sensitivity analysis of genetic

8/12/2019 Sensitivity analysis of genetic

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Sensitivity Analysis

of Genetic Algorithm BasedCamera Calibration

Project done by

A. Abinesh (05G01)

M. Arul Ram (05G03)

B.E. Final Year

Guided by

Mr. C. Muruganantham,

Associate Professor,

Mechanical Engineering Department


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ABSTRACT

In this project, a robust camera calibration based on thelinear camera model is performed. In this approachGenetic algorithm is used as a tool for performingcamera calibration. Also the distorted image plane error

is calculated as a performance measure, it providesaccuracy for camera calibration.

Sensitivity analysis of the parameters is performed inwhich a known percentage of error is given to theparameters, to find which parameter has greater

influence on the camera calibration. The results indicate that the principal point is highly

sensitive in machine vision applications and all otherparameters are relatively less sensitive.


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CAMERA CALIBRATION

Camera Calibration is the process ofdetermining the internal geometricand optical

characteristics (Intrinsic parameters) and/or the3-D position and orientationof camera relative toa chosen world coordinate system (Extrinsic

parameters)

INTRODUCTION


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EXTRINSIC PARAMETERS

TRANSLATION ALONG XAXIS

TRANSLATION ALONG Y

AXIS TRANSLATION ALONG Z

AXIS

ROTATION ALONG X AXIS

ROTATION ALONG Y AXIS ROTATION ALONG Z AXIS

INTRINSIC PARAMETERS

FOCAL LEGTH OF THE LENS

SCALING FACTOR ALONG X AXIS

SCALING FACTOR ALONG Y AXIS

CENTRE OF IMAGE PLANE(U0,V0)


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Coordinate Systems

World Coordinate System: Its a known

reference coordinate system with respect to

which we calibrate the camera.

Camera Coordinate System:Its a coordinatesystem with its origin at the optical center of the

camera.


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Geometry of image formation


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MATHEMATICAL FORMULATION Projection from world to pixel reference frame using all

camera parameters without considering distortion is givenby:

u 1/sx 0 uo f 0 0 r1 r2 r3 tX X

v = 0 1/sy vo 0 f 0 r4 r5 r6 tY Y1 0 0 1 0 0 1 r7 r8 r9 tZ Z

0 0 0 1 1

Rotation &

TranslationProjection

Scaling &

TranslationWorld

pointImage point


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Need for calibration

The basic need is to find an equation linking thecamera reference frame with the imagereference frame (Link I), and another equationlinking the world reference frame with the

camera reference frame (Link II). Solving this system, results in linking the link

between world reference frame and imagereference frame.

Finding Link I and Link II are equivalent tofinding the camera's characteristics, also knownin computer vision as the camera's extrinsic andintrinsic parameters.


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DEFINITION OF THE GENETIC

ALGORITHM

The genetic algorithmis a probabilistic search

algorithm that iteratively transforms a set of

mathematical objects each with anassociated fitness value, into a new

population of offspring objects using the

Darwinian principle of natural selection and

using operations that are patterned afternaturally occurring genetic operations, such

as crossover and mutation.


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Selection is done on the basis of relative fitness

and it probabilistically culls solutions from the

population that have relatively low fitness.

Crossover is a structured yet stochastic operatorthat allows information exchange between

candidate solutions.

Mutation insures against the permanent loss of

genetic material during the selection process.


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SPECIFICATIONS

CCD Camera

Scanning System Progressive ScanPixel Clock 20.25 MHz

CCD Sensor Monochrome 2/3 IT CCD

Sensing Area 8.7 mm X 6.9 mm

Picture Elements 1300 X 1030

Cell Size 0.0067 mm X 0.0067 mm

Resolution 1040 X 1030 TV Lines


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Result For Calibration


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Results for various population

PARAMETER POP=25 POP=50 POP=75 POP=100 POP=125 POP=1000

z-axis -1.112 -1.112 -1.112 -1.112 -1.112 -0.854

x-axsis -0.339 -0.339 -0.339 -0.339 -0.339 -2.336

y-axis 1.997 2.518 -2.518 -2.518 -2.518 -2.405

focal length 24.942 24.354 24.354 24.354 24.354 24.354

scale factor, x 1.19 1.19 1.19 1.19 1.19 0.376

scale factor, y 0.678 0.678 0.678 0.678 0.678 1.352

translation, x 0.9 -0.9 -0.836 -0.836 -0.836 -0.836

translation, y -0.846 0.652 -0.634 -0.634 -0.634 -0.634

translation, z 50.88 50.88 50.88 50.88 50.88 36.48

uo 649.86 649.86 641.4 641.4 641.4 645.62

vo 518.063 518.063 508.547 508.547 508.547 519.844


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DISTORTED IMAGE PLANE ERROR:

One of the first questions about cameramodel is how accurately it captures the

cameras imaging behavior. This information is necessary both for

measuring progress during model

calibration and estimating the performanceor accuracy of any application the model isused in.


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Distorted image plane error gives how

accurately camera model captures the cameras

imaging behavior.

Given the measured coordinates of a point in theobject space (xw, yw, zw ) and the measured

position of the points image in the frame

grabber (Xf, Yf) we can define an error metric for

the model anywhere along the models chain ofcoordinate transformations.


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One obvious error metric is the difference

between the position of a points image we

measure and the position the camera

predicts. Thus it can be defined thedistorted image plane error (DIPE) as

DIPE= 2 2( ) ( )f f f fX X Y Y


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POPULATION SIZE DIPE

25 43.667

50 43.546

75 31.949

100 31.949

125 31.949

1000 39.903


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SENSITIVITY ANALYSIS:

In LP, the parameter of the model can

change within certain limits without

causing the optimum solution to change.

This is referred to as sensitivity analysis

In LP models, the parameters are usually

not exact. With sensitivity analysis, we can

ascertain the impact of the uncertainty onthe quality of the optimum solution.


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Sensitivity analysis is used to determine

how sensitive a model is to changes in

the value of the parameters of the model

and to changes in the structure of themodel. By showing how the model

behavior responds to changes in

parameter values, sensitivity analysis is auseful tool in model building as well as in

model evaluation.


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For example, for an estimated unit profit of

a product, if sensitivity analysis reveals

that the optimum remains the same for a

10% change in the unit profit, we canconclude that the solution is more robust

than in the case where the difference

range is only 1%.


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Sensitivity analysis helps to build

confidence in the model by studying the

uncertainties that are often associated with

parameters in models

Sensitivity analysis allows one to

determine what level of accuracy is

necessary for a parameter to make themodel sufficiently useful and valid.


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SENSITIVITY ANALYSIS

31.4

31.5

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31.7

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31.932

32.1

32.2

32.3

-10 -8 -6 -4 -2 0 2 4 6 8 10

error

DIPE

sx

tx

tytz

za

xa

ya

focal


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SENSITIVITY ANALYSIS

0

10

20

30

40

50

60

70

80

90

100

-10 -8 -6 -4 -2 0 2 4 6 8 10

ERROR

DIPE cx

cy


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CONCLUSION

Using genetic algorithm JAI-CV M1 ProgressiveScan camera is calibrated and the cameraparameters are estimated.

At the completion of sensitivity analysis it is foundthat except the principal point (u0, v0), all the

parameters are less sensitive as their variation isminimum of 1% of the DIPE.

Thus it is concluded that while calibrating acamera or in the application of camera calibrationutmost care should be taken in finding theprincipal point,


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REFERENCE

Fabio Remondino, Niclas Boerlin, Photogrammetric Calibration OfImage Sequences Acquired With A Rotating Camera,International

Archives Of Photogrammetry Remote Sensing And Spatial InformationScience, Vol. 34, Part 5, 19-22 Feb 2004

Motilal Agrawal and Larry S. Davis ,Camera calibration usingspheres: A semi-definite programming approach,Ninth IEEEInternational Conference on computer vision, Vol. 2, 2003, pp-782

Qiang Ji &Yongmian Zhang, Camera calibration with genetic

algorithms, IEEE Transactions on Systems, Man and Cybernetics,Part A, Vol. 31 Issue: 2, (March 2001), pp. 120130.

Roger Y. Tsai, An efficient and accurate camera calibrationtechnique for3D machine vision, Proceedings of IEEE Conference onComputer Vision and Pattern Recognition, Miami Beach,February,1986, pp.364-374.

Savii.G.G,Camera Calibration Using Compound Genetic-Simplex,Journal of Optoelectronics and Advanced Materials Vol.6,No. 4,December 2004,p. 12551261.

Zhang, Z.,A flexible new technique for camera calibration,IEEETransactions on Pattern Analysis and Machine Intelligence, Vol. 22,no. 11, (Nov. 2000), pp. 13301334.


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Xiaoqiao Q. Meng and Z Y. Hu, A new easy camera calibrationtechnique based on circular points, (2003). Pattern Recognition.Vol. 36, No. 5, pp. 1155-1164.

Kalyanmoy DebOptimization for Engineering Design Algorithmsand Examples,. Prentice-Hall of India Private Limited, ISBN-81-203-0943-X. pp 290-319.

Reg G. Wilson Modeling and Calibration of Automated ZoomLenses, proceeding of the SPIE #2350: videometricsIII, October,1944, PP.170-186.

Hamdy A. Taha,Operations Research An Introduction,Prentice-Hall of India Private Limited. ISBN-978-81-203-3034-6.pp 123-130.

Hati S. & Sengupta S., Robust Camera Parameter Estimationusing Genetic Algorithm, Proceedings of the IEEE InternationalConference on Systems, Man and Cybernetics, vol.4, (12-15 oct,1999), pp.943-947.

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