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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—

- Input movie – relatively low distortion
- Plan view: red is structure, blue is motion

(a) (b)

- 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]

[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....

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

p

p

x

x

Known

Unknown

- Single-parameter model
- Radial term only
- Assumes distortion centre is at centre of image

A one-parameter model suffices

A direct solution for k

>> 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...]

>>

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

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

True

Fitted: 10 trials

Data

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

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In: magnifying glass moving over background

Out: same magnifying glass, new background

Learn

light-transport properties of complex optical elements

Previously

Ray tracing geometric models

Calibrated acquisition

Here

Acquisition in situ

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.

Input:

Mosaic:

Clean plate:

Point tracks:

Input:

Input: Two images

Moving camera

Planar background

- Need priors

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

- 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

Raw distance

Transform-invariant distance

Extend to feature selection, texton clustering etc

Remove face detector