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?

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

2. A single distortion parameter is enough for mapping and SFX accuracy

3. Including the parameter in the multiview relations changes the 8-point algorithm from

4. You can solve such “Polynomial Eigenvalue Problems”

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

Even small amounts of lens

distortion can upset uncalibrated structure from motion—

A map-building problem

(a) Input movie – relatively low distortion(b) Plan view: red is structure, blue is motion

(a) (b)

Effects of Distortion

(a) Input movie – relatively low distortion(b) Recovered plan view, uncorrected distortion

(a) (c)

Does distortion do that?

Distortion of image plane is conflated with focal lengthwhen the camera rotates

[From: Tordoff & Murray, ICPR 2000]

Distortion correction in man-made scenes

Distortion correction in natural scenes

In natural images, distortion introduces correlations in frequency domain

Choose distortion parameters to minimize correlations in bispectrum

Less effective on man-made scenes....

[Farid and Popescu, ICCV 2001]

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 scenesDisadvantages:• Iterative solutions|require initial estimates

A single distortion parameter

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

Modelling lens distortion

x: xeroxednoxious

experimental artifax

p: perfect pinhole

perspective pure

xp p

x

Known Unknown

Single-parameter models

Single-parameter modelling power

Single-parameter model

Radial term onlyAssumes distortion

centre is at centre of image

A one-parameter model suffices

A direct solution for

Look at division model again

>> 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...]

>>

A quick matlab session

Algorithm

T his is as stable as

computation of the fundamental matrix, so you can use it all the time—

Performance: Synthetic data

0 0.2 0.4 0.6 0.8 1-0.4

-0.3

-0.2

-0.1

0

Noise (pixels)

Com

pu

ted

• Stable – small errorbars• Biased – not centred on true value

Analogy: Linear ellipse fitting

True

Data

Fitted: 10 trials

Best-fit line

Performance: Synthetic data

Performance: Real sequences

-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.30

10

20

30

40

50

• 250 pairs• Low distortion• Linear estimate used to initialize nonlinear• Number of inliers changes by [-25..49]

Conclusions

Environment matting

In: magnifying glass moving over background

Out: same magnifying glass, new background

Environment matting: why?

• Learn– light-transport

properties of complex optical elements

• Previously– Ray tracing

geometric models– Calibrated

acquisition

• Here– Acquisition in situ

Image formation model

• 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

Step 1: Computing backgroundInput:

Mosaic:

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

A more subtle exampleInput: Two images

Moving cameraPlanar background

- Need priors

Window example

Discussion

• 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

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

Clustering: pairwise distances

Raw distance

Clustering: pairwise distances

Transform-invariant distance

Clusters: “tangent distance”

Clusters: Bayesian tangent distance

Conclusions

• Extend to feature selection, texton clustering etc

• Remove face detector