3-D Scene

u

u’

Study the mathematical relations between corresponding image points.

“Corresponding” means originated from the same 3D point.

Objective

Two-views geometryOutline

Background: Camera, Projection models Necessary tools: A taste of projective geometry Two view geometry:

Planar scene (homography ). Non-planar scene (epipolar geometry).

3D reconstruction (stereo).

A few words about Cameras

Camera obscura dates from 15th century First photograph on record shown in the book – 1826 The human eye functions very much like a camera

History Camera Obscura

"Reinerus Gemma-Frisius, observed an eclipse of the sun at Louvain on January 24, 1544, and later he used this illustration of the event in his book De Radio Astronomica et Geometrica, 1545. It is thought to be the first published illustration of a camera obscura..." Hammond, John H., The Camera Obscura, A Chronicle

6

The first “photograph”www.hrc.utexas.edu/exhibitions/permanent/wfp/

Joseph Nicéphore Niépce.View from the Window at Le Gras.

A few words about Cameras

Current cameras contain a lens and a recording device (film, CCD, CMOS)

Basic abstraction is the pinhole camera

A few words about LensesIdeal Lenses

Lens acts as a pinhole (for 3D points at the focal depth).

Regular LensesE.g., the cameras in our lab.

To learn more on lens-distortion see Hartley & Zisserman Sec. 7.4 p.189.Not part of this class.

Modeling a Pinhole Camera (or projection)

Single View Geometry

f

X

P Y

Z

x

p y

f

∏

Modeling a Pinhole Camera (or projection)

Perspective Projection

f Xx

Zf Y

yZ

Origin (0,0,0) is the Focal center X,Y (x,y) axis are along the image axis (height / width). Z is depth = distance along the Optical axis f – Focal length

Projection

x y f

X Y Z

f

y

Z

X

Y

f Xx

Zf Y

yZ

P=(X,Y,Z)

f

Projection

x y f

X Y Z

f

y

Z

X

Y

f Xx

Zf Y

yZ

P=(X,Y,Z)

f

Orthographic Projection

•Projection rays are parallel•Image plane is fronto-parallel(orthogonal to rays)

•Focal center at infinity

x X

y Y

Scaled Orthographic ProjectionAlso called “weak perspective”

x sX

y sY

0

fsZ

Pros and Cons of Projection Models Weak perspective has simpler math.

Accurate when object is small and distant. Useful for object recognition.

When accuracy really matters (SFM), we must model the real camera (Pinhole / perspective ): Perspective projection, calibration parameters (later), and

all other issues (radial distortion).

Two-views geometryOutline

Background: Camera, Projection Necessary tools: A taste of projective geometry Two view geometry:

Planar scene (homography ). Non-planar scene (epipolar geometry).

3D reconstruction from two views (Stereo algorithms)

Hartley & Zisserman: Sec. 2 Proj. Geom. of 2D.Sec. 3 Proj. Geom. of 3D.

Reading

Hartley & Zisserman:

Sec. 2 Proj. Geo. of 2D:• 2.1- 2.2.3 point lines in 2D• 2.3 -2.4 transformations • 2.7 line at infinity

Sec. 3 Proj. Geo. of 3D. • 3.1 – 3.2 point planes & lines. • 3.4 transformations

Euclidean Geometry is good for

questions like:

what objects have the same shape (= congruent)

Same shapes are related by rotation and translation

Why not Euclidian Geometry(Motivation)

Where do parallel lines meet?

Parallel lines meet at the horizon (“vanishing line”)

Why Projective Geometry (Motivation)

Coordinates in Euclidean Line R1

0 1 2 3 ∞

Not in space

Coordinates in Projective Line P1

-1 0 1 2 ∞

k(0,1)

k(1,0)

k(2,1)k(1,1)k(-1,1)

Realization: Points on a line P1

“Ideal point”

Take R2 –{0,0} and look at scale equivalence class (rays/lines trough the origin).

Coordinates in Projective Plane P2

k(0,0,1)

k(x,y,0)

k(1,1,1)

k(1,0,1)

k(0,1,1)

“Ideal point”

Take R3 –{0,0,0} and look at scale equivalence class (rays/lines trough the origin).

z

y

x

z

y

x

Projective Line vs. the Real Line

-1 0 1 2 ∞

k(0,1)

k(1,0)

k(2,1)k(1,1)k(-1,1)

“Ideal point”

SymbolRP1

SpaceThe real lineR^2 – {0,0}

Objects (points)pointsEquivalence classes (2D “rays”)

RealizationIntersection with line y=1

Projective Plane vs Euclidian plane

k(0,0,1)

k(x,y,0)

k(1,1,1)

k(1,0,1)

k(0,1,1)“Ideal line”

SymbolR2P2

SpaceThe real planeR3 – {0,0,0}

Objects (points)pointEquivalence classes (3D rays)

RealizationIntersection with plane z=1

2D Projective Geometry: Basics A point:

A line:

we denote a line with a 3-vector

Line coordinates are homogenous

Points and lines are dual: p is on l if

Intersection of two lines/points

2 2( , , ) ( , )T Tx yx y z P

z z

0 ( ) ( ) 0x y

ax by cz a b cz z

0Tl p

1 2 ,l l 1 2p p

( , , )Ta b c

ll

Cross Product

1 2 1 2 1 2

1 2 1 2 1 2

1 2 1 2 1 2

x x y z z y

y y z x x z

z z x y y x

0T Tw u v w u w v

Every entry is a determinant of the two other entries

w Area of parallelogram bounded by u and v

Hartley & Zisserman p. 581

Cross Product in matrix notation [ ]x

0

0

0

xy

xz

yz

x

tt

tt

tt

t1 2 1 2 1 2

1 2 1 2 1 2

1 2 1 2 1 2

x x y z z y

y y z x x z

z z x y y x

0

0

0

x y z z y

y z x z x

z x y y x

t x t z t y t t x

t y t x t z t t y

t z t y t x t t z

Hartley & Zisserman p. 581

ptpt x

Example: Intersection of parallel lines

00

)(

0

)(

)(

2122

21

21 a

b

a

b

cccca

ccb

ll

1 2 1 2 1 2

1 2 1 2 1 2

1 2 1 2 1 2

x x y z z y

y y z x x z

z z x y y x

Q: How many ideal points are there in P2?A: 1 degree of freedom family – the line at infinity

),,( ),,( 2211 cbalcbal

Projective Transformations

u

u’

Transformations of the projective line

1P

Pencil of raysPerspective mapping

A perspective mapping is a projective transformation T:P1 P1

Perceptivity is a special projective mapping. Hartley & Zisserman p. 632Lines connecting corresponding points are “concurrent”

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Perspectivities Projectivities Perspectivities are not a group

ityperspectiv a is 1 Ll ityperspectiv a is 2lL

ityperspectiv anot is 21 ll

L

l1 l2

Projective transformations of the projective line

dycx

byax

y

xG dc

ba

:

1/

//

dc

dbda

dc

ba

11'

''

1 xc

bxax G

Given a 2D linear transformation G:R2 R2 Study the induced transformation on the Equivalents classes.

1'

''

xc

bxax GOn the realization y=1 we get:

Properties:1'

''

xc

bxax T

dc

baT

1. Invertible (T-1 exists) 2. Composable (To G is a projective transformation)3. Closed under composition

• Has 4 parameters • 3 degrees of freedom • Defined by 3 points

TT Every point defines 1 constraint

Ideal points and projective transformations

Projective transformation can map ∞ to a real point

Plane Perspective

2P

cos sin, , det 1

sin cosTR R R I R

Rotation:

Translation:x

y

tt

t

2 2, 1, (2)a b

R a b R SOb a

Euclidean Transformations (Isometries)

q Rp t

Rotation:Translation:

Hierarchy of 2D Transformations

Rigid (Isometry)

Similarity

Affine

Projective

Scale

Hartley & Zisserman p. Sec. 2.4