IEEE Signal Processing - July 2018 - 50
Parameter setting for nearly optimal-robust
matrix completion [17]
The parameter setting here is very similar to that of AltProj.
However, in addition to these parameters, NO-RMC also requires an estimate of the incoherence parameter n, and an
estimate of the largest singular value of L, v. Both of these estimates are required to set the number of iterations for the inner
loop and ensure that the rank k q estimate is "good enough." For
default parameters, please see their code.
Parameter setting for recursive projected
compressive sensing [27], [29]
The first step for setting algorithm parameters for ReProCSbased approaches is to obtain an initialization. There are two
approaches that can be used for this. If there is outlier-free
training data (e.g., for long surveillance videos obtained from
closed-circuit television cameras), one can use simple SVD to
obtain the initial subspace estimate: compute the top singular
vectors that contain at least a certain percentage, e.g., 95%, of
the total energy (sum of squares of all singular values). Another
approach is to use a few iterations of a batch-RPCA method such
as PCP or AltProj applied to the first t train = Cr frames of data.
PCP requires no tuning parameters but is slower. AltProj is faster but needs to know r. By cross-validation, we conclude that
r = 40 and t train = 400 = 10r suffices in all of our video experiments. We set K = C log (1/e) = 3, and a = Cr log n = 60
as required by the guarantee. Since the dependence of a on
n is only logarithmic, for all practically used values of n
(n = 1, 000 to n = 320 * 240) a = 60 suffices. The parameter m + is computed as the square of the largest singular value
of Lt init t train ; and m - is the square of its rth largest singular
value. The two parameters for the projected CS step are p and
~ supp . The required theoretical values of these depend on S min:
the theorem requires p = s min 15 and ~ supp = s min 2. One
way to estimate s min is to let it be time varying and set it as
min i ! Tt t - 1 (ts t - 1) i at t. An approach that works better for the
video application is to set p t = W,t t - 1 , ~ supp, t = m t 2 n .
The former relies on the idea that p t needs to be a bound W, t .
The latter is the root mean square value of the measurement
vector at time t. If there are not too many outliers, most outlier
magnitudes will be larger than this threshold (the problematic outliers being large magnitude corruptions), and, hence,
this is a useful support estimation threshold. Finally the
subspace change detection threshold ~ evals can be set equal
to a very small fraction of m +, ~ evals = 0.0007m + is used in
all video experiments.
Experimental comparisons for video layering
(foreground-background separation)
We evaluated the performance of the current state-ofthe-art RPCA-based methods for foreground-background
separation (background subtraction) using the CDnet 2012
data set [61]. We compared a total of 26 existing methods
comprising 16 batch methods and eight online methods.
These methods can be classified into the following three
main categories:
50
Provable methods comprise PCP [4], [5], nonconvex
AltProj-based RPCA [8], NO-RMC [17], RPCA-GD [10],
ReProCS [7], [12], simple ReProCS (simple-ReProCS)
[29], and modified-PCP [30].
■ Heuristics methods include GRASTA [50], OR-PCA
[47], three-term decomposition (3TD) [62], two-pass
RPCA (2PRPCA) [63], GoDec [37], pROST [51], and
PRMF [64].
■ Heuristics methods with application-specific constraints
consist of modified-ReProCS [12], incremental PCP (incPCP) [52], motion-assisted spatiotemporal clustering of low
rank (MSCL) [65], detecting contiguous outliers in the
low-rank representation (DECOLOR) [66], low-rank structured-sparse decomposition (LSD) [67], total variation
RPCA (TVRPCA) [68], spatiotemporal RPCA (SRPCA)
[69], robust motion-assisted matrix restoration (RMAMR)
[70], generalized fused lasso (GFL) [71], GOSUS [53],
contiguous outliers representation via online low-rank
approximation (COROLA) [72], online mixture of
Gaussians for matrix factorization with total variation
(OMoGMF+TV) [73], and OR-PCA with illumination
constraints (OR-PCA-illum) [74].
We used the authors' implementations of TVRPCA, 2PRPCA,
GRASTA, GoDec, DECOLOR, 3TD, LSD, SRPCA, GFL,
GOSUS, RMAMR, OMoGMF+TV, simple-ReProCS, modified-ReProCS, AltProj, NO-RMC, and RPCA-GD, where we
reported the results that were obtained in their respective studies for the remaining methods. The implementation of all the
methods is also available in the low-rank and sparse library
[26]. The execution times required by all of the algorithms
were compared on a machine with a 3:0 GHz Intel core i5 processor and 4GB of random access memory.
The CDnet 2012 data set [61] is the real-world region-level
benchmark obtained by human experts. This data set contains
almost 31 video sequences that are divided into the following
six different video categories:
■ baseline
■ DB
■ IOM
■ thermal
■ camera jitter
■ shadows.
The resolution of the videos also varies from 320 # 240 to
480 # 720 with hundreds to thousands of frames.
Visual results were reported using 15 challenging sequences
from CDnet data set for comparison purpose. This contained
■ two sequences highway and office from the baseline
category
■ three sequences canoe, boats, and overpass from the
DB category
■ two sequences traffic and badminton from the camera jitter
category
■ three sequences winterDriveway, sofa, and streetLight
from the IOM category
■ three sequences backdoor, copyMachine, and cubicle from
the shadows category
■
IEEE Signal Processing Magazine
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July 2018
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Table of Contents for the Digital Edition of IEEE Signal Processing - July 2018
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