Announcements

Project 2 extension: Friday, Feb 8

Project 2 help session: today at 5:30 in Sieg 327

Projective geometry

Readings

Mundy, J.L. and Zisserman, A., Geometric Invariance in Computer Vision, Appendix: Projective Geometry for Machine Vision, MIT Press, Cambridge, MA, 1992,
(read 23.1 - 23.5, 23.10)

available online: http://www.cs.cmu.edu/~ph/869/papers/zisser-mundy.pdf

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

Measurements on planes

Image rectification

Solving for homographies

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

Projective lines

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

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

Ideal points and lines

Ideal point (“point at infinity”)

p @ (x, y, 0) – parallel to image plane

It has infinite image coordinates

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

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

3D to 2D: “perspective” projection

Matrix Projection:

Vanishing points (2D)

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

Vanishing lines

Multiple Vanishing Points

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

The union of all of these vanishing points is the horizon line

also called vanishing line

Vanishing lines

Multiple Vanishing Points

Different planes define different vanishing lines

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_¥

Computing vanishing lines

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

Slide 22

Fun with vanishing points

Perspective cues

Comparing heights

Measuring height

Computing vanishing points (from lines)

Intersect p₁q₁ with p₂q₂

Measuring height without a ruler

The cross ratio

A Projective Invariant

Something that does not change under projective transformations (including perspective projection)

Measuring height

Computing (X,Y,Z) coordinates

Okay, we know how to compute height (Z coords)

how can we compute X, Y?

3D Modeling from a photograph

Camera calibration

Goal: estimate the camera parameters

Version 1: solve for projection matrix

Vanishing points and projection matrix

Calibration using a reference object

Place a known object in the scene

identify correspondence between image and scene

compute mapping from scene to image

Chromaglyphs

Estimating the projection matrix

Place a known object in the scene

identify correspondence between image and scene

compute mapping from scene to image

Direct linear calibration

Direct linear calibration

Advantage:

Very simple to formulate and solve

Disadvantages:

Doesn’t tell you the camera parameters

Doesn’t model radial distortion

Hard to impose constraints (e.g., known focal length)

Doesn’t minimize the right error function

Alternative: multi-plane calibration

Some Related Techniques

Image-Based Modeling and Photo Editing

Mok et al., SIGGRAPH 2001

http://graphics.csail.mit.edu/ibedit/

Single View Modeling of Free-Form Scenes

Zhang et al., CVPR 2001

http://grail.cs.washington.edu/projects/svm/

Tour Into The Picture

Anjyo et al., SIGGRAPH 1997

http://koigakubo.hitachi.co.jp/little/DL_TipE.html


		Project 2 extension: Friday, Feb 8
		Project 2 help session: today at 5:30 in Sieg 327


Readings
	Mundy, J.L. and Zisserman, A., Geometric Invariance in Computer Vision, Appendix: Projective Geometry for Machine Vision, MIT Press, Cambridge, MA, 1992, (read 23.1 - 23.5, 23.10)
		available online: http://www.cs.cmu.edu/~ph/869/papers/zisser-mundy.pdf


	Uses of projective geometry
		Drawing
		Measurements
		Mathematics for projection
		Undistorting images
		Focus of expansion
		Camera pose estimation, match move
		Object recognition


	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


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


		A line l is a homogeneous 3-vector
		It is ^ to every point (ray) p on the line: l p=0


	Ideal point (“point at infinity”)
		p @ (x, y, 0) – parallel to image plane
		It has infinite image coordinates


	Computed by 3x3 matrix multiplication
		To transform a point: p’ = Hp
		To transform a line: lp=0 ® l’p’=0


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


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


Multiple Vanishing Points
	Any set of parallel lines on the plane define a vanishing point
	The union of all of these vanishing points is the horizon line
		also called vanishing line


	Multiple Vanishing Points
		Different planes define different vanishing lines


	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_¥


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


	A Projective Invariant
		Something that does not change under projective transformations (including perspective projection)


	Okay, we know how to compute height (Z coords)
		how can we compute X, Y?


	Goal: estimate the camera parameters
		Version 1: solve for projection matrix


	Place a known object in the scene
		identify correspondence between image and scene
		compute mapping from scene to image


	Advantage:
		Very simple to formulate and solve

	Disadvantages:
		Doesn’t tell you the camera parameters
		Doesn’t model radial distortion
		Hard to impose constraints (e.g., known focal length)
		Doesn’t minimize the right error function


	Image-Based Modeling and Photo Editing
		Mok et al., SIGGRAPH 2001
		http://graphics.csail.mit.edu/ibedit/

	Single View Modeling of Free-Form Scenes
		Zhang et al., CVPR 2001
		http://grail.cs.washington.edu/projects/svm/

	Tour Into The Picture
		Anjyo et al., SIGGRAPH 1997
		http://koigakubo.hitachi.co.jp/little/DL_TipE.html