Priors for the Ball Position in Football
Match using Contextual Information
CARLOS
DUQUE
Master of Science Thesis
Stockholm, Sweden 2010
Priors for the Ball Position in Football
Match using Contextual Information
CARLOS
DUQUE
Master’s Thesis in Computer Science (30 ECTS credits)
at the School of Electrical Engineering
Royal Institute of Technology year 2010
Supervisor at CSC was Josephine Sullivan
Examiner was Stefan Carlsson
TRITA-CSC-E 2010:048
ISRN-KTH/CSC/E--10/048--SE
ISSN-1653-5715
Royal Institute of Technology
School of Computer Science and Communication
KTH CSC
SE-100 44 Stockholm, Sweden
URL: www.kth.se/csc
Abstract
Due to the growth of media services in football broadcasting, a Swedish company
called TRACAB is modelling the reconstruction of football matches in 3-D.
TRACAB has developed a system to track players’ positions for the whole match
using cameras installed in each stadium. However, this system is less reliable when
applied to tracking the football. Therefore this Master Thesis focuses on estimating the ball’s position from the players’ trajectories. In particular we investigate
exploiting the contextual information to localize the ball.
The estimation of the position of the ball is calculated using different supervised
learning methods from feature vectors which are built with players’ positions. At
first, exact estimated coordinates are researched in general situations where no
priors are given. After looking at the results, this work selects regions which are
more suitable to find the ball. Another scenario is investigated where it is assumed
that we know which player has the ball. Then we try to predict to which region he
will pass the ball to. Consequently, TRACAB could find the ball efficiently on the
pitch.
Results are reported from data provided by TRACAB. These data are the coordinates of the ball and players’ positions of a particular whole match. Histograms
and percentages of success illustrate results and error aproximations in each case.
Contents
1 Introduction
1.1 Background . . . . . . . . . . . . . . . . . . . . . .
1.2 Problem this project addresses . . . . . . . . . . .
1.3 Overview of how to exploit contextual information
1.4 Supervised Learning Methods . . . . . . . . . . . .
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2 Data utilized in this work
2.1 Collection of needed data . . . . . . . . . . . . . . . . . . . . . . . .
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3 Estimation of ball position
3.1 Nearest Neighbour Regression . . . . . . . . . . . . . . . . . . . . . .
3.2 Nearest Neighbour regression with players in grid . . . . . . . . . . .
3.3 Nearest Neighbour with players in grid and temporal window . . . .
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4 Heat Maps: Scenario given no prior information
4.1 Scenario 1: General Situations . . . . . . . . . . . . . . . . . . . . .
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5 Heat Maps: Scenario where players pass the
5.1 Scenario 2: Particular Situations . . . . . . .
5.1.1 Scenario 2: Feature Vector . . . . . .
5.1.2 Results and Conclusions . . . . . . . .
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6 Conclusions and Future Work
43
References
49
Acknowledgement
After attending to the course called Image Analysis and Computer Vision, Javier
Romero González, who was teaching at the laboratory lectures, introduced the
CVAP department in KTH. Therefore, my first gratitude is to him. I met Josephine
Sullivan there and she offered me a project I could not refuse. It was firstly called,
“Action recognition for football players”. Hence, thanks to Josephine Sullivan, who
has become my supervisor since then, and who has dedidated all the time I needed
to answer my questions and has taught me extra mathematical theories when I
needed. She has guided me, step by step, working on this Master Thesis.
I cannot forget Mohammad Rastegari, who was my workmate in the department
at first, and later who has become a real friend. I have to express my gratitude to
him for his patience replying me my queries about programming in MATLAB.
My infinite thankfulness is to my parents, who have let me finish my career
abroad. Thanks also to Ricardo Oña, Sergio García, Guillermo Tato, Víctor Herrero,
who are my football mates, and added their points of view about this project, and
to Umi Morita and Helena Pérez who have helped me with this report.
During my stay in Sweden, I have met lots of friends from many countries.
Thanks to them to make this year the best and the most meaningful one in my
life. Day by day, they have made me laugh and I have spent great time with them.
Besides, I would like to wish them the best and I hope that we will see each other
here again or in other parts of the world.
Chapter 1
Introduction
1.1
Background
Nowadays, the development of the technology in image processing and the possibility
of broadcasting huge amount of sporting events, have increased the demand of
multiple services requested by customers. Spectators are not content to only watch
the game. They also want more information on the screen like measures to check
statistics, names and data, replays from different angles, etc.
An important company in this field which provides such services is TRACAB
[1]. This firm has been selected recently by FIFA to provide these multiple services
in the biggest football competition, the world cup in South Africa in 2010.
Figure 1.1: TRACAB: example of services
Since 2003, TRACAB has been improving its technology in order to develop
new products. One of its products is to provide virtual representations of games
in real-time, for helping sport industries and media creating new products for the
end consumer. SAAB and several universities are involved in this exciting project.
In particular KTH, which is one of Sweden’s leading technical universities, has
1
CHAPTER 1. INTRODUCTION
close collaborations with TRACAB. This Master’s Thesis is a part of this on-going
collaboration.
1.2
Problem this project addresses
One of TRACAB’s ambitious goals is to reconstruct a real match in 3-D in real-time.
This Thesis is a small part of this huge project and is devoted to the estimation
of the position of the ball. There are many possible approaches to ball detection.
The best and more utilized method is via usual tracking. TRACAB has 8 pairs
of cameras to cover the whole pitch during matches and by using stereo image
processing which they capture the movements of all objects such as players, officials
and the ball at a rate of 25 images per second.
Figure 1.2: TRACAB: Covering the whole pitch
Therefore, image and video data are utilized to track the players and the ball
at every moment. TRACAB’s tracking of people is reliable but not in the case of
the ball. Reasons why the ball tracking is not reliable will be explained later. If its
algorithm fails at a certain moment, frequently, knowing where the ball is, makes it
easy to predict where the ball will be in the near future. With this prediction, it is
possible to search and find the ball. This method is sufficient in many situations.
Moreover, improvements have been made with tracking algorithms and ball detection algorithms [2]. However, these algorithms will lose track of the ball at some
points and they have to look at the whole image to refind it. This is potentially a
computationally expensive task and there is no certainty of finding the ball during
this search.
There are often situations where detecting the ball is hard, and as it has been
commented before, sometimes tracking the ball is not reliable. For example, it may
be occluded by a player if he is in front of the ball or even its appearance may be
similar to a player’s sock or other different objects like a bag that may be in the
2
1.2. PROBLEM THIS PROJECT ADDRESSES
Figure 1.3: Sixteen cameras are installed in the stadium.
pitch. Lighting is usually another problem experienced with this technique. In some
stadiums, depending on the time of the match, some parts of the pitch have solar
light and other parts may have shadow. Moreover, the fact that ball is small and
can move very quickly, faster than players, is a problem that affects to the methods
which analyze images. Therefore, predicting its position from one frame to another
may require a large search area. This phenomenon can be confirmed in Figure 1.4.
Besides, the problem of tracking the ball could be also due to the fact that, when
ball moves quickly, its shape become oval. The appearance of the ball may vary
alot due to motion blur. Figure 1.5 illustrates some examples of situations where
some of these problems could appear.
This Thesis explores methods for highlighting where the football ball could be.
These new algorithms are based on exploiting the contextual information. This
means that the ball position will be estimated by only looking at the position of
the players over time on the pitch that is provided. This information, in this case,
is formed with the one concerning the players. Intuitively, many people could think
that whatever human acts, it is unpredictable because human behavior is based on
impulses. Although many patterns could be detected, human being may behave
3
CHAPTER 1. INTRODUCTION
Figure 1.4: Ball, in green, moves faster than players, in red and blue. Then, predicting the position of the ball may require a large area
(a) Ball appearance may be
similar to player’s sock
(b) Ball appearance may be
blurred and oval because of
motion
(c) The ball may be occluded
behind the players
(d) The ball could be in the solar light part or in the shadow
one
Figure 1.5: Different problems may appear in the scene causing unreliable ball
tracking
4
1.3. OVERVIEW OF HOW TO EXPLOIT CONTEXTUAL INFORMATION
without any predictable sense anytime. On the other hand, patterns are found
easily when players behave similar as they usually play, and the ball position could
be exactly defined with a minimum error approximation. Hence, this work does not
aim to find the exact result but to provide to a large extent to make easier for ball
detector and assist that program to locate faster where the ball position is.
The methods that this Thesis utilized are applied in two different scenarios.
The first one is: given no information about the situation, using the position of the
players in a time window to predict the plausible regions of where the ball could
be. The second one is: given that the ball was in the possession of player x at time
t, and the position of players in the subsequent time window, to locate where is
the ball. The goal in both cases is to find ‘priors’ for the ball location. In other
words, imagine that the ball detector has lost the ball while looking for the ball
in the picture. Furthermore, this report will construct priorities given the general
situation by spotting an area where the ball could be with most probability.
1.3
Overview of how to exploit contextual information
As it has been commented above, this work uses data provided by TRACAB. This
data s collection of both position of the ball and that of all players classified by the
number of each frame of football matches’ video. The means of how this data is
exploited is as follows: it is collected and organized in order to set a list of training
data. By machine learning, any new query that arrives is passed to the training
data and the machine will generate the most accurate answer of the ball position.
It will be more or less precise based on the algorithm of the machine learning. This
Thesis utilizes one part of the provided data as learning and the other part for
testing to check later how far the prediction of the ball position can be. A detailed
explanation of how machine learning performs is described in the next part of this
report.
1.4
Supervised Learning Methods
Basically, methods utilized in this work try to learn a mapping from inputs to
outputs. Besides, the function that is being searched is too hard to specify explicity
so that it is no a real regression method. In order to learn this mapping, there
are examples which are stored as a database where every input has its own output.
Afterwards, each new input will be compared with the database and depending on
the utilized algorithm, the result will be an output more or less precise than an
expected value. It is impossible to store infinity database because it might waste so
much time, but on the other hand, it cannot keep a simple list of examples because
then the result of mapping will be very far from the real value.
Mechanisms used in this paper for describing this mapping are nearest neighbour
and distance-weighted nearest neighbour. A first introduction is described in [6] .
The theory of this chapter has been studied from the book called Machine Learning
5
CHAPTER 1. INTRODUCTION
Figure 1.6: Inputs and outputs from the database are in black. New input and new
output in red. This is a mapping from space X to the space O
[3]. Nearest neighbour is a method that checks the most suitable input from the
database depending on the new input and gives its corresponding output as a result
to value of the output of the new input. The most suitable input is chosen because
it has the minimum distance between it and the new input. Additional mechanisms
have been studied in order to learn efficiently this mapping as [7] and as Weber,
Schek, Blott explore in [8] and [9].
On the other hand, distace-weighted nearest neighbour is a method that introduces a weight for each input from the database depending on the distance from
them to the new input. Therefore, more weight is provided to the closer neighbours.
Moreover, the most suitable input in this case is the one with minimum distance
once this weight is given. Additionally, other alternatives are taken from [10]. Besides, a full review of metric space has been studied in [5] . Euclidean distance is
utilized to compute all the distance in all the metric spaces used in this work.
6
Chapter 2
Data utilized in this work
2.1
Collection of needed data
TRACAB has provided the data from the match AIK Solna against Örgryte that
corresponds to the Swedish football league and it took place in Rasunda Stadium
on 21 September 2009 in Sweden. This data shows the X and Y coordinates of
players and referees, and the X, Y and Z coordinates of the ball position on the
pitch for every frame that they could extract from the algorithm and software that
they utilize. Although they recorded 90 minutes of the match, ball data was tracked
for only 66 minutes of these 90. In total there are 98511 frames since they utilize
video of 25 frames per second. The lack of ball data can be seen in Figure 2.1. This
is normal performance of the ball tracker since the ball is much time outside.
Figure 2.1: Presence of ball data
The thick white line in the middle of the picture represents the break between
the first and the second half. Thin white lines are the moments where the ball
is missed in the frames of the match. Thin white lines separate sequences of the
match. The data useful for our supervised learning task is the players’ positions
in the frames where ball data exists, too. Although there are many frames where
the ball is not tracked, the reason most common is that the ball is out of the pitch.
Therefore, it is necessary to re-initilizate the tracking algorithm. The program has
to search the ball on the pitch with the possibility that some problems may appear,
as it has been mentioned before, in section 1.2 . Figures 2.2a and 2.2b represent
the last tracked frame and the first one of the next sequence. Thus, the algorithm
need a re-initialization to start tracking the ball when it is on the pitch again.
Some examples of how TRACAB provides this data are represented in the fol7
CHAPTER 2. DATA UTILIZED IN THIS WORK
(a) In the last tracked frame from a sequence, the ball is almost outside
(b) First tracked frame from the next sequence, when the ball is on the pitch again
Figure 2.2: The tracking program needs to be re-initialized after the ball goes out.
The ball is represented as the big green point in both images
lowing tables.
Frame
163923
163923
163923
163923
163923
163923
163923
163923
163923
163923
163923
163923
163923
163923
163923
163923
163923
163923
163923
163923
163923
163923
Team 0/1
0
0
1
1
0
1
1
1
0
1
1
1
1
0
0
0
1
0
0
0
1
0
Player number
22
24
27
20
18
4
11
3
5
10
19
7
5
40
17
7
30
30
6
44
18
19
X coordinate
34,5660
43,2075
99,5190
73,4475
52,7835
71,5575
54,5475
73,3320
55,1985
52,9725
57,5505
58,2645
63,2205
52,9515
31,2585
44,2155
61,2465
0
46,6515
30,4185
72,6810
52,3215
Player Data
8
Y coordinate
51,0544
31,2596
33,9864
31,8104
32,3680
15,9120
42,2076
42,0648
62,7708
25,0512
50,3268
12,8860
35,8496
34,1224
21,7600
12,3692
28,9816
33,5648
25,5816
39,2020
55,9912
12,2876
2.1. COLLECTION OF NEEDED DATA
Frame
163923
163924
163925
163926
X coordinate
52,8885
52,7520
52,5840
52,2165
Y coordinate
31,2188
31,3004
31,3956
31,5724
Z coordinate
0,3000
0,3000
0,3000
0,3000
Ball Data
These coordinates have the origin in the top right of the screen. What is more,
a transformation to obtain the coordinates as the match is watched in the video
is necessary. This is a simple transformation, where the X position is computed
as X − 105 and the Y position with Y − 68 because the length and width of the
pitch are 105 and 68 meters respectively. The example of player and ball data is
illustrated in Figure 2.3 .
Figure 2.3: Illustrated example of how TRACAB provides player and ball data.
Red and blue points represent both teams and the green point is the position of the
ball
Besides, they provided videos of the match. TRACAB installed other four
cameras covering the whole pitch to record the match. Their images are in Figure
2.4 . These videos are very useful to prove, in some points, what is really happening
9
CHAPTER 2. DATA UTILIZED IN THIS WORK
(a) Camera 1
(b) Camera 2
(c) Camera 3
(d) Camera 4
Figure 2.4: TRACAB installed HD cameras to record the whole pitch and get a
higher resolution video then that used for tracking system
during the match. Additionally it confirms if the position of the players is true, or
if the algorithm of TRACAB has some mistakes in a particular case.
10
Chapter 3
Estimation of ball position
3.1
Nearest Neighbour Regression
The simplest method that is applied in this Master’s Thesis is collecting a database
and then search which frame is the most similar to our test frame. The database
consists of coordinates of each player for each frame from the learning data, and
it is called training data, too. Thus, the algorithm has to learn a mapping from
players’ positions to ball position. Therefore, the feature vector is made up by two
coordinates of each player at one frame.
~xi,t ∈ ℜ2 i = 1, ..., 22 t = 1, ..., T
~ t = (~xT , ~xT , ..., ~xT )T ∈ ℜ44
X
22,t
1,t 2,t
f : ℜ44 → ℜ2
~ t ) → ~xb,t ∈ ℜ2 (3.1)
f (X
~xi,t means the position of player i at time t on the pitch, coordinates X and
~ t)
Y, and it is a matrix with dimension 2x1. ~xb,t is ball position. Besides, f (X
is defined via nearest neighbours and it is a matrix with dimension 44x1. The
euclidean distance is utilized to obtain the training data which has the minimum
distance to the test data. The euclidean distance is defined in 3.2.
2
~ a, X
~ b) = ~a − X
~ b
d(X
X
(3.2)
This method has some problems. First of all, we may have a problem if a
player changes his position with another player. For example, if a defender changes
11
CHAPTER 3. ESTIMATION OF BALL POSITION
his position with a midfielder, the distance between the team formations at these
different time instances will be increased. This problem may also appear if the
coach orders players to change. It is also necessary to do a transformation in the
coordinates in the second half in order to have teams in the same position in the
field. Thus, it is able to work with both halfs easily.
The first results of this method were not so encouraging: as it can be seen in
the figure below, the players are almost in the same position but the ball is very far
from its real position.
Figure 3.1: Test data against selected training data. It is selected because it has the
minimum euclidean distance to the test frame. Blue and red points represent both
teams from one test frame. Magenta and cyan points are players’ positions selected
as the nearest neighbour from database. Although the players are approximately in
the same position, in one case the ball is at the big green point while in the other
case it is at the big black point. Hence, the estimation of the ball position is very
far from the real position.
3.2
Nearest Neighbour regression with players in grid
In order to solve some problems that appeared in the last section, a rectangular
grid is placed over the pitch. This idea means that the pitch is divided into several
regions, and then, the number of players is counted in each region. This method
solves many problems. For example, it is neither necessary to transform coordinates
for the second half, nor to consider about changes in position of players because now
each player is not associated with a number. Therefore, the algorithm does not check
the number of each player. The grid defines a histogram which shows where the
12
3.2. NEAREST NEIGHBOUR REGRESSION WITH PLAYERS IN GRID
players are generally. It is possible to solve the problem that could appear too, if
any player receives a red card in the match because the histogram is normalized
with the total number of players present on the pitch.
Grid defines N rectangular regions: R1 , ..., RN
~ t = (h1,t , ..., hn,t )T where hi,t ∈ [0, 1]
H
hi,t =
1
NtP
PNtP
ℑ [x~j,t ∈ Ri ]
j=1
2
(3.3)
~ a, H
~ b) = ~a − H
~ b
Euclidean Distance : d(H
H
NtP is the number of player on the pitch in t . ℑ is the indicate function which
~ t is a 2-D histogram of the players’ positions. It
adds 1 if x~j,t ∈ Ri . In addition, H
is necessary to blur this histogram in order to spread players’ positions and smooth
the borders of the grid. Therefore, each histogram is convolved with itself to smooth
the borders of the regions, as it can be seen in figure 3.2. The players’ positions
related with this histograms can be seen in figure 2.3. The following tables represent
the number of players in each region before and after blurring.
0
0
1
0
0
2
2
8
6
2
6
6
6
2
3
0
0
0
0
0
4
12
12
8
2
0
1
1
0
0
0
0
0
1
0
2
1
1
0
0
1
1
3
2
1
7
10
9
10
6
16
16
18
10
4
17
20
25
18
12
0
1
2
1
1
0
0
0
0
0
6
14
22
18
14
0
0
0
0
0
1
4
6
8
5
0
0
1
0
0
4
2
2
0
0
2
2
6
4
2
This idea is implemented in MATLAB, and new results are obtained. How
methods are implemented in MATLAB have been studied from [11]. These results of
13
CHAPTER 3. ESTIMATION OF BALL POSITION
(a) Histogram of players’ positions
(b) Convolved histogram of players’ positions
Figure 3.2: These figures illustrate histograms that summarize the players’ positions
on the pitch
nearest neighbour classfication are represented in a histogram, showing the average
of the error, in meters, from the estimated ball position and the real ball position.
The chosen grid is made of 10 x 5 bins to cover the whole pitch, and the database is
collected from all the available frames of the second half. In total, there are 39627
frames. There are 1005 test frames from the first half and they have been taken not
randomly but with a distance between them enough to have a representation of the
whole first half. First histogram can be seen in Figure 3.3.
The election of the size of the grid has been chosen to minimize the average
error in the estimation of the ball position. A validation set is used to estimate this
14
3.3. NEAREST NEIGHBOUR WITH PLAYERS IN GRID AND TEMPORAL
WINDOW
Figure 3.3: The histogram of errors with parameters: bins 10 x 5. Average Error:
22,54 meters. The error is measured by the distance from the estimated ball position
to the real ball position
quantity as opposed to the test set. The validation set uses performance on this set
to estimate free parameters and the test set uses parameter settings learned from
the validation set. Figure 3.4 shows the same histogram as in 3.3 but a grid of 20 x
10 bins is utilized. The average of the error is 24,04 meters, but if the size of 10 x 5
bins is utilized, the error was 22,54 meters, as it can be seen in the last histogram.
It is essential to say that test frames are the same in both cases. Therefore, different
methods are able to be compared.
3.3
Nearest Neighbour with players in grid and temporal
window
As has been seen before, at different time instances, the players can be in very
similar positions but the ball at very different locations. In order to try to solve
this problem, a temporal window is created. This temporal window introduces the
concept of a sequence. Now, the histograms that have been compared to find the
nearest neighbours have been added to the histograms of a forward step of time
and a backward step of time. The selected nomenclature takes grids of the frames
following this rule:
15
CHAPTER 3. ESTIMATION OF BALL POSITION
Figure 3.4: The histogram of errors with parameters: bins 20 x 10. Average Error:
24,04 meters.
t - K * S, ..., t - 2 * S, t - S, t, t + S, t + 2 * S, ..., t + K * S
T
T
T
T
T
~ t,K = (H T
H
t−KS , Ht−(K−1)S , ..., Ht , ..., Ht+(K−1)S , Ht+KS )
(3.4)
2
~ a,K , H
~ b,K ) = ~ a,K − H
~ b,K Euclidean Distance : d(H
H
Where K is the width of the temporal window, S is the step of the temporal
window and t means the "present" of the tested frame. K equals to zero would be
the experiment that has been executed before.
The distance that has to be minimized is written with this expression:
Distance ≡
P
t∈−K∗S,...,+K∗S
PN
test
i=1 ||Xtj +t
training 2
|| (3.5)
− Xi,t
j +t
First of all, K = 1 and S = 10 frames are selected as parameters. This means
that a sequence of 20 frames will be utilized in total, almost one second. The
distance that is aimed to be minimized is in this situation:
16
3.3. NEAREST NEIGHBOUR WITH PLAYERS IN GRID AND TEMPORAL
WINDOW
Distance ≡
P
t∈−10,0,+10
PN
test
i=1 ||Xtj +t
training 2
|| (3.6)
− Xi,t
j +t
Hence, the same 1005 frames are utilized for testing and the same 39627 frames
from the database that the computer will use for training data. Results are shown
in this histogram, where the average of the error is 23,03 meters in this case.
Figure 3.5: The histogram of errors with parameters: K = 1, bins 10 x 5. Average
Error: 23,02 meters
Therefore, the temporal window has to be longer. The next experiment is to
explore the effect of varying K > 1. The reason why the training data is exactly
39627 frames is going to be explained now. Although the whole match is composed
of 98511 frames, 49279 frames of the first half are dedicated for testing, and 49232
frames for learning data. The problem is that, as it has been explained before, there
are holes in the data, lack of ball data in some frames consequently, and actually
not all frames are useful for training because for each frame it is necessary that
from −K ∗ S to +K ∗ S the frames exist, related with each particular frame. In
this way, choosing K = 10 as most and S = 10 frames, the total number of useful
frames are 39627 in the second half and 38185 in the first half. There are less useful
frames because the whole 66 minutes of ball data is not continuous, as it has been
described before in Figure 2.1.
Next histogram has been calculated utilizing K = 10, that means, a temporal
window from −100 + f ramei to +100 + f ramei . In seconds, the temporal window
has duration of 8 seconds.
17
CHAPTER 3. ESTIMATION OF BALL POSITION
Figure 3.6: The histogram of errors with parameters: K = 5, bins 10 x 5. Average
Error: 22,02 meters
The improvement is almost 3 meters regarding the average of the error. Errors
between 0 and 20 meters have the most number of frames, too. In conclusion, there
has been an improvement of the method in general.
Consequently, to watch the improvement of this method using different values
of K, another experiment has been proved. It consists of using different values of K
for a certain frame. Errors, that decrease as long as K is increased, are represented
in this table. All experiments correspond to the same frame.
K
1
2
3
4
5
6
7
8
9
10
Error (meters)
17,07
34,54
4,93
7,29
6,06
7,29
6,06
3,97
2,29
3,97
18
3.3. NEAREST NEIGHBOUR WITH PLAYERS IN GRID AND TEMPORAL
WINDOW
Figure 3.7: The histogram of errors with parameters: K = 10, bins 10 x 5. Average
Error: 20,81 meters
The improvement that is shown in the table could be visualized by looking at
the next figures. The estimation of the position of the ball (red point) and the real
position of the ball (green point) are drawn in each pitch.
In all these cases, 1005 frames are utilized for testing. Surely, the average of the
error will decrease by using more test frames.
19
CHAPTER 3. ESTIMATION OF BALL POSITION
(a) K = 1, Error: 17,07 m.
(b) K = 2, Error: 34,54 m.
(c) K = 3, Error: 4,93 m.
(d) K = 4, Error: 7,29 m.
(e) K = 5, Error: 6,06 m.
(f) K = 6, Error: 7,29 m.
(g) K = 7, Error: 6,06 m.
(h) K = 8, Error: 3,97 m.
(i) K = 9, Error: 2,29 m.
Figure 3.8: Figures showing the distance between the estimated (in red) and the
real position of the ball (in green) for K = 1, ..., 9 on the pitch
20
Chapter 4
Heat Maps: Scenario given no prior
information
In all these previous situations, it has been learnt that actually the ball could be in
a range near the real position of the ball. Heat maps are divided in two chapters
which describe two different football situations. In both of them, the objective is
to apply the learning of a mapping to select and decide which zones have more
probability to find the ball inside. Therefore, the same supervised learning methods
are utilized in both scenarios, and only the feature vector will change in each case.
4.1
Scenario 1: General Situations
This part of the Master’s Thesis describes, what it has been called, a kernel density
estimation. A grid of bins is created, but this time for the ball. In this new grid,
each bin obtains score depending on the distance of the players from a test frame
to the whole learning data. This score is shown as a probability in this expression:
p( ball in region j ) ∝
2
wi = e(−λkxi −xtest k
)
P
i s.t. x~b,i in region j
f or i = 1, ..., Ntraining
wi
examples
(4.1)
Scores are normalized for each frame in favor of helping the visualization of the
experiment. Therefore, by mapping these score to a grey scale, the most white ones
correspond to the highest scored bins and the most black ones to the opposite. This
method is used to see if the minimum distance of the position of players in bins is
also the most scored ones or could be another bin nearer the real position of the
ball. In other words, this chapter has the purpose to produce no explicit results of
21
CHAPTER 4. HEAT MAPS: SCENARIO GIVEN NO PRIOR INFORMATION
the estimation of the ball, but to select zones where the ball is located with higher
probability. Examples of these heat maps are added below:
Figure 4.1: Examples of Heat Map. Green point is the real position of the ball and
red point is the estimated position of the ball utilizing the method explained in 3.3
. The most white bin is the most scored one which is the one suppose to be the
most probable zone where the real position of the ball could be. Therefore, heat
map workd perfectly in this case
Figure number 4.1 displays a heat map where the real position of the ball is
located at the big green point and is inside the white bin, which is the most scored
bin. This bin is suppose to be the most probable zone where the real position of the
ball could be. Therefore, this method called heat map works perfectly, in this case.
The estimation of the position of the ball, the big red point, is located 10 meters
far from the real position of the ball. Hence, heat map works better than nearest
neighbour regression method, in this case, too. In addition, Figure 4.2 is the same
figure as 3.8, but heat maps are included. By utilizing heat maps, it can be seen
how much the estimation of the zones with most priority is closer as the value of
K, defined in 3.3, is increased.
However, heat maps do not give an exact position of the ball with particular x
and y coordinates. Nevertheless, since the length and the width of the bins are 5,25
and 6,8 meters respectively, the error estimating the position of the ball with this
method would be ± 2,625 meters for the x coordinate and ± 3,4 meters for the y
coordinate as Figure 4.3 shows. Those approximations of the error of the position of
the ball depend on the number of bins that are utilized to divide the pitch. In this
case, as it has been commented before, 20 x 10 bins are selected for this experiment
to simplify the possible position of the ball into regions.
22
4.1. SCENARIO 1: GENERAL SITUATIONS
(a) K = 1, Error: 17,07 m.
(b) K = 2, Error: 34,54 m.
(c) K = 3, Error: 4,93 m.
(d) K = 4, Error: 7,29 m.
(e) K = 5, Error: 6,06 m.
(f) K = 6, Error: 7,29 m.
(g) K = 7, Error: 6,06 m.
(h) K = 8, Error: 3,97 m.
(i) K = 9, Error: 2,29 m.
Figure 4.2: Heat maps showing the distance between the estimated and the real
position of the ball for K = 1, ..., 9 on the pitch. Changes can be seen in gray scale
as the density of players change with the team formation. The real ball position
is situated at the big green point, and the big red point represents the estimation
of the ball position utilizing the method studied in 3.3. Moreover, the most scored
region, is very near to the real position of the ball. It is represented as the most
white region, below the green point.
The experiment checks if the real position of the ball corresponds to the most
scored region. Moreover, the results prove that this is not valid in most cases. It
is represented as a percentage of how near the real position of the ball is from the
most scored bin. After all, the aim of this chapter is to introduce an idea of where
23
CHAPTER 4. HEAT MAPS: SCENARIO GIVEN NO PRIOR INFORMATION
Figure 4.3: Heat maps show a region where the ball could be found, therefore, the
error on the estimation is ± 2,625 meters for the x coordinate and ± 3,4 meters
for the y coordinate.
the location of the ball is with more probability. In conclusion, heat maps represent
changes in zones where the ball is more suitable to be located as the density of
players change.
Sometimes, more than one bin could be selected as the most scored bin. Therefore, there could be more than one bin totally white. Besides, the bin in the second
position (or in the third, or fourth position) of the most scored bin table, could be
a perfect candidate to be selected and perhaps it corresponds to the bin where the
real position of the ball is located. For this reason, histograms that represent the
values that the first five most scored bins get have been calculated, see Figure 4.5.
As the most scored bin has always value equal to 1 which means that it will be
totally white, its histogram is obvious and it is not represented.
Figure 4.4: Heat maps show a larger region because the neighbours have been included. The ball could be found in this region, therefore, the error on the estimation
is ± 7,875 meters for the x coordinate and ± 10,2 meters for the y coordinate.
To obtain these results, it is reasonable to set a threshold below the value 1. If
the real position of the ball is located in a bin with value above this threshold, the
result of the test will announce that the estimation of the bin has been calculated
successfully. Finally, the eight neighbours of the bin, where the ball is located, are
included to compute again a percentage. An important improvement is obtained
then, although the error estimating the position of the ball is increased because the
zone where the ball could be is now larger. Using neighbours in the calculation
makes the predicted region larger. The error in this case is ± 7,875 meters for the
x coordinate and ± 10,2 meters for the y coordinate, as it is illustrated in Figure
4.4.
24
4.1. SCENARIO 1: GENERAL SITUATIONS
(a) Second position
(b) Third position
(c) Fourth position
(d) Fifth position
Figure 4.5: Histograms that represent the value of the score of the second, third,
fourth and fifth bin in the most scored bins table.
The evolution of the histogram that shows the probability to obtain a correct
selected bin or region can be seen in Figure 4.6 .
Therefore, percentages of how this method works are computed and that is
illustrated in Figure 4.7 .
25
CHAPTER 4. HEAT MAPS: SCENARIO GIVEN NO PRIOR INFORMATION
(a) Histogram from 0,0.1,...,0.8,0.9,1
(b) Histogram with a threshold in 0.75
(c) Histogram with a threshold in 0.75
and neighbours of the bin are included
Figure 4.6: Histograms showing the value of the bin where the real position of the
ball is located.
26
4.1. SCENARIO 1: GENERAL SITUATIONS
Figure 4.7: This figure illustrates percentages of the score of the large region, once
its neighbours are included, where the real ball position is located. As it can be
seen, only the 21 % of the cases are successed
27
Chapter 5
Heat Maps: Scenario where players pass
the ball
5.1
Scenario 2: Particular Situations
The second scenario investigated for ball prediction is whenever a pass happens in
the match. This section describes a method that can determine, when there is a pass,
who the most suitable player to receive the ball is. Therefore, the most probable
zone where the ball is going to be located in the near future can be predicted. Then,
the prior information given is that a player has the ball in a certain frame and he
is going to pass the ball.
First of all, we define when a completed pass occurs. This is the case when:
1. A player, the passer, has the current possession of the ball.
2. A player receives the ball when he obtains the possession of the ball.
3. The player, who receives the ball, has to be a teammate of the player who has
passed the ball.
Summary of a pass:
1. Therefore, a pass is computed and it is possible to calculate all these parameters of that pass:
a) The team which completed the pass.
b) The position and the identity of the player who passes.
c) The position and the identity of the player who receives.
d) ρ : the distance in meters of the pass.
e) θ : the angle in degrees of the pass. It is considered that horizontal
passes, orthogonal to the goal, have zero degrees.
f) The length of the pass. Computed in frames and in seconds.
29
CHAPTER 5. HEAT MAPS: SCENARIO WHERE PLAYERS PASS THE BALL
g) The velocity of the ball. Computed as an average in meters and in
kilometers per second.
h) In which part the pass has been computed. First or second half. Very
useful in order to represent the pass in a figure.
2. If the player who receives the ball belongs to the opponent team, the pass is
canceled but he obtains the possession of the ball.
Detection of passes in a game:
1. A player has the possession of the ball when the following conditions are
fulfilled:
a) The ball is inside an imaginary circle around himself.
b) The radius of this circle could be flexible, but it has been fixed to 1.5
meters in this work.
c) The height of the position of the ball, coordinate Z, must be less than 2
meters because the ball often goes above the player and he actually cannot touch it. Hence, it has been considered that he has not in possession
of the ball.
d) If there are two players whose circles involve the ball, the player who has
the minimum distance to the ball has the possession of the ball.
i. In this case, some problems will appear. If both players belong to
the same team and one has the ball in possession but when he loses it
in favour of his teammate, a problem appear. The pass is computed,
but then, velocity and time length of the ball cannot be taken into
account because the pass length would be only one frame. Therefore,
the velocity becomes huge and physically impossible to occur.
2. The velocity of the pass has to be less than 40 meters per second, that means
around 140 kilometers per hour.
Moreover, those passes with a velocity of more than this threshold are not computed because they will not be really true passes. The reason to select such a
conservative speed threshold is because sometimes a player kicks a ball strongly but
it goes to a teammate who may catch and get the ball in his possession. Experiments estimating the velocity kicking a ball can be seen in [14]. Besides, in the
situation of throwing a ball, the coordinates of the ball are not tracked accurately
and a player receives the ball quite fast.
Therefore, MATLAB code is written to obtain the information and the parameters of all the completed passes of the match. It is based on loops that keep the
number of the player and which team has the possession of the ball or if anybody
has the possession of the ball for every frame. In addition basically, if a possession
of the ball changes from one player to another of the same team, a pass will be
30
5.1. SCENARIO 2: PARTICULAR SITUATIONS
recorded. Following the instructions given before about how a completed pass is
defined, all the distances from all players to the ball have to be computed in order
to decide who has the possession of the ball if two players are very close to the ball.
However, there are many situations where this method could fail. For example, if a
player has the ball and one opponent player runs behind him to kick the ball, the
machine will not recognize that it is actually the opponent who kicks the ball and
not the player in possession as the machine has decided. Otherwise, the pass detector works quite well, with very coherent parameters of the pass, and with a very
logical determination. For every frame, a picture illustrates the actual situation on
the pitch. An example of this can be seen in Figure 5.2 .
Figure 5.1: Parameters of each completed pass are saved in a text file
Moreover, the parameters of each completed pass is saved in a text file. An
example of how these parameters are saved can be seen in figure 5.1 .
31
CHAPTER 5. HEAT MAPS: SCENARIO WHERE PLAYERS PASS THE BALL
(a) Part 1: Player 30 of red team has the possession of the ball
(b) Part 2: Ball goes towards the position of player 4
Figure 5.2: In these pictures, ball is the green point, teams are represented with red
and blue dots and their circles of possession are illustrated, too. Sequence shows a
pass from player 30 to player 4 in red team. In 5.2a player 30 has the possession of
the ball, and ball goes to player 4 in 5.2b. The second part of this sequence can be
seen in 5.3. Besides, at the top of each picture which player has the possession of
the ball can be seen
32
5.1. SCENARIO 2: PARTICULAR SITUATIONS
(a) Part 3: Ball continues going towards the position of player 4
(b) Part 4: Player 4 obtains the possession of the ball
Figure 5.3: These figures are the second part of the sequence seen in 5.2. Sequence
shows a pass from player 30 to player 4 in red team. Ball goes to player 4 in 5.3a.
Player 4 gets the ball in 5.3b and the pass is completed. Besides, at the top of each
picture which player has the possession of the ball can be seen
33
CHAPTER 5. HEAT MAPS: SCENARIO WHERE PLAYERS PASS THE BALL
In this particular match, 838 completed passes are computed. At first, they are
classified by their distance and angle. Hence, polar coordinates are utilized in order
to help this classification. A review of polar coordinates has been studied from [12]
and extra knowledge has been acquired from [13]. It is necessary to compute a
coordinate transformation for passes located in the second half and also depending
on the team that they belong to. Following this rule, forward passes will be always
have zero degrees, that is to the right direction, although they happen in the second
half or they belong to the team that actually goals to the left.
(a) Scatter plot of red team. Forward passes to the right
(b) Scatter plot of blue team. Forward passes to the left.
Figure 5.4: These pictures describe the distance and the direction of the passes
for each team. Red team has completed 546 passes and blue team has only 292.
Furthermore, most of the passes are forward
Figure 5.4 represents two histograms, one per team. All the passes are classified
34
5.1. SCENARIO 2: PARTICULAR SITUATIONS
by their distance and angle. The analysis of passes may reflect the summary of the
match. For example, in this match 546 passes have been computed for red team
and 292 for the blue one. The final result of the match was a winning for the red
team by 3 goals.
5.1.1
Scenario 2: Feature Vector
The feature vector for this scenario consists of a grid of bins in polar coordinates. 10
regions are chosen in radial direction, from 1 meter to 100 meters irregularly, spaced
non-linearly and 16 regions in angle direction regularly spaced. Players’ positions
relative to the player in possession are recorded in the corresponding bin. Moreover,
the reason to select an irregular spaced radial direction in the composition of bins,
is that there may be more players around the player who passes than far away. An
example of relative positions can be seen in Figure 5.5.
Figure 5.5: Relative positions of every player to the player who has passed the ball.
This player is situated at the center of this figure. In particular, this is a pass that
belongs to the blue team. The pass is described with the green line
A problem may appear if a player is located between two bins or near some
borders, is solved smoothing players’ positions. In order to smooth the border that
each bin has with its neighbours, it cannot be possible to do a convolution of the
array as it has been done before in the other scenario. This time, the grid has
polar coordinates and it means that the last element of each row is neighbour of
the first one. To solve this problem, each players’ position is represented with a
white point in a black background and each image is blurred with a gaussian kernel.
Furthermore, each player is represented as a number of points around its particular
position and all of these points have a particular weight that the gaussian function
35
CHAPTER 5. HEAT MAPS: SCENARIO WHERE PLAYERS PASS THE BALL
gives. This idea can be better understood by seeing Figure 5.6 and 5.7. All these
points are kept in their corresponding bin creating the feature vector for each pass.
(a) Original image not blurred
(b) Blurred image with gaussian function
Figure 5.6: Players’ relative positions to the player who has passed the ball. Both
pictures represent the same situation, but in 5.6a there are only single white points
as players and in 5.6b each player is represented as blurred points. This method
will solve the problem that could appear if a player is located in the middle of two
or more bins
Once the relative positions of all the players for each completed pass are kept in
an array, the next step is to learn a mapping. This mapping is learnt from inputs,
36
5.1. SCENARIO 2: PARTICULAR SITUATIONS
(a) Weight of players in bins without blurring their positions
(b) Weight of players in bins after blurring
their positions
Figure 5.7: The matrix that stores the weight of players in bins and represents the
feature vector, is transformed into a 1-D array in order to be able to see this figure.
The effect of blurring players’ positions relative to the passer can be seen in 5.7b.
If a player is between two or more bins, he gives weight to both bins. This cannot
happend in 5.7a where players gives weight to only one bin
the own relative positions of the players, to outputs, the length and direction of
each pass. In this case, all passes are independent from each other. Moreover, in
this mapping each pass can be utilized as one test. Then, the number of computed
tests are 838 since this is the total number of recorded completed passes for this
match.
In addition, supervised learning methods are used to learn this mapping and
the distance from the test feature vector to all the training examples, that have
their own feature vectors, has been calculated. Therefore, the most suitable passes
for each situation can be selected. They are also called as the most similar nearest
neighbours. The mathematical expression can be checked in the following equation.
The distance can be calculated by minimizing this expression and X represents
the space of the feature vector. This distance is defined as euclidean distance.
Therefore, the expresion is:
Distance ≡ argmin
1 ≤i ≤ N ||X
37
test
− Xitraining ||2
(5.1)
CHAPTER 5. HEAT MAPS: SCENARIO WHERE PLAYERS PASS THE BALL
The purpose of this chapter is to compute from the list of passes with their
corresponding relative positions of the players, the probability that the ball will be
located in each region in the near future. These probabilities are easy to determine
since it is known where the teammates of the player who passes the ball are and the
most common passes that the player may do for each situation. Therefore, training
data is chosen such that the distance and angle of each pass from the passer is in a
region containing a teammate. Hence, only a subset of training data is utilized to
compute the score for each region following this expression:
score region j = score region j + e−λ||X
test −X training ||2
i
(5.2)
To illustrate them, a heat map is created where a grid of 15x8 bins cover the whole
pitch. Bins where the teammates are located are in a gray scale where the most
white region is related with the most probable zone where the ball could go. The
following Figure 5.8 summarizes this concept.
Figure 5.8: Heat map of a particular pass. In this case, the blue team has the
possession of the ball. The player who has the possession of the ball is represented
at the yellow point. The green point represents where the ball finally went in
this situation. Players do not pass the ball directly to the other players in some
occasions. As it can be seen in the figure, bins where the teammates are located
have a gray scale. This scale depends on the probability of each player to receive
the ball. Therefore, this is a great example because the ball has been passed to the
zone with the most probability, the most white
Finally, the last step of the method applies these percentages to the teammates.
Teammates obtain the percentage that they have to get the possession of the ball
38
5.1. SCENARIO 2: PARTICULAR SITUATIONS
in the near future for each pass. It depends on how far they are from the player
who has currently the possession of the ball and how the opponents are placed on
the pitch. Hence, teammates are classified by this percentage. In cases where two
players are in the same region, the percentage is split so that both players obtain the
split percentage that has this region. Which player is going to receive the ball is not
analyzed in this work. Therefore, Figures 5.9 and 5.10 describe two examples where
a player has the possession of the ball and the percentages of his teammates are
analyzed. A situation where there are some players without an explicit percentage
is owing to the fact that they have obtained zero percentage.
Figure 5.9: In this example, red team has the possession of the ball. In particular,
the yellow point marks the position of the player that currently has the possession
of the ball. His teammates are classified by his number and percentage receiving
the ball in the near future. This percentage depends on the colour of his region. A
list of these percentages is organized from the highest percentage to the lowest one.
This example is selected because it has all the particular situations. Players with
number 18, 4 and 27 have the same percentage although they are not in the same
regions. Players with number 10 and 19 have the split percentage of their region.
Actually the ball is located at the green point in the near future. Player number
7 is situated in that region. He had the most percentage to receive the ball. In
conclusion, this method has worked perfectly because it estimates the region where
the ball could be found in the near future correctly
39
CHAPTER 5. HEAT MAPS: SCENARIO WHERE PLAYERS PASS THE BALL
Figure 5.10: Another selected example is illustrated in this figure. In this case,
player 30 obtains the most probability to receive the ball. Moreover, he belongs to
the blue team so that the numbers of his teammates are represented in the figure.
It is very reasonable because he is by far the closest to the player who has the ball.
Therefore, he obtains almost 90 % of probability to receive the ball and actually the
ball went to his region as it can be seen at the green point. Some players are not
in the percentage list because they have not received any score to obtain the ball
5.1.2
Results and Conclusions
In conclusion, players nearer the ball have more probability to obtain the ball in
the near future. In cases when a defender, who has the current possession of the
ball, passes the ball to the strikers without any priority, this method does not work
efficiently. Despite of selecting the most similar situations with supervised learning
methods, short passes are more common than long ones. Therefore, teammates
closer to the player who is going to pass acquire more percentages.
Thus, results are represented in percentages. To demostrate how this method
could be successful, the number of times, that the prediction of the most white bin
which was actually the region where the player passed the ball, is counted. Almost
in 60 % of the cases, the ball went to the first or to the second most probable
region, as it can be checked in Figure 5.11. Therefore, this method could be utilized
to obtain an alternative solution when tracking is not enough to find the ball. This
method will decide successfully, in most of the cases, where the most suitable zone
40
5.1. SCENARIO 2: PARTICULAR SITUATIONS
is to find the ball if there has been a reinitialization of the system that analyzes the
image.
Figure 5.11: This figure illustrates the percentage of the cases where in the most
probable zone the ball actually went, and also to the second, the thrid, the fourth
and the fifth one
41
Chapter 6
Conclusions and Future Work
As it has been seen in histogram errors in the first scenario, the results are not as
good as expected. There are different ways to improve these results.
Firstly, a better database has to be collected. This means not only to store more
ball data but also have clearer data. For example, fake data can appear when the
referee whistles and players stop and the ball may not be in any expected position
because the situation is invalid. This could be improved by adding sound to videos
or only selecting the data carefully.
Inserting the height of the ball on the pitch could add more information to the
feature vector, but then, our database has to be huge. The height of the ball is only
used in the second scenario to see if the player could touch the ball and have the
possession of the ball or he is not able to touch it.
The most important improvement could be done if the feature vector is complemented with the velocity and acceleration of the players, but in order to estimate
where the ball is for every moment, our database has to be computed with many
scenarios and many different matches.
On the other hand, in the second scenario, it could be more useful if the probabilities are calculated taking into account whether any opponent is near to a teammate
or not. In that case, his pertentage of receiving the ball would be less. Velocity and
acceleration in the feature vector could add more information about the situation
because in this scenario, there is not a temporal sequence.
Other suggestion to improve in the second scenario is to use irregular, circle or
oval regions where teammates could receive the ball from the player who has the
ball. By utilizing rectangle regions, there might be some variations in cases that a
player is near to the border of a region. This improvement has not been realized in
the project because there may appear problems in order to decide how those regions
will be when two teammates are very near from each other. It could be a geometric
problem, therefore it has not been implemented. In this case, the heat maps could
seem as in Figure 6.1.
In addition, as the length of the passes recorded from the match are known, it
could be possible to predict where the player who is going to receive the ball could
43
CHAPTER 6. CONCLUSIONS AND FUTURE WORK
Figure 6.1: How heat maps look when using circle regions where teammates may
receive the ball from the player who has currently the possession of the ball. In this
case, regions with more probability to have the ball in the near future are painted
in black
be. Then, it may predict how long the player could move depending on the length
of passes. Length of passes would depend on the distance from each teammate to
the player who has the ball. In this work the positions where the teammates are
when the ball is currently with the player are taken into account.
Additionally, the data provided by TRACAB is not 100 % accurate. This project
would help them to locate the better position of the ball, but at the same time, it has
to deal with their tracked ball data. Sometimes this data is wrong and impossible
situations may appear when the ball is in a particular position, and it appears 2 or
3 meters far in the next frame. It is impossible to check frame by frame looking for
mistakes because there are almost 100000 frames. An example of this event can be
seen in Figure 6.2 and 6.3 .
44
(a) Player 22 has the possession of the ball in frame 2070
(b) Player 22 has the possession of the ball in frame 2071
Figure 6.2: In this sequence it can be seen how the possession of the ball changes
from player 22 to player 44 in one frame, although they are far from each other.
The second part of this sequence can be seen in 6.3
45
CHAPTER 6. CONCLUSIONS AND FUTURE WORK
(a) Player 44 has the possession of the ball in frame 2072
(b) Player 44 continues with the possession of the ball in frame 2075
Figure 6.3: This figure is the second part of the sequence seen in 6.2. Player 44
obtains the possession of the ball in one frame, although they are far from each
other. The third part of this sequence can be seen in 6.4
46
(a) Nobody has the possession of the ball in frame 2076
(b) Nobody has the possession of the ball in frame 2077
Figure 6.4: This figure is the third part of the sequence seen in 6.2 and 6.3. Suddenly
the ball appears near player 22 again and it seems that he has passed the ball to
the goalkeeper
47
References
[1]
TRACAB: http://www.tracab.com 2003.
[2]
Rafael Osorio. “Ball detection via Machine Learning”. Master Thesis in KTH,
Stockholm, Sweden 2009.
[3]
Tom M. Mitchell. “Machine Learning”. pages 230-249 ,1997.
[4]
Richard Hartley and Andrew Zisserman. “Multiple View Geometry in computer
vision”. Second Edition, 2003.
[5]
Pavel Zezula, Giuseppe Amato, Vlastislav Dohnal and Michal Batko. ‘Similarity Search - The Metric Space Approach”. Advances in Database Systems, Vol.
32, 2006.
[6]
Beyer, K., Goldstein, J., Ramakrishnan, R., and Shaft, “When is nearest neighbor meaningful?” Proceedings of the 7th ICDT, Jerusalem, Israel, 1999.
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