h005-07-halcon-calibration-theory(基础学习资料)

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Software

Gmb

HCamera

Calibration

TheoryModel

for

a

Pinhole

CameraxwywsycxcysxPwzwCamera

coordinate

system

(CCS)

(x,y,z)World

coordinate

system

(WCS)

(xw,yw,zw)Image

coordinates

(Pixels)

(r,c)Image

plane

coordinate

system

(r,c)R

+

TzxyfcPuvrCamera

Model:

Virtual

Image

PlanesycxcysxsysxcycxPwvfrcuPyxzrcvuPfProjection

from

World

to

Camera:

PoseTransformation

from

the

world

coordinate

system

(WCS)

into

the

cameracoordinate

system

(CCS):

rigid

mapping

with

6

degrees

of

freedom

(3rotations,

3

translations)Outer

orientation

of

the

camera

relative

to

the

WCSPose

and

RotationszxcrbyagxzxzyyabgThe

pin

hole

camera

is

used

as

model:

perspective

projectionmodels

the

real

focal

length

of

the

camera

(i.e.,

it

isnot

an

exactrepresentation

of

the

real

world)is

the

distance

from

the

cameraProjection

for

cameras

with

telecentric

lenses:

ParallelprojectionNo

focal

length

for

telecentric

lensesDistance

from

the

camera

has

no

influence

on

the

position

in

the

image

Nperspective

distortionsProjection

from

Camera

to

Image

PlaneDivision

Model

for

the

Radial

DistortionsDistortions

are

modeled

as

a

transformation

within

the

image

planecoordinate

systemIn

most

cases

of

2D

sensors

the

radial

distortions

are

a

good

approximatiofor

typical

lensesAdvantage

of

this

definition:

It

allows

a

fast

calculation

by

inverting

thfunction:The

simpler

divisionmodel

only

models

radial

distortions

as

they

are

knownas

"barrel"

and

"pin

cushion"

distortion.The

division

modelis

suitable

for

simple

radial

distortionsBarrelPin

cushionIf

the

lens

distortions

are

more

complex

than

just

"barrel"

or

"pin

cushion"

distortions,

the

divisionmodel

is

no

longer

valid.

This

figure

shows

theeffects

of

complexdistortions

that

can

be

modeled

withthe

polynomial

model.The

polynomial

modelis

suitable

for

complex

distortionsThis

figure

shows

the

transformationfrom

the

distorted

image

coordinates

to

corrected

image

coordinates

(the

transformationto

v

is

analog,

fordetails

refer

to

the

reference

of

camera_calibration

or

the

Solution

Guide

IIIc

on

3D

visionchapter

2.2).As

you

can

easily

see,

the

division

model

only

uses

asingle

parameter

(kappa),

while

the

polynomial

model

uses5

parameters,

where

K1,

K2,K3

model

the

radial

distortion,

and

P1,

P2

model

the

decentering

distortion.Comparison

of

the

distortion

modelslDlDivision

mPolynomiafodel:KRadial

distortion

paral

model:metersDecentering

distortion

parameterscorrected

image

pointdistorted

image

pointImage

Coordinate

SystemTransformation

from

the

image

plane

to

image

coordinates:

Principal

pointCenterof

the

radial

distortionsPoint

where

the

line

of

sight

and

the

image

plane

are

perpendicular:

Scaling

factorsStandard

lenses:

Distance

of

CCD

elements

on

the

sensorTelecentric

lenses:

Pixel

size

in

world

coordinates:

Camera

parameters

with

standard

lenses:

Camera

parameters

with

telecentric

lensesCalibration

Process

with

One

Image

IDetermine

the

correspondence

between

the

circles

of

the

model

and

thecircles

in

the

imageLet

be

the

3D-position

of

circle

and

its

2D-image

coordinate,

i.e.forstandardlensesfor

telecentric

lensis

determined

as

the

solution

of

the

following

nonlinearminimizationproblemCalibration

Process

with

one

Image

IIForstandard

lenses

,

and

cannot

be

determined

at

the

same

timebecause,

e.g.,

a

change

of

can

be

compensated

by

andhas

to

be

determined

from

the

specification

of

the

CCD-Sensorcan

be

assumed

fixedbecausethe

frame

grabber

digitizes

the

signal

linsynchronouslycannot

be

used

from

the

specification

with

analog

frame

grabbersbecause

the

frame

grabberdoesnot

digitize

the

signal

pixel-synchronouslyProblem:

with

oneplanarcalibration

platenotall

parameters

of

the

camerparameters

and

the

pose

can

be

determinedMultiple

calibration

images

are

needed

todetermineall

parameters

at

tsame

timeExample

for

the

Indeterminationfo2o2fMulti

Image

Calibration

ProcessCalibration

uses

images,

where

the

calibration

plate

uses

all

degrees

ofreedom

(positions

and

angles)Especially

all

poses

must

not

be

parallelsets

of

posesfor

standardlensesfor

telecentric

lensesis

determined

as

the

solution

of

the

following

nonlinearminimizationproblemCalculation

of

World

Coordinates

Prerequisite

for

the

reconstruction

with

standard

lenses:

All

pointsto

be

in

the

same

plane

(e.g.,

a

conveyor

belt)Only

x-y-positions

in

plane

can

be

determinedIdea:

Intersection

of

the

line

of

sight

with

the

reference

planeThe

referenceplanecan

be

derived

from

the

poseFor

telecentric

lenses:

The

points

do

not

have

to

be

within

one

planeOnly

the

x-y-positions

within

the

CCS

can

be

determinedCalculation

of

World

CoordinatesFor

telecentric

lenses

the

distance

of

the

object

cannot

be

determinedThe

x-

and

y-Coordinates

of

an

object

in

CCS

can

easily

be

determined

Typically

the

objects

are

aligned

parallel

to

the

camera

coordinatethe

CCS

are

treated

as

world

coordinates

The

calculation

of

coordinates

in

the

CCS

is

done

by

inverting

themappings:·

The

calculation

corresponds

to

the

intersection

of

the

line

of

sightwith

theplaneCalculation

of

World

Coordinates

For

standard

cameras

the

same

principle

of

intersecting

a

line

of

sighwiththe

reference

plane

is

appliedIn

this

case

the

lines

of

sight

have

the

following

form

The

plane

is

defined

by

the

parameters

of

the

referencepose

Its

equation

in

the

CCS

is

very

complicated,

but

in

WCS

it

isvery

simple:

To

calculate

the

intersection,

the

line

of

sight

must

be

transformed

ithe

WCS

The

transformation

is

defined

as

the

inversion

of

the

mapping

from

theWCS

into

the

CCS:Calculation

of

World

CoordinatesTransformation

of

the

optical

centerTransformation

the

point

into

the

image

planeThe

Equation

of

the

line

of

sight

in

the

WCSIntersection

of

the

line

of

sight

with

the

plane

in

the

WCSExamples

for

World

CoordinatesExamples

for

World

CoordinatesCalibration

process

with

one

Image

Problem:

Not

all

parameters

can

be

determined

when

using

only

onecalibration

image

In

the

plane

defined

by

the

poseof

the

reference

image

one

can

measureaccurately

becausefor

each

combination

of

parameters

of

the

cameraparameters

and

poses

theplaneis

consistently

defined

But

in

this

case

one

cannotdeterminea

stable

calibrationwith

thestandard7 7

circles

on

a

calibration

plate

becausetoo

few

points

areavailable

Therefore

it

is

important

to

have

much

more

points,

e.g.,

15

15,

todetermine

all

parameters

reliablyMultiple

Image

CalibrationTestApply

the

calibration

witha

varying

number

of

imagesUse

different

combinations

of

calibration

images

Check

the

variation

of

the

calibration

parameters

depending

on

thenumber

andcombinationof

calibration

images

Display

the

relation

between

the

number

of

images

and

theparameters

asgraphsResultsThe

deviation

becomes

smallerwith

increasing

number

of

images

The