GeometryCalibration_Ioana2

7/31/2019 GeometryCalibration_Ioana2

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10/16/200810/16/2008 CS 441, Copyright G.D. Hager

Computer VisionProjective Geometry

and Calibration

Professor Hagerhttp://www.cs.jhu.edu/~hager

Ioana Fleminghttp://lcsr.jhu.edu/IoanaFleming

.


Camera parameters

Summary: points expressed in external frame points are converted to canonical camera coordinates points are projected points are converted to pixel units

=

T

Z

Y

X

W

V

U

parametersextrinsic

ngrepresenti

tionTransforma

modelprojection

ngrepresenti

tionTransforma

parametersintrinsic

ngrepresenti

tionTransforma

point in cam.coords.

point in metricimage coords.

point in pixelcoords.

point inworld coords.


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Intrinsic Parameters + Projection Model

U V W

=

f 0 0 00 f 0 00 0 1 0

X Y Z T

U V W

pix

=

fs u 0 o u 0 fs v o v 0 0 1

U V W

mm

=

u 00 v 00 0 1

U V W

Ap

cameraimage (mm)

U V W

pix

=

s u 0 o u 0 s v o v 0 0 1

U V W

mm


Putting it All Together

Now, using the idea of homogeneous transforms , we can write:

p ' = R T

0 0 0 1

p

R and T both require 3 parameters. These correspond

to the 6 extrinsic parameters needed for camera calibration

Finally, we can write:

U V W

pix

=

u 00 v 00 0 1

X Y Z T

world


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Compute the camera intrinsic (4 or 5) and extrinsicparameters (6) using only observed camera data.

Camera Calibration: Problem Statement

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A Quick Aside: Least Squares

= the pseudoinverse of A Ainv = pinv(A) in Matlab

Least Squares courtesy of Rene Vidal


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A Quick Aside: Least Squares

Least Squares courtesy of Rene Vidal


Camera CalibrationMethods: Photogrammetric Calibration Self Calibration Multi-Plane Calibration

Think of it like of a least squares problem (error minimization):

U V W

pix

=

u 00 v 00 0 1

X Y Z T

world

General strategy: view calibration object identify image points obtain camera matrix by

minimizing error obtain intrinsic parameters from

camera matrix

Most modern systems employthe multi-plane method avoids knowing absolute

coordinates of calibration points


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Photogrammetric Calibration

Calibration is performed through imaging a pattern whosegeometry in 3d is known with high precision.

PRO:Calibration can be performed very efficiently

CON:Expensive set-up apparatus is required;Multiple orthogonal planes.

Approach 1: Direct Parameter Calibration Approach 2: Projection Matrix Estimation


Direct Calibration - Basic Equations

From world to cameracoordinates

(extrinsic parameters)

From world to imagemetric coordinates

(projection model +extrinsic parameters)


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Basic Equations

From image metric toimage pixel coordinates

(intrinsic parameters)

, where s x and s y = horizontal, vertical effectivepixel size

f x = focal length inhorizontal pixels;

= aspect ratio


Basic Equations

one of these for each point


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Basic Equations

N x 8

, where m is defined up to scale.


Properties of SVD

Recall the singular values of a matrix are related to its rank.

Recall that Am = 0 can have a nonzero m as solution only if A issingular.

Finally, note that the matrix V of the SVD is an orthogonal basisfor the domain of A; in particular the zero singular values are thebasis vectors for the null space.

Putting all this together, we see that A must have rank 7 (in thisparticular case) and thus m must be a vector in this subspace.

Clearly, m is defined only up to scale.m = * m


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Basic Equations

We now know R x and R y up to a sign and , and R z = R x x R yHave to use another SVD to orthogonalize this system

(R = U D V; set D to I and multiply).

m defined up to a scale factor ,||t|| -> retrieve ||r|| -> retrieve


Last Details We still need to compute the correct sign.

note that the denominator of the original equations must be positive(points must be in front of the cameras)

Thus, the numerator and the projection must disagree in sign. We know everything in numerator and we know the projection, hence

we can determine the sign.

We still need to compute T z and f x we can formulate this as a least squares problem on those two

values using the first equation.


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Direct Calibration: The Algorithm

Compute the A matrix Compute solution with SVD: get m Compute gamma and alpha from m Compute R (and normalize) Compute f x and and T z If necessary, estimate distortion parameters


Indirect Calibration: The Basic Idea

We know that we can also just write u h = M p h x = (u/w) and y = (v/w), u h = (u,v,1) As before, we can multiply through (after plugging in for u,v, and w)

Once again, we can write A m = 0

Once again, we use an SVD to compute m up to a scale factor.


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Self-Calibration

Calculate the intrinsic parameters solely from pointcorrespondences from multiple images.

Static scene and intrinsics are assumed. No expensive apparatus. Highly flexible but not well-established.


Multi-Plane Calibration

Hybrid method: Photogrammetric and Self-Calibration. Uses a planar pattern imaged multiple times (inexpensive). Used widely in practice and there are many implementations. Based on a group of projective transformations called

homographies.

m be a 2d point [u v 1] and M be a 3d point [x y z 1].

Projection is


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Planar HomographiesFirst Fundamental Theorem of Projective Geometry: There exists a uniquehomography H that performs a change of basis between two projectivespaces of the same dimension.


Estimating A Homography


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Estimating A Homography

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Planar Homographies

First Fundamental Theorem of Projective Geometry: There exists a unique homography that performs a change of basis

between two projective spaces of the same dimension.

Projection Becomes

Notice that the homography H is defined up to scale (s).


We know that

From one homography, how many constraints on the intrinsicparameters can we obtain?

Extrinsics have 6 degrees of freedom. The homography supplies 8 values. Thus, we should be able to obtain 2 constraints per homography.

Use the constraints on the rotation matrix columns



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Computing Intrinsics

Rotation Matrix is orthogonal.

Write the homography in terms of its columns


Computing Intrinsics

Derive the two constraints:


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Closed-Form Solution

Notice B is symmetric, 6 parameters can be written as a vector b. From the two constraints, we have h 1T B h 2 = v 12 T b

Stack up n of these for n images and build a 2n*6 system. Solve with SVD. Extrinsics fall-out of the result easily.


Computing Extrinsics

Orthogonalize using SVD to get rotation


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10/16/2008

In general, lens introduce minor irregularities into images, typically radial distortions:

x = x d(1 + k 1r2 + k2r4)y = y d(1 + k 1r2 + k2r4)r2 = xd2 + yd2

Distortion Correction: make lines straight.

The values k 1 and k 2 are additional parameters that must be estimated in order tohave a model for the camera system.

Lens Distortion

f (r) xd

yd

r

xy

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Lens Distortion


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Calibration Summary

Two groups of parameters: internal (intrinsic) and external (extrinsic)

Many methods direct and indirect, flexible/robust

The form of the equations that arise here and the way they aresolved is common in vision: bilinear forms Ax = 0 Orthogonality constraints in rotations

Most modern systems use the method of multiple planes (matlabdemo) more difficult optimization over a large # of parameters more convenient for the user

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