Camera Geometry and Single View Geometry

7/25/2019 Camera Geometry and Single View Geometry

1/36

1-1Professur Digital- und SchaltungstechnikProf. Dr.-Ing. Gangolf Hirtz

Computer Vis ionVersion vom 11.05.2015

Chapter 3

Camera Geometry and

Single View GeometryVersion 11.05.2015


2/36



Content

3.1 Geometrical Imaging Process

3.1.1 Introduction

3.1.2 Extrinsic Model3.1.3 Extrinsic ModelMatlab Example

3.1.4 Intrinsic ModelProjection

3.1.5 Intrinsic ModelAffine Transformation

3.1.6 Summary

3.2 Pinhole Camera Model

3.2.1 The Geometrical Model of a Pinhole Camera

3.2.2 The Geometrical Model of a Pinhole CameraMatlab

Example

3.2.3 Intrinsic ModelCamera Calibration Matrix K

3.2.4 Extrinsic Model

3.2.5 Extrinsic ModelMatlab Example

3.3 Projective Camera Model

3.3.1 Intrinsic ModelCamera Calibration Matrix K3.3.2 The Projection Matrix P

3.3.2 The Projection Matrix PMatlab Example

3.3.3 The Projection Matrix PExample Basler acA630-

100gc


3/36




An imaging device or camera usually maps an observed

3D scene to a 2D image by projection.

Several projection principles are known and vitally

characterize a camera: parallel projection, perspective

projection, orthographic projection, oblique projection,

Projection principles can be mathematically approximated

by an appropriate model which is commonly known as

camera modelorprojection model.

Most commonly known imaging devices follows the

perspective projection principle and is known as pinhole

camera model

3.1.1 Introduction

Principle Point / Camera Center / Optical Center / Pinhole

Sensor Plane/Image Plane

Focal length f

Camera (box)

Optical axis/principle axis


4/36




Mathematically the image that has been projected upside-

down can be flipped using the principle point

3.1.1 Introduction

Focal length f

Sensor Plane/Image Plane

Principle Plane


5/36




The principle of the pinhole camera was already

discovered by Aristoteles (384-322 B.C.).

First steps with the pinhole camera was taken by the Arab

Alhazen in 980 as he investigates the human eye.

At the end of the 13th century astronomics used the

pinhole camera to observe sunspots.

3.1.1 Introduction


6/36




The first published picture of a pinhole camera obscurais apparently a drawing in Gemma Frisius' De Radio Astronomica et

Geometrica(1545).

The very term camera obscura("dark room") was coined by Johannes Kepler (15711630).

The first photography was taken by a pinhole camera in 1816 by Joseph Nicphore Nipcecht. Unfortunately he wasnt

able to fix it. But a few months later he presents the first photo of the world, a view out of his studyroom.

3.1.1 Introduction


7/361-7Professur Digital- und Schaltungstechnik

Prof. Dr.-Ing. Gangolf HirtzComputer Vis ion

Version vom 11.05.2015


The geometrical relationship between 3D scene points

Xwrldand the resulting 2D image points Ximgis often quite

complex due to the physical construction of the imaging

device.

By employing a finite amount of numerical parameters,

the actual process of projection can be replaced by an

appropriate mathematical relationship of bearable

computational cost.

3.1.1 Introduction

Observed scene

Camera

World coordinate system

wrldX

Image

imgX

zwrld

xwrldywrld

3D scene point

2D

Image point

Geometrical imaging process

wrld

wrld

wrld

wrld

wrld

x

y

z

w

X

img

img img

img

x

y

w

X






The overall imaging process that transfers a scene point

Xwrld to an image point Ximg successively utilizes an

extrinsicand an intrinsic model

3.1.1 Introduction

Observed scene

Camera

wrldX imgX

zwrld

xwrldywrld

Extrinsic model Intrinsic model

xcamzcam

ycam

Camera coordinate system

yimg

ximg

Image coordinate system

camX

wrld

wrld

wrld

wrld

wrld

x

y

z

w

X

img

img img

img

x

y

w

X

cam

cam

cam

cam

cam

x

y

z

w

X






Transformation from world points Xwrldto camera points Xcamin multiple camera environments

Incorporates the distinct geometrical relationship Hiof each camera with respect to WCS


Observed scene

Camera

wrldX

zwrld

xwrldywrld

xcamzcam

Camera

xcam

zcam

cam,0X

cam,1X

0 0R C

1 1R C

i i ii

1

R R CH0

1

cam,0 0 wrld

1

cam,1 1 wrld

X H X

X H X

1H

0H

Ci origin of the camera coordinate

frame provided in world frame

coordinates

Ri 3x3 rotation matrix representing the

orientation of the cameracoordinate frame with respect to

the world frame

R results out of three angles

discribing a sequence of three

elementary rotations around x, y, z-

axis respectively (each performed

clockwise)






Example: Optical sensor network to observe a complex indoor environment (e.g. domestic home environment)

3.1.3 Extrinsic Model - Matlab Example

WCS

Camera 1

Camera 2

Camera 3


11/36





0001 clear all; close all; clc;

0002

0003 %% Extrinsic Camera Parameter0004 % Camera centres

0005 C1 = [ 2.6607; 0.1006; 2.4089]; % Smart Sensor 605

0006 C2 = [-1.6157; 2.7995; 2.4119]; % Smart Sensor 601

0007 C3 = [-1.8482; -1.7443; 2.3955]; % Smart Sensor 602

0008

0009 % Rotation matrices

0010 R1 = [ -0.7770 0.6286 0.03390011 0.3719 0.5018 -0.7809

0012 -0.5079 -0.5942 -0.6237];

0013

0014 R2 = [ -0.5233 -0.8521 0.0080

0015 -0.5388 0.3236 -0.7778

0016 0.6602 -0.4113 -0.6285];

0017

0018 R3 = [ -0.4861 -0.8670 0.1097

0019 -0.6689 0.2884 -0.6852

0020 0.5624 -0.4065 -0.7201];

0021


12/36





0022 % Translation vectors

0023 T1 = -R1*C1;

0024 T2 = -R2*C2;0025 T3 = -R3*C3;

0026

0027 %% Setup Homographies

0028 H1 = [R1 T1; 0 0 0 1];

0029 H2 = [R2 T2; 0 0 0 1];

0030 H3 = [R3 T3; 0 0 0 1];

00310032 %% Plot WCS

0033 h1 = figure(1); hold on; grid on; axis equal;

0034 xlabel('x [m]'); ylabel('y [m]'); zlabel('z [m]'); view(70, 20);

0035 axis([-3 +3 -3 +3 0 3]);

0036

0037 i = [1; 0; 0];

0038 j = [0; 1; 0];

0039 k = [0; 0; 1];

0040 o = [0; 0; 0];

0041

0042 plot3([o(1) i(1)], [o(2) i(2)], [o(3) i(3)], 'r-', 'MarkerSize', 5);

0043 plot3([o(1) j(1)], [o(2) j(2)], [o(3) j(3)], 'g-', 'MarkerSize', 5);

0044 plot3([o(1) k(1)], [o(2) k(2)], [o(3) k(3)], 'b-', 'MarkerSize', 5);


13/36





0045

0046 %% Plot camera coordinate systems

0047 i1 = H1^-1*[i; 1];0048 j1 = H1^-1*[j; 1];

0049 k1 = H1^-1*[k; 1];

0050 o1 = H1^-1*[o; 1];

0051

0052 plot3([o1(1) i1(1)], [o1(2) i1(2)], [o1(3) i1(3)], 'r-', 'MarkerSize', 1);

0053 plot3([o1(1) j1(1)], [o1(2) j1(2)], [o1(3) j1(3)], 'g-', 'MarkerSize', 1);

0054 plot3([o1(1) k1(1)], [o1(2) k1(2)], [o1(3) k1(3)], 'b-', 'MarkerSize', 1);0055

0056 i2 = H2^-1*[i; 1];

0057 j2 = H2^-1*[j; 1];

0058 k2 = H2^-1*[k; 1];

0059 o2 = H2^-1*[o; 1];

0060



0063 plot3([o2(1) k2(1)], [o2(2) k2(2)], [o2(3) k2(3)], 'b-', 'MarkerSize', 1);


14/36





0064

0065 i3 = H3^-1*[i; 1];

0066 j3 = H3^-1*[j; 1];

0067 k3 = H3^-1*[k; 1];0068 o3 = H3^-1*[o; 1];

0069



0072 plot3([o3(1) k3(1)], [o3(2) k3(2)], [o3(3) k3(3)], 'b-', 'MarkerSize', 1);

1H

0H

2H

Observed Indoor Scene

Floor (z == 0 m)

WCS


15/36




The Projection process from Xcam to normalized sensor

coordinates Xnorm is modelled by a projection function F

The cameras sensor plane is assumed to have unit

length from the principle point

3.1.4 Intrinsic Model - Projection

Observed scene

ynorm

xnorm

xcam

ycamcamX

zcam

Normalized sensor coordinate system

normXF

Camera centre/Principle point

Camera sensor plane

Focal length f normalizedto unit length

Projection

norm

norm norm

1

x

y

X

cam

cam

camcam

cam

x

y

z

w

X


16/36




The camera coordinate Xcam is considered as incident

light ray(,,)

The radial projection of an incoming light ray (,,)

onto a virtual image plane with a distance of f= 1 to the

projection center can be modelled with the radial

projection function F():


Observed scene

ynorm

xnorm

xcam

ycamcam X

zcam

elevation

azimuth

()

cam

cam cam

cam

x

y

z

X

norm

norm r

norm

r

( ) ( )

coswithsin

xF

y

X u

u

normXF

distance

ur


17/36




Projection

Type

Function F Properties and Applications

Perspective

projection

= tan Applied in conventional

cameras that follow the pinhole

principle with <

Stereographic

projection = 2 tan

Commonly used for the

generation of stellar maps and

photography. angles are

preserved (equal angles) and

circles are imaged without

distortions (equal circles).

Equidistanceprojection

= It is the most popular model fordescribing fisheye projection

lenses. Incidence angles of rays

are imaged linearly.

Equisolid

angle

projection

= 2 sin

2

Each pixel corresponds to one

distinct solid angle which makes

the image look like the reflection

by a specular ball.

Orthogonal

projection = sin This camera principle models

the projection onto a sphere.

The projection radius is

increasing with increasing

incidence angle

until reachespi/2. Than it is

declining again.



18/36





Equiangular Projection

Applied by e.g. fish-eye cameras

FOV > 180

Perspective Projection

Most commonly applied projection method

Limited field of view (FOV) to approx. 120()

= tan =


19/36




3.1.5 Intrinsic Model Affine Transformation

The transformation from normalized sensor coordinates to

image coordinates is finally done by means of an affine

transformation matrixK

Kis called camera calibration matrix

imgX

Affine

Transformation

yimg

ximg


img

img img

img

x

y

w

X

norm

norm norm

1

x

y

X

ynorm

xnormnormX

Normalized sensor coordinate system

K

img norm X K X


20/36




3.1.6 Summary

Geometrical imaging process is characterized by an

extrinsic and intrinsic model

Intrinsic model (camera model) is split into different

projection models Fand an affine transformation K

Relevant camera models are

Pinehole camera model

Projective camera model


21/36



Central projection is mapping a point Xcamin 3D space to

a point Ximgin 2D space:

Using similar triangles one verifies that:

Using homogeneous coordinates, one can put that

transformation in a matrix expression:

The 3x4 transformation matrix used is called a camera

projection matrixP

Pcan be decomposed in the following manner:

The Diagonal Matrix is called the camera calibration

matrixK


3.2.1 The Geometrical Model of a Pinhole Camera

x

object point in

3D world frameimage point in 2D

image frame

imaging process

cam cam cam img img, , ,TT

x y z x y

f

fcam

z

cam cam/f y z

cam cam cam cam cam cam cam, , / , /T T

x y z f x z f y z

cam

img camcam

img cam

camcam cam

cam

normalized sensor coordinatestan

01 1

0

1 1 01

xx f x f

yy f y f

zz zz

img cam

X P X

diag( , ,1) |f f P I 0

( , ,1)diag f f K

camX

camXimgX


22/36




3.2.2 The Geometrical Model of a Pinhole Camera - Matlab Example

0001 %% Camera Matrix

0002 % Intrinsic Camera Parameter

0003 f = 4; % Focal Length

00040005 % Setup Projection matrix

0006 P = diag([f f 1])*[eye(3) zeros(3, 1)];

0007

0008 %% Project World Point

0009 Xw = [4; 5; 9]; % Object Point (World Frame)

0010 % X = 4, Y = 5, Z = 9

0011 Xw_h = [Xw; 1]; % Transform to homogeneous Coordinates

0012 Xc_h = P*Xw_h; % Project!

00130014 Xc = 1/Xw(3)*Xc_h;% Image Point (Camera Frame)

0015

0016 %% Plot Principal Point

0017 figure_01 = figure; hold on; grid on;

0018 plot3(0, 0, 0, 'm.', 'Markersize', 20);

0019

0020 %% Draw Image Plane

0021 [X, Y, Z] = meshgrid(-5:0.1:5, -5:0.1:5, -5:0.1:5);

0022

0023 data = 0*X+0*Y+1*Z-f; % Normal from for a plane

0024 p = patch(isosurface(X, Y, Z, data, 0)); hold on;

0025 set(p, 'FaceColor', 'red', 'EdgeColor', 'none')

0026 alpha(0.5); view(17,24);

0027 axis([-5 5 -5 5 0 10]);

0028


23/36




3.2.2 The Geometrical Model of a Pinhole Camera - Matlab Example

0029 %% Draw Object and Image Point

0030 plot3(Xw(1), Xw(2), Xw(3), 'b*', 'Markersize', 10);

0031 plot3(Xc(1), Xc(2), f, 'b*', 'Markersize', 5);

0032 plot3([0 Xw(1)], [0 Xw(2)], [0 Xw(3)], 'b--');0033

0034 %% Add some text

0035 text(0.5, 0.5, 0,...

0036 'Principal Point', 'FontSize', 10,...

0037 'FontWeight', 'bold', 'Color', 'magenta');

0038 text(Xw(1) - 3.0, Xw(2) - 3.0, Xw(3),...0039 'Object Point', 'FontSize', 10, 'FontWeight', 'bold', 'Color', 'blue');

0041 text(Xc(1) + 0.5, Xc(2) + 0.5, f,...

0042 'Image Point', 'FontSize', 10,...0043 'FontWeight', 'bold', 'Color', 'blue');

0044 text(-2, -2, f,...

0045 'Image Plane', 'FontSize', 10,...

0046 'FontWeight', 'bold', 'Color', 'red');

3 2 C


24/36



It is common for images to have only positive coordinates

(image coordinates start at the upper left corner of the

image), so a further adaption has to be done

The principal point (cx, cy) is the offset with respect to the

image frame and enhances the camera calibration matrix

Kis an upper triangle matrix with 3 dof

A camera is denoted as acalibrated cameraso soon as

Kis known

The pinhole camera model is a ideal one assuming that

image coordinates are Euclidean coordinates having

equal scales in both axial directions


3.2.3 Intrinsic Model - Camera Calibration Matrix K

img img img imgx yx x c y y c

1

x

y

f cf c

K

imgX


yimg

ximg

yimg

ximg

Principal Point (cx, cy)

3 2 Pi h l C M d l


25/36



Every single cameras position C is described by a

rotation Rand a translation Twith respect to the world

coordinate system

A Point Xwrld can be translated into a point Xcamby the

following mathematical relationship

Homogeneous coordinates let us use this relation in a

more compact matrix form

The overall imaging process is now as follows:



With the new relationship one can determine the image

coordinates of a point given in world coordinates as

follows:

Pcan be computed from point correspondences Xwrld

Ximg

cam wrld X R X C

cam wrld1

R RCX X

img wrld with | X P X P KR I C

img cam

img cam

img wrld

img wrld

Translation vector

img wrld

|

|1

|

|

T

X P X

X K I 0 X

R RCX K I 0 X

X K R RC X

X K R I C X

1H

0H

2H

WCS

3 2 Pi h l C M d l


26/36




3.2.5 Extrinsic Model Matlab Example




0004 K = diag([f f 1]); % Intrinsic camera Parameter0005

0006 % Extrinsic Camera Parameter

0007 C1 = [-3; +4; -3]; % Camera Centre Cam 1

0008 C2 = [3; -4; 1]; % Camera Centre Cam 2

0009 C3 = [3; -5; -5]; % Camera Centre Cam 2

0010

0011 % Rotate Cam 1 by 0, Cam2 by 45 and Cam3 by 900012 % arround the x-axis

0013 R1 = [1.0000 0 00014 0 1.0000 0

0015 0 0 1.0000];

0016 R2 = [1.0000 0 0

0017 0 -0.7071 0.7071

0018 0 0.7071 0.7071];

0019 R3 = [1.0000 0 0

0020 0 0 1.0000

0021 0 1.0000 0];

00220023 t1 = -R1*C1; % Translation vector with respect to R and C



0026

3 2 Pi h l C M d l


27/36






0028 P1 = K*[R1 t1];

0029 P2 = K*[R2 t2];

0030 P3 = K*[R3 t3];0031

0032 %% Plot Principal Points

0033 figure_01 = figure; hold on; grid on;

0034 plot3(C1(1), C1(2), C1(3), 'r.', 'Markersize', 20);

0035 plot3(C2(1), C2(2), C2(3), 'g.', 'Markersize', 20);

0036 plot3(C3(1), C3(2), C3(3), 'b.', 'Markersize', 20);

0037

0038 %% Draw Image Plane Camera 1

0039 [X, Y, Z] = meshgrid( (C1(1)-3):0.1:(C1(1)+3),...0040 (C1(2)-3):0.1:(C1(2)+3),...

0041 (C1(3)-3):0.1:(C1(3)+3));

0042 plane = P1(3, :);0043 data = plane(1)*X + plane(2)*Y + plane(3)*Z + (plane(4)-f)*1;


0045 set(p, 'FaceColor', 'red', 'EdgeColor', 'none');


0047 [X, Y, Z] = meshgrid( (C2(1)-3.0):0.1:(C2(1)+3.0),...

0048 (C2(2)-4.2):0.1:(C2(2)+4.2),...0049 (C2(3)-4.2):0.1:(C2(3)+4.2));

0050 plane = P2(3, :);

0051 data = plane(1)*X + plane(2)*Y + plane(3)*Z + (plane(4)-f)*1;


0053 set(p, 'FaceColor', 'green', 'EdgeColor', 'none');

0054

3 2 Pi h l C M d l


28/36






0056 [X, Y, Z] = meshgrid( (C3(1)-3):0.1:(C3(1)+3),...

0057 (C3(2)-3):0.1:(C3(2)+3),...

0058 (C3(3)-3):0.1:(C3(3)+3));0059 plane = P3(3, :);



0062 set(p, 'FaceColor', 'blue', 'EdgeColor', 'none');

0063

0064

0065 alpha(0.5); view(140, 35);

0066 axis([C(1)-10 C(1)+10 C(2)-10 C(2)+10 C(3)-10 C(3)+10]);

3 3 Projecti e Camera Model


29/36



In the case of CCD cameras there is the possibility to

have non-square pixels and hence having unequal scale

factors for both image directions.

Imagine a pixel density(no. of pixels per unit length) of

mx(x-direction) and my(y-direction), the calibration matrix

Kchanges to

wherexandyrepresent the focal length with respect

to the pixel dimensions in both directions.

Although it is very seldom for some rare cases it is

necessary to introduce the skew factor s, defining the

geometrical dependency of the y-dimension in pixel

frame from the x-axis:

A camera having Pas described above is known as finite

projective camera.and has 11 degrees of freedom


3.3.1 Intrinsic Model - Camera Calibration Matrix K

Aspect Ratio

The ratio ofxandy is often calledaspect ratioand is

very close to 1.

mxno. of pixel per m along the x-axis, hor. pixel density,

[px/m]

myno. of pixel per m along the y-axis,vert. pixel density,

[px/m],

Shear angle between the image axes; is encoded

within s

(x,y) focal length, with respect to pixel dimensions

with , ,

1

x x

y y x x y y

s c

c f m f m s f

K

s>0 s=0

x/y = =1x/y =


30/36



The imaging process of a camera is completely described

by the projection matrix P

The projection matrix P is a 3x4 matrix performing the

transformation from projective space in 3D to a 2D plane

Note that Phas only eleven degrees of freedom, though

it has twelf components. In other words, P is determined

up to an unknown scale factor

Pcan be decomposed in the following manner


3.3.2 The Projection Matrix P

img wrld X P X

1

1 11 12 13 14

2

2 21 22 23 24

3

3 31 32 33 34

4

xx p p p p

xx p p p p

xx p p p p

x

| | | |

| | | |

1R

2R 1C 2C 3C 4C

3R

p

P p p p p p

p

3 3 Projective Camera Model


31/36



Rows

RowsofPcan be seen as planes in the projective space

P3 (Principal Plane, Planes orthogonal to x-, y-axis

containing principal point and origin of image coordinatesystem)

The line formed by the intersection of the rows p2Rand

p3R connects the principal point and the origin of the

image coordinate system

Columns

Columns ofPdenote special points within the projection

plane

p1C, p2C, p3Care the projections of the axis directions of

x-, y- and z-axis.

p4C is the projection of the origin of the world frame.

Projection Centre/Principal Point


3.3.2 The Projection Matrix P

P C 0



32/36




3.3.3 The Projection Matrix P - Matlab Example




0004 K = diag([f f 1]); % Intrinsic camera Parameter0005 K(1, 3) = 0;

0006 K(2, 3) = 0;

0007


0009 C = [0; 0; 0]; % Camera Centre

0010 R = [1 0 0

0011 0 0 1

0012 0 1 0]; % We change x and y coordinates

0013 t = -R*C; % Translation vector with respect to R and C


0016 P = K*[R t];

0017

0018 %% Graphical Sketch

0019 figure_01 = figure; hold on; grid on;0020 plot3(C(1), C(2), C(3), 'm.', 'Markersize', 20);

0021

0022 %% Draw all Planes encoded in P

0023 [X, Y, Z] = meshgrid( (C(1)-5):0.1:(C(1)+5),...

0024 (C(2)-0):0.1:(C(2)+5),...0025 (C(3)-5):0.1:(C(3)+5));

0026

0027 % The x-axis Plane is the 1rd row of P!

0028 plane = P(1, :);

0029 data = plane(1)*X + plane(2)*Y + plane(3)*Z + plane(4)*1;


0031 set(p, 'FaceColor', 'yellow', 'EdgeColor', 'none');

00320033 % The y-axis Plane is the 2rd row of P!

0034 plane = P(2, :);




0038

0039 % The Principal Plane is the 3rd row of P!

0040 plane = P(3, :);




0044

0045 % The Image Plane is the Principal Plane with an offset of f!

0046 plane = P(3, :);


0048 p = patch(isosurface(X, Y, Z, data, 0)); hold on;0049 set(p, 'FaceColor', 'blue', 'EdgeColor', 'none');

0050

0051 alpha(0.5); view(110, 15);

0052 axis([C(1)-5 C(1)+5 C(2)-0 C(2)+5 C(3)-5 C(3)+5]);0053

0054 % Save Image

0055 print(figure_01, '-r600', '-dtiff',...

0056 '..\Images\16_Example_Projection_Matrix_01.tif');

0057

0058 % Convert document to rtf

0059 highlight('m05_Example_Camera_Matrix.m', 'rtf', ...

0060 'm05_Example_Camera_Matrix.rtf')

P =

2 0 0 0

0 0 2 0

0 1 0 0



33/36




3.3.3 The Projection Matrix P - Matlab Example




0004 K = diag([f f 1]); % Intrinsic camera Parameter0005 K(1, 3) = 3;

0006 K(2, 3) = 2;

0007


0009 C = [3; 2; 1]; % Camera Centre

0010 R = [1 0 0

0011 0 0 1

0012 0 1 0]; % We change x and y coordinates

0013 t = -R*C; % Translation vector with respect to R and C


0016 P = K*[R t];

0017

0018 %% Graphical Sketch

0019 figure_01 = figure; hold on; grid on;0020 plot3(C(1), C(2), C(3), 'm.', 'Markersize', 20);

0021

0022 %% Draw all Planes encoded in P

0023 [X, Y, Z] = meshgrid( (C(1)-5):0.1:(C(1)+5),...

0024 (C(2)-0):0.1:(C(2)+5),...0025 (C(3)-5):0.1:(C(3)+5));

0026

0027 % The x-axis Plane is the 1rd row of P!

0028 plane = P(1, :);



0031 set(p, 'FaceColor', 'yellow', 'EdgeColor', 'none');

00320033 % The y-axis Plane is the 2rd row of P!

0034 plane = P(2, :);




0038

0039 % The Principal Plane is the 3rd row of P!

0040 plane = P(3, :);




0044

0045 % The Image Plane is the Principal Plane with an offset of f!

0046 plane = P(3, :);


0048 p = patch(isosurface(X, Y, Z, data, 0)); hold on;0049 set(p, 'FaceColor', 'blue', 'EdgeColor', 'none');

0050

0051 alpha(0.5); view(110, 15);

0052 axis([C(1)-5 C(1)+5 C(2)-0 C(2)+5 C(3)-5 C(3)+5]);0053

0054 % Save Image

0055 print(figure_01, '-r600', '-dtiff',...

0056 '..\Images\16_Example_Projection_Matrix_01.tif');

0057

0058 % Convert document to rtf

0059 highlight('m05_Example_Camera_Matrix.m', 'rtf', ...

0060 'm05_Example_Camera_Matrix.rtf')

P =

2 3 0 -12

0 2 2 -6

0 1 0 -2



34/36




3.3.4 The Projection Matrix P - Example Basler acA640-100gc



35/36




3.3.4 The Projection Matrix P - Example Basler acA640-100gc

0001 %% Example Projection Matrix (Computed)

0002

0003 fx = 4.5e-3; % Focal Length 4.5mm

0004 fy = 4.5e-3; % Focal Length 4.5mm

0005 mx = 1/6.5e-6; % no of pixels per mm

0006 my = 1/6.5e-6; % no of pixels per mm

0007 x0 = 640/2;

0008 y0 = 480/2;

0009 ax = fx*mx;

0010 ay = fy*my;

0011 s = 0; % skew factor

00120013 K = [ ax s x0;

0014 0 ay y0;

0015 0 0 1 ];

0016

0017 C = [0; 0; 0];

0018 R = eye(3);

0019 t = -R*C;

0020

0021 P = K*[R t];0022 disp(P);

P (estimated by calibration procedure)

8.275244814e+02 0 3.2435919943e+02

0

0 8.2706873781e+02 2.4445479377e+020

0 0 1 0

P (computed)

692,3076923076 0 320 0

0 692,3076923076 240 0

0 0 1 0

Literature


36/36

Professur Digital und Schaltungstechnik Computer Vis ion

Literature

[1]Richard Hartley, Andrew Zisserman. Multiple View Geometry in

computer vision. Cambridge university press, 2003

Documents

Camera Geometry and Single View Geometry