Some problems
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Some problems. Lens distortion. Uncalibrated structure and motion recovery assumes pinhole cameras Real cameras have real lenses How can we correct distortion , when original calibration is inaccessible?. Even small amounts of lens distortion can upset uncalibrated structure from motion

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Some problems...


Lens distortion

Uncalibrated structure and motion recovery assumes pinhole cameras

Real cameras have real lenses

How can we correct distortion, when original calibration is inaccessible?


Even small amounts of lens distortion can upset uncalibrated structure from motion

A single distortion parameter is enough for mapping and SFX accuracy

Including the parameter kin the multiview relations changes the 8-point algorithm from

You can solve such “Polynomial Eigenvalue Problems”

This is as stable as computation of the Fundamental matrix, so you can use it all the time.


E

ven small amounts of lens distortion can upset uncalibrated structure from motion—


A map-building problem

  • Input movie – relatively low distortion

  • Plan view: red is structure, blue is motion

(a) (b)


Effects of Distortion

  • Input movie – relatively low distortion

  • Recovered plan view, uncorrected distortion

(a) (c)


Distortion of image plane is conflated with focal length

when the camera rotates

[From: Tordoff & Murray, ICPR 2000]

Does distortion do that?


Distortion correction in man-made scenes


Distortion correction in natural scenes

[Farid and Popescu, ICCV 2001]

  • In natural images, distortion introduces correlations in frequency domain

  • Choose distortion parameters to minimize correlations in bispectrum

  • Less effective on man-made scenes....


Distortion correction in multiple images

Multiple views, static scene

  • Use motion and scene rigidity [Zhang, Stein, Sawhney, McLauchlan, ...]

    Advantages:

  • Applies to man-made or natural scenes

    Disadvantages:

  • Iterative solutions|require initial estimates


A

single distortion parameter is accurate enough for map-building and cinema post production—


x:xeroxednoxious experimental artifax

p:perfect pinhole perspective pure

Modelling lens distortion

p

p

x

x

Known

Unknown


Single-parameter models


Single-parameter modelling power

  • Single-parameter model

  • Radial term only

  • Assumes distortion centre is at centre of image

A one-parameter model suffices


A direct solution for k


Look at division model again


A quick matlab session

>> help polyeig

POLYEIG Polynomial eigenvalue problem.

[X,E] = POLYEIG(A0,A1,..,Ap) solves the polynomial eigenvalue problem

of degree p:

(A0 + lambda*A1 + ... + lambda^p*Ap)*x = 0.

The input is [etc etc...]

>>


Algorithm


T

his is as stable as computation of the fundamental matrix, so you can use it all the time—


Stable – small errorbars

Biased – not centred on true value

Performance: Synthetic data

0

-0.1

Computed l

-0.2

-0.3

-0.4

0

0.2

0.4

0.6

0.8

1

Noise s (pixels)


Best-fit line

Analogy: Linear ellipse fitting

True

Fitted: 10 trials

Data


Performance: Synthetic data


Performance: Real sequences


250 pairs

Low distortion

Linear estimate used to initialize nonlinear

Number of inliers changes by [-25..49]

50

40

30

20

10

0

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3


Conclusions


In: magnifying glass moving over background

Out: same magnifying glass, new background

Environment matting


Learn

light-transport properties of complex optical elements

Previously

Ray tracing geometric models

Calibrated acquisition

Here

Acquisition in situ

Environment matting: why?


Purely 2D-2D

Optical element performs weighted sum of (image of) background at each pixel

suffices for many interesting objects

separate receptive field for each output pixel

Environment matte is collection of all receptive fields—yes, it’s huge.

Image formation model


Image formation model


Input:

Mosaic:

Step 1: Computing background

Clean plate:

Point tracks:


Step 2: Computing w...

Input:


Computing w(x,y,u,v) at a single (x,y)


Assume wi independent


Composite over new background


Input: Two images

Moving camera

Planar background

- Need priors

A more subtle example


Window example


Works well for non-translucent elements

need to develop for diffuse

Combination assumes independence

ok for large movements: “an edge crosses the pixel”

Need to develop for general backgrounds

Discussion


A Clustering Problem

  • Watch a movie, recover the cast list

    • Run face detector on every frame

    • Cluster faces

  • Problems

    • Face detector unreliable

    • Large lighting changes

    • Changes in expression

    • Clustering is difficult


A sample sequence


Detected faces


Face positions


Lighting correction


Raw distance

Clustering: pairwise distances


Transform-invariant distance

Clustering: pairwise distances


Clusters: “tangent distance”


Clusters: Bayesian tangent distance


Extend to feature selection, texton clustering etc

Remove face detector

Conclusions


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