Calibration - Department of Computer Science

600. 445 Fall 2001Copyright © R. H. Taylor 1999, 2001

Calibration

Calibrate (vt) : 1. to determine the caliber of (as a thermometer tube); 2. to determine, rectify, or mark the gradations of (as a thermometer tube); 3. to standardize (as a measuring instrument) by determining the deviation from a standard so as to ascertain the proper correction factors; 4. ADJUST, TUNE


Calibration

Reality

Measuring device

Model of Reality

+

-Error


Calibration

Reality

Actuation device

Model of Reality

+

-Error


Basic Techniques

• Parameter Estimation

• Mapping the space


Parameter Estimation

• Compare observed system performance to reference standard (“ground truth”)

• Compute parameters of mathematical model that minimizes residual error.

Reality

System

Model of reality

+-

Error

Parameters Calibration process


Pointing device calibration

F R pi i i= ( , )

p t (unknown)p pivot



F R pi i i= ( , )

tpp pivot



i t i pivot+ =R p p p

F R pi i i= ( , )




t

j pivot j

≅ − −

pR I p p

F R pi i i= ( , )



Linear Parameter Estimation

p f q q

p f q q

f q q

f q q qf

nom n

i

jn

q q

fq

q

q

J

= =

= +

≈ +∂∂

L

N

MMMM

O

Q

PPPP

L

NMMM

O

QPPP

≡ +

( ) [ ,...

( )

( ) ( )

( ) ( )

where ] are parametersT

*

1

1

∆

∆

∆

∆


Linear Parameter Estimation

•Given f(q), and a set of observations pk* corresponding to nominal parameter values qk*, solve the least squares problem

J k k kf q q p f q( ) ( )*L

NMMM

O

QPPP

≈ −L

NMMM

O

QPPP

∆


Example: 2 link robot calibration

*kp

1a

2a



p

p

=+

+

L

NMMM

O

QPPP

= +

=+ + + + + +

+ + + + + +

L

NMMM

O

QPPP

a a

a a

a a a a

a a a a

1 1 2 12

1 1 2 12

12 1 2

1 1 1 1 2 2 12 1 2

1 1 1 1 2 2 12 1 2

0

0

sin sin( )

cos cos( )

( ) sin( ) ( ) sin( )

( cos( ) ( ) cos( )

*

θ θ

θ θθ θ θ

θ θ θ θ θ

θ θ θ θ θ

where

∆ ∆ ∆ ∆ ∆

∆ ∆ ∆ ∆ ∆



p f q f

f q f q f q f q py

k k

k k

k k

k k k k k

a aa a

a a

a a

aa a Rot

= = =+

+

L

NMMM

O

QPPP

= +

∂∂

∂∂

∂∂

∂∂

L

N

MMMM

O

Q

PPPP

L

N

MMMM

O

Q

PPPP≈ −

( ) ( , , , )sin sin( )

cos cos( )

( ) ( ) ( ) ( )( ,

, ,

, ,

* ,

1 2 1 2

1 1 2 12

1 1 2 12

12 1 2

1 2 1 2

1

2

1

2

1 1

0θ θθ θ

θ θθ θ θ

θ θ θθ

θ

where

so we solve the least squares problem

∆∆∆∆

k

ka Rot)

( , ),+LNM

OQP

L

N

MMMMM

O

Q

PPPPP2 1y θ



Here

so

Ja a

a a

a aa a

aa

k

k k k k k

k k k k k

k k k k k

k k k k k

f q( )sin sin (cos cos ) cos

cos cos (sin sin ) cos

sin sin (cos cos ) coscos cos (sin sin ) cos

, , , , ,

, , , , ,

, , , , ,

, , , , ,

=+

− + −

L

NMMM

O

QPPP

+− + −

L

N

MMMM

O

Q

PPPP

θ θ θ θ θ

θ θ θ θ θ

θ θ θ θ θθ θ θ θ θ θ

θ

1 12 1 1 12 2 12

1 12 1 1 12 2 12

1 12 1 1 12 2 12

1 12 1 1 12 2 12

1

2

1

0 0 0 0

∆∆∆∆ 2

1 1 2 12

1 1 2 12

L

N

MMMM

O

Q

PPPP≈

− +− +

L

N

MMMM

O

Q

PPPPx a az a ak k k

k k k

*, ,

*, ,

sin sin( )cos cos( )

θ θθ θ


Example: Robodoc Wrist Calibration

• Basic robot had very accurate calibration

• Custom wrist was less accurate

• Crucial goal was to determine position of cutter tip

Cutter

Calibration post


Kinematic Model

p p R z x vtool wrist distal= + + • +( , ) ( )θ θ α4 4∆

v R x R y v vdistal c c= • + +( , ) [ ( , )( )]β θ θ5 5∆ ∆


Linearization

p p R R v vpost wrist c c≈ + + +[ ( )] .....4 5 ∆


Linear Least Squares

• Most commonly used method for parameter estimation

• Many numerical libraries

• See the web site• Here is a quick

review

Microsoft PowerPoint Presentation


Example: Undistorted fluoroscope calibration


Calibration if no distortion (version 1)

{ }0 1

Assume no distortion. For the moment alsoassume that you have N point calibrationfeatures (e.g., small steel balls) at knownpositions , , relative to the detector.Assume further that the points

N−a a

{ }0 1

create imagesat corresponding points , , on the detector. Estimate the position of the x-raysource relative to the detector

N−d ds


Approach

s

kd

ka

Calibration objectobjF

Detector


Projection of a point feature

s

d

a

( )ss

λ

λ

µ

µ

= − +

=

s = (a - d) + d(a - d) • (s - d)=(a - d) • (a - d)

d a s s(a - ) • (d - s)(a - ) • (a - s)


Approachs

d

a( ) ( )

( ) ( )

00

0

z z y y

z z x x

y y x x

skew s− × − =

− • = − ×= × − ×= ×

− − − − = × − −

a d s d 0a d a d d

a d d da d

d a a da d d a s a dd a a d


Approach

s

d

a

0 0 0 0

1 1 1 1

Solve least squares problem

( )

( )

x

y

z

N N N N

skew

skew − − − −

− × ≅

− ×

sa d a dss

a d a d


What if pose of calibration object is imprecisely known?

• This is a hairier problem, but solvable• In fact, it makes a great homework

assignment ….


Mapping the space

• Compare observed system performance to reference standard (“ground truth”)

• Interpolate residual errors

Reality

System

Model of reality

+-

Error

Lookup TableModel Correction


Example: Fluoroscope calibration


Projection of a point feature with distortion

s

u

a

λ

λ

s = (a - d) + d(a - d) • (s - d)=(a - d) • (a - d)

d

ν= ( , )u f d


Experimental Setup

C-Arm Detectorand dewarp plate

Robot Arm

Surgical Cutter

Corkscrew

Phantom


Dewarping Method


Intrinsic Image Calibration• Intrinsic imaging parameters (Schreiner et. al.)• Image Warping (Checkerboard Based Method)

4/16”1/16” 3/16”

4/16”

Top View Side View


Step 0: Acquire Image


Step 1: Find groove points

•Find image points corresponding to the centerline of each vertical and horizontal groove


Step 2: Fit 5’th order Bernstein Polynomial Curves

• Fit least square smooth curve to each vertical and horizontal groove

• 5’th order Bernstein Polynomial

( ) ( )5

50 5

0

5, , ; 1 k k

kk

B a a v a v vk

−

=

= −

∑…


Step 3: Dewarp

• Employ a two pass scan line algorithm to dewarp the image


Advantages

• Fast– ≈ 2 seconds on Pentium II 400

• Robust– works well even with overlaid objects

• Sub-pixel Accuracy– mean error 0.12 mm on the central area

• Does not completely obscure the image– trades off image contrast depth for image area


Two Plane Method

Plane 1 pattern Plane 2 pattern

• E.g., Lavallee• E.g., Helm


Two Plane MethodPlane 1 pattern Plane 2 pattern

Given a point in image coordinates,determine the points the point on grid 1 corresponding to

the point on grid 2 corresponding to

The desired ray in space passes through

and

q

f qf q

ff

=

=

=1

2

1

2

*

*

*

*.


Photos: Sofamor Danek


Interpolation

• Ubiquitous throughout CIS research and applications

• Many techniques and methods

• Here are a few more notes

Microsoft PowerPoint Presentation


Two plane calibration• Again, the essential problem is to

determine the coordinates in the two planes at which the source-to-detector ray passes through the plane.

• Many methods for this. E.g., – Find the four surrounding bead locations

on each plane and use bilinear interpolation

– Fit a general spline model for the distortion on each plane and then directly interpolate

Calibration - Department of Computer Science

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