Objective
This presentation is the property of its rightful owner.
Sponsored Links
1 / 39

Objective PowerPoint PPT Presentation


  • 86 Views
  • Uploaded on
  • Presentation posted in: General

Objective. 3-D Scene. u ’. u. Study the mathematical relations between corresponding image points. “Corresponding” means originated from the same 3D point. Two-views geometry Outline. Background: Camera, Projection models Necessary tools: A taste of projective geometry

Download Presentation

Objective

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Objective

Objective

3-D Scene

u’

u

Study the mathematical relations between corresponding image points.

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


Two views geometry outline

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

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

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

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

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


A few words about cameras1

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 lenses ideal lenses

A few words about LensesIdeal Lenses

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


Regular lenses

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)


Single view geometry

Single View Geometry

f


Modeling a pinhole camera or projection1

Modeling a Pinhole Camera (or projection)


Perspective 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


Projection

Projection

P=(X,Y,Z)

y

f

Y

X

f

Z


Projection1

Projection

P=(X,Y,Z)

y

f

Y

X

f

Z


Orthographic projection

Orthographic Projection

  • Projection rays are parallel

  • Image plane is fronto-parallel

  • (orthogonal to rays)

  • Focal center at infinity


Scaled orthographic projection

Scaled Orthographic Projection

Also called “weak perspective”


Pros and cons of projection models

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 geometry outline1

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

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


Objective

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


Objective

Why Projective Geometry (Motivation)

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

Where do parallel lines meet?


Coordinates in euclidean line r 1

Coordinates in Euclidean Line R1

Not in space

0 1 2 3 ∞


Coordinates in projective line p 1

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 p 2

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)


Projective line vs the real line

Projective Line vs. the Real Line

“Ideal point”

k(1,1)

k(0,1)

k(-1,1)

k(2,1)

-1 0 1 2 ∞

k(1,0)


Projective plane vs euclidian plane

k(0,1,1)

k(1,1,1)

“Ideal line”

k(0,0,1)

k(1,0,1)

k(x,y,0)

Projective Plane vs Euclidian plane


2d projective geometry basics

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


Cross product

Area of parallelogram bounded by u and v

Cross Product

Every entry is a determinant of the two other entries

Hartley & Zisserman p. 581


Cross product in matrix notation x

Cross Product in matrix notation [ ]x

Hartley & Zisserman p. 581


Example intersection of parallel lines

Example: Intersection of parallel lines

Q: How many ideal points are there in P2?

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


Projective transformations

u’

u

Projective Transformations


Transformations of the projective line

Perspective mapping

Pencil of rays

Transformations of the projective line

A 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”


Perspectivities projectivities perspectivities are not a group

Perspectivities Projectivities Perspectivities are not a group

L

l2

l1


Projective transformations of the projective line

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

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

Ideal points and projective transformations

Projective transformation can map ∞ to a real point


Plane perspective

Plane Perspective


Euclidean transformations isometries

Euclidean Transformations (Isometries)

Rotation:

Translation:


Hierarchy of 2d transformations

Hierarchy of 2D Transformations

Projective

Affine

Similarity

Rigid (Isometry)

Translation:

Rotation:

Scale

Hartley & Zisserman p. Sec. 2.4


  • Login