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Engineering, Technology & Applied Science Research Vol. 12, No. 4, 2022, 8872-8877 8872 
 

www.etasr.com Ozturk & Cayiroglu: A Real-Time Application of Singular Spectrum Analysis to Object Tracking with … 

 

A Real-Time Application of Singular Spectrum 

Analysis to Object Tracking with SIFT 
 

Ali Ozturk 

Department of Mechatronics 

Vocational School of Technical Sciences 

Mus Alparslan University 

Mus, Turkey 

a.ozturk@alparslan.edu.tr 

Ibrahim Cayiroglu 

Department of Mechatronics 

Faculty of Engineering 

Karabuk University 

Karabuk, Turkey 

icayiroglu@karabuk.edu.tr 
 

Received: 27 April 2022 | Revised: 31 May 2022 | Accepted: 1 June 2022 

 

Abstract- This study combined SIFT and SSA to propose a novel 

algorithm for real-time object tracking. The proposed algorithm 

utilizes an intermediate fixed-size buffer and a modified SSA 

algorithm. Since the complete reconstruction step of the SSA 

algorithm was unnecessary, it was considerably simplified. In 

addition, the execution time of a Matlab implementation of the 

SSA algorithm was compared with a respective C++ 

implementation. Moreover, the performance of the two different 

matching algorithms in the detection, the FlannBasedMatcher 

and Brute-Force matcher algorithms of the OpenCV library, was 

compared. 

Keywords-object tracking; object detection; computer vision; 

SIFT; SSA 

I. INTRODUCTION  

Object tracking or visual tracking has been employed, 
studied, and commercialized in a broad spectrum of 
applications such as Simultaneous Localization And Mapping 
(SLAM), traffic surveillance, pedestrian counting, hand gesture 
recognition, tracking the path of protein stress granules in cells, 
and video compression [1]. Object tracking can be defined as 
dynamic object detection [2]. The main problems of object 
tracking are common with object detection, such as occlusion, 
clutter in the background, viewpoint changes due to affine 
transformation such as translation and rotation, scale changes 
(zoom in and out), and photometric deformation such as 
distortion, blur, and illumination changes. The most important 
challenge is still the speed or real-time performance of the 
online tracker, such as the tracker used in visual servoing in 
robotics [3]. On the other hand, object tracking has some 
problems of its own. An optimal algorithm for visual tracking 
can vary depending on the camera’s position, the number of 
cameras, and the number of moving objects. In the case of a 
stationary camera, background removal can simplify the task 
by eliminating the stationary pixels [4]. If the number of 
moving objects and the target object increase, then the task 
becomes complicated. Object tracking consists of two 
important steps: detecting the object of interest and tracking it 
in the following frames [5]. Object detection can be defined as 
template matching in which a template or a model is detected 
within a scene. Typically, a template is the picture of an object 

of interest. Detection can be defined as enclosing a given 
template in the scene in a rectangle. In object tracking, the 
basic task is to track the object as long as it exists in the scene. 

Scale Invariant Feature Transform (SIFT) is one of the most 
robust local descriptors which can be used in object tracking 
[6-8]. It has been utilized in a broad spectrum of applications, 
such as face recognition, gender recognition [9], and smoke 
detection [10]. It is a good candidate to tackle the problems of 
tracking, such as translation, rotation, occlusion, and scale. A 
method was proposed in [11] to track objects by combining 
SIFT with mean shift. In [12], SIFT was utilized with Kanade-
Lucas-Tomasi Tracker (KLT) [13] for UAV-based 
applications. In this approach, the object was detected with 
SIFT, and its location was computed with KLT. Since the 
algorithms are not perfectly robust to all image transformations 
and deformations, the object of interest can be lost or detected 
with low accuracy during any object tracking task. In such 
cases, the pose can be recovered or improved by using filters or 
smoothing. Singular Spectrum Analysis (SSA) is a method of 
time series analysis based on multivariate statistics. It is also 
known as the principal component analysis of time series [14]. 
SSA is considered to be a non-parametric (model-free) filtering 
technique. However, it can also be used for modeling [15]. It 
has been mainly used in applications that require post-
processing [16, 17]. SSA was combined with the Kalman filter 
to smooth real-time positioning data obtained from the Global 
Navigation Satellite System (GNSS) in [18]. However, there is 
no description of how SSA was implemented in real-time. A 
rank-one eigenvector update was proposed in a recursive 
framework in [19] for the identification of structural damage 
detection in real-time by utilizing First-Order Perturbation 
(FOP) for the Henkel matrix. 

This paper proposes an object-tracking algorithm utilizing 
SIFT and SSA. To the best of our knowledge, this is the first 
study that employs SIFT and SSA in object tracking. As SSA 
always requires a batch of data, its real-time implementation 
was carried out using a fixed-sized buffer. Since the real-time 
implementation of SAA does not require a full reconstruction 
step by diagonal averaging, the reconstruction algorithm was 
considerably simplified. 

Corresponding author: Ali Ozturk 



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II. METHOD 

This section investigates the SSA algorithm, provides the 
real-time implementation, describes object tracking with a 
template matching algorithm, and provides the details of the 
experiments. 

A. Singular Spectrum Analysis 

The SSA algorithm can be summarized in four steps: 
embedding, Singular Value Decomposition (SVD), grouping, 
and finally, reconstruction. In the embedding step, a Henkel 
matrix was constructed from the input signal. Each column of 
the Henkel matrix is a portion of the signal obtained by 
windowing with a selected window length. This process is also 
called a caterpillar since it behaves as such. If the length of the 
raw signal is n and the window length is l, the size of the 
Henkel matrix is l×k, where l is the rows, � is the columns, and 
k=n-l+1. Figure 1 shows an example of the embedding step. In 
this example, a Henkel matrix is created from an input signal. 
The size of the signal is 10, and the window length is 4. 
Therefore, the size of the resulting matrix is 7×4. 

 

 

Fig. 1.  An example of the embedding step. 

In the SVD step, the singular value decomposition of the 
trajectory matrix is computed. If X is the Henkel matrix, then 
the trajectory matrix is XX

T
, where X

T
 is the transpose of X. The 

Henkel matrix can be written as a sum of different and 
independent matrices with the aid of SVD. This decomposition 
can be stated as follows: 

� � �� � �� � ⋯ � ��    (1) 
where each Xi is called an elementary matrix. Each elementary 
matrix Xi is calculated as follows: 

�	 � 
�	 �	 
	� , � � 1, ⋯ , �    (2) 
where Ui denotes the eigenvectors of SVD, and each vector Vi 
is calculated as follows: 


	 � �� ��
�� , � � 1, ⋯ �    (3) 
Each elementary matrix �	  corresponds to one of the 

eigenvalues of SVD. It should be noted that: 

‖�	 ‖� � �	 ,     � � 1, ⋯ , �    (4) 

The contribution of each elementary matrix to X is 
proportional to its corresponding eigenvalue λi. The 
eigenvalues are sorted in decreasing order. In SSA, elementary 
matrices corresponding to small eigenvalues are considered to 
be the noise portion of the signal. This portion is excluded in 
the grouping step: 

�� � ∑ �	�	��     (5) 
where r<m. The value of r is decided by looking at the ratio of 
the sum of the first r eigenvalues to the sum of all eigenvalues. 
If the value of this ratio is greater than a certain threshold, the 
sum of the first r elementary matrices (5) is considered to be an 
optimal approximation of the Henkel matrix X. Finally, in 
reconstruction, the optimal approximation of the Henkel 

Matrix, ��  is converted to a time series. The reconstruction 
operation should be considered to be the inverse operation of 
the construction of the Henkel matrix from the raw signal. This 
inverse operation is performed by diagonal averaging on the 
approximated Henkel matrix (Figure 1). The diagonal 
averaging is realized in three steps: top diagonalization, center, 
diagonalization, and tail diagonalization on the approximated 
Henkel matrix as follows: 

�� �
⎩⎪
⎨
⎪⎧

�
�"� ∑ #	,�$	"��"�	�� , �% 0 ' ( ) *∗ , 1�
-∗ ∑ #	,�$	"�-

∗
	�� , �% *∗ , 1 ' ( ) . ∗

�
/$� ∑ #	,�$	"�/$0

∗"�	��$0∗"� , �% . ∗ ' ( ) 1
    (6) 

where #	,2 ,  1 ' � ' * , 1 ' 3 ' .  denote entries of the 
approximated Henkel matrix and, �� ,  0 ' ( ) 1, are entries of 
the reconstructed time series.  

Two parameters affect the result of SSA: the window length 
L and the reconstruction parameter r. The last one is 
determined by the eigenvalues as explained above. In general, 
the first or at least the sum of two eigenvalues constitutes the 
largest portion of the sum of all eigenvalues. However, there is 
no rule on how to select L. The choice of L affects the result of 
the filter. If L increases, the smoothing effect of the filter 
increases. This value can be determined by test and trial. 

B. Real-Time Implementation of SSA 

The real-time implementation of SSA was conducted by 
using a fixed-size buffer (Figure 2). At first, this buffer is filled 
with data. The size of the buffer is determined beforehand. So, 
this method requires a time delay to start filtering. Once the 
buffer is full, the filtering operation starts. As new frames 
arrive, the buffer is updated. However, for the real-time 
implementation of SSA, there is no need for full reconstruction 
by diagonal averaging. Since the last frame is written on the 
last element of the buffer, it is sufficient to recover only the last 
element of the approximated trajectory matrix, as follows: 

% � ��45 , 1, � , 16    (7) 
where f is the data filtered and the size of the approximated 

Henkel matrix ��  is 5 7 � . It is assumed that the first index 
starts from zero. Thus, using (7), the reconstruction algorithm 
is significantly simplified by omitting diagonal averaging, 
compared to (6). This method is called Real-Time SSA (RT-
SSA). 



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Fig. 2.  Fixed-size buffer used for real-time implementation of SSA. 

C. Object Tracking with Template Matching 

Template matching is the task of detecting a given template 
within a given scene. Typically, a template is the picture of an 
object of interest. Detection can be defined as enclosing a given 
template in the scene in a rectangle. Figures 3 and 4 show the 
template used in this study and the detection task respectively. 

 

 

Fig. 3.  A matched sample frame from the video. 

 
Fig. 4.  A detection example from the video. 

This study used SIFT for object detection. SIFT is a local 
descriptor, typically invariant to local occlusion [20]. SIFT has 
two main parts: keypoint detection and feature description. 
Keypoints (or salient points) are detected on the Gaussian 
scale-space pyramid [21, 22]. This kind of keypoint detection 
makes the algorithm scale (zoom in and out) invariant. A 
dominant orientation of each keypoint is computed using the 
Histogram of Oriented Gradients (HOG). This computation 
makes the algorithm orientation invariant. Finally, a feature 
vector for each keypoint is calculated using HOG [23]. The 
normalization of the feature vector makes it invariant to 
changes in illumination. In summary, SIFT is invariant to 
translation, rotation, scale change (zoom in and out), partial 
occlusion, and illumination change. The runtime performance 
of SIFT mainly depends on the size of the image and the 
number of the extracted keypoints. In this study, SIFT was 
implemented using OpenCV 4.5.5. 

Once the keypoints and features are computed both on the 
template and scene, the following task is matching. For the 
matching algorithm, the FlannBasedMatcher of OpenCV can 
be utilized. The Fast Library for Approximated Nearest 
Neighbor (FLANN) speeds up the matching task by using the 
k-nearest neighbor [24]. If the number of the extracted 
keypoints is low, the Brute-Force matcher of OpenCV can 
alternatively be utilized. Figure 3 shows the SIFT keypoints on 
both the template and the scene, as well as the matched 
keypoints. Once the matching pairs between the template and 
the scene are detected, the template’s corners should be 
projected on the scene. A homography matrix is needed for this 
projection. The OpenCV findHomography function was used 
to obtain the homography matrix using the RANSAC 
algorithm. Finally, the OpenCV perspetiveTransform function 
provides four corners of the rectangular polygon that encloses 
the template in the scene. 

D. Experimental Setup 

An Everest SC-HD03 webcam was used to capture the 
video. In the video, the template (a penholder) was moved by 
hand. It was mobilized, moved away from the camera, zoomed 
in, rotated, and twisted in different directions. During this 
process, the pixel coordinates of the center point of the detected 
four corners of the template in the scene were considered to be 
the pose of the object. This center point is shown by a blue 
circle in Figure 4. The video consists of 1439 frames. The size 
of each frame was 640×480. 

A C/C++ code was written for SSA and RT-SSA, using the 
Eigen library [25], with floating-point arithmetic.The C/C++ 
code is available in [26]. On the other hand, Matlab utilizes 
double-point arithmetic by default. The C/C++ code was 
compiled with GCC 9.4.0 on Ubuntu 20.04 LTS and run on an 
Intel i9-10850K CPU @3.60GHZ with 16GB RAM. The 
parameters, notations, and abbreviations used in the results are 
shown in Table I. As was observed, the first eigenvalue always 
constituted more than 90% of all. So, the reconstruction 
parameter r of SSA was taken as 1 for all tests. 

TABLE I.  NOTATIONS AND ABBREVIATIONS USED IN RESULTS 

Notation/ 

Abbreviation 
Explanation 

Pose Position of the object (x and y coordinates) 

FLANN FlannBasedMatcher of OpenCV 

Brute-Force Brute-Force matcher of OpenCV 

Filter-30-3 RT-SSA with a buffer of size 30, and window length of 3 

Filter-30-9 RT-SSA with a buffer of size 30, and window length of 9 

 

III. RESULTS 

This section, initially, presents the results of template 
matching according to two different matching methodologies, 
FLANN and Brute-Force. Then, the results of the execution 
time of SAA are provided. Finally, the results of the application 
of real-time SAA to object tracking are summarized.  

A. Template Matching 

Figure 5 shows the results of FLANN and Brute-Force 
matching. As shown, FLANN lost the pose of the object in 
frame 562. However, the Brute-Force graph shows no loss at 



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any frame. The average number of keypoints on the scene is 
67, which is considered low. Therefore, it is more feasible to 
use the Brute-Force matcher. FLANN should be preferred 
when the number of keypoints is high. The average matching 
time with the Brute-Force matcher was about 3ms, while the 
average detection time of SIFT was about 33ms. 

 

 
Fig. 5.  Results of the FlannBasedMatcher and Brute-Force matching 
algorithms. 

B. Execution Time of SSA 

Figure 6 shows the speed of the C++ and Matlab SSA 
implementations. As shown, when the data length is 1K, the 
C++ implementation of SSA reaches 1000FPS. This speed 
meets the requirements of many real-time applications.  

 

 
Fig. 6.  Speed of C++ and Matlab implementations of SSA algorithm for 
data length (1K=1024). Execution time is expressed in FPS. 

 

Fig. 7.  Speed-up ratio of a C++ over a Matlab implementation of SSA. 

Figure 7 shows the speed-up ratio of a C++ over the Matlab 
implementation of SSA. The C++ implementation of SSA is up 
to 18 times faster than Matlab. However, Matlab code can still 
meet the speed requirements of many applications. 

C. Application of SSA to Real-time Object Tracking 

Figures 8 and 9 show the x and y coordinates of the pose of 
the object when it is filtered with the real-time SSA. Figures 
10-12 show the scaled points A, B, and C marked in Figure 8. 
These Figures show that as the window length increases, the 
filter smoothing effect increases, and fluctuations smooth out. 
However, when the object changes its directions (Figure 10), 
the filter may considerably move away from the actual pose 
because of over smoothing. Therefore, the extensive 
parameters of the filter can be avoided. 

 

 
Fig. 8.  Application of real-time SSA to object’s x coordinate obtained 
with Brute Force matcher.  

 
Fig. 9.  Application of real-time SSA to object’s y coordinate obtained 
with Brute Force matcher. 

 

Fig. 10.  Zoom in point A of Figure 8. 

 

Fig. 11.  Zoom in point B of Figure 8. 



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Fig. 12.  Zoom in point C of Figure 8. 

IV. CONCLUSION 

SIFT and SSA are two different methods that have become 
popular in the last two decades. SIFT is a local object detection 
algorithm. On the other hand, SSA is a filtering technique that 
utilizes some methods of multivariate statics, such as PCA. 
This study combined SIFT with SSA for real-time object 
tracking. The main contributions of this paper can be 
summarized as follows: 

• The results of two different feature matching methods, 
FlannBasedMatcher and Brute-Force matcher, were 
compared. 

• SIFT and SSA were used in a real-time object tracking 
application for the first time. 

• A C++ implementation of SSA was coded using the Eigen 
library. 

• The performance (execution time) of C++ and Matlab 
implementations of SSA were compared. 

• A novel method for the real-time implementation of SSA 
was presented, simplifying the reconstruction step. 

Performance comparison showed that the classic SSA 
algorithm has enough speed for many real-time 
implementations and that the C/C++ code was faster than 
Matlab. As the proposed real-time method did not require full 
reconstruction, it simplifies the SSA reconstruction step. 
Moreover, instead of updating the raw data buffer, the Henkel 
matrix can be updated by omitting the embedding step of SSA, 
which could be future work. On the other hand, it was also 
observed that the performance of the object detection algorithm 
heavily depends on the matching algorithm. It was concluded 
that if the number of keypoints is low, then the Brute-Force 
matcher should be preferred instead of the FlannBasedMatcher. 

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