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May 20, 2023 115 likes •496 views

Camera Calibration (Compute Camera Matrix P) Shao-Yi Chien Department of Electrical Engineering National Taiwan University Fall 2019 1 Outline Camera calibration [Slides credit: Marc Pollefeys] 2 Resectioning X P ? x i

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Camera Calibration (Compute Camera Matrix P)

簡韶逸 Shao-Yi Chien Department of Electrical Engineering National Taiwan University Fall 2019

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Outline

Camera calibration

[Slides credit: Marc Pollefeys]

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i i

x X 

? P

Resectioning

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i i

PX x 

 

i i PX

x



Ap 

Basic Equations

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Ap 

minimal solution Over-determined solution  5½ correspondences needed (say 6) P has 11 dof, 2 independent eq./points

n  6 points Ap

minimize subject to constraint

1 p  1 p ˆ 3 

p ˆ

 P

Basic Equations

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More complicate than 2D case (i) Camera and points on a twisted cubic (ii) Points lie on plane or single line passing through projection center

Degenerate Configurations

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Less obvious (i) Simple, as before (ii) Anisotropic scaling

3 2

Data Normalization

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Extend DLT to lines

l PT  

i i 1 TPX

(back-project line)

i i 2 TPX

(2 independent eq.)

Line Correspondences

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Geometric Error

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Gold Standard Algorithm

Objective Given n≥6 2D to 2D point correspondences {Xi↔xi’}, determine the Maximum Likelyhood Estimation of P Algorithm (i) Linear solution: (a) Normalization: (b) DLT: (ii) Minimization of geometric error: using the linear estimate as a starting point minimize the geometric error: (iii) Denormalization:

i i

UX X ~ 

i i

Tx x ~ 

U P ~ T P



~ ~ ~

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(i) Canny edge detection (ii) Straight line fitting to the detected edges (iii) Intersecting the lines to obtain the images corners typically precision <1/10 (HZ rule of thumb: 5n constraints for n unknowns

Calibration Example

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Errors in the image and in the world

i i

X P x  

Errors in the World

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 

) x ˆ , x ( ˆ



i i i id

w

 

i i i i

y x w PX 1 , ˆ , ˆ ˆ 

P) depth(X; p ˆ ˆ

 

w ) X ˆ , X ( ~ ) x ˆ , x ( ˆ

i i i i i

fd d w then 1 p ˆ if therefore,

3 

Geometric Interpretation of Algebraic error

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note that in this case algebraic error = geometric error

Estimation of Affine Camera

Last row = (0, 0, 0, 1)

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Objective Given n≥4 2D to 2D point correspondences {Xi↔xi’}, determine the Maximum Likelyhood Estimation of P (remember P3T=(0,0,0,1)) Algorithm (i) Normalization: (ii) For each correspondence (iii) solution is (iv) Denormalization:

i i

UX X ~ 

i i

Tx x ~ 

U P ~ T P



b p A

8 8

 b A p

8 8 



Gold Standard Algorithm

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Minimize geometric error impose constraint through parametrization Image only 9  2n, otherwise 3n+9  5n

Find best fit that satisfies

skew s is zero
pixels are square
principal point is known
complete camera matrix K is known

Minimize algebraic error assume map from param q  P=K[R|-RC], i.e. p=g(q) minimize ||Ag(q)||

Restricted Camera Estimation

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One only has to work with 12x12 matrix, not 2nx12

p A ~ Ap A p Ap

T T

 

Reduced Measurement Matrix

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Initialization

Use general DLT
Clamp values to desired values, e.g. s=0, x= y

Note: can sometimes cause big jump in error Alternative initialization

Use general DLT
Impose soft constraints
gradually increase weights

Restricted Camera Estimation

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Calibrated camera, position and orientation unkown  Pose estimation 6 dof  3 points minimal (4 solutions in general)

Exterior Orientation

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ML residual error Example: n=197, =0.365, =0.37

Covariance Estimation

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short and long focal length

Radial Distortion

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෤ 𝑦, ෤ 𝑧: non-distorted projection 𝑦𝑒, 𝑧𝑒: distorted projection

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Choice of the distortion function and center

Computing the parameters of the distortion function (i) Minimize with additional unknowns (ii) Straighten lines (iii) …

Correction of Distortion

: interior parameters

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Correction of Distortion

After radial correction

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Another Method of Calibration

Notation
Homography between the model plane and its

image

Ref: Zhengyou Zhang, “Flexible camera calibration by viewing a plane from unknown orientations,” ICCV1999.

≡

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Another Method of Calibration

Constraints on the intrinsic parameters

r1 and r2 are orthonormal 

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Another Method of Calibration

Close-form solution
Let

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Another Method of Calibration

Close-form solution
From the two constraints on the intrinsic parameters
V is a 2n x 6 matrix, if 𝑜 ≥ 3, we will have in general a