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Description of 2D and 3D Coordinate Systems and

Derivation of their Rotation Matrices

Conventions:

In a 3D coordinate system, Xs, Ys, Zs will be used for object coordinates in the scanner

coordinate system. This is the coordinate system from which the transformation is made.

Xc ,Yc ,Zc will be used for the object coordinates expressed in the camera coordinate system

after they have been scaled by the camera lens. The Zc value for the object will be the

location of the image plane in the camera, the CCD sensor and will be equal to the focal

length of the camera, f (at CCD sensor, Zc = f). This is the coordinate system to which the

scanner coordinates are transformed. Xi, Yi, Zi will be used to designate the object coordinates

after they are transformed to a coordinate system which has as its origin, the camera

coordinate system origin but before scaling by the lens occurs. Xi, Yi, Zi are unaffected by the

camera lens optics. x, y, z will be used for the translation needed to move the origin of the

scanner coordinate system to the origin of the camera coordinate system.

The derivation will first be explained used a 2D example. In a 2D planar coordinate system

Xs and Ys will be used for the coordinate system that corresponds to the scanner coordinates,

that is the system from which the transformation is made. Xc and Yc will be used for the

coordinate system that corresponds to the camera coordinates, that is the system to which the

X and Y coordinates are transformed. The terms camera and scanner are used here only to

maintain continuity between the 2D and the 3D derivations. A 2D transformation does not

apply to an actual transformation between scanner and camera coordinates.

In a 3D coordinate system Omega (ω) will describe rotation about the X-axis, Phi (Ф) will

describe rotation about the Y-axis, and Kappa (κ) will describe rotation about the Z-axis.

Theta (θ) will describe rotation in a 2D planar coordinate system.

Derivation of 2D transformation In a 2D planar coordinate system a counter-clockwise rotation from the scanner coordinates to

the camera coordinates can be accomplished with the following transformation matrix

assuming that the origins of the two coordinate systems are located at the same spot. In a

counter-clockwise system positive rotation is in the counter-clockwise direction. Negative

rotation is in the clockwise direction.

Counter-clockwise rotation will be used in all transformations. This appears to be the most

commonly used convention, although it is possible to perform the transformations equally as

well with a convention that uses a clockwise rotation. This may be the preferable convention

for rotation because the cross-product of two vectors which have been defined relative to

each other in a counter-clockwise convention has the positive direction of the result of the

cross-product pointing toward the viewer. That is, with the two vectors in the horizontal

plane, the cross-product is positive if it points up.

The black coordinate axes in Figure 1 are rotated counter-clockwise to align them with the

blue coordinate axes.

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Figure 1. Counterclockwise rotation of coordinate systems,

transformation of X, Y to x, y. Point P is transformed from the

black coordinate system to the blue coordinate system.

Conversion from one coordinate system to the other is derived below.

Y = Xb + bP

x = oa + ab + bx

= X*cosθ + Xb*sinθ + bP*sinθ

= X*cosθ + (Xb + bP)*sinθ

= X*cosθ + Y*sinθ

y = -cY + od = -aX + od

= -X*sinθ + Y*cosθ

The above equations for x and y are expressed below in matrix notation.

−=

Y

X

y

x

θθθθ

cossin

sincos

If the two coordinate systems have their origins separated by ∆X and ∆Y distances, then a

translation of the origin of the XY coordinate system must be made to the xy origin by adding

the difference between their locations.

∆+

∆+

−=

YY

XX

y

x

θθθθ

cossin

sincos

The transformation matrix

−=

θθθθ

cossin

sincosM is a 2X2 matrix.

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A clockwise rotation from the scanner coordinates to the camera coordinates will use the

following transformation matrix. The only difference is the signs for sinθ are reversed.

∆+

∆+=

∆+

∆+

−=

YY

XXM

YY

XX

y

x

θθθθ

cossin

sincos

Derivation of the 3D transformation matrix

In a 3D coordinate system the terms right and left hand coordinate systems are used. This

method of defining a 3D coordinate system has the positive direction of the X-axis aligned

with the thumb, the positive direction of the Y-axis aligned with the index finger, and the Z-

axis aligned with the middle finger. The thumb and index finger are spread at 90o and aligned

with the plane of the palm. The middle finger is bent at 90o to the palm.

If the coordinate system aligns with the appropriate fingers of the left hand it is called a left-

hand coordinate system. And vice-versa for the right-hand coordinate system. There is no

rotation possible that will align a left-hand coordinate system with a right-hand coordinate

system. However, the problem is solved by simply negating (multiplication by -1) the values

of one of the axes of one of the coordinate systems. It makes no difference, which system or

which axis.

Note: The Riegel Laser Scanner uses a right-hand coordinate system. A camera with the

positive direction of the Z-axis exiting the lens toward the object, X being the horizontal

dimension with positive to the right, and Y being the vertical dimension with positive being

up, uses a left-hand coordinate system. The camera can be converted to a right-hand system

by negating the values on the Z-axis.

Positive rotation about an axis is determined by aligning the thumb of the hand (right or left)

in the positive direction of the axis and curling the fingers. The direction of the curled fingers

is the direction of positive rotation. A right hand rotation has the same matrix form as a

counter-clockwise rotation in a 2D coordinate system. A left hand rotation has the same

matrix form as a clockwise rotation in a 2D coordinate system.

Having three axes, a 3D coordinate system will require a 3X3 transformation matrix. The

matrices to transform the scanner coordinates to the camera coordinates are described below.

Right-hand rotation will be used. The equations could as well be derived using a left-hand

rotation.

−

=

ωωωω

cossin0

sincos0

001

Mx Rotation about the camera x-axis

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−

=

φφ

φφ

cos0sin

010

sin0cos

My Rotation about the camera y-axis

−=

100

0cossin

0sincos

κκ

κκ

Mz Rotation about the camera z-axis

Note: Below the negation of the Z-axis value is disregarded, the camera and the scanner

coordinates are assumed to use the same coordinate system. This will be corrected at the end

of the derivation to reflect the reality of the Riegl scanner and the camera coordinate systems.

A single rotation about the X-axis would have the form shown below.

+

+

+

=

zZ

yY

xX

Mx

Z

Y

X

s

s

s

i

i

i

* �

+

+

+

−

=

zZ

yY

xX

Z

Y

X

s

s

s

i

i

ωωωω

cossin0

sincos0

001

Rotation about the three camera axes can be done in any order, but must then be done

consistently throughout the computations in the same order. Generally, matrix multiplication

is not transitive, that is [M] * [N] does not equal [N] * [M]. Rotation about the X-axis

followed by rotations about the Y and Z-axes is the common sequence that is used. Thus, a

transformation from the scanner coordinate system to the camera coordinate system is

accomplished with the following matrix operations.

The full transformation would have the form shown below.

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+

+

+

=

zZ

yY

xX

MzMyMz

Z

Y

X

s

s

s

i

i

i

** or, expressed in complete form below

+

+

+

−

−

−=

zZ

yY

xX

Z

Y

X

s

s

s

i

i

i

ωωωω

φφ

φφ

κκ

κκ

cossin0

sincos0

001

cos0sin

010

sin0cos

100

0cossin

0sincos

The combined transformation matrix M can be expressed as MxMyMzM **= . And

the transformation from scanner to camera coordinates can be expressed as

+

+

+

=

zZ

yY

xX

M

Z

Y

X

s

s

s

i

i

i

Multiplying out the rotation matrices Mz, My, and Mx is shown below

−

+−−

−+

=

φωφωφκφωκωκφωκωκφ

κφωκωκφωκωκφ

coscoscossinsin

sinsincoscossinsinsinsincoscossincos

cossincossinsincossinsinsincoscoscos

M

Each element of the matrix is named according to its row/column position.

φωφω

φκφωκωκφωκω

κφκφωκωκφωκω

κφ

coscos33

cossin32

sin31

sinsincoscossin23

sinsinsincoscos22

sincos21

cossincossinsin13

cossinsinsincos12

coscos11

=

−=

=

+=

−=

−=

−=

+=

=

m

m

m

m

m

m

m

m

m

+

+

+

=

zZ

yY

xX

mmm

mmm

mmm

Z

Y

X

s

s

s

i

i

i

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In order to bring the camera and the scanner coordinate systems into the right-hand coordinate

system of the scanner, it is necessary to multiply Zi by -1.

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+

+

+

−

=

− zZ

yY

xX

mmm

mmm

mmm

Z

Y

X

s

s

s

i

i

i

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100

010

001

In order to scale the coordinates of the camera to that of the image plane (Xc, Yc) in the

camera, Xi and Yi must be scaled by the ratio of the focal length (f) to that of the Zi value for

each point. The first and second rows of the matrix equation are multiplied by the scaling

factor f/Zi. The third row of the matrix equation is left unchanged.

+

+

+

−

=

−

=

− zZ

yY

xX

mmm

mmm

mmm

Z

f

Z

f

Z

YZ

f

XZ

f

Z

Y

X

s

s

s

i

i

i

i

i

i

i

i

c

c

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100

00

00

Refer to Derivation of Tranformation Parameter Computation for a 2D/3D Coordinates

System From Laser Scanner Coordinates to Camera Coordinates for a more complete

explanation of the transformation to camera focal plane coordinates.

Note: The magnification of a camera lens system is the ratio of the focal length to the distance

to the object. If a camera lens has a focal length of 25 mm and this is considered to be 1X,

then replacing the 25 mm lens with a 50 mm lens will increase the magnification by a factor

of two and reduce the area that was imaged by a factor of ½. Assume that the CCD sensor

has a width of 25 mm (this is very close to the actual width of high quality Digital Single Lens

Reflex (DSLR) cameras. A lens with a focal length of 25 mm is used. The distance to the

outcrop is 100 m. The picture that is taken will cover 100 m of width of the outcrop,

assuming that the camera is aligned perpendicular to the outcrop and that the outcrop runs

straight left to right. If a lens with a focal length of 50 mm is put on the camera, the

magnification will be about 2X and will only cover 50 m of the outcrop. Likewise, if a lens

with a focal length of 100 mm is put on the camera, the image will appear to be magnified by

4X the first image and will cover only 25 m of the outcrop.

Derivation of the Collinearity Equations (basis for scanner to camera transformation)

Below is a 3D transformation matrix without scaling by the camera lens.

+

+

+

=

zZ

yY

xX

mmm

mmm

mmm

Z

Y

X

s

s

s

i

i

i

333231

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131211

The camera coordinate system has the origin located at the rear nodal point of the lens system.

The Z-axis of the camera is parallel to the axis of the lens system and is perpendicular to the

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CCD sensor in the camera. The positive direction of the Z-axis is toward the object being

photographed. Since the CCD sensor is located behind the focal point the CCD sensor is

located at a Z-coordinate of (–f ), [f: focal length of the lens]. The CCD sensor represents

the X-Y plane of the camera coordinate system and is orthogonal to the Z-axis. Note that with

the CCD sensor (X-Y plane) located behind the focal point, inversion in the X and Y

dimensions occurs. This is corrected by virtually moving the CCD sensor a distance (f) in

front of the focal point. This places the image on the CCD sensor in the same orientation that

the object has in space. It would be the same as taking a picture of the object and then

holding the picture up in front of the camera. What is on the left side of the object appears on

the left side of the picture. What is on the upper side of the object appears on the upper side

of the picture.

The above transformation matrix transforms the scanner coordinates to camera coordinates

but does not scale the dimensions to the scale of the CCD sensor. That is, the dimensional

reduction that occurs due to the camera lens is not reflected in the transformation matrix at

this point.

An image dimension is related to the object dimension by the ratio of the focal length of the

lens and the distance to the object. Using the relationship of similar triangles as shown below,

Xc = (f/Zi)*Xs, where f is the focal length of the camera, Zc is the distance from the rear nodal

point of the camera lens to the object and Xs is the dimension of the object.

The Xi and Yi values will be scaled by the ratio f/Zi yielding Xc and Yc which are the

coordinates on the CCD sensor. Thus, the matrix equation above when translated to the plane

of the CCD sensor can be expressed as

+

+

+

=

=

zZ

yY

xX

mmm

mmm

mmm

Z

f

Z

f

Z

Z

fY

Z

fX

Z

Y

X

s

s

s

c

c

i

i

i

i

i

i

c

c

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100

00

00

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The above complete transformation matrix is expressed in equation form as

( ) ( ) ( )( )

( ) ( ) ( )( )

( ) ( ) ( )zZmyYmxXmZ

zZmyYmxXmZ

fY

zZmyYmxXmZ

fX

sssi

sss

i

c

sss

i

c

+++++=

+++++=

+++++=

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What are called the collinearity equations are formed by substituting for Zi in the equations

for Xc and Yc. The collinearity equations are

( ) ( ) ( )( )( ) ( ) ( )( ) ( ) ( )( )( ) ( ) ( )zZmyYmxXm

zZmyYmxXmfY

zZmyYmxXm

zZmyYmxXmfX

sss

sssc

sss

sss

c

+++++

+++++=

+++++

+++++=

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The scanner and camera systems are not compatible, one is left handed the other is right

handed. They are made compatible by negating the values of one of the axes. By negating

the value of the Z axis, the equations become

( ) ( ) ( )( )( ) ( ) ( )( ) ( ) ( )( )

( ) ( ) ( )zZmyYmxXm

zZmyYmxXmfY

zZmyYmxXm

zZmyYmxXmfX

sss

sssc

sss

sss

c

+++++

+++++−=

+++++

+++++−=

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The following assignments are made in order to simplify the notation later, refer to the

numerators and denominators of the above equations.

( ) ( ) ( )( )( ) ( ) ( )( )

( ) ( ) ( )zZmyYmxXmW

zZmyYmxXmfV

zZmyYmxXmfU

sss

sss

sss

+++++=

+++++−=

+++++−=

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Xc and Yc are shown below expressed in terms of U, V, W.

W

VY

W

UX

c

c

=

=

Xc and Yc are the coordinates of a point on the camera CCD sensor with the origin of the

coordinate system at the center of the CCD sensor. These are not UV coordinates. UV

coordinates have the origin at the upper left corner of a photograph and have values between 0

and 1.0. Xc and Yc have dimensions which relate to the size of the CCD sensor and range

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from -1/2 the sensor width to +1/2 the sensor width and from -1/2 the sensor height to +1/2

the sensor height. In solving for the transformation parameters (ω, Φ, κ, x, y, and z) four

points are identified common between the photograph and the 3D model. Thus we have four

sets of values for Xc, Yc, Xs, Ys, and Zs. We do not have values for Zi in the camera

coordinate system. By substituting for Zi in the equations for Xc and Yc, the need to solve for

Zi is eliminated.

A different derivation of the collinearity equations is provided by Mikhail and is

described below.

Starting with the 3D transformation below, the system is turned into a 2D/3D transformation

by adding a uniform scaling factor (a) and setting the image Z value, Zi, to –f, the focal length

of the camera.

+

+

+

=

zZ

yY

xX

mmm

mmm

mmm

Z

Y

X

s

s

s

i

i

i

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becomes

+

+

+

=

− zZ

yY

xX

mmm

mmm

mmm

a

f

Y

X

s

s

s

c

c

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Writing this in equation form

( ) ( ) ( )( )( ) ( ) ( )( )( ) ( ) ( )( )zZmyYmxXmaf

zZmyYmxXmaY

zZmyYmxXmaX

sss

sssc

sssc

+++++=−

+++++=

+++++=

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Division of the first two equations by the third equation eliminates the scaling factor a.

( ) ( ) ( )( )( ) ( ) ( )( ) ( ) ( )( )

( ) ( ) ( )zZmyYmxXm

zZmyYmxXmfY

zZmyYmxXm

zZmyYmxXmfX

sss

sssc

sss

sss

c

+++++

+++++−=

+++++

+++++−=

333231

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333231

131211

The least squares solutions for these equations are provided in the other documents.

References:

Mikhail, E.M., Bethel, J.S., McGlone, J.C.; Introduction to Modern Photogrametry, 479

pages, John Wiley and Sons, 2001.

Larson, R.E., Edwards, B.H.; Elementry Linear Algebra, 568 pages, D.C. Heath and

Company, 3rd

Edition, 1996.

Wolf, P.R., Dewitt, B.A.; Elements of Photogrammetry with Applications in GIS, Edition,

McGraw Hill, Inc., 3rd

Edition, 2000,