Engineering, Mississippi State University, Starkville, MS 39762 USA · National Research Council, Institute of Methodologies for Environmental Analysis (CNR-IMAA), 85050 Tito, Italy, and also with National Biodiversity Future Center (NBFC), 90133 Palermo, Italy

Abstract

Robust principal component analysis (RPCA) seeks a low-rank component and a sparse component from their summation. Yet, in many applications of interest, the sparse foreground actually replaces, or occludes, elements from the low-rank background. To address this mismatch, a new framework is proposed in which the sparse component is identified indirectly through determining its support. This approach, called robust principal component completion (RPCC), is solved via variational Bayesian inference applied to a fully probabilistic Bayesian sparse tensor factorization. Convergence to a hard classifier for the support is shown, thereby eliminating the post-hoc thresholding required of most prior RPCA-driven approaches. Experimental results reveal that the proposed approach delivers near-optimal estimates on synthetic data as well as robust foreground-extraction and anomaly-detection performance on real color video and hyperspectral datasets, respectively. Source implementation and Appendices are available at https://github.com/WongYinJ/BCP-RPCC.

Figures & tables

Panel of Figure 1 — Fig. 1: Visual comparison of various quantities of the RPCA formulation for an image.

Panel of Figure 2 — Fig. 2: B-Unfolding: The reorganization of each tensor block into a single column of a matrix.

	$\displaystyle\mathbf{\mu}^{(n)}_{i_{n}}\leftarrow\bar{\mathbf{a}}^{(n)}_{i_{n}},\,\,\mathbf{\Sigma}_{i_{n}}^{(n)}\leftarrow\mathbf{I}_{R},\,\,a_{r}\leftarrow a_{0},\,\,b_{r}\leftarrow b_{0},$
	$\displaystyle c_{k}\leftarrow c_{0},\,\,d_{k}\leftarrow d_{0},\,\,\bar{z}_{k}\leftarrow z_{0},\,\,\alpha\leftarrow\alpha_{0},\,\,\beta\leftarrow\beta_{0}$

Algorithm 1 The BCP-RPCC Algorithm

Panel of Figure 4 — Fig. 3: Box plots of RRSE and IoU on synthetic data. RRSE is standardized via Z-score transformation. The statistics in blue in the upper area are the original mean (mn), standard deviation (std) and median (md) of RRSE for each group of 20 runs.

Panel of Figure 5 — Fig. 4: Example frames, along with ground-truth foreground masks and ROIs, from the four CDnet videos used for the foreground-extraction experiments.

Panel of Figure 6 — Fig. 5: Hyperparameter tuning for foreground extraction using $\operatorname{F1}$ and $\operatorname{IoU}$ . Row 1: Tuning $\sigma$ when $R=25$ . Row 2: Tuning $R$ when $\sigma=10^{-3}$ .

Panel of Figure 7 — Fig. 6: Foreground extraction for the Highway dataset. Row 1: Frame 10. Row 2: Frame 25. Row 3: Frame 50.

Panel of Figure 8 — Fig. 7: Foreground extraction for the Turnpike dataset. Row 1: Frame 10. Row 2: Frame 25. Row 3: Frame 50.

Panel of Figure 9 — Fig. 8: Foreground extraction for the Crossroad dataset. Row 1: Frame 10. Row 2: Frame 45. Row 3: Frame 50.

Panel of Figure 10 — Fig. 9: Foreground extraction for the Busstation dataset. Row 1: Frame 1. Row 2: Frame 25. Row 3: Frame 50.

Methods	Metrics
Methods	$\text{AUC}_{\operatorname{F1}}\uparrow$	$\text{AUC}_{\operatorname{IoU}}\uparrow$	$\text{AUC}_{\operatorname{Fa}}\downarrow$
PCP	0.2395	0.1543	0.0292
KBR	0.3919	0.2862	0.0060
TRPCA	0.3905	0.2779	0.0138
ETRPCA	0.3607	0.2498	0.0197
t-CTV	0.3470	0.2310	0.0455

TABLE I: Foreground Extraction for Highway. Best results are in boldface.

Methods	Metrics
Methods	$\text{AUC}_{\operatorname{F1}}\uparrow$	$\text{AUC}_{\operatorname{IoU}}\uparrow$	$\text{AUC}_{\operatorname{Fa}}\downarrow$
PCP	0.2501	0.1527	0.0713
KBR	0.2466	0.1561	0.0394
TRPCA	0.2791	0.1765	0.0591
ETRPCA	0.2758	0.1736	0.0648
t-CTV	0.2253	0.1389	0.0706

TABLE II: Foreground Extraction for Turnpike. Best results are in boldface.

Methods	Metrics
Methods	$\text{AUC}_{\operatorname{F1}}\uparrow$	$\text{AUC}_{\operatorname{IoU}}\uparrow$	$\text{AUC}_{\operatorname{Fa}}\downarrow$
PCP	0.2202	0.1361	0.0412
KBR	0.2319	0.1554	0.0095
TRPCA	0.2854	0.1949	0.0214
ETRPCA	0.2657	0.1752	0.0328
t -CTV	0.2807	0.1870	0.0510

TABLE III: Foreground Extraction for Crossroad. Best results are in boldface.

Methods	Metrics
Methods	$\text{AUC}_{\operatorname{F1}}\uparrow$	$\text{AUC}_{\operatorname{IoU}}\uparrow$	$\text{AUC}_{\operatorname{Fa}}\downarrow$
PCP	0.1658	0.1013	0.0631
KBR	0.2105	0.1404	0.0124
TRPCA	0.2031	0.1335	0.0265
ETRPCA	0.1923	0.1229	0.0391
t-CTV	0.1461	0.0874	0.0925

TABLE IV: Foreground Extraction for Crossroad. Best results are in boldface.

Panel of Figure 15 — Fig. 10: Foreground-extraction performance in terms of $\operatorname{F1}(\tau)$ , $\operatorname{IoU}(\tau)$ , and $\operatorname{Fa}(\tau)$ : Highway (row 1), Turnpike (row 2), Crossroad (row 3), and Busstation (row 4).

Panel of Figure 16 — Fig. 11: Hyperspectral datasets for anomaly detection; second row is ground truth.

Panel of Figure 17 — Fig. 12: Hyperparameter tuning for hyperspectral anomaly detection. Row 1: Tuning $\sigma$ when $R=25$ . Row 2: Tuning $R$ when $\sigma=4\times 10^{-3}$ for Belcher and Beach, $\sigma=9\times 10^{-3}$ for Urban and $\sigma=3\times 10^{-2}$ for Salinas.

Methods	Metrics
Methods	$\text{AUC}_{\operatorname{F1}}\uparrow$	$\text{AUC}_{\operatorname{IoU}}\uparrow$	$\text{AUC}_{\operatorname{Fa}}\downarrow$
PCP	0.1717	0.0992	0.0101
KBR	0.2135	0.1297	0.0088
TRPCA	0.2084	0.1218	0.0329
LRTFR	0.2785	0.1751	0.0167
PTA	0.0921	0.0490	0.1093

TABLE V: Anomaly Detection for Belcher. Best results are in boldface.

Methods	Metrics
Methods	$\text{AUC}_{\operatorname{F1}}\uparrow$	$\text{AUC}_{\operatorname{IoU}}\uparrow$	$\text{AUC}_{\operatorname{Fa}}\downarrow$
PCP	0.1656	0.0931	0.1365
KBR	0.2672	0.1715	0.1052
TRPCA	0.2987	0.1956	0.1504
LRTFR	0.2354	0.1453	0.1058
PTA	0.1403	0.0792	0.3321

TABLE VI: Anomaly Detection for Urban. Best results are in boldface.

Methods	Metrics
Methods	$\text{AUC}_{\operatorname{F1}}\uparrow$	$\text{AUC}_{\operatorname{IoU}}\uparrow$	$\text{AUC}_{\operatorname{Fa}}\downarrow$
PCP	0.0960	0.0520	0.0030
KBR	0.1391	0.0770	0.0284
TRPCA	0.1285	0.0714	0.0411
LRTFR	0.2469	0.1552	0.0275
PTA	0.0703	0.0378	0.1100

TABLE VII: Anomaly Detection for Beach. Best results are in boldface.

Methods	Metrics
Methods	$\text{AUC}_{\operatorname{F1}}\uparrow$	$\text{AUC}_{\operatorname{IoU}}\uparrow$	$\text{AUC}_{\operatorname{Fa}}\downarrow$
PCP	0.4369	0.3056	0.0090
KBR	0.5613	0.4311	0.0221
TRPCA	0.4733	0.3537	0.0583
LRTFR	0.5533	0.4317	0.0222
PTA	0.2946	0.2094	0.2352

TABLE VIII: Anomaly Detection for Salinas. Best results are in boldface.

Panel of Figure 22 — Fig. 13: Anomaly detection on Belcher (row 1), Urban (row 2), Beach (row 3), and Salinas (row 4).

Panel of Figure 23 — Fig. 14: Anomaly-detection performance in terms of $\operatorname{F1}(\tau)$ , $\operatorname{IoU}(\tau)$ and $\operatorname{Fa}(\tau)$ : Belcher (row 1), Urban (row 2), Beach (row 3) and Salinas (row 4).

Principal Component Analysis

Principal component analysis (PCA) finds orthogonal directions of greatest variance for data compression, visualization, denoising, and feature discovery.

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Abstract

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