3D Vision

3D Computer Visionand Video Computing 3D Vision

Topic 2 of Part IICalibration

CSc I6716Fall 2010

Zhigang Zhu, City College of New York [email protected]

3D Computer Visionand Video Computing Lecture Outline

n Calibration: Find the intrinsic and extrinsic parametersl Problem and assumptionsl Direct parameter estimation approachl Projection matrix approach

n Direct Parameter Estimation Approachl Basic equations (from Lecture 5)l Homogeneous Systeml Estimating the Image center using vanishing pointsl SVD (Singular Value Decomposition) l Focal length, Aspect ratio, and extrinsic parametersl Discussion: Why not do all the parameters together?

n Projection Matrix Approach (…after-class reading)l Estimating the projection matrix Ml Computing the camera parameters from Ml Discussion

n Comparison and Summaryl Any difference?

3D Computer Visionand Video Computing Problem and Assumptions

n Given one or more images of a calibration pattern, n Estimate

l The intrinsic parametersl The extrinsic parameters, orl BOTH

n Issues: Accuracy of Calibrationl How to design and measure the calibration pattern

n Distribution of the control points to assure stability of solution – not coplanar

n Construction tolerance one or two order of magnitude smaller than the desired accuracy of calibration

n e.g. 0.01 mm tolerance versus 0.1mm desired accuracyl How to extract the image correspondences

n Corner detection?n Line fitting?

l Algorithms for camera calibration given both 3D-2D pairs

n Alternative approach: 3D from un-calibrated camera

3D Computer Visionand Video Computing Camera Model

n Coordinate Systemsl Frame coordinates (xim, yim) pixelsl Image coordinates (x,y) in mml Camera coordinates (X,Y,Z) l World coordinates (Xw,Yw,Zw)

n Camera Parametersl Intrinsic Parameters (of the camera and the frame grabber): link the

frame coordinates of an image point with its corresponding camera coordinates

l Extrinsic parameters: define the location and orientation of the camera coordinate system with respect to the world coordinate system

Zw

Xw

Yw

Y

X

Zx

yO

Pw

P

p

xim

yim

(xim,yim)

Pose / Camera

Object / World

Image frame

Frame Grabber

3D Computer Visionand Video ComputingLinear Version of Perspective Projection

n World to Cameral Camera: P = (X,Y,Z)T

l World: Pw = (Xw,Yw,Zw)T

l Transform: R, T n Camera to Image

l Camera: P = (X,Y,Z)T

l Image: p = (x,y)T

l Not linear equationsn Image to Frame

l Neglecting distortionl Frame (xim, yim)T

n World to Framel (Xw,Yw,Zw)T -> (xim, yim)T

l Effective focal lengthsn fx = f/sx, fy=f/sy

zT

yT

xT

zwww

ywww

xwww

T

TT

TZrYrXr

TZrYrXrTZrYrXr

w

w

w

w

PR

PRPR

TRPP

3

2

1

333231

232221

131211

yyim

xximsoyysoxx)()(

) ,(),(ZYf

ZXfyx

zwww

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zwww

xwwwxxim

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foy

TZrYrXrTZrYrXrfox

333231

232221

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131211

3D Computer Visionand Video Computing Direct Parameter Method

n Extrinsic Parametersl R, 3x3 rotation matrix

n Three angles a,b,gl T, 3-D translation vector

n Intrinsic Parametersl fx, fy :effective focal length in pixel

n a = fx/fy = sy/sx, and fxl (ox, oy): known Image center -> (x,y) knownl k1, radial distortion coefficient: neglect it in the basic algorithm

n Same Denominator in the two Equationsl Known : (Xw,Yw,Zw) and its (x,y)l Unknown: rpq, Tx, Ty, fx, fy

zwww

ywwwyyim

zwww

xwwwxxim

TZrYrXrTZrYrXr

foyy

TZrYrXrTZrYrXr

foxx

333231

232221

333231

131211

'

'

)(')(' 131211232221 xwwwxywwwy TZrYrXrfyTZrYrXrfx

'/)('/)( 131211232221 xTZrYrXrfyTZrYrXrf xwwwxywwwy

3D Computer Visionand Video Computing Linear Equations

n Linear Equation of 8 unknowns v = (v1,…,v8)

l Aspect ratio: a = fx/fyl Point pairs , {(Xi, Yi,, Zi) <-> (xi, yi) } drop the ‘ and subscript “w”

)(')(' 131211232221 xwwwywww TZrYrXryTZrYrXrx a

0)()()()( 131211232221 xiiiiiiiyiiiiiii TyrZyrYyrXyTxrZxrYxrXx aaaa

087654321 vyvZyvYyvXyvxvZxvYxvXx iiiiiiiiiiiiii

),,,,,,,(),,,,,,,(

131211232221

87654321

xy TrrrTrrrvvvvvvvv

aaaa

3D Computer Visionand Video Computing Homogeneous System

n Homogeneous System of N Linear Equations l Given N corresponding pairs {(Xi, Yi,, Zi) <-> (xi, yi) }, i=1,2,…Nl 8 unknowns v = (v1,…,v8)T, 7 independent parameters


0Av

NNNNNNNNNNNNNN yZyYyXyxZxYxXx

yZyYyXyxZxYxXxyZyYyXyxZxYxXx

22222222222222

11111111111111

A

n The system has a nontrivial solution (up to a scale) l IF N >= 7 and N points are not coplanar => Rank (A) = 7l Can be determined from the SVD of A

3D Computer Visionand Video Computing

n Homework #3 online, due October 25 (Monday) before class


n Homogeneous System of N Linear Equations l Given N corresponding pairs {(Xi, Yi,, Zi) <-> (xi, yi) }, i=1,2,…Nl 8 unknowns v = (v1,…,v8)T, 7 independent parameters


0Av

NNNNNNNNNNNNNN yZyYyXyxZxYxXx

yZyYyXyxZxYxXxyZyYyXyxZxYxXx

22222222222222

11111111111111

A

n The system has a nontrivial solution (up to a scale) l IF N >= 7 and N points are not coplanar => Rank (A) = 7l Can be determined from the SVD of A

3D Computer Visionand Video Computing SVD: definition

n Singular Value Decomposition:l Any mxn matrix can be written as the product of three

matricesTUDVA

n Singular values si are fully determined by An D is diagonal: dij =0 if ij; dii = si (i=1,2,…,n)

n s1 s2 … sN 0

n Both U and V are not uniquen Columns of each are mutual orthogonal vectors

nnnn

n

n

n

mmmm

m

m

mnmm

n

n

vvv

vvvvvv

uuu

uuuuuu

aaa

aaaaaa

21

22212

121112

1

21

22221

11211

21

22221

11211

000

000

s

ss

V1U1

Appendix A.6

3D Computer Visionand Video Computing SVD: properties

n 1. Singularity and Condition Numberl nxn A is nonsingular IFF all singular values are nonzerol Condition number : degree of singularity of A

n A is ill-conditioned if 1/C is comparable to the arithmetic precision of your machine; almost singular

n 2. Rank of a square matrix Al Rank (A) = number of nonzero singular values

n 3. Inverse of a square Matrixl If A is nonsingularl In general, the pseudo-inverse of A

n 4. Eigenvalues and Eigenvectors (questions)l Eigenvalues of both ATA and AAT are si

2 (si > 0)l The columns of U are the eigenvectors of AAT (mxm)l The columns of V are the eigenvectors of ATA (nxn)

TUDVA

nC ss /1

TUVDA 11 TUVDA 1

0

iiT

iuuAA 2s

iiT

ivAvA 2s

3D Computer Visionand Video Computing SVD: Application 1

n Least Squarel Solve a system of m equations for n unknowns x(m >= n)l A is a mxn matrix of the coefficients l b (0) is the m-D vector of the data l Solution:

l How to solve: compute the pseudo-inverse of ATA by SVDn (ATA)+ is more likely to coincide with (ATA)-1 given m > nn Always a good idea to look at the condition number of ATA

bAx

bAAxA TT bAAAx TT )(

nxn matrix Pseudo-inverse


n Homogeneous Systeml m equations for n unknowns x(m >= n-1)l Rank (A) = n-1 (by looking at the SVD of A)l A non-trivial solution (up to a arbitrary scale) by SVD:l Simply proportional to the eigenvector corresponding to the

only zero eigenvalue of ATA (nxn matrix)n Note:

l All the other eigenvalues are positive because Rank (A)=n-1

l For a proof, see Textbook p. 324-325l In practice, the eigenvector (i.e. vn) corresponding to

the minimum eigenvalue of ATA, i.e. sn2

0Ax

iiT

ivAvA 2s


n Problem Statementsl Numerical estimate of a matrix A whose entries are not

independentl Errors introduced by noise alter the estimate to Â

n Enforcing Constraints by SVDl Take orthogonal matrix A as an examplel Find the closest matrix to Â, which satisfies the constraints

exactlyn SVD of Â n Observation: D = I (all the singular values are 1) if A is

orthogonaln Solution: changing the singular values to those expected

TUDVA ˆ

TUIVA


n Homogeneous System of N Linear Equations l Given N corresponding pairs {(Xi, Yi,, Zi) <-> (xi, yi) },

i=1,2,…Nl 8 unknowns v = (v1,…,v8)T, 7 independent parameters

n The system has a nontrivial solution (up to a scale) l IF N >= 7 and N points are not coplanar => Rank (A) = 7l Can be determined from the SVD of Al Rows of VT: eigenvectors {ei} of ATAl Solution: the 8th row e8 corresponding to the only zero

singular value l8=0

n Practical Considerationl The errors in localizing image and world points may make

the rank of A to be maximum (8)l In this case select the eigenvector corresponding to the

smallest eigenvalue.

0Av

TUDVA

8ev c

3D Computer Visionand Video ComputingScale Factor and Aspect Ratio

n Equations for scale factor g and aspect ratio a

n Knowledge: R is an orthogonal matrix

n Second row (i=j=2):

n First row (i=j=1)

),,,,,,,( 131211232221 xy TrrrTrrr aaaagv

T

T

T

ijrrrrrrrrr

r

3

2

1

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33RRR

R

ji if 0ji if 1

jTi RR

1222232221

rrr 23

22

21|| vvv g

1222131211

rrr 27

26

25|| vvv ga

a

||g

v1 v2 v3 v4 v5 v6 v7 v8

3D Computer Visionand Video Computing Rotation R and Translation T

n Equations for first 2 rows of R and T given a and |g|

n First 2 rows of R and T can be found up to a common sign s (+ or -)

n The third row of the rotation matrix by vector product

n Remaining Questions :l How to find the sign s?l Is R orthogonal?l How to find Tz and fx, fy?

),,,,,,,( || 131211232221 xy TrrrTrrrs aaaagv

yxTT sTsTss ,,, 21 RR

TTTTT ss 21213 RRRRR

T

T

T

ijrrrrrrrrr

r

3

2

1

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33RRR

R

3D Computer Visionand Video Computing Find the sign s

n Facts:l fx > 0l Zc >0l x knownl Xw,Yw,Zw known

n SolutionÞ Check the sign of Xc

Þ Should be opposite

to x zwww

ywwwyy

zwww

xwwwxx

TZrYrXrTZrYrXr

fZcYcfy

TZrYrXrTZrYrXrf

ZcXcfx

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Zw

Xw

Yw

Y

X

Zx

yO

PwP

p

xim

yim

(xim,yim)

3D Computer Visionand Video Computing Rotation R : Orthogonality

n Question: l First 2 rows of R are calculated

without using the mutual orthogonal constraint

n Solution: l Use SVD of estimate R

TTTTT ss 21213 RRRRR

T

T

T

ijrrrrrrrrr

r

3

2

1

333231

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33RRR

R

?ˆˆ IRR T

TUDVR ˆ TUIVR

Replace the diagonal matrix D with the 3x3 identity matrix

3D Computer Visionand Video Computing Find Tz, Fx and Fy

n Solutionl Solve the system of N linear

equations with two unknownn Tx, fx

l Least Square method

l SVD method to find inverse

zwww

xwwwx TZrYrXr

TZrYrXrfx

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x

zfT

)()( 333231131211 wwwxxwwwz ZrYrXrxfTZrYrXrxT

ai2ai1 bi

bAAA TT

x

z

f

T 1)(ˆ

ˆ

3D Computer Visionand Video ComputingDirect parameter Calibration Summary

n Algorithm (p130-131)1. Measure N 3D coordinates (Xi, Yi,Zi)2. Locate their corresponding image

points (xi,yi) - Edge, Corner, Hough3. Build matrix A of a homogeneous

system Av = 0 4. Compute SVD of A , solution v5. Determine aspect ratio a and scale |g|6. Recover the first two rows of R and the

first two components of T up to a sign7. Determine sign s of g by checking the

projection equation8. Compute the 3rd row of R by vector

product, and enforce orthogonality constraint by SVD

9. Solve Tz and fx using Least Square and SVD, then fy = fx / a

Yw

Xw

Zw


n Homework #3 online, due October 25 before class

3D Computer Visionand Video Computing Discussions

n Questions

l Can we select an arbitrary image center for solving other parameters?

l How to find the image center (ox,oy)?

l How about to include the radial distortion?

l Why not solve all the parameters once ?

n How many unknown with ox, oy? --- 20 ??? – projection matrix method

zwww

ywwwyyim

zwww

xwwwxxim

TZrYrXrTZrYrXr

foyy

TZrYrXrTZrYrXrfoxx

333231

232221

333231

131211

3D Computer Visionand Video Computing Estimating the Image Center

n Vanishing points:l Due to perspective, all parallel lines in 3D space appear to meet in

a point on the image - the vanishing point, which is the common intersection of all the image lines




VP1


n Vanishing points:l Due to perspective, all parallel lines in 3D space appear to meet in a point

on the image - the vanishing point, which is the common intersection of all the image lines

n Important property:l Vector OV (from the center of projection to the vanishing point)

is parallel to the parallel lines

O

VP1

Y

X

Z




VP1

VP2


n Orthocenter Theorem:l Input: three mutually

orthogonal sets of parallel lines in an image

l T: a triangle on the image plane defined by the three vanishing points

l Image center = orthocenter of triangle T

l Orthocenter of a triangle is the common intersection of the three altitudes VP1

VP2

VP3






l Orthocenter of a triangle is the common intersection of the three altitudes VP1

VP2

VP3






l Orthocenter of a triangle is the common intersection of the three altitudes

n Orthocenter Theorem:l WHY?

VP1

VP2

VP3

h3

h1

h1

(ox,oy)


n Assumptions:l Known aspect ratiol Without lens distortions

n Questions:l Can we solve both

aspect ratio and the image center?

l How about with lens distortions?

VP1

VP2

VP3

h3

h1

h1

(ox,oy) ?

3D Computer Visionand Video ComputingDirect parameter Calibration Summary

n Algorithm (p130-131)0. Estimate image center (and aspect ratio)1. Measure N 3D coordinates (Xi, Yi,Zi)2. Locate their corresponding image (xi,yi) -

Edge, Corner, Hough3. Build matrix A of a homogeneous system

Av = 0 4. Compute SVD of A , solution v5. Determine aspect ratio a and scale |g|6. Recover the first two rows of R and the first

two components of T up to a sign7. Determine sign s of g by checking the

projection equation8. Compute the 3rd row of R by vector product,

and enforce orthogonality constraint by SVD

9. Solve Tz and fx using Least Square and SVD , then fy = fx / a

Yw

Xw

Zw

3D Computer Visionand Video ComputingRemaining Issues and Possible Solution

n Original assumptions:l Without lens distortionsl Known aspect ratio when estimating image centerl Known image center when estimating others including aspect ratio

n New Assumptionsl Without lens distortion l Aspect ratio is approximately 1, or a = fx/fy = 4:3 ; image center about

(M/2, N/2) given a MxN image

n Solution (?)1. Using a = 1 to find image center (ox, oy)2. Using the estimated center to find others including a3. Refine image center using new a ; if change still significant, go to step

2; otherwise stop

l Projection Matrix Approach



3D Computer Visionand Video Computing Linear Matrix Equation of

perspective projectionn Projective Space

l Add fourth coordinate n Pw = (Xw,Yw,Zw, 1)T

l Define (u,v,w)T such thatn u/w =xim, v/w =yim

n 3x4 Matrix Eextl Only extrinsic parametersl World to camera

n 3x3 Matrix Eintl Only intrinsic parametersl Camera to frame

n Simple Matrix Product! Projective Matrix M= MintMext

l (Xw,Yw,Zw)T -> (xim, yim)T

l Linear Transform from projective space to projective planel M defined up to a scale factor – 11 independent entries

1ww

w

ZYX

wvu

extintMM

zT

yT

xT

z

y

x

ext

T

TT

TTT

rrrrrrrrr

3

2

1

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R

RR

M

1000

0

int yy

xxofof

M

wvwu

yx

im

im//

3D Computer Visionand Video Computing Projection Matrix M

n World – Frame Transforml Drop “im” and “w” l N pairs (xi,yi) <-> (Xi,Yi,Zi)l Linear equations of m

n 3x4 Projection Matrix Ml Both intrinsic (4) and extrinsic (6) – 10 parameters

1ii

i

i

i

i

ZYX

wvu

M

34333231

24232221

34333231

14131211

mZmYmXmmZmYmXm

wuy

mZmYmXmmZmYmXm

wux

iii

iii

i

ii

iii

iii

i

ii

z

zyyy

zxxx

yyyyyy

xxxxxx

TToTfToTf

rrrrorfrorfrorfrorfrorfrorf

333231

332332223121

331332123111M

0Am

3D Computer Visionand Video ComputingStep 1: Estimation of projection matrix

n World – Frame Transforml Drop “im” and “w” l N pairs (xi,yi) <-> (Xi,Yi,Zi)

n Linear equations of m l 2N equations, 11 independent variablesl N >=6 , SVD => m up to a unknown scale

34333231

24232221

34333231

14131211

mZmYmXmmZmYmXm

wuy

mZmYmXmmZmYmXm

wux

iii

iii

i

ii

iii

iii

i

ii

0Am

1111111111

11111111111000000001

yYyYyXyZYXxZxYxXxZYX

A

Tmmmmmmmmmmmm 343332312423222114131211m

3D Computer Visionand Video ComputingStep 2: Computing camera parameters

n 3x4 Projection Matrix Ml Both intrinsic and extrinsic

n From M^ to parameters (p134-135)l Find scale |g| by using unit vector R3

T

l Determine Tz and sign of g from m34 (i.e. q43)l Obtain R3

T

l Find (Ox, Oy) by dot products of Rows q1. q3, q2.q3, using the orthogonal constraints of R

l Determine fx and fy from q1 and q2 (Eq. 6.19) Wrong???)l All the rests: R1

T, R2T, Tx, Ty

l Enforce orthognoality on R?

z

zyyy

zxxx

yyyyyy

xxxxxx

TToTfToTf

rrrrorfrorfrorfrorfrorfrorf

333231

332332223121

331332123111M

MM gˆ

43

42

41

3

2

1ˆ

qqq

qqq

M

3D Computer Visionand Video Computing Comparisons

n Direct parameter method and Projection Matrix method

n Properties in Common:l Linear system first, Parameter decomposition secondl Results should be exactly the same

n Differencesl Number of variables in homogeneous systems

n Matrix method: All parameters at once, 2N Equations of 12 variables

n Direct method in three steps: N Equations of 8 variables, N equations of 2 Variables, Image Center – maybe more stable

l Assumptionsn Matrix method: simpler, and more general; sometime projection

matrix is sufficient so no need for parameter decompostionn Direct method: Assume known image center in the first two steps,

and known aspect ratio in estimating image center

3D Computer Visionand Video Computing Guidelines for Calibration

n Pick up a well-known technique or a fewn Design and construct calibration patterns (with known 3D)n Make sure what parameters you want to find for your cameran Run algorithms on ideal simulated data

l You can either use the data of the real calibration pattern or using computer generated data

l Define a virtual camera with known intrinsic and extrinsic parametersl Generate 2D points from the 3D data using the virtual cameral Run algorithms on the 2D-3D data set

n Add noises in the simulated data to test the robustness n Run algorithms on the real data (images of calibration target)n If successful, you are all set n Otherwise:

l Check how you select the distribution of control pointsl Check the accuracy in 3D and 2D localizationl Check the robustness of your algorithms againl Develop your own algorithms NEW METHODS?

3D Computer Visionand Video Computing Next

n 3D reconstruction using two cameras

Stereo Vision

& project discussions


3D Vision

Documents

d images

d models

d points

d projections

d scene

d visiontopic

d transformation

camera calibration