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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 structure from motion

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

A map-building problem structure from motion

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

(a) (b)

Effects of Distortion structure from motion

- Input movie – relatively low distortion
- Recovered plan view, uncorrected distortion

(a) (c)

Distortion of image plane is structure from motionconflated with focal length

when the camera rotates

[From: Tordoff & Murray, ICPR 2000]

Does distortion do that?Distortion correction in man-made scenes structure from motion

Distortion correction in natural scenes structure from motion

[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 structure from motion

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 structure from motion

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

x structure from motion:xeroxed noxious experimental artifax

p:perfect pinhole perspective pure

Modelling lens distortionp

p

x

x

Known

Unknown

Single-parameter models structure from motion

Single-parameter modelling power structure from motion

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

A one-parameter model suffices

A direct solution for structure from motionk

Look at division model again structure from motion

A quick matlab session structure from motion

>> 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 structure from motion

T structure from motion

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

Stable structure from motion – small errorbars

Biased – not centred on true value

Performance: Synthetic data0

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Computed l

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Noise s (pixels)

Performance: Synthetic data structure from motion

Performance: Real sequences structure from motion

250 pairs structure from motion

Low distortion

Linear estimate used to initialize nonlinear

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

50

40

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Conclusions structure from motion

In structure from motion: magnifying glass moving over background

Out: same magnifying glass, new background

Environment mattingLearn structure from motion

light-transport properties of complex optical elements

Previously

Ray tracing geometric models

Calibrated acquisition

Here

Acquisition in situ

Environment matting: why?Purely 2D-2D structure from motion

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 modelImage formation model structure from motion

Step 2: Computing structure from motionw...

Input:

Computing structure from motionw(x,y,u,v) at a single (x,y)

Assume structure from motionwi independent

Composite over new background structure from motion

Input: Two images structure from motion

Moving camera

Planar background

- Need priors

A more subtle exampleWindow example structure from motion

Works well for non-translucent elements structure from motion

need to develop for diffuse

Combination assumes independence

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

Need to develop for general backgrounds

DiscussionA Clustering Problem structure from motion

- 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 structure from motion

Detected faces structure from motion

Face positions structure from motion

Lighting correction structure from motion

Raw distance structure from motion

Clustering: pairwise distancesTransform-invariant distance structure from motion

Clustering: pairwise distancesClusters: “tangent distance” structure from motion

Clusters: Bayesian tangent distance structure from motion

Extend to feature selection, texton clustering etc structure from motion

Remove face detector

Conclusions
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