Computer Vision and Measurements in

Aerospace Applications

Tianshu Liu

Department of Mechanical and Aeronautical Engineering

Western Michigan University

Kalamazoo, MI 49008

Objective

To build a unified theoretical framework for

quantitative image-based measurements of

morphology and motion fields of deformable

bodies like fluids

Geometric Structures:

Points, Curves, Surfaces

Motion Fields:

Points, Curves, Surfaces (Geometric Flow)

Complex Continuous Patterns

Background

• Computer Vision • Photogrammetry

Human Vision Computer Vision

Analogy

Optical Info

Processing

Principles of Physics Mathematical Models

Emission Radiance

in Images

Geometric and Physical-Based Processing

Geometric Features

In Images

Perspective

Transformation

Emission Radiance

in Images

Geometrical, Kinematical

and Physical Representations

in 3D object Space

Mapping of Geometry

and Motion

Mapping of Physical

Quantities

(1) Pressure (2) Temperature (3) Skin Friction

(4) Velocity Field (5) Attitude and Kinematics

(6) Shape

Important Quantities in Aerodynamics

Molecular Sensors!

Combination of Approaches

• Perspective Geometry

• Differential Geometry

• Continuum Kinematics & Dynamics

• Radiometry

Needs for New Methods

Quantitative, Dynamic, Universal, Applicable

to Complex Patterns & Motions of Deformable

Bodies

Relevant Topics

(1) Perspective Projection Transformation

(2) Projective Developable Conical Surface

(4) Perspective Projection of Motion Field

Constrained on Surface

(5) The Correspondence Problem

(6) Composite Image Space and Object Space

(3) Perspective Projection under Surface Constraint

(7) Perspective Invariants of 3D Curve

(8) Reflection and Shape Recovery

(9) Motion Equations of Image Intensity

Perspective Projection Transformation

)P,P,K,K,x,xc,,X,X,X,,,( 21212p

1p

3c

2c

1cΠ

Camera parameters

exterior interior distortion

Formulations of Perspective Transformation

)(

)(cxxx

)(

)(cxxx

T

T

22p

2

T

T

11p

1

c3

c2

c3

c1

XXm

XXm

XXm

XXm

hhh XPx

(1) Collinearity Equations

(2) Homogenous Coordinate Form

(3) W-Vector Form

0)(

0)(

T

T

c2

c1

XXW

XXW

Geometric Image Measurements

Projection 3D 2D

Recover the Lost Dimension

Perspective Developable Conical Surface

0)s()( Dc NXX

0ds/)s(d)( Dc NXX

Conical Surface Equations

Plane parallel to image 3D curve

s)d(s

0

321

tPxPXX 0hcCP

Ray Equation

Reconstruction of 3D Curves and Surfaces

Using Projective Conical Surfaces

3D Space Curve Surface

Reconstruction of 3D Displacement Vectors

Providing a rational

and general method for

Stereoscopic Particle Image

Velocimetry (SPIV) and

Scalar Image Velocimetry

(SIV)

Motion Field of 3D Space Curve

min||t)(|| St XUX

XSt

Variational Problem:

0)]([Gi XU

Physical and Geometric Constraints:

2,1,i

time 1

time 2

Perspective Projection under Surface Constraint

dxdxg|d|dS 2 X

Geometric Structures

Quantities: tangents, normal,

length, angle, area

topology

2

1

1

2

1

dx

dxQ)(

dX

dXSc

T

3 fXm

One-to-One Differential Relation

)X,X(FX 213

Motion Field

)]([U

)]([U

)(

Q

x

x

dt

d

2

1

T2

1

xf

xf

fXmu

S

S

Sc3

Optical Flow

Surface-Constrained Motion Field

Motion Field on Surface

Image

The Point Correspondence Problem

Generalized Longuet-Higgins Relation:

Image 1 Image 2

Epipolar Line

h(1)x

h(2)x

Epipolar Line

0)xx(Q)x(x )1(hh(1))2(hh(2)

Determining Point Correspondence

Four Images or Cameras

Six Longuet-Higgins

Equations

for

Six Unknowns

1

2

3

4

1

2

1(1)m

2(1)m3(1)m

Composite Image Space and Object Space

One-to-One Relation

Provide additional

dimensions

Reconstruct 3D

Displacement Vectors

from Composite Image

Coordinates

Perspective Invariants of 3D Curve

2212,obj

2121,obj

2212,im

2121,im

D

D

d

d

Torsion

)3,'2,2,1(ID

)3,2,'1,1(ID

)3,'2,2,1(id

)3,2,'1,1(id

2211,obj

2122,obj

2211,im

2122,im

Curvatures

2341

4321

2341

4321

DD

DD

dd

dd

Distances

Rectifying plane

Osculating plane

new

new

(Brill, et al.

1992)

Geometrical Flow

Problem

Shape and Reflection

)]aa(p[Ec

Ec)(I

LNsdlssys

aasys

VLVNLN

x

ss

Motion Equations of Image Intensity

)t(Lc)t(I g,a,X,x,

Image Intensity and Radiance

TN21 )p,,p,(p pPhysical parameters:

TM21 )q,,q,(q qGeometric parameters:

Radiance: L(X, p, q)

Generic Motion Equations of Image Intensity

L

dt

dL

dt

dL

t

LcI

t

IqpXsysx

qpUu

TN21 )p,,p,(p pPhysical parameters:

TM21 )q,,q,(q qGeometric parameters:

Radiance: L(X, p, q)

Optical Flow Velocity Field

Governing Equation

Emitting Passive Scalar Transport

Luminescence

2XD

tdt

d

U )t,(c)t,(L XX

Perspective Projection onto Image Plane

Radiance

Motion Equation of Image Intensity

for Emitting Passive Scalar Transport

xx

Ih

x

IhD

x

Iu

t

I 2

)]([U

)]([U

)(x

x

dt

d

2

1

2

1

xf

xf

fXm

Gu

S

S

Sc3

Optical Flow and Velocity Field on Surface

Governing Equation

Light Transmitting Scalar Transport

2XD

tdt

d

U

)dsexp(LLs

0ext0

Perspective Projection onto Image Plane

x

I

x

II

xx

ID

x

Iu

t

I 12

2

Ix

Iu 1212

U

Optical Flow and Path-Averaged Velocity

Motion Equation of Image Intensity

for Light Transmitting Scalar Transport

2

1

2

1

3

3

12

12

Xd

Xd

U

Uwhere

Motion Equations of Image Intensity

Schlieren Image of Density-Varying Flows

0]dX)II([dX)II(t

2

20

2

20

X

X

2K1212

X

X

2K

U

2

1

3

2schl

K

K dXX

CI

II

Image Intensity and Density Gradient

2

1

2

1

3

3

12

12

Xd

Xd

U

UT21

12 )X/,X/( where

0)]II([)II(t

T2

121212T2

12

U

Motion Equations of Image Intensity

Shadowgraph Image of Density-Varying Flows

Image Intensity and Second-Order Density Derivative

2

1

3212shad

T

T dXCI

II

)II( T2

12

T212 II

solution

where

0)]II([)II(t

T1212T

U

Motion Equations of Image Intensity

Transmittance Image of Density-Varying Flows

Image Intensity and Density

2

1

3trans

T

T dXCI

II

Typical Applications

• Aerodynamic measurements: pressure and

temperature sensitive paints, videogrammetric

attitude measurement, stereoscopic PIV, laser-

tagging technique, schlieren, shadow and

transmittance imaging, oil-film/liquid-crystal skin

friction measurements.

• Metrology and kinematics of large inflatable space

structure.

Unification of Measurement Systems

Conventional Techniques Image-Based Techniques

Force Balances Accelerometers

Pressure Taps & Probes

Hot-Wires & Films

Temperature & Heat

Transfer Gauges

PSP & TSP Videogrammetry

Global Velocimetries

(PIV, DGV)

Schlieren/Shadowgraph

Oil & LC -Film Skin

Friction Meters

Data Fusion and Understanding

Measurements Data Base & Models

Integrated data & data

at discrete locations

Distributions on surfaces

Fields in 3D space

Aerodynamics

data base

Theoretical models

CFD

“SuperAerodynamicist”:

Intelligent Expert System!?

Conclusions We will see unified image-based instrumentation providing

non-contact, global measurements of important physical,

geometric and dynamical quantities in wind tunnel testing.

Unified Measurement

Techniques

Wind Tunnel Testing

CFD

Theories

Aerospace

System

Design

Ideal Aerodynamics Lab