arXiv:1603.04026v1 [cs.CV] 13 Mar 2016
A COMPREHENSIVE STUDY OF SPARSE CODES ON ABNORMALITY DETECTION
Huamin Ren1, Hong Pan2, Søren Ingvor Olsen3, Thomas B. Moeslund4
1,4Department of Architecture, Design and Media Technology, Aalborg University, Denmark
2,3Department of Computer Science, University of Copenhagen, Denmark
Sparse representation has been applied successfully in abnor-
mal event detection, in which the baseline is to learn a dic-
tionary accompanied by sparse codes. While much empha-
sis is put on discriminative dictionary construction, there are
no comparative studies of sparse codes regarding abnormal-
ity detection. We comprehensively study two types of sparse
codes solutions - greedy algorithms and convex L1-norm so-
lutions - and their impact on abnormality detection perfor-
mance. We also propose our framework of combining sparse
codes with different detection methods. Our comparative ex-
periments are carried out from various angles to better un-
derstand the applicability of sparse codes, including compu-
tation time, reconstruction error, sparsity, detection accuracy,
and their performance combining various detection methods.
Experiments show that combining OMP codes with maxi-
mum coordinate detection could achieve state-of-the-art per-
formance on the UCSD dataset .
Index Terms—Sparse representation, sparse codes, ab-
normal event detection
Sparse representation has gained a great deal of attention
since being applied effectively in many image analysis ap-
plications, such as image denoising  and action recogni-
tion . Sparse representation ﬁnds the most compact repre-
sentation of a signal in terms of linear combination of atoms
in an overcomplete dictionary. As is pointed out in , re-
search has focused on three aspects of sparse representation:
pursuit methods for solving the optimization problem, such
as matching pursuit , orthogonal matching pursuit , and
basis pursuit ; dictionary design, such as the K-SVD 
method and the BSD algorithm ; and the applications of
the sparse representation for different tasks, such as abnor-
mal event detection. Abnormal event detection is the core of
video surveillance applications, which could assist people in
various situations, such as monitoring patients/children, ob-
serving people and vehicles within a busy environment, or
preventing theft and robbery. The aim of the method is to
learn normal patterns or behaviors through training and de-
tect any abnormal or suspicious behaviors in test videos.
Research on sparse representation can be generally
divided into dictionary learning  and sparse cod-
ing   . Dictionary learning aims to obtain atoms (or
basis vectors) for a dictionary. Such atoms could be either
predeﬁned, e.g., undecimated Wavelets, steerable Wavelets,
Contourlets, Curvelets, and more variants of Wavelets, or
learned from the data itself. Sparse coding, on the other hand,
attempts to ﬁnd sparse codes (or coefﬁcients) by giving a dic-
tionary, i.e., ﬁnding the solution to the underdetermined sys-
tem of equations y=Dx either by greedy algorithms or con-
vex algorithms. Through sparse coding, input features can be
approximately represented as a weighted linear combination
of a small number of (unknown) basis vectors.
When applying sparse representation on abnormal event
detection, much emphasis is put on dictionary learning. A
common procedure is: ﬁrst, visual features are extracted ei-
ther on a spatial or temporal domain. A dictionary Dis then
learned based on these visual features, which consists of basis
vectors capturing high-level patterns in the input features, as
in [9, 8]. A sparse representation of a feature is a linear com-
bination of a few elements or atoms from a dictionary. Math-
ematically, it can be expressed as y=Dx, where y∈Rpis a
feature of interest, D∈Rp×mis a dictionary, and x∈Rmis
the sparse representation of yin D. Typically m≫presults
in an overcomplete or redundant dictionary. During the de-
tection procedure, each testing feature can be determined as
normal or an anomaly based on its reconstruction error.
However, most approaches use only an approximate re-
construction error to save computation; for example, the least
square error. This means the sparse codes are actually not
taken into consideration during the detection. In fact, the
impact of sparse codes generated by different approaches is
still unclear. Therefore, we offer a comprehensive study of
the sparse codes, in terms of their performance on abnormal
event detection. Among the huge research of codes represen-
tations, we put special attention on two major types: greedy
algorithms and L1-norm minimization algorithms.
Greedy algorithms rely on an interactive approximation
of the feature coefﬁcients and supports, either by iteratively
identifying the support of the feature until a convergence cri-
terion is met, or by obtaining an improved estimate of the
sparse signal at each iteration that attempts to account for the
mismatch with the measured data. Compared to L1-norm
minimization methods, greedy algorithms are much faster,
and thus are more applicable to very large problems.
Meanwhile, L1-norm minimization has become a popu-
lar tool to solve sparse coding, which beneﬁts both from efﬁ-
cient algorithms and a well-developed theory for generaliza-
tion properties and variable selection consistency . We
list two common L1-norm minimization formulations in E.q.
1 and E.q. 2. Since the problem is convex, there are efﬁcient
and accurate numerical solvers.
xkxk1subject to kDx −yk2≤ǫ(2)
Our main contributions are: 1) we offer a comprehen-
sive study of sparse codes, in terms of their reconstruction
error, sparsity, computation time and detection performance
on anomaly datasets; 2) we propose a framework to detect
abnormality, which combines sparse representation with var-
ious detection methods; and 3) we provide insights into the
impact of sparse representation and their detection methods.
The remainder of this paper is organized as follows. We
give a brief review of greedy algorithms and L1-norm solu-
tions in Sec.2 and propose our framework of abnormal event
detection in Sec.3, which combines sparse codes with various
detection methods. We show our comparative results in Sec.4
and concludes the paper with discussions and future work in
2. SPARSE CODES REPRESENTATION
There are various ways of generating sparse codes through
optimization solutions. We introduce two categorized solu-
tions: greedy algorithms and L1-norm approximation solu-
2.1. Greedy Algorithms
We review two broad categories of greedy methods to recon-
struct y, which are called ‘greedy pursuits’ and ‘threshold’
algorithms. Greedy pursuits can be deﬁned as a set of meth-
ods that iteratively build up an estimate x. They contains
three basic steps. First, the xis set to a zero vector. Sec-
ond, these methods estimate a set of non-zero components of
xby iteratively adding new components that are deemed to
be non-zeros. Third, the values for all non-zeros components
are optimized. In contrast, thresholding algorithms alternate
between element selection and element pruning steps.
There is a large and growing family of greedy pursuit
methods. The general framework in greedy pursuit tech-
niques is 1) to select an element and 2) to update the coefﬁ-
cients. Matching Pursuit (MP)  discusses a general method
for approximate decomposition in E.q. 3, which addresses
the sparsity issue directly. The algorithm selects one column
from Dat a time and only the coefﬁcient associated with the
selected column is updated at each iteration. More concretely,
it starts from an initial approximation x(0) = 0 and residual
R(0) = x, then builds up to a sequence of sparse approxima-
tions stepwise. At stage k, it identiﬁes the dictionary atom
that best correlates with the residual and then adds to the cur-
rent approximation a scalar multiple of that atom. After m
steps, one has a sparse code in E.q. 3 with residual R=R(m).
Orthogonal Matching Pursuit (OMP) , updates xin
each iteration by projecting yorthogonally onto the columns
of Dassociated with the current support atoms. Different
from MP, OMP never reselects an atom and the residual at any
iteration is always orthogonal to all currently selected atoms
in the dictionary. Another difference is that OMP minimizes
the coefﬁcients for all selected atoms at iteration k, while
MP only updates the coefﬁcient of the most recently selected
atom. In order to speed up pursuit algorithms, it is necessary
to select multiple atoms at a time; therefore, the algorithms
are proposed to keep computational costs low enough for ap-
plying to large-scale problems, such as Stagewise Orthogonal
Matching Pursuit (StOMP) . These algorithms choose the
element that meets some threshold criterion at the atom selec-
tion step and has demonstrated both theoretical and empirical
effectiveness for the large system.
Greedy algorithms are easy to implement and use and can
be extremely fast. However, they do not have recovery guar-
antees, i.e., how well each sample can be reconstructed by
the dictionary and their sparse codes, compared to L1-norm
2.2. L1-norm Approximation
L1-norm approximation replaces the L0 constraint with a re-
laxed L1-norm. For example, in the Basis Pursuit method
(BP) , an almost everywhere differentiable and often con-
vex cost function is applied, while in the Focal Underdeter-
mined System Solver (FOCUSS) algorithm , a more gen-
eral model is optimized.
Donoho and etc.  sugguest that for some measurement
matrices D, the generally NP-Hard problem (L0 norm) should
be equivalent to its convex relaxation: L1 norm, see E.q. 1.
The convex L1 problem can be solved using methods of lin-
ear programming. Representative work includes Basis Pur-
suit (BP). Instead of seeking sparse representations directly,
it seeks representations that minimize the L1 norm of the
coefﬁcients. Furthermore, BP can compute sparse solutions
in situations where greedy algorithms fail. The Lasso algo-
rithm  is quite similar to BP and is, in fact, know as Basis
Pursuit De-Noising (BPDN) in some areas. Rather than trying
to minimize the L1-norm like BP, the Lasso places a restric-
tion on its value.
The FOCUSS algorithm has two integral parts: a low-
resolution initial estimate of the real signal and the iteration
process that reﬁnes the initial estimate to the ﬁnal localized
energy solution. The iterations are based on the weighted
norm minimization of the dependent variable with the weights
acting as a function of the preceding iterative solutions. The
algorithm is presented as a general estimation tool usable
across different applications. In general, L1-norm methods
offer better performance in many cases, but they are also more
demanding with respect to computation.
3. SPARSE CODE BASED DETECTION
Feature Extraction Dictionary Learning Algorithm
Sparse Codes Represnetation
Abnormal FeaturesAbnormal Frames
Fig. 1. Our framework of combining sparse codes with dif-
ferent detection methods.
In addressing the detection of abnormal behaviors based on
sparse codes, two issues should be addressed: 1) how to gen-
erate the sparse codes, i.e., the solution of x; and 2) how to
determine whether the testing code is normal or anomalous.
For the ﬁrst issue, various sparse codes discussed in Sec.2
could be adopted; while for the second issue, we take various
detection methods into consideration; our proposed abnormal
event detection framework is shown in Fig. 1.
After a testing feature is represented by a sparse code, the
detection method determines whether it is normal or abnor-
mal. There are two commonly used detection methods: the
reconstruction error (RE) and the approximated reconstruc-
tion error (ARE). In terms of sparse codes, the high response
of dictionary atoms, or concentrated non-zeros in coefﬁcients,
may indicate a connection to a possible normality. Unfortu-
nately, these codes property and their connection with nor-
mality or abnormality are not explored yet. Therefore, we
also introduce maximum coordinate (MC) and the non-zero
concentration (NC) as two new detection methods.
Reconstruction Error (RE): Most existing approaches
treat dictionary learning and detection as two separate pro-
cesses, i.e., a dictionary is typically learned based on the train-
ing data, and then different measurements are adopted to de-
termine whether the testing sample is an anomaly. More so-
phisticated approaches unify these two processes into a mixed
reconstructive and discriminative formulation. Nevertheless,
a basic measurement that is widely used in both cases is re-
construction error. The reconstruction error of the testing
sample y, according to the dictionary D, is represented as:
2, where αis the sparse code of y.
Approximate Reconstruction Error (ARE): To speed
up detection, reconstruction error is sometimes not calculated
based on sparse codes through an optimization solution; in-
stead it is approximated by the least squares ; thus, the re-
construction error is calculated as: ky−D(DTD)−1DTyk.
Maximum Coordinate (MC): Given a testing sample y,
its sparse code is denoted as α. Ideally, all non-zero entries
in the estimate αwould be associated with the columns of the
dictionary from a normal pattern (note that only normal data is
used during the training). Then we could detect yas a normal
feature if a single largest entry in αwere found; otherwise, it
would be detected as an anomaly.
Non-zero Concentration (NC): Inspired by , the dis-
tribution of non-zeros is more important to the detection than
the location of non-zero elements. Thus, we propose a detec-
tion measurement called non-zero concentration. Based on
the dictionary proposed in , a normal code should have a
non-zero concentration property, i.e., non-zeros concentrated
in the dictionary that has the smallest reconstruction error.
Anomalies can be detected if no concentration is found on
any of the existing dictionaries.
4. EXPERIMENTAL RESULTS
We provide a comprehensive study the abnormality detection
performance of sparse codes. Our experiments are carried out
on the UCSD  Ped1 dataset, due to that it is a popular
abnormal event detection dataset, and many detection results
are reported. We start by evaluating the performance of var-
ious sparse codes, especially comparing sparse codes gener-
ated by two types of algorithms: greedy algorithms and L1-
norm approximation algorithms. The following aspects are
highlighted: computation time, reconstruction error, the ra-
tio of sparsity in codes, and their performance on abnormal
event detection. Next, we use the OMP algorithm to generate
sparse codes, and then we combine the codes with different
detection methods concluding by evaluating their detection
performance with state-of-the-art algorithms.
4.1. Dataset and Settings
UCSD Ped1 dataset  is a frequently used public dataset
for detecting abnormal behaviors. It includes clips of groups
of people walking towards and away from the camera with
some perspective distortion. There are 34 training videos and
36 testing videos with a resolution of 238 ×158. Training
videos contain only normal behaviors. Testing videos are ab-
normal behaviors where there are either non-pedestrian enti-
ties in the walkways or anomalous pedestrian motion patterns.
We use the spatial-temporal cubes, in which 3D gradient
features are computed, which mimics the setting in . Each
frame is divided into patches with a size of 23 ×15. Consec-
utive 5 frames are used to form 3D patches, and gradients fea-
tures are extracted in each patch. See details in . Through
this, we obtain 500-dimensional visual features and reduce
them to 100 dimension by using the PCA algorithm.
4.2. Comparison of Sparse Codes
We evaluate sparse codes from four perspectives: computa-
tion time, reconstruction error, the ratio of sparsity in codes,
and the codes’ performance on abnormal event detection
based on their reconstruction error.
We randomly select 1% of the training features (238,000
features in total), use the K-SVD algorithm  to construct
a dictionary consisting of 1000 atoms, and generate sparse
codes by applying various algorithms. There are many al-
gorithms available; we select only representative greedy al-
gorithms (OMP, MP, StOMP) and compare them with repre-
sentative L1-norm solutions (BP and Lasso algorithm). The
reconstruction error is calculated by Re =ky−Dxk2
also calculate the mean ratio of sparsity in the codes, i.e., the
average percentage of non zeros in the dimension of the codes
(1000). We report these results as well as computation time
in Tab. 1. Greedy algorithms need far less time to compute,
and the OMP achieves the fastest computation, followed by
the StOMP algorithm. OMP is approx. 180 times faster than
the Lasso algorithm. Both OMP and StOMP could achieve
sparser solutions, while BP could obtain an extremely dense
solution with an exact recovery.
To measure the accuracy of abnormality detection, we
calculate the reconstruction error of each feature and regis-
ter features with large reconstruction errors as anomalies. A
frame with an abnormal feature is considered a positive frame.
To compare performance, we adopt two popular evaluation
criteria in abnormality detection: frame-level evaluation and
pixel-level evaluation, which are deﬁned in . We follow
precisely their setting in our evaluation, which is to say that
in the frame-level evaluation, a frame is considered abnormal
if it contains at least one anomaly feature. In contrast, for
the pixel-level evaluation, a frame is marked as a correctly
detected abnormality if at least 40% of the truly abnormal
pixels are detected. Ground truth on frame-level and pixel-
level annotation is available, and we calculate the true posi-
tive and false positive rates to draw ROC curves, and report
the Area Under the Curve (AUC). Following , we obtain
the value when the false positive number equals the missing
value. These are called the equal error rate (EER) and equal
detected rate (EDR) in the frame and pixel-level evaluations,
respectively. See Tab. 2 for details. In the frame-level evalu-
ation, the MP algorithm achieves the best results with a mod-
erate computation time. The StOMP algorithm is relatively
fast, and the AUC is satisfactory.
It is worth noting that the pixel-level AUC is lower than
the frame-level AUC in general because the pixel-level eval-
uation is stricter and takes location into consideration. In the
frame-level evaluation, there could be a coincidental detec-
tion - a normal feature could be erroneously detected as an
anomaly in an abnormal frame, and this erroneous detection
could end up with a correct detection of that frame. In pixel-
level evaluation in contrast, a frame is marked as a correctly
detected abnormality only if a sufﬁcient number of anomaly
features has been found. Compared to the MP algorithm,
the StOMP algorithm can achieve a competitive detection re-
sult in the pixel-level evaluation, but it is three times faster
than the MP algorithm. The BP algorithm also performs well
on pixel-level detection; however, its high computation cost
hampers its application in real detection problems.
In summary, greedy algorithms compute quickly, but their
reconstruction errors are larger than L1-norm solutions. Con-
vex relaxations, such as the BP and the Lasso algorithm, have
better theoretical guarantees and recovery ability, but they are
more time consuming. Surprisingly, greedy algorithms, espe-
cially the StOMP algorithm, seem to perform better on pixel-
level detection, which means that they could more accurately
localize anomaly features.
4.3. Comparison of Combining Sparse Codes with Detec-
We choose the OMP algorithm to generate sparse codes due to
computation considerations, combine them with four types of
detection methods (RE, ARE, MC, NC), and compare their
detection performance. We draw comparative frame-level
AUC curves that corresponds to the detection methods. Fur-
thermore, we compare these combinations with state-of-the-
art methods on abnormality detection.
As displayed in Fig. 2, abnormality detection by com-
puting the real reconstruction error outperforms the estimated
reconstruction error on frame-level evaluation, which fur-
ther validates the idea that the decomposition of real co-
efﬁcients is necessary. Among all of these approaches,
OMP+RE achieves the best AUC score on frame-level evalu-
ation (0.6603), followed by MC (0.6340), NC (0.5697) and
ARE (0.5013). We give further insight into how accurate
the detection is in an even stricter pixel-level evaluation. We
ﬁnd that OMP+MC achieves the best result, with a AUC of
0.5433. This is because that the high response in the code
Table 1. Comparison of greedy algorithms and L1-norm solutions on sparse code generation.
ALGORITHMS COMPUTATION TIME (S) RECONSTRUCTION ERROR SPARSITY (%)
MP 166.00 0 31.8%
OMP 1.83 0.4236 1.9%
STOMP 15.79 0 10%
BP 114.20 0 100%
LASSO 333.49 0.0005 9.9%
Table 2. Comparative results on UCSD Ped1: frame-level evaluation results (AUC and EER) and pixel-level evaluation results
(AUC and EDR) are reported.
ALGORITHMS AUC (FRAME-LEVEL) EER AUC (PIXEL-LEVEL) EDR COMPUTATION TIME (S)
MP 0.6956 0.3547 0.3898 0.5716 13342
OMP 0.5003 0.5052 0.2849 0.6637 527
STOMP 0.5415 0.465 0.3494 0.6190 4668
BP 0.5454 0.4764 0.3057 0.6479 38949
LASSO 0.5305 0.5173 0.3132 0.6383 56400
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
False positive rate
True positive rate
Fig. 2. Combining sparse codes generated by the OMP al-
gorithm with various detection methods: non-zero concen-
tration (NC), approximate reconstruction error (ARE), recon-
struction error (RE) and maximum coordinate (MC).
means there is a strong connection between the testing fea-
ture with some atoms in the dictionary. This happens when
the features have the similar pattern as those atoms convey.
Therefore, the high response also implies that the testing
feature is normal. However, we also notice that NC detec-
tion, which also considers the non-zeros distribution in sparse
codes, performs relative poorly. This may be due to the type
of dictionary being adopted, or due to the principle of how the
OMP code is generated, which are based on the reconstruction
error of the chosen atoms, rather than the concentrated atoms.
Finally we compare combining OMP codes and various
detection methods with state-of-the-art abnormality detection
algorithms. Comparison of AUC in the frame-level evalua-
tion of UCSD Ped1 is shown in Fig. 3, and quantized eval-
uations are shown in Tab. 3 Compared with state-of-the-art
algorithms, combining OMP codes with detection methods
outperforms other methods on two criteria evaluation, which
veriﬁes the effectiveness of sparse codes generated by greedy
algorithms; furthermore, maximum coordinate detection out-
performs other methods, which implies that a high response
(large code value) could contribute to the detection.
0 0.2 0.4 0.6 0.8 1
False positive rate
True positive rate
UCSD Ped1 Frame-level ROC
Fig. 3. Comparison with state-of-the-art abnormality detec-
Table 3. Comparative values of AUC, EER and EDR on UCSD Ped1 dataset.
Method AUC (frame-level) EER AUC (pixel-level) EDR
SF-MPPCA  0.5900 0.3200 0.2130 0.3200
MDT  0.8180 0.2500 0.4610 0.2500
Lu13 0.5842 0.4413 0.3622 0.5826
OMP+RE 0.6603 0.3823 0.5386 0.5113
OMP+ARE 0.5013 0.5081 0.5317 0.5113
OMP+NC 0.5697 0.5055 0.5397 0.5113
OMP+MC 0.6339 0.4016 0.5433 0.5113
5. DISCUSSIONS AND CONCLUSION
In this paper, we givea comprehensive study of sparse codes,
in respect to their performance in abnormal event detection.
We compare two category sparse codes: codes generated by
greedy algorithms and those generated by L1-norm solutions.
Various aspects are covered: computational cost, recovery
ability, sparsity, and their detection performance. Further-
more, we explore into the sparse codes, and compare different
methods to determine whether a testing code is an anomaly or
not.Experimental results show that greedy algorithms can ob-
tain good detection results with fewer computations. Among
the top three best detection results, two are greedy algo-
rithms. Considering the computation requirement, which
limits some L1-norm algorithms from being applied in real
surveillance applications, greedy algorithms are promising.
When combining OMP codes with various detection measure-
ments, maximum coordinate measurement outperforms other
methods, which implies that the high response in the code
could help the detection result.
We have also found that due to the large amount of video
data only the OMP code is acceptable, which is mainly for
computational reasons. Despite the great progress being made
in the optimization ﬁeld, the applicability of various optimiza-
tion solutions is still unknown. Therefore, one line of future
work could focus on more practical sparse code algorithms.
Another line of work may fall into discriminative feature se-
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