Ames Room Projective Geometry - University of Washington3D projective geometry These concepts...

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Projective Geometry CSE 455, Winter 2010 January 27, 2010 Ames Room

Transcript of Ames Room Projective Geometry - University of Washington3D projective geometry These concepts...

Page 1: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Projective Geometry

CSE 455, Winter 2010

January 27, 2010

Ames Room

http://www.illusionworks.com/html/ames_room.html
Page 2: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

Review from last time

Page 3: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

Lenses

A lens focuses parallel rays onto a single focal point

Thin lens equation:

Page 4: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

Depth of field

Changing the aperture size affects depth of field

A smaller aperture increases the range in which the object is approximately in focus

http://en.wikipedia.org/wiki/Image:Jonquil_flowers_at_f32.jpg
http://en.wikipedia.org/wiki/Image:Jonquil_flowers_at_f5.jpg
http://en.wikipedia.org/wiki/Depth_of_field
Page 5: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

Modeling projection

Projection equations

• We get the projection by throwing out the last coordinate:

Distant objects are smaller

Page 6: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

Projection

Perspective Projection

divide by third coordinate

Orthographic Project

f f

f f

Page 7: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

Distortions/Artifacts

Page 8: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

Digital Cameras

Basic process:

photons hit a detector

the detector becomes charged

the charge is read out as brightness

Sensor types:

CCD (charge-coupled device)

CMOS

Page 9: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

Today

More on Cameras

Projective Geometry

Page 10: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

Issues with digital cameras Noise

big difference between consumer vs. SLR-style cameras

low light is where you most notice noise

Compression

creates artifacts except in uncompressed formats (tiff, raw)

Color

color fringing artifacts from Bayer patterns

Blooming

charge overflowing into neighboring pixels

In-camera processing

oversharpening can produce halos

Interlaced vs. progressive scan video

even/odd rows from different exposures

Are more megapixels better?

requires higher quality lens

noise issues

More info online, e.g.,

http://electronics.howstuffworks.com/digital-camera.htm

http://www.dpreview.com/

http://www.dpreview.com/learn/?/key=noise
http://www.dpreview.com/learn/?/key=jpeg
http://www.dpreview.com/learn/?/Glossary/Optical/chromatic_aberration_01.htm
http://electronics.howstuffworks.com/digital-camera4.htm
http://www.dpreview.com/learn/?/key=blooming
http://www.dpreview.com/learn/?/key=sharpening
http://www.axis.com/products/video/camera/progressive_scan.htm
http://electronics.howstuffworks.com/digital-camera.htm
http://electronics.howstuffworks.com/digital-camera.htm
http://electronics.howstuffworks.com/digital-camera.htm
http://electronics.howstuffworks.com/digital-camera.htm
http://www.dpreview.com/
Page 11: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

Large Aperture (Shallow Depth of Field)

Page 12: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

Synthetic Aperture [Vaish et al.]

Page 13: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

Bullet Time (from the Matrix)

Page 14: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

Projective geometry—what’s it good for?

Uses of projective geometry

Drawing

Measurements

Mathematics for projection

Undistorting images

Focus of expansion

Camera pose estimation, match move

Object recognition

Page 15: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

Applications of projective geometry

Vermeer’s Music Lesson

Reconstructions by Criminisi et al.

Page 16: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

3D Modeling from a photograph

Page 17: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

Measurements on planes

Approach: unwarp then measure

What kind of warp is this?

Page 18: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

Image rectification

To unwarp (rectify) an image

• solve for homography H given p and p’

• solve equations of the form: wp’ = Hp

– linear in unknowns: w and coefficients of H

– H is defined up to an arbitrary scale factor

– how many points are necessary to solve for H?

pp’

Page 19: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

How many points?

Page 20: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

Solving for homographies

Page 21: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

Solving for homographies

A h 0

Defines a least squares problem:

2n × 9 9 2n

• Since h is only defined up to scale, solve for unit vector ĥ

• Solution: ĥ = eigenvector of ATA with smallest eigenvalue

• Works with 4 or more points

Page 22: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

(0,0,0)

The projective plane

Why do we need homogeneous coordinates?

represent points at infinity, homographies, perspective projection, multi-view relationships

What is the geometric intuition?

a point in the image is a ray in projective space

(sx,sy,s)

• Each point (x,y) on the plane is represented by a ray (sx,sy,s)

– all points on the ray are equivalent: (x, y, 1) (sx, sy, s)

image plane

(x,y,1)

-y

x-z

Page 23: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

Projective lines

What does a line in the image correspond to in projective space?

• A line is a plane of rays through origin

– all rays (x,y,z) satisfying: ax + by + cz = 0

z

y

x

cba0 :notationvectorin

• A line is also represented as a homogeneous 3-vector l

l p

Page 24: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

l

Point and line duality

A line l is a homogeneous 3-vector

It is to every point (ray) p on the line: l p=0

p1p2

What is the intersection of two lines l1 and l2 ?

• p is to l1 and l2 p = l1 l2Points and lines are dual in projective space

• given any formula, can switch the meanings of points and lines to get another formula

l1

l2

p

What is the line l spanned by rays p1 and p2 ?

• l is to p1 and p2 l = p1 p2

• l is the plane normal

Page 25: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

Ideal points and lines

Ideal point (“point at infinity”)

p (x, y, 0) – parallel to image plane

It has infinite image coordinates

(sx,sy,0)-y

x-z image plane

Ideal line

• l (a, b, 0) – parallel to image plane

(a,b,0)

-y

x

-z image plane

• Corresponds to a line in the image (finite coordinates)

– goes through image origin (principle point)

Page 26: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

Homographies of points and lines

Computed by 3x3 matrix multiplication

To transform a point: p’ = Hp

To transform a line: lp=0 l’p’=0– 0 = lp = lH-1Hp = lH-1p’ l’ = lH-1

– lines are transformed by postmultiplication of H-1

Page 27: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

3D projective geometry

These concepts generalize naturally to 3D

Homogeneous coordinates

Projective 3D points have four coords: P = (X,Y,Z,W)

Duality

A plane N is also represented by a 4-vector

Points and planes are dual in 3D: N P=0

Projective transformations

Represented by 4x4 matrices T: P’ = TP, N’ = N T-1

Page 28: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

3D to 2D: “perspective” projection

Matrix Projection: ΠPp

1************

ZYX

w

wywx

What is not preserved under perspective projection?

What IS preserved?

Page 29: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

Geometric properties of perspective projection

Geometric properties of perspective projection

Points go to points

Lines go to lines

Planes go to whole image or half-plane

Polygons go to polygons

Angles & distances not preserved

Degenerate cases:

line through focal point yields point

plane through focal point yields line

Page 30: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

Vanishing points

Page 31: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

Vanishing points

Properties

Any two parallel lines have the same vanishing point v

The ray from C through v is parallel to the lines

An image may have more than one vanishing point

in fact every pixel is a potential vanishing point

Page 32: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

Vanishing lines

Multiple Vanishing Points

Any set of parallel lines on the plane define a vanishing point

The union of all of vanishing points from lines on the same plane is the vanishing line

For the ground plane, this is called the horizon

Page 33: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

Vanishing lines

Multiple Vanishing Points

Different planes define different vanishing lines

Page 34: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

Computing vanishing points

DPP t 0

1

ZZ

YY

XX

ttDP

tDP

tDP

P

Page 35: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

0/1

/

/

/

1

Z

Y

X

ZZ

YY

XX

ZZ

YY

XX

tD

D

D

t

t

DtP

DtP

DtP

tDP

tDP

tDP

PP

Computing vanishing points

Properties

P is a point at infinity, v is its projection

They depend only on line direction

Parallel lines P0 + tD, P1 + tD intersect at P

DPP t 0

ΠPv

Page 36: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

Computing the horizon

Properties

l is intersection of horizontal plane through C with image plane

Compute l from two sets of parallel lines on ground plane

All points at same height as C project to l

points higher than C project above l

Provides way of comparing height of objects in the scene

l

Page 37: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

Above or Below?

Page 38: Ames Room Projective Geometry - University of Washington3D projective geometry These concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords:

Neel Joshi, CSE 455, Winter 2010

Fun with vanishing points


Computer Vision cmput 499/615 Lecture 8: 3D projective geometry and it’s applications 3D projective geometry and it’s applications Martin Jagersand.

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