Uncalibrated Epipolar - Calibration

Jana KoseckaCS223b

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Uncalibrated Camera

pixelcoordinates

calibratedcoordinates

Linear transformation

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Overview

Calibration with a rig

Uncalibrated epipolar geometry

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Uncalibrated Camera

• Pixel coordinates

• Projection matrix

Uncalibrated camera

• Image plane coordinates • Camera extrinsic parameters

• Perspective projection

Calibrated camera

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Taxonomy on Uncalibrated Reconstruction

is known, back to calibrated case

is unknown Calibration with complete scene knowledge (a rig) – estimate

Uncalibrated reconstruction despite the lack of knowledge of

Autocalibration (recover from uncalibrated images)

Use partial knowledge Parallel lines, vanishing points, planar motion, constant intrinsic

Ambiguities, stratification (multiple views)

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Calibration with a Rig

Use the fact that both 3-D and 2-D coordinates of feature points on a pre-fabricated object (e.g., a cube) are known.

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Calibration with a Rig

• Eliminate unknown scales

• Factor the into and using QR decomposition

• Solve for translation

• Recover projection matrix

• Given 3-D coordinates on known object

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More details• Direct calibration by recovering and decomposing the projection matrix

2 constraints per point

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More details

• Factor the into and using QR decomposition

• Solve for translation

• Recover projection matrix

• Collect the constraints from all N points into matrix M (2N x 12)

• Solution eigenvector associated with the smallest eigenvalue

• Unstack the solution and decompose into rotation and translation

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Calibration with a planar pattern

To eliminate unknown depth, multiply both sides by

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Calibration with a planar pattern

Because are orthogonal and unit norm vectors of rotation matrixWe get the following two constraints

• Unknowns in K (S)

Skew is often close 0 -> 4 unknowns

• We want to recover S

• S is symmetric matrix (6 unknowns) in general we need at least 3 views• To recover S (2 constraints per view) - S can be recovered linearly • Get K by Cholesky decomposition of directly from entries of S

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Alternative camera models/projections

Orthographic projection

Scaled orthographic projection

Affine camera model

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Barrel and Pincushion Distortion

telewideangle

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Models of Radial Distortion

)1( 42

21 rkrk

y

x

y

x

d

d ++⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛

distance from center

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Tangential Distortion

cheapglue

cheap CMOS chipcheap lens image

cheap camera

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Barrel distortion

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Distorted Camera Calibration

Set k1k2, solve for undistorted case

Find optimal k1k2 via nonlinear least squares

Iterate

Tends to generate good calibrations

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Calibration Software: Matlab

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Calibration Software: OpenCV

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Calibration by nonlinear Least Squares

Least Mean Square

Gradient descent:

J

},,,,]},[{]},[{]},[{]},[{{ yxyx oossfkTkkkX ψϕφ=

0X

0X

J

∂∂

)(1.0 11 −− ∂∂

⋅−← kkk XXJ

XX

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The Calibration Problem Quiz

Given Calibration pattern with N corners

K views of this calibration pattern

How large would N and K have to be?

Can we recover all intrinsic parameters?

N 1 3 1 3 4 4 6

K 1 1 3 3 3 4 6

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Constraints

N points K images 2NK constraints

4 intrinsics (distortion: +2) 6K extrinsics need 2NK ≥ 6K+4

(N-3)K ≥ 2

Hint: may not be co-linear

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The Calibration Problem Quiz

N 1 3 1 3 4 4 6

K 1 1 3 3 3 4 6

No No No No Yes Yes Yes

need (N-3)K ≥ 2

Hint: may not be co-linear

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Problem with Least Squares

Many parameters (=slow) Many local minima! (=slower)