Nick 2D Tran

7/27/2019 Nick 2D Tran

1/33

2D Coordinate Transformations

Nick Battjes, Senior Student

Bluelake

GreenPark

Dark

Rive

rRedHills

WGStation

Spatial data without coordinates Control Points

Bluelake

Green Park

Dark

Rive

r

Red Hills

WG Station


2/33

Integrating of maps and spatial data in local coordinate system into aworld database system.

Note how the vector data (USGS Road layer) should be transformedto match the raster data (Ikonos 1 meter res. Image)

Applications


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Three different transformation primitives for theSimilarity transformation:

Translation-origin is moved, axes do not rotate i.e.:

Xn = X0 DX0 Yn = Y0 DY0

Scaling -both origin and axes are fixed, scale change

Xn = sXX0 Yn = sY Y0

Rotation - origin fixed, axes move (rotate about origin)

Xn = X0 cos() + Y0 sin(); Yn = - X0 sin() + Y0 cos()

2D Spatial Transformations


4/33

Two-Dimensional

Geographic Transformations

Xn

Yn

X0

Y0

Xn

Yn

X0

Y0

Xn

Yn

X0

Y0

Translation Scaling Rotation


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Moves and rotates objects in 2D and 3D space. Additionally,

you can scale the objects based on alignment points whenusing the 2D option.

Conformal Transformation


6/33

Isogonal: having equal angles

Impose additional condition of equal scale (S = Cx =Cy) yielding 4 parameters: S, , DX0, DY0

Moves and rotates and scale objects in 2D space.

Isogonal Affine Transformation

or Conformal/SimilarityTransformation

cos sin* * *

sin cos

Xn a b Xo c Xo DXoS

Yn b a Yo d Yo DYo

4 parameters: Sscale, rotation, DX0, DY0 shifts in X and Y.

Xn,Yn are the transformed coordinates.

Xo, Yo are the original coordinates.

Two given points are required ( X1,Y1 and X2,Y2)


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Moves and rotates and scale objects in 2D space.

Often called similarity Transformation since the basic shaperemain similar after the transformation


8/33

The formulas can be written in different forms1. To compute the parameters given the coordinates

2. To compute the new coordinates given the parameters

0 0 0

0 0 0

cos sin

sin cos

Xn S X S Y DX

Yn S X S Y DY

0 0

0 0

Xn a X b Y c

Yn b X a Y d

0 0

0 0

1 0

0 1

a

X YXn b

Y XYn c

d



9/33

The formulas can be written in different forms3. To compute the old coordinates given the parameters and new

coordinates (back substitution)

0 0

0 0

Xn a X b Y c

Yn b X a Y d

Conformal/SimilarityTransformation

0 0

0 0

( )

( )

Xn c a X b Y

Yn d b X a Y

2

0 0

2

0 0

( )( )

a Xn c a X ab Y b Yn d b X ab Y

Multiply Eq. 1 by a, Eq 2 by b

0 2 2

0 2 2

a Xn c b Yn d

X a b

b Xn c a Yn d Y

a b

Add equations to get X0

The same operation to

obtain Y0


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Moves, rotates and scale objects in 2D space.

Cy

Cx

Yo

Xo

ab

ba

Yn

Xn*

y

x

x

Cab

Cba

Cyab

Cba

6080300

6080250

1010190

1010350

No Xo

(map)

Yo

(map)

Xn

(ground)

Yn

(ground)

1 10 10 350 190

2 80 60 250 300

y

x

C

C

ba

*

108060

016080

101010011010

300

250

190350

Conformal Transformation: Example


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Transform a given point X0=121.48, Y0=22.78

The point is transformed to Xn=310.59, Yn=373.6773

865.174

189.369

716.1203.0

300

250

190350

*

108060

016080

101010011010

1

y

x

C

C

ba

6773.37359.310

865.174189.369

78.2248.121*

203.0716.1716.1203.0

YnXn

Conformal Transformation:

Example


12/33

Used in photogrammetry for:

Transform comparator coordinates to photo coordinates and

used for correcting film distortion

Transform model coordinates to survey coordinates

Property

Carry parallel lines

into parallel lines

Does not have to

preserve orthogonality

y1

y2

x1

x2

t2

t1

Affine Transformation


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Physical interpretation:

6 parameters: Cx, Cy, , , Dx0, Dy0, and in linear form:

0 0 0

0 0 0

cos( ) sin

sin cos

x y

x y

Xn C X C Y DX

Yn C X C Y DY

0 0

cos sin

sin cos

x x

y y

a C d C

b C e C

c Dx f Dy

Xn

Yn

X0Y0

DX DY

Cy

Cx

Yo

Xo

dc

ba

Yn

Xn*

Affine Transformation


14/33

2-D Affine Transformation

The formulas for an affine transformation:

Ifn control points are measured, this Equation is reorganized asfollows:

sT A A A

sT A A A

xx a b c

yy d e f

1 1 1

1 1 1

1 0 0 0

0 0 0 1

1 0 0 0

0 0 0 1

A

T s s

A

T s s

A

A

Tn sn sn

A

Tn sn sn

A

ax x y

by x y

c

dx x y

ey x y

fy A


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Projective or Polynomial Transformation

Instead of 4 or 6 parameters we have many parameters at

least 8 (8 would be the Bi-linear or projective

transformation)

With more parameters we need more known points to solve

the equations N-equations and N unknowns.

...

...2

052

0400302010

205

20400302010

YbXbXYbYbXbbYn

YaXaXYaYaXaaXn


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General form

Alternatively,

xybybxbybbxbb'y

xyayaxayaxaa'x

5

2

4

2

3210

5

2

4

2

3210

xy2AyxAyAxAB'y

xy2AyxAyAxAA'x

3

22

4120

422

3210

Polynomial Transformation


17/33

Frequently used in photogrammetry

General form:

1 0 2 0 3

1 0 2 0

1 0 2 0 3

1 0 2 0

1

1

a X a Y aXn

d X d Y

b X b Y bYn

d X d Y

Projective Transformation


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Impose condition of orthogonality ( = 0)yielding 5 parameters: Cx, Cy, , x, y

Orthogonal Affine Transformation

0 0 0

0 0 0

cos sin

sin cosx y

x y

Xn C X C Y DX

Yn C X C Y DY


19/33

Condition: orthogonality and no scale change (Cx =

Cy = 1)

3 parameters: , x, y

Rigid Body Transformation

0 0 0

0 0 0

cos sinsin cos

Xn X Y DXYn X Y DY


20/33

Least Squares Adjustment ReviewModel of adjustment of indirect observation ( Gauss-Markov model)

n: # of observations

m: # of parameters

f is a vector of given observation

Vis the vector of residuals

B is the matrix of coefficients

W it the matrix of weights (VarianceCovariance matrix)

0 is the reference variance

is the vectors of parameters to be estimated

Number of observation is larger than number of parameters (redundant

observations). The solution that minimize the least-squares criterion

(vTwv) is:

),0(~ 20 WVVBf

nurkB

fWBBWBTT

1

)(


21/33

Four fiducial marks (1 - 4) and two image points (a and

b) were measured on a comparator. The comparator

photo observations and the known values from the

camera calibration report are given in the following

spreadsheet.

Photo Coordinates Known Values

Point No. x y X Y

1 -111.734 -114.293 -113.007 -112.997

2 111.734 114.293 113.001 112.9893 -114.289 111.699 -112.997 113.004

4 114.280 -111.749 112.985 -112.997

a 74.794 12.202

b -67.123 53.432

General 2D Conformal Transformation,

Example


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4-Parameter Coordinate Transformation Program______________________________________________________________________________________

Solution

Forming the B-matrix and f-matrix:

B

x1

y1

x2

y2

x3

y3

x4

y4

y1

x1

y2

x2y3

x3

y4

x4

1

0

1

0

1

0

1

0

0

1

0

1

0

1

0

1

f

X1

Y1

X2

Y2

X3

Y3

X4

Y4

General 2D Conformal Transformation,

Example


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N B

T

B

1

The variance-covariance matrix is: QXX N

QXX

9.787E-006

0E+000

22.02E-009

122.332E-009

0E+000

9.787E-006

122.332E-009

22.02E-009

22.02E-009

122.332E-009

250E-003

0E+000

122.332E-009

22.02E-009

0E+000

250E-003

t BT

f

t

102157.371

1161.611

0.018

0.001

General 2D Conformal Transformation, Example


24/33

The solution vector is:

N t

0.99977

0.01137

0.00211

0.01222

The resisuals are V B f

V

0.002

0.013

0.004

0.019

0.002

0.020

0.004

0.013



25/33

The reference variance for the adjustment is

V

TV

4 0.0003( )

The Transformed coordinates become:

Xa 1 xa 2 ya 3 Xa 74.913

Ya 2 xa 1 ya 4 Ya 11.361

Xb 1 xb 2 yb 3 Xb 66.502

Yb 2 xb 1 yb 4 Yb 54.195



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Normally shown as

Unique solution if

0 0

0 0

Xn a X b Y c

Yn d X e Y f

0a b

d e

General 2D Affine Transformation

0 0

0 0

1 0 0 0

0 0 0 1

a

b

X YXn c

X YYn d

e

f


27/33

Four fiducial marks (1 - 4) and two image points (a andb) were measured on a comparator. The comparator

photo observations and the known values from the

camera calibration report are given in the following

spreadsheet.

Photo Coordinates Known Values

Point No. x y X Y

1 -111.734 -114.293 -113.007 -112.997

2 111.734 114.293 113.001 112.9893 -114.289 111.699 -112.997 113.004

4 114.280 -111.749 112.985 -112.997

a 74.794 12.202

b -67.123 53.432

General 2D Affine Transformation,

Example


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_____________________________________________________________________________________

yb 53.432xb 67.123

ya 12.202xa 74.794The measured points are:

Y4 112.997X4 112.985y4 111.749x4 114.280

Y3 113.004X3 112.997y3 111.699x3 114.289

Y2 112.989X2 113.001y2 114.293x2 111.734

Y1 112.997X1 113.007y1 114.293x1 111.734

Input Values:Note that low er case values represent observed comparator coordinates w hile the uppercase represents the know n camera calibration coordinates for the respective f iducial values

______________________________________________________________________________________

6-Parameter Coordinate Transformation Program

General Affine Transformation, Example


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B

x1

0

x2

0

x3

0

x4

0

y1

0

y2

0

y3

0

y4

0

1

0

1

0

1

0

1

0

0

x1

0

x2

0

x3

0

x4

0

y1

0

y2

0

y3

0

y4

0

1

0

1

0

1

0

1

f

X1

Y1

X2

Y2

X3

Y3

X4

Y4

N BT

B 1


Solution: forming the B matrix and f vector


30/33

The variance-covariance matrix is: QXX N

QXX

19.573E-006

1.603E-009

44.019E-009

0E+000

0E+0000E+000

1.603E-00919.573E-006

244.661E-009

0E+000

0E+0000E+000

44.019E-009

244.661E-009

250E-003

0E+000

0E+0000E+000

0E+000

0E+000

0E+000

19.573E-006

1.603E-00944.019E-009

0E+000

0E+000

0E+000

1.603E-009

19.573E-006244.661E-009

0E+000

0E+000

0E+000

44.019E-009

244.661E-009250E-003



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t BT

f

t

51079.018

583.52

0.018

578.092

51078.353

0.001

The solution vector is: N t

0.99977

0.01134

0.00211

0.01140

0.99977

0.01222



32/33

The resisuals are V B f

V

0.001

0.016

0.001

0.016

0.001

0.016

0.001

0.016

The reference variance for the adjustment is

V

TV

2

0.001( )



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The Transformed coordinates become:

Xa 1 xa 2 ya 3 Xa 74.913

Ya 4 xa 5 ya 6 Ya 11.359

Xb 1 xb 2 yb 3 Xb 66.504

Yb 4 xb 5 yb 6 Yb 54.197


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