Measurements in Fluid Mechanics
058:180:001 (ME:5180:0001)
Time & Location: 2:30P - 3:20P MWF 218 MLH
Office Hours: 4:00P – 5:00P MWF 223B-5 HL
Instructor: Lichuan Gui
lichuan-gui@uiowa.edu
http://lcgui.net
Lecture 39. Stereo High-speed Motion Tracking
2
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
3
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
4
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
5
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
6
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
7
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
8
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



9
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
10
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, z, L, R, )
- unknown variables: (xc, yc, zc, , , )
- multiple surface markers required
11
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
12
13
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
14
15
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
16
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
17
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


18
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  
19
20
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
21
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
Stereo High-speed Motion Tracking
100
– Experimental results
Model center position [mm]
- 7” test model 0.7” in diameter
- drop & bounce motion
- 2000 fps w. 1024x512 pixels
- playback at 30 fps
80
xc
yc
zc
60
40
20
0
-20
50
(a)
75
100 125 150 175 200 225 250 275 300 325
Time [ms]
80



Angles []
60
- 551 frames in 275 ms
- left camera with view angle =16 , =-0.3
- right camera w. view angle =-15.3 , =-0.1
40
20
0
-20
50
(b)
75
100 125 150 175 200 225 250 275 300 325
Time [ms]
Stereo High-speed Motion Tracking
– 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
23
24
Phase-resolved Stereo PIV Measurement
- Experimental setup
• Schematic illustration of the measurement system
• PCO 2000 cameras , 26.7 fps @ 10241024 pixels
• Nd:YAG laser up to 30 Hz
• Air flow visualized with fog particles of micro-meters in diameter
• 1271 stereo PIV recordings in each camera shot
• 10,000 instantaneous 3D velocity maps for each test cases
25
Phase-resolved Stereo PIV Measurement
- Experimental setup
• Measurement planes
26
Phase-resolved Stereo PIV Measurement
- Experimental setup
• Picture of the measurement system
27
Phase-resolved Stereo PIV Measurement
- PIV image processing
Background image
Phase information
Low-pass filtering
Original image
Compare to guidelines
Images pair correlation
Particle image
High-pass filtering
Velocity vector map
28
Phase-resolved Stereo PIV Measurement
- Test results
• Side view test results for a male fire ant
Velocity distribution
Vorticity distribution
29
Phase-resolved Stereo PIV Measurement
- Test results
• Rear view test result for a male fire ant
Wing motion
Velocity distribution
30
Phase-resolved Stereo PIV Measurement
- Test results
• Rear view test result for a male fire ant
Wing motion
Vorticity contours
Exercises for final exam
1. How many gray value levels are there in a 8-bit grayscale digital image?
What are the minimal and maximal gray value?
2. Please estimate the minimal file size in bytes of a uncompressed true color image of 1024x1024 pixels.
3. What is the look-up table (LUT) of a digital color image?
4. What is the pixel operation and what is the filter operation in digital image processing?
5. Please describe two pixel operations that can be used to increase the contrast of digital images.
6. Please describe two digital filters that can be used to reduce the low frequency background noise
in digital PIV recordings.
7. Please list basic components of standard 2D PIV system.
8. Please explain how to obtain a double exposed PIV recording and a single exposed PIV recording pair.
9. What is the traditional evaluation method for a double exposed PIV recording in positive photo film?
10. Please explain how to use auto-correlation algorithm to evaluate a double exposed digital PIV recording.
11. Please explain how to use cross-correlation algorithm to evaluate a single exposed digital PIV recording pair.
12. What are limitations of the correlation-based interrogation algorithm?
13. Please list advantages and disadvantages of the correlation-based tracking algorithm when compared to the
correlation interrogation algorithm.
14. Please explain how to enable arbitrarily sized interrogation window when using radix-2 FFT to accelerate the
correlation interrogation algorithm.
15. Please explain how to accelerate the correlation-based tracking algorithm with radix-2 FFT .
16. Please briefly describe the discrete and continuous window shift method and their advantages.
17. Please briefly describe the central difference interrogation (CDI) method and explain why it is better than the
forward difference interrogation (FDI) method.
18. Please briefly describe the central difference image correlation (CDIC) method and its advantages.
19. Please explain how to determine the sub-pixel displacement.
Exercises for final exam
20. Please list two methods that can be used to identify evaluation errors in a vector map with regular grid.
21. Please explain how to use target vector method to correct evaluation errors.
22. What is the peak-locking effect?
23. What are the sources of the peak-locking effect?
24. Please suggest an algorithm with the least peck-locking.
25. Please list two methods that can be used to enable a phase-separated measurement with PIV.
26. Please list at least 5 particle image parameters that are determined with particle image identification and
used to track particle images between two frames.
27. Please explain how to evaluate low image number density recording with high accuracy.
28. Please explain why micro PIV recordings usually have lower signal-to-noise ratio, and list at least two
methods that can be used to increase the accuracy of the micro PIV.
29. Please briefly explain how the stereo PIV determines the particle image displacement component
perpendicular to the measurement plane.
30. Please list advantages and disadvantages of the translation system and rotational system of stereo PIV.