computer graphics - 3-dimensional transformations - applied to surveying

8/7/2019 Computer Graphics - 3-Dimensional Transformations - Applied to Surveying

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3-D Transformations

Brian Romsek

Senior Student

Surveying Engineering Department


2/23

Three-Dimensional Conformal

Coordinate Transformation Converting from one three-dimensional system to another,

while preserving the true shape.

This type of coordinate transformation is essential in analytical

photogrammetry to transform arbitrary stereo model

coordinates to a ground or object space system.

It is often used in Geodesy to convert GPS coordinates in

WGS84 to State Plane Coordinates.

YAxis

ZAxis

X Axis


3/23

Applications of 3D Conformal

Coordinate Transformations

Mobile mapping systemsRelations between different coordinate frames

Sensor frameBody frameMapping frame


4/23

Applications of 3D Conformal

Coordinate Transformations

Homeland securityE.G., facial pattern recognitionImage processing


5/23

3D Conformal Coordinate

Transformation

Also known as the 7 Parameters transformation since itinvolves: Three rotation angles omega ( ), phi ( ), and kappa ( );

Three translation parameters (TX, TY,TZ) and

a scale factor, S

X-axisY-a

xis

Z-axis

Omega

( )

Kappa ( )

Phi ( )

(X,Y,Z)


6/23

Rotation angles Omega

In general form:

In matrix form:

More concisely

++=

++=

++=

cosZ)sin(Y0XZ

sinZcosY0XY

0Z0YXX

1112

1112

1112

=

1

1

1

2

2

2

Z

Y

X

cossin0

sincos0

001

Z

Y

X

2C M C

=


7/23

Rotation angles Phi

Omega

( )

Kappa

( )

Phi ( )

In general form:

In matrix form:

More concisely X-axis

Z-ax

is

++=

++=++=

cosZ0YsinXZ

0ZY0XY

)sin(Z0YcosXX

2223

2223

2223

=

2

2

2

3

3

3

Z

Y

X

cos0sin

010

sin0cos

Z

Y

X

23 CMC =


8/23

Rotation angles Kappa

Omega

( )

Kappa

( )

Phi ( )

In general form:

In matrix form:

More concisely X-axis

Z-ax

is

( )

333

333

333

Z0Y0X'Z

0ZcosYsinX'Y

0ZsinYcosX'X

++=

++=

++=

=

3

3

3

Z

Y

X

100

0cossin

0sincos

'Z

'Y

'X

3CM'C =


9/23

Combined Rotation Matrix

If we combine all the rotation matrices

MG becomes, after multiplication

=

=

1

1

1

1

1

1

G

Z

Y

X

MMM

Z

Y

X

M

'Z

'Y

'X

++

=coscoscossinsin

sinsincoscossinsinsinsincoscossincos

cossincossinsincossinsinsincoscoscos

M G

11 12 13

21 22 23

31 32 33

m m m

M M M M m m m

m m m

= =


10/23

COMPUTING ROTATION ANGLES

If rotation matrix

known, rotation

angles can be

computed as shown

on the right

11

21

31

33

32

m

mtan

msin

m

mtan

=

=

=


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Properties of rotation matrix

The rotation matrix is an orthogonal matrix,

which has the property that its inverse is equal

to its transpose, or

This can be used for inverse relationship

T

RR =1


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Three-Dimensional Conformal

Coordinate Transformation

Finally the 3D Conformal Transformation is derived

by multiplying the system by a scale factors and adding

the translation factors TX, TY, and TZ.

Where:

YAxis

ZAxis

X Axis

'C s M C T = +

=

Z

Y

X

T

T

T

T '

X

C Y

Z

=

11 12 13

21 22 23

31 32 33

m m mM m m m

m m m

=


13/23

BURSA-WOLF TRANSFORMATION

Geodesy assumption rotation angles small

cos = 1

sin = (radians)

Product of two sines = 0

Rotation matrix R becomes:

=

1

11

RR

RR

RR

R


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3D similarity transformation

Observation Equation:

+

+

=

=

Z

Y

X

T

T

T

z

y

x

s

z

y

x

RR

RR

RR

Z

Y

X

h(x)

g(x)

f(x)

0

0

0

fBV

=


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Coefficient matrix, B:

Vector of parameters, , and

discrepancy vector, f

=

0100

0001

0100

0010

0001

222

111

111

111

nnnxyz

yzx

xyz

xzy

yzx

B

[ ]T

RRRsTTTZYX

=

=

Z

Y

X

f


16/23

Three Dimensional Coordinates

Transformation

General polynomial approach: transformation is not

conformal2 2 2

0 1 2 3 4 5 6 7 8

2 2 29 10 11 12

2 2 2

0 1 2 3 4 5 6 7 8

2 2 29 10 11 12

2 2 2

0 1 2 3 4 5 6 7 8

2 2

9 10 11

Xn a a x a y a z a x a y a z a xy a yz

a zx a xy a x y a xz

Yn b b x b y b z b x b y b z b xy b yz

b zx b xy b x y b xz

Zn c c x c y c z c x c y c z c xy c yz

c zx c xy c x

= + + + + + + + +

+ + + + +

= + + + + + + + +

+ + + + +

= + + + + + + + +

+ + +

L

L

2

12y c xz + +L


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Transformation

Alternative that is conformal in the three planes

( )

( )

( )

2 2 2

0 1 2 3 5 7 6

2 2 2

0 2 1 4 6 7 5

2 2 20 3 4 1 7 6 5

0 2

2 0 2

2 2 0

Xn A A x A y A z A x y z aA zx A xy

Yn B A x A y A z A x y z A yz A xy

Zn C A x A y A z A x y z A yz A zx

= + + + + + + + +

= + + + + + + + +

= + + + + + + + +

L

L

L


18/23


Transformation

Polynomial

projectivetransformation, 15

parameters

1 2 3 4

1 2 3

1 2 3 4

1 2 3

1 2 3 4

1 2 3

1

1

1

a x a y a z aXn

d x d y d z

b x b y b z bYn

d x d y d z

c x c y c z cZn

d x d y d z

+ + +=

+ + +

+ + +=

+ + +

+ + +=

+ + +


19/23

Testing 4 Methods

Bursa Wolf Linear model

assume smallrotation angles

Best for satelliteto globalsystemtransformations

Bazlov et al:determined PX90 to WGS 84parameters

GeneralizedBursa Wolf Linear model

errors in both

observations andmodel parameters

Usefultransforming

classical to space-borne (Kashani,2006)


20/23

Testing 4 Methods

Polynomial

1st order

Useful when

coordinatesystems not

uniform in

orientation or

scale

Rubber-sheeting

Expanded Full- Model

Photogrammetric

approach

Angles not consideredsmall

Non-linear: requires a

priori estimate of

parameters


21/23

Employed method shown inPhotogrammetric Guide by Abertz &Kreiling

X, Y, Z coordinates translated torelative values based in meancoordinates

Expanded Full-Model


22/23

Data include a set of know control points, transformed from

WGS84 system to State Plane Coordinates.

3D Transformations Testing3D Transformations Testing


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Test Results

Method

Photo Guide

Bursa-Wolf

Refer

ence

Variance

un

2

0 = WVV

T

nZZYYXXRMSE /)()()( 222 ++=

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