Robotica - Politecnico di Milanohome.deib.polimi.it/restelli/Lezione1.pdf · 06/10/2006 Corso di Robotica A 06/07 Davide Migliore - [email protected] 2 Today Introduction Camera

Robotica Robotica

Anno accademico 2006/2007

Davide [email protected]

06/10/200606/10/2006 Corso di Robotica A 06/07Corso di Robotica A 06/07Davide Migliore - [email protected] Migliore - [email protected]

22

TodayToday

● Introduction ● Camera Model

– Survey on homogeneous coordinate rappresentation


33

Why these lessons?Why these lessons?


44


??


55


● What from sensors?

Camera Laser


66



77


??


88


● What from sensors?

Disparity Image Stereo Camera


99

Possible solutionPossible solution

Sensing

Simultaneous Localization And Mapping

(SLAM)

Computer Computer VisionVision

ProbabilisticProbabilisticRoboticsRobotics


1010

Why study Computer Vision?Why study Computer Vision?

● Images and movies are simple to acquire.● Fast-growing collection of useful applications

– building representations of the 3D world from pictures

– automated surveillance, object tracking– obstacle detection and robot localization

● Now cameras are inexpensive


1111

What can I do with C.V.?What can I do with C.V.?


1212

Why Probabilistic Robotics?Why Probabilistic Robotics?

● A new and growing area in robotics, concerned with perception and control in the face of uncertainty.

● Uncertainty arises from four major factors:– Environment stochastic, unpredictable– Robot stochastic– Sensor limited, noisy– Models inaccurate

● Key idea: Explicit representation of uncertainty


1313

What can I do with P.R.?What can I do with P.R.?Indoors

Space

Undersea

Underground


1414

Purpose of these lessonsPurpose of these lessons

● To provide an introduction to the uses of visual sensing and probabilistic methods for mobile robotic tasks.

● A survey of the mathematical and algorithmic problems that recur in its application.


1515

Date Lesson

06/10/06

13/10/06 2D and 3D geometry


27/10/06 Features extraction

03/11/06 Introduction and state estimation

01/12/06 Gaussian Filters

15/12/06 Non parametric Filters

12/01/06 Conclusion and demos

Computer Vision & Probabilistic Robotics introduction.Homogeneous Coordinate and Pin-Hole Camera Model


1616

Date Lesson

06/10/06



27/10/06 Features extraction

03/11/06 Introduction and state estimation

01/12/06 Gaussian Filters

15/12/06 Non parametric Filters

12/01/06 Conclusion and demos

Computer Vision & Probabilistic Robotics introduction.Homogeneous Coordinate and Pin-Hole Camera Model


1717

ReferencesReferences

● “Multiple View Geometry”, R.Hartley and A.Zisserman, ed. Cambridge University Press, 2nd edition 2003.● “An Invitation to 3D Vision”, Y.Ma, S.Soatto, J.Kosecka, S.S.Sastry, ed. Springer, 2000. ● “Probabilistic Robotics”, S.Thrun, W.Burgard, D.Fox, ed. MIT Press 2005.● “Robotica”, G.Gini and V.Caglioti, ed. Zanichelli, 2003.


1818

The Camera The Camera andand

Homogeneous RepresentationHomogeneous Representation


1919

How do we see the world?How do we see the world?

Let’s design a camera● Idea: put a piece of film in front of an object● Do we get a reasonable image?


2020

Pinhole cameraPinhole camera

Add a barrier to block off most of the rays– This reduces blurring– The opening known as the aperture– How does this transform the image?


2121

Pinhole camera modelPinhole camera model

Pinhole model:– Captures pencil of rays – all rays through a single point– The point is called Center of Projection (COP)– The image is formed on the Image Plane– Effective focal length f is distance from COP to Image

Plane


2222

Dimensionality Reduction Machine Dimensionality Reduction Machine (3D to 2D)(3D to 2D)

Point of observation

3D world 2D image

What have we lost?• Angles• Distances (lengths)


2323

Funny things happen…Funny things happen…


2424

Parallel lines aren’t…Parallel lines aren’t…


2525

Distances can’t be trusted...Distances can’t be trusted...


2626

……but humans adopt!but humans adopt!

Müller-Lyer Illusion

http://www.michaelbach.de/ot/sze_muelue/index.html

We don’t make measurements in the image plane

http://www.michaelbach.de/ot/sze_muelue/index.html


2727

Building a real cameraBuilding a real camera


2828

Camera ObscuraCamera Obscura

The first camera● Known to Aristotele● Depth of the room is the effective focal length


2929

Aperture tuningAperture tuning

Why not make the aperture as small as possible?● Less light gets through● Diffraction effects…


3030

Aperture tuningAperture tuning


3131

From Pinhole to lensesFrom Pinhole to lenses


3232

Image Formation using LensesImage Formation using LensesIdeal Lens: Same projection as pinhole but gathers more light!

foi

111 =+

i o

Lens Formula:

• f is the focal length of the lens – determines the lens’s ability to bend (refract) light

• f different from the effective focal length f discussed before!

P

P’

f


3333

Focus and DefocusFocus and Defocus

A lens focuses light onto the film– There is a specific distance at which objects are “in focus”

● other points project to a “circle of confusion” in the image– How can we change focus distance?

“circle of confusion”


3434

Depth of FieldDepth of Field


3535



3636



3737

Aperture controls Depth of Aperture controls Depth of FieldField

Changing the aperture size affects depth of field– A smaller aperture increases the range in which the

object is approximately in focus– But small aperture reduces amount of light – need to

increase exposure


3838

Varying the aperture Varying the aperture

Large apeture = small DOF Small apeture = large DOF


3939

Modeling projectionModeling projection

The coordinate system– We will use the pin-hole model as an approximation– Put the optical centre or camera centre(Centre) at the origin– Put the image plane in front of the COP

● Why?– The camera looks up the positive z axis


4040

Modeling projectionModeling projection

Projection equations– Compute intersection with PP of ray from (x,y,z) to COP– Derived using similar triangles (on board)

– Ignoring the final image coordinate: (from 3D to 2D)

x= f XZ

y= f YZ

X ,Y , Z T f XZ, fYZT


4141

Homogeneous coordinatesIs this a linear transformation?

Trick: add one more coordinate (w=1):

homogeneous image coordinates

homogeneous scene coordinates

Converting from homogeneous coordinates

● no—division by z is nonlinear

x , yT⇒ xy1 X ,Y , Z T⇒XYZ1

xyw ⇒x /w , y /wT XYZW

⇒X /W ,Y /W , Z /W T


4242

Central ProjectionCentral Projection

● Projection is a matrix multiply using homogeneous coordinates:

N.b. w = ZXYZ1 fXfYZ =[ f 0 0

0 f 00 0 1][1 0 0 0

0 1 0 00 0 1 0]XYZ

1

fXfYZ =[ f 0 0 00 f 0 00 0 1 0 ] XYZ

1

x=PX P=diag f , f ,1[ I∣0]


4343

Principal Point OffsetPrincipal Point Offset

X ,Y , Z T f XZ px , f

YZ p y

T

● The origin oh coordinates in image planes is not at the principal point p x , p yT

XYZ1 fXZpx

fYZp yZ =[ f 0 px 0

0 f p y 00 0 1 0 ] XYZ1


4444

Camera Calibration Matrix (K)Camera Calibration Matrix (K)

K=[ f 0 px0 f py0 0 1 ]

x=K [ I∣0] X cam● Rewriting the equation:

● The camera is assumed to be located at the origin of Euclidean coordinate system, with principal axis of camera pointing down the Z-axis: this system may be called camera coordinate frame.


4545

Trasformation and matricesTrasformation and matrices

[ x 'y ' ]=[ab ][ xy ]● Traslation:

● Homogeneous rappresentation:

[ x 'y '1 ]=[1 0 a0 1 b0 0 1][ xy1 ]


4646


[ x 'y ' ]=[cos −sinsin cos ][ xy ]

● Rotation around origin 2D (anticlockwise is positive):

● Homogeneous rappresentation:

[ x 'y '1 ]=[cos −sin 0sin cos 00 0 1][ xy1 ]


4747


● In 3D:

RR=[1 0 00 cosR −sin R0 sin R cosR ]

P P=[ cosP 0 sinP0 1 0

−sinP 0 cosP]Y Y =[cosY −sinY 0

sin Y cosR 00 0 1 ]

Homoneneous

R R=[1 0 0 00 cosR −sinR 00 sinR cosR 00 0 0 1

]P P=[ cosP 0 sinP 0

0 1 0 0−sinP 0 cosP 00 0 0 1

]Y Y =[cosY −sinY 0 0

sin Y cosR 0 00 0 1 00 0 0 1

]


4848

Camera Rotation and TraslationCamera Rotation and Traslation

X cam=R X− C is an inhomogeneus point in world coordinate frame

represents the same point in camera coordiante frame

represents the coordinate of the camera centre in world coordinate frame

X cam=[R −R C0 1 ][ XY1 ]

x=KR [ I∣− C ] X

X

X cam

C


4949

Camera Rotation and TraslationCamera Rotation and Traslation

● It is often convenient not to make the camera centre explicit:

X cam=R Xt t=−R C

P=K [R∣t ] Camera MatrixCamera Matrix


5050

End of first lesson

● Any question?

Documents

Robotica - Politecnico di Milanohome.deib.polimi.it/restelli/Lezione1.pdf · 06/10/2006 Corso di Robotica A 06/07 Davide Migliore - [email protected] 2 Today Introduction Camera