Single view metrology
Download
1 / 34

Single-view metrology - PowerPoint PPT Presentation


  • 155 Views
  • Uploaded on

Single-view metrology. Magritte, Personal Values , 1952. Many slides from S. Seitz, D. Hoiem. Review: Camera projection matrix. camera coordinate system. world coordinate system. i ntrinsic parameters. extrinsic parameters. Camera parameters. Intrinsic parameters

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Single-view metrology' - tacey


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
Single view metrology
Single-view metrology

Magritte, Personal Values, 1952

Many slides from S. Seitz, D. Hoiem


Review camera projection matrix
Review: Camera projection matrix

camera coordinate system

world coordinate system

intrinsic parameters

extrinsic parameters


Camera parameters
Camera parameters

  • Intrinsic parameters

    • Principal point coordinates

    • Focal length

    • Pixel magnification factors

    • Skew (non-rectangular pixels)

    • Radial distortion


Camera parameters1
Camera parameters

  • Intrinsic parameters

    • Principal point coordinates

    • Focal length

    • Pixel magnification factors

    • Skew (non-rectangular pixels)

    • Radial distortion

  • Extrinsic parameters

    • Rotation and translation relative to world coordinate system



Camera calibration1

Xi

xi

Camera calibration

  • Given n points with known 3D coordinates Xi and known image projections xi, estimate the camera parameters


Camera calibration linear method
Camera calibration: Linear method

Two linearly independent equations


Camera calibration linear method1
Camera calibration: Linear method

  • P has 11 degrees of freedom

  • One 2D/3D correspondence gives us two linearly independent equations

  • Homogeneous least squares: find p minimizing ||Ap||2

    • Solution given by eigenvector of ATA with smallest eigenvalue

    • 6 correspondences needed for a minimal solution


Camera calibration2

Vertical vanishing

point

(at infinity)

Vanishing

line

Vanishing

point

Vanishing

point

Camera calibration

  • What if world coordinates of reference 3D points are not known?

  • We can use scene features such as vanishing points

Slide from Efros, Photo from Criminisi


Recall vanishing points

vanishing point v

Recall: Vanishing points

image plane

camera

center

line in the scene

  • All lines having the same direction share the same vanishing point


Computing vanishing points
Computing vanishing points

  • X∞ is a point at infinity, vis its projection: v = PX∞

  • The vanishing point depends only on line direction

  • All lines having direction D intersect at X∞

v

X0

Xt


Calibration from vanishing points
Calibration from vanishing points

  • Let us align the world coordinate system with three orthogonal vanishing directions in the scene:

v2

v1

.

v3


Calibration from vanishing points1
Calibration from vanishing points

  • p1= P(1,0,0,0)T – the vanishing point in the x direction

  • Similarly, p2 and p3 are the vanishing points in the y and z directions

  • p4= P(0,0,0,1)T– projection of the origin of the world coordinate system

  • Problem: we can only know the four columns up to independent scale factors, additional constraints needed to solve for them


Calibration from vanishing points2
Calibration from vanishing points

  • Let us align the world coordinate system with three orthogonal vanishing directions in the scene:

  • Each pair of vanishing points gives us a constraint on the focal length and principal point


Calibration from vanishing points3
Calibration from vanishing points

Can solve for focal length, principal point

Cannot recover focal length, principal point is the third vanishing point


Rotation from vanishing points
Rotation from vanishing points

  • Thus,

  • Get λ by using the constraint ||ri||2=1.


Calibration from vanishing points summary
Calibration from vanishing points: Summary

  • Solve for K (focal length, principal point) using three orthogonal vanishing points

  • Get rotation directly from vanishing points once calibration matrix is known

  • Advantages

    • No need for calibration chart, 2D-3D correspondences

    • Could be completely automatic

  • Disadvantages

    • Only applies to certain kinds of scenes

    • Inaccuracies in computation of vanishing points

    • Problems due to infinite vanishing points


Making measurements from a single image
Making measurements from a single image

http://en.wikipedia.org/wiki/Ames_room


Recall measuring height
Recall: Measuring height

5.3

5

Camera height

4

3.3

3

2.8

2

1


Measuring height without a ruler
Measuring height without a ruler

Z

O

ground plane

  • Compute Z from image measurements

    • Need more than vanishing points to do this


The cross ratio
The cross-ratio

  • A projective invariant: quantity that does not change under projective transformations (including perspective projection)


The cross ratio1
The cross-ratio

  • A projective invariant: quantity that does not change under projective transformations (including perspective projection)

    • What are invariants for other types of transformations (similarity, affine)?

  • The cross-ratio of four points:

P4

P3

P2

P1


Measuring height

scene cross ratio

C

image cross ratio

Measuring height

T (top of object)

t

r

R (reference point)

H

b

R

B (bottom of object)

vZ

ground plane



Single view metrology

t

v

H

image cross ratio

vz

r

vanishing line (horizon)

t0

vx

vy

H

R

b0

b


2d lines in homogeneous coordinates
2D lines in homogeneous coordinates

  • Line equation: ax + by + c = 0

  • Line passing through two points:

  • Intersection of two lines:

    • What is the intersection of two parallel lines?


Single view metrology

t

v

H

image cross ratio

vz

r

vanishing line (horizon)

t0

vx

vy

H

R

b0

b


Measurements on planes

1

4

2

3

4

3

2

1

Measurements on planes

Approach: unwarp then measure

What kind of warp is this?


Image rectification
Image rectification

p′

p

  • To unwarp (rectify) an image

  • solve for homographyH given p and p′

  • how many points are necessary to solve for H?


Image rectification example
Image rectification: example

PierodellaFrancesca, Flagellation, ca. 1455


Application 3d modeling from a single image
Application: 3D modeling from a single image

J. Vermeer,Music Lesson, 1662

A. Criminisi, M. Kemp, and A. Zisserman, Bringing Pictorial Space to Life: computer techniques for the analysis of paintings, Proc. Computers and the History of Art, 2002

http://research.microsoft.com/en-us/um/people/antcrim/ACriminisi_3D_Museum.wmv


Application 3d modeling from a single image1
Application: 3D modeling from a single image

D. Hoiem, A.A. Efros, and M. Hebert, "Automatic Photo Pop-up", SIGGRAPH 2005.

http://www.cs.illinois.edu/homes/dhoiem/projects/popup/popup_movie_450_250.mp4


Application image editing
Application: Image editing

Inserting synthetic objects into images: http://vimeo.com/28962540

K. Karsch and V. Hedau and D. Forsyth and D. Hoiem, “Rendering Synthetic Objects into Legacy Photographs,” SIGGRAPH Asia2011


Application object recognition
Application: Object recognition

D. Hoiem, A.A. Efros, and M. Hebert, "Putting Objects in Perspective", CVPR 2006