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Some problems...

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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?

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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.

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

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A map-building problem Input movie – relatively low distortion Plan view: red is structure, blue is motion (a) (b)

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Effects of Distortion Input movie – relatively low distortion Recovered plan view, uncorrected distortion (a) (c)

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Does distortion do that? Distortion of image plane is conflated with focal length when the camera rotates [From: Tordoff & Murray, ICPR 2000]

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Distortion correction in man-made scenes

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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]

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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

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

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Modelling lens distortion x: x ero x ed no x ious e x perimental artifa x p: p erfect p inhole p erspective p ure x p p x KnownUnknown

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Single-parameter models

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Single-parameter modelling power Single-parameter model Radial term only Assumes distortion centre is at centre of image A one-parameter model suffices

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A direct solution for

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Look at division model again

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>> 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

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Algorithm

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

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Performance: Synthetic data 00.20.40.60.81 -0.4 -0.3 -0.2 -0.1 0 Noise (pixels) Computed Stable – small errorbars Biased – not centred on true value

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Analogy: Linear ellipse fitting True Data Fitted: 10 trials Best-fit line

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Performance: Synthetic data

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Performance: Real sequences

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

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Conclusions

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Environment matting In: magnifying glass moving over background Out: same magnifying glass, new background

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Environment matting: why? Learn –light-transport properties of complex optical elements Previously –Ray tracing geometric models –Calibrated acquisition Here –Acquisition in situ

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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.

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Image formation model

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Step 1: Computing background Input: Mosaic: Clean plate: Point tracks:

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Step 2: Computing w... Input:

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

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Assume w i independent

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Composite over new background

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A more subtle example Input: Two images Moving camera Planar background - Need priors

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Window example

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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

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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

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A sample sequence

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Detected faces

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Face positions

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Lighting correction

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Clustering: pairwise distances Raw distance

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Clustering: pairwise distances Transform-invariant distance

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Clusters: “tangent distance”

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Clusters: Bayesian tangent distance

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Conclusions Extend to feature selection, texton clustering etc Remove face detector

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