Imaging Techniques for Flow and Motion Measurement
Lecture 20
Stereo High-speed Motion Tracking
Lichuan Gui
University of Mississippi
2011
1
Stereo High-speed Motion Tracking
– Stereo high-speed imaging system in wind tunnel test
Test model
Mesurement volume
Glass window
Strobe light
High-speed cameras
Test model
- length: 7 inches (178 mm)
- diameter: 0.7 inches (0.18 mm)
High-speed cameras
- lenses: 60mm Nikon Micro-Nikkor
- 30 view angle difference
- frame rate: up to 4000 fps
- resolution: 1024X512 pixels
Measurement volume
- width: 305 mm
- height: 152 mm
- maximal depth: 104 mm
Strobe light
- Synchronized with camera
2
Stereo High-speed Motion Tracking
– Stereo system coordinates
Physical coordinates: (x, y, z)
Y
Image coordinates: (x*, y*)
Camera coordinates: (x’, y’, H)




 x  x H  x  x z  0



*
*
 y  y H  y  y z  0

*


*

H


Target
(x’,y’)
(x,y)
Z
(x*,y*)
z
Camera view angles: (, )
x *  x'

tan



H



y *  y'
 tan  
H

X
Traverse
3
Stereo High-speed Motion Tracking
– Calibrate stereo system with target shift
Y
Target
H
(x’,y’)
(x,y,0)
Z
X
1. Image calibration target at z=0
Traverse
4
Stereo High-speed Motion Tracking
– Calibrate stereo system with target shift
Y
Target
H
(x’,y’)
Z
(x,y,z)
(x1*,y1*)
X
2. Forward shifted target at zs /2
Traverse
5
Stereo High-speed Motion Tracking
– Calibrate stereo system with target shift
Y
Target
H
(x’,y’)
(x,y,-z)
Z
(x2*,y2*)
(x1*,y1*)
X
3. Backward shifted target at -zs /2
Traverse
6
Stereo High-speed Motion Tracking
– Calibrate stereo system with target shift
Geometrical relations:







*
*
 x  x1 H  x  x1

 y  y* H  y  y*
1
1

 z2  0
 z2  0







x  x2* H  x  x2*



 y  y* H  y  y*
2
2


s
s
 z2  0
 z2  0
s
s
Reduced equations for calibration points k=1,2,3,, N :
 *
x2*,k
*
 x2,k  x1,k H  z s x 


y 2*, k
 *
*
y 2,k  y1, k H  z s y  






 x1*, k
2
 y1*, k
2
zs  0
zs  0
Sum square difference function:
*
*
 *

x

x
2, k
1, k
*


DH , x     x2, k  x1, k H  zs x 
zs 
2
k 1


N


2
7
Stereo High-speed Motion Tracking
– Calibrate stereo system with target shift
Conditions for achieve a minimal sum square difference:

D H , x '  0 ,
H

D H , x '  0
x'
Linear equation system to determine H and x’ :







N
zs N *
 N *
* 2
*
*

x

x

H

z
x

x

x

 x2,k  x1*,k x2*,k  x1*,k  0
1, k
s  2, k
1, k
  2, k
2 k 1
k 1
 k 1

2
 z N x *  x *  H  z 2  x  z s N x *  x *  0
 2,k 1,k
s  2, k
1, k
s

2 k 1
k 1





Equation to determine y’ :
y2*, k  y1*, k 
1 N *
*
y 

  y2, k  y1, k H 
zs N k 1
2



8
Stereo High-speed Motion Tracking
– Stereo coordinate reconstruction
Camera coordinates: (x’a, y’a, Ha) for left camera, (x’b, y’b, Hb) for right camera
Image coordinates: (xa, ya) for left camera, (xb, yb) for right camera
Reconstructed physical coordinates: (x, y, z)
xa  xa
x  xb
ya  ya
y  yb
xb  b
xa
yb  b
ya
xa  xb
Ha
Hb
Ha
Hb
x
, y
, z
xa  xa xb  xb
ya  ya yb  yb
xa  xa xb  xb



Ha
Hb
Ha
Hb
Ha
Hb
Camera view angle at image frame center (x0, y0, z0):
 x0  x 
1  y  y 
 ,   tan  0

H
H




  tan 1
9
Stereo High-speed Motion Tracking
– 3D motion tracking
Tracking variables
- model center: (xc, yc, zc)
- roll angle:

- pitch angle:

- yaw angle:

Surface marker local coordinates
- L: axial coordinate
- R: radius coordinate
- : angular coordinate
Surface marker coordinates (x, y, z)
- image pattern tracking results
Geometrical relations
- three equations
- known variables: (x, y, c, L, R, )
- unknown variables: (xc, yc, zc, , , )
- multiple surface markers required
10
Stereo High-speed Motion Tracking
– Least square approach
Available data
- surface markers (Ln, Rn, n)
- tracked position (xn, yn, zn)
- n=1, 2, 3, …,M
First step
- determine ,  at minimum of D1(,  )
- yc determined accordingly
Second step
- determine  at the minimum of D2( )
- xc determined accordingly
Third step
- determine zc with known variables
11
12
Stereo High-speed Motion Tracking
– Simulated 3D motion
- 7-inch revolving surface model, 120 frames
- red image from left camera with view angle =15 , =3
- green image for right camera with view angle =-22.5 , =-2
(300mmx150mm, =0-45, =0-20, =0-10)
Stereo High-speed Motion Tracking
– Tracked surface makers
- spherical dots & cross-sections of grid lines
- combination of 18 surface markers for 9 test cases
13
14
Stereo High-speed Motion Tracking
– Simulation results
- 4-point results agree well with given values
- coordinate biases < 0.5 mm
- angular biases < 1
Stereo High-speed Motion Tracking
– Simulation results
- minimum of 3 surface
marker required
- 4 surface markers sufficient
to achieve high accuracy
- more markers not help
because of add-in noises
- discussion limited in high
image quality cases
15
Stereo High-speed Motion Tracking
– 4-point tracking method
1. Distribution of markers “1”, “2”, “3” and “4”
y
- Plane “2-4-c” perpendicular to model axis
(“c” on axis, may not be at center)
3
x
- Point “2” and “4” at the same radius R
z
- Sufficient angular difference between
line “c-2” and “c-4”

1
2
R
mc
4


- Line “1-3” parallel to model axis
- When line “1-3” not parallel to model axis, plane “1-c-3” line “2-4”
4-point method less sensitive to image noises than multi-point least square approach
16
Stereo High-speed Motion Tracking
– 4-point tracking method
2. Pitch and yaw angle determined with line “1-3”
that parallel to model axis

2
2
1 
   tan  y3  y1  / x3  x1   z 3  z1  



1
   tan z 3  z1  x3  x1 

y
3
x
z

1
2
R
mc
4


17
Stereo High-speed Motion Tracking
– 4-point tracking method
3. Roll angle and “c” position determined in “2-4-c” plane
y
Define midpoint “m” on line “2-4”:
x  x4
y  y4
z  z4
xm  2
ym  2
zm  2
2
2
2
Line “c-m” determined with “c-m”“1-3” & “c-m”“2-4”:



3
x3  x1 x  xm    y3  y1  y  ym   z3  z1 z  z m   0
x4  x2 x  xm    y4  y2  y  ym   z 4  z 2 z  z m   0
x
z
x  xm y  ym z  zm


o
p
q
2

1
 o   y3  y1 z4  z2    y4  y2 z3  z1 

 p  z3  z1 x4  x2   z4  z2 x3  x1 
q  x  x  y  y   x  x  y  y 
3
1
4
2
4
2
3
1

4


2
Length of “m-c”:
R
mc
2
 x  x2   y4  y2   z4  z2 
l  R  4
 
 

 2   2   2 
2
2
 x  x  l  o / o2  p 2  q 2
c
m


Model position:  yc  ym  l  p / o 2  p 2  q 2
 z  z  l  q / o2  p 2  q 2
m

 c
1  y  yc 

Roll angle:   sin  m
 R cos  
18
19
Stereo High-speed Motion Tracking
– Experimental results
- 80mm cylindrical model, 20mm diameter, 2000 fps, 1024x512 pixels
- left image from left camera with view angle =16.0 , =-0.3
- right image from right camera with view angle =-15.3 , =-.1
20
Stereo High-speed Motion Tracking
– Experimental results
dx/dt
= -0.00 m/s
- roll angle:
- y-motion: parabolic, dy/dt2
= -9.25 m/s2
- pitch angle: linear, d/dt = -0.02 r/s
- z-motion: linear,
= 0.16 m/s
- yaw angle: linear, d/dt = 0.05 r/s
dz/dt
120
20
100
0
x
y
z
80
60
-60
20
-80
(a)
0
10
20
30
40
50
Time [ms]
60
70
80



-40
40
0
linear, d/dt = -3.00 r/s
-20
Angles []
Model center position [mm]
- x-motion: linear,
90
-100
(b)
0
10
20
30
40
50
Time [ms]
60
70
80
90
Homework
– References
• Lichuan Gui, Nathan E. Murray and John M. Seiner (2010) Tracking an
aerodynamic model in a wind tunnel with a stereo high-speed imaging
system. The 3rd International Congress on Image and Signal
Processing (CISP’10), October 16-18, Yantai, China
– Practice with EDPIV
• Application example #a
21