Predicting pool ball trajectories
Faiz Punjani1, Ishan Shah2, Mehul Gada3, Ruhina Karani4
1
UG Student, Computer Department, D.J. Sanghvi College of Engineering, Mumbai, India
UG Student, Computer Department, D.J. Sanghvi College of Engineering, Mumbai, India
3
UG Student, Computer Department, D.J. Sanghvi College of Engineering, Mumbai, India.
4
Assistant Professor, Computer Department, D.J. Sanghvi College of Engineering, Mumbai, India.
2
1faizpunjani@gmail.com
2ishanvshah@outlook.com
3mehulgada08@gmail.com
4ruhina.karani@djsce.ac.in

Abstract—
Pool
ball
trajectory
predictor
is
designed to assist unskilled or amateur pool, or snooker or
billiards players in predicting trajectories. This augmented
reality tool utilizes a camera , placed above the table ,
which acquires and processes the game at the interleaved time
interval after playing each round. The system recognizes the
pool table border, the white ball’s position and the
cue
direction
in
order
to
compute
the
predictable trajectory of the cue ball and the ball directly in
its trajectory. The result is then forwarded to a projector,
placed above the table. The projector then projects onto
the snooker playable field. Various edge detection algorithms and
motion detection techniques are reviewed in the paper. An entire
flow of the process is explained with the techniques used at each
step.
Keywords— edge detection, motion detection, pool game
I. INTRODUCTION
A pool game is a very challenging endeavor for an amateur
player. An ardent effort is required to excel in the game of pool.
The game has basic a classic mechanism.
A conceptual tool is introduced to assist amateur pool players.
The pool trajectory predictor will predict the trajectories just
before a player hits the white ball with the cue at a particular
angle. The computed trajectory will be displayed on the table
by a projector. In this paper, the complete process of this project
is explained. Edge detection is required to extract the pool table
boundaries, which are to be used for calculating the reflection
of the trajectories [1].Various edge detection algorithms are
introduced in brief. The on the fly computation is made only
after motion is not detected. A detailed comparison for motion
detection algorithms is made. The various computations include
trajectory detection, ball interactions and projection of these
computed trajectories [2].
II. LITERATURE REVIEW
A. Edge Detection
For finding the boundaries of objects within images edge
detection techniques are used. These techniques identify points
at which the image brightness changes acutely or, more
formally, has discontinuities. The points at which image
brightness changes acutely are typically organized into a set of
curved line segments termed edges. Significant local changes
of intensity in an image are characterized by edges. Edges
typically occur on the boundary between two differentregions
in an image.
The goal of edge detection is to show a line drawing of a scene
from an image of that scene. Edges of an image can be used to
extract important attribute (like corners, lines, curves). These
attributes are used by higher-level computer vision algorithms
(e.g., recognition) [1].
The four steps of edge detection
(1) Smoothing: Without ruining the true edges, suppress as
much noise as possible.
(2) Enhancement: The quality of the edges in the image are
enhanced by applying a filter.
(3) Detection: determine which edge pixels should be retained
and which should be discarded as noise (usually, thresholding
provides the criterion used for detection).
(4) Localization: The exact location of an edge is determined
(sub-pixel resolution might be required for some applications,
that is, estimate the location of an edge to better than the spacing
between pixels). Edge thinning and linking are usually required
in this step
There are many ways to perform edge detection. However, the
most may be grouped into two categories, gradient and
Laplacian [3].
(1) The gradient method detects the edges by looking for the
maximum and minimum in the first derivative of the image.
(2) The Laplacian method searches for zero crossings in the
second derivative of the image to find edges.
A gradient based edge detector method includes Roberts, B. Motion Detection technique
Prewitt and Sobel operator [5].
There are three techniques that can be used for motion
1. Robert Edge Detection:
detection. These three are explained as follows:
In Robert Edge detection, a gradient of the image is obtained. i.
Basic Blank Background technique
This image is approximated through discrete differentiation. A
2 X 2 matrix called Robert mask is obtained. The entire image First, set the first frame as the background frame. Second,
is convolved using horizontal and vertical Robert masks to compare the current frame with the background frame. Third,
detect edges in x and y direction respectively.
move the background frame to the current frame on the
specified amount. Then, move the background frame slightly in
gx = (z4 - z1) for horizontal derivative approximate
the direction of the present frame - changing colors of pixels in
gy = (z3 - z2) for vertical derivative approximate
the background frame by one level per frame. Detect regions
where the two frames differ, using difference or threshold
2. Prewitt Edge Detection:
filters. Then, image is displayed with white pixels on place
This algorithm uses a discrete differentiation operator called where the difference is present on the specified threshold value.
Prewitt mask. Two 3 X 3 masks are used by this operator which Count the pixels and if the number of pixels are greater than a
calculate values for horizontal and vertical directions.
predefined value then the motion is detected [4].
gx = (z3 + z6 + z9) - (z1 + z4 + z7)
for horizontal derivative approximate
ii.
Previous – Current comparison technique
gy = (z7 + z8 + z9) - (z1 + z2 + z3)
for vertical derivative approximate
First, detect regions where the two frames differ, using
difference or threshold filters. Image is displayed with white
3. Canny Edge Detection:
pixels on place where the difference is present on the specified
In Canny algorithm, the image is made smooth using two- threshold value. Then count the pixels and if the number of
dimensional Gaussian function. It uses a mask of 2 X 2 matrix pixes are greater than a predefined value then the motion is
to calculate magnitude and direction of image gradient
detected [4].
Ex[i,j]=(I[i+1,j]-I[i,j]+I[i+1,j+1]-I[i,j+1])/2
Ey[i,j]=(I[i,j+1]-I[i,j]+I[i+1,j+1]-I[i+1,j])/2
iii.
Using Pixellate filter technique
Image gradient magnitude is calculated as:
||M(i, j)|| = √E𝑥 [i, j]2 + E𝑦 [i, j]2
To position the edges, perform non maximum suppression
which makes the edges one-pixel width. Compare gradient
magnitude of the pixels, if they’re above high-threshold, they’re
marked as edge-points. Points below low-average are marked
non-edge points and rest are marked as candidate-edge points.
If candidate edge points meet with edge points, they’re
connected and made edge-points as well [8].
First, set the first frame as the background frame. Second, apply
Pixellate filter to both the frames. Third, compare the current
frame with the background frame. Then, move the background
frame towards the current frame. Detect regions where the two
frames differ, using difference or threshold filters. Then, image
with white pixels on place where the difference is present on
the specified threshold value. Then, count the pixels and if the
number of pixels are greater than a predefined value then the
motion is detected [7].
4. Hough Transform:
In edge detection technique the edge is rarely characterized by
the resulting pixels completely because of noise, breaks in the
edge from uneven illumination and other effects that introduce
false intensity discontinuities. Thus edge detection technique
algorithms are typically followed by linking procedures to
assemble edge pixels into useful edges. Hough transforms are
one of the methods used for this type of problem.
Steps for this:
1. Peak detection-Peak detection is the first step for line
detection and linking. The first step is to find the location of all
nonzero pixels for each peak in the image that contribute to that
peak.
2. Line detection and linking-Once a set of peaks has been
identified in the Hough transform, it is to be determined if there
is association of line segments with those peaks, as well as their
start and end [6] [9].
A comparison is made as follows in the Table 2.1
Table 2.1: Motion Detection Techniques
Basic Blank
Background
technique
Comparison
between the
current frame and
the first frame
with empty
background.
Advantages:
Simple to
implement
Useful in video
compression when
you need to
estimate changes
and to write only
the changes, not
the whole frame.
Previous – current
comparison
technique
Comparison
between the
current frame and
the previous
frame.
Using Pixellate
filter technique
Advantages
It's possible to
compare the
current frame not
with the previous
one but with the
first frame in the
video sequence.
So, if there were
no objects in the
initial frame,
comparison of the
current frame with
the first one will
give us the whole
moving object
independently of
its motion speed.
Advantages:
Performance
optimization
possible
Disadvantages:
Since most
cameras produce a
noisy image,
motion is detected
in such places,
where there is no
motion at all. To
remove random
noisy pixels, an
Erosion filter can
be used.
If the object is
moving smoothly,
small changes are
received from
frame to frame.
So, it's impossible
to get the whole
moving object.
Things become
worse, when the
object is moving
so slowly, when
the algorithms will
not give any result
at all.
Disadvantages:
If there’s an object
in the first frame
and then it’s gone,
motion will be
detected. To
avoid this, initial
frame should be
renewed.
Disadvantages:
Boundaries are not
accurate.
Comparison
between pixelated
current frame and
the background
frame
III. PROPOSED SYSTEM
This augmented reality tool comprises of a camera and a
projector. The camera provides an input and the projector
outputs the predicted trajectories onto the surface of the pool
table. The proposed system goes through a series of steps
before processing each image. After processing the image, the
computed trajectories are projected. A detailed algorithm of
the system is as follows.
1. Capture the image of the empty background (pool table
without balls).
2. Reduce noise from the image.
3. Apply edge detection algorithm.
4. Apply movement detection algorithm to check if any of the
balls are in movement.
5. Detect the cue ball using image transformations by
selecting the ball with the brightest area.
6. Project a circle around the white ball.
7. Detect the cue by comparing with a reference frame.
8. Detection of balls: Compare the current average frame
with the initial average frame. Each blob detected is a ball,
if and only if the following parameters are met:
a) Ratio between the blob’s height and width is
approximately one.
b) Blob’s area and circle’s area should be equal.
c) Ratio between circle (πr2) and square’s area (2r)2 be
equal to π/4.
9. Represent Cue stick by calculating line equation using two
points in the queue.
10. Calculate point of intersection of the cue and the contour
𝑃𝑥,± =
(√−𝑏 2 −2𝑏(𝑚𝑥0 −𝑦0 )+𝑚2 (𝑟 2 −𝑥02 )+2𝑚𝑥0 𝑦0 −𝑦02 +𝑟 2 )−𝑏𝑚2 𝑦0 +𝑥0
±
𝑃𝑦,± =
𝑚2 +1
(√−𝑏 2 −2𝑏(𝑚𝑥0 −𝑦0 )+𝑚2 (𝑟 2 −𝑥02 )+2𝑚𝑥0 𝑦0 −𝑦02 +𝑟 2 )𝑚+𝑏+𝑚(𝑚𝑦0 +𝑥0 )
±
𝑚2 +1
11. Calculate Ball collisions using vectors at tangent points of
that ball.
12. Convert the ball’s points and calculated trajectory to the
table co-ordinates using transformation matrix and project
them on the table.
IV. CONCLUSION AND FUTURE SCOPE
In this paper, a system is presented that predicts and displays
trajectories of a cue ball on a pool table. A projector is mounted
and a camera on top of the pool table. The camera is used to
detect the objects and motion of the balls while the projector
displays the trajectories on the table itself. This system involves
two stages, (a) the setup which is done when the system is
mounted. In this all the necessary parameters are adjusted. The
second stage is (b) the running stage. Here no parameters are
changed and the system works by using the table as an interface.
The projector should be calibrated using the projector menu to
make sure that it covers the entire playable area. The projection
should not affect the detection of balls and motion. No
trajectories are projected while the balls are in motion or when
a move is made, when the movement stops the trajectories are
projected again. The enhancement of the ball detection
algorithm can be done so as to enable it to detect balls close to
each other. The cue detection algorithm can be enhanced so as
to minimize the angular error of the system while playing the
game. A further future goal is to enable the player to save the
setup of his move so that he can practice it in future.
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