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Objective

3-D Scene

u’

u

Study the mathematical relations between corresponding image points.

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

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

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 Lenses

E.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)

Modeling a Pinhole Camera (or projection)

Perspective Projection

- 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

Orthographic Projection

- Projection rays are parallel
- Image plane is fronto-parallel
- (orthogonal to rays)
- Focal center at infinity

Scaled Orthographic Projection

Also called “weak perspective”

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

Why not Euclidian Geometry(Motivation)

- Euclidean Geometry is good for
questions like:

what objects have the same shape (= congruent)

Same shapes are related by rotation and translation

Why Projective Geometry (Motivation)

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

Where do parallel lines meet?

Coordinates in Projective Line P1

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

Realization: Points on a line P1

“Ideal point”

k(1,1)

k(0,1)

k(-1,1)

k(2,1)

-1 0 1 2 ∞

k(1,0)

Coordinates in Projective Plane P2

Take R3 –{0,0,0} and look at scale equivalence class

(rays/lines trough the origin).

k(0,1,1)

k(1,1,1)

“Ideal point”

k(0,0,1)

k(1,0,1)

k(x,y,0)

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

Area of parallelogram bounded by u and v

Cross ProductEvery entry is a determinant of the two other entries

Hartley & Zisserman p. 581

Cross Product in matrix notation [ ]x

Hartley & Zisserman p. 581

Example: Intersection of parallel lines

Q: How many ideal points are there in P2?

A: 1 degree of freedom family – the line at infinity

Pencil of rays

Transformations of the projective lineA perspective mapping is a projective transformation T:P1 P1

Perceptivity is a special projective mapping. Hartley & Zisserman p. 632

Lines connecting corresponding points are “concurrent”

Projective transformations of the projective line

Given a 2D linear transformation G:R2 R2

Study the induced transformation on the Equivalents classes.

On the realization y=1 we get:

Properties:

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

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

Every point defines 1 constraint

Ideal points and projective transformations

Projective transformation can map ∞ to a real point

Hierarchy of 2D Transformations

Projective

Affine

Similarity

Rigid (Isometry)

Translation:

Rotation:

Scale

Hartley & Zisserman p. Sec. 2.4

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