calibração de câmeras cap. 6 trucco & verri. câmera segue um modelo simples plano de...

Calibração de Câmeras

Cap. 6 Trucco & Verri

Câmera segue um modelo simples

plano de projeção

centro de projeção

Projeção cônica

caixa

filme

objetopinhole

raios de luz

imagem

Câmera

“pinhole”

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parâmetros intrínsecos

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Calibração de câmera

• Problema: obter os parâmetros extrínsecos (R, T) e intrínsecos (K) da transformação projetiva de câmera.

• Dados: n pares de pontos correspondentes (Pi, pi) na cena e na imagem.

Calibração de câmeras

• Calibração estimação de parâmetros otimização

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pontos da cena

pontos da imagem

projeção (função não linear)

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Compute v by SVD decomposition of A=UDVT (The solution vector is the column of V corresponding to null (or smallest) singular value) in D.

Estimativa dos parâmetros da câmera

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Passo 2 do Tsai

Computing Image Center yx oo ,

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escolha um sinal

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Implementation of “A Flexible New Technique for Camera

Calibration”

based on the report of:

Zhengyou Zhang

December 2, 1998

Notação do artigo de Zhang:

1

0100

0

13210

0 Y

X

v

u

v

u

s trrr

MtRAm~~ s

1100

0

1210

0 Y

X

v

u

v

u

s trr

X

Y

x

y

s

vy

s

ux

Determinação de uma Homografia

MtRAm~~ s MHm

~~ s

1876

543

210

i

i

i

ii

ii

Y

X

hhh

hhh

hhh

s

ys

xs

tRAH

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543 hYhXhv iii

876 hYhXhs iii

876

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hYhXh

hYhXhx

ii

iii

876

543

hYhXh

hYhXhy

ii

iii

210876 hYhXhxhYhXh iiiii

543876 hYhXhyhYhXh iiiii

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Y

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0100 876543210 hxhYxhXxhhYhXhhh iiiiiii

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543876 hYhXhyhYhXh iiiii

2 equações por ponto. 9 (8) incógnitas

int homography(int nPoints, double* modelPoints, double* imagePoints, double* H){ int k; double* L=(double*)malloc(2*nPoints*9*sizeof(double)); /* L is a 2nx9 matrix where Lij is in L[9*i+j] */

/* Assembles coeficiente matrix L */ for(k=0; k<nPoints; k++) { double X=modelPoints[3*k+0]; /* X coord of model point k */ double Y=modelPoints[3*k+1]; /* Y coord of model point k */ double W=modelPoints[3*k+2]; /* W coord of model point k */ double x=imagePoints[2*k+0]; /* x coord of image point k */ double y=imagePoints[2*k+1]; /* y coord of image point k */

int i=2*k; /* line number in matrix L */ L[9*i+0] = X; L[9*i+1] = Y; L[9*i+2] = W; L[9*i+3] = 0; L[9*i+4] = 0; L[9*i+5] = 0; L[9*i+6] = -x*X; L[9*i+7] = -x*Y; L[9*i+8] = -x*W; i=2*k+1; L[9*i+0] = 0; L[9*i+1] = 0; L[9*i+2] = 0; L[9*i+3] = X; L[9*i+4] = Y; L[9*i+5] = W; L[9*i+6] = -y*X; L[9*i+7] = -y*Y; L[9*i+8] = -y*W; } solveAx0(2*nPoints,9,L,H); /* solves the system [L]{h}={0} */

free(L); return 0;}

Minimização (Levenberg-Marquardt)

N

i i

ii

i

ii s

vy

s

ux

0

22

1876

543

210

i

i

i

i

i

Y

X

hhh

hhh

hhh

s

v

uonde:

Tome H como solução inicial

Minimize:

int homography(int nPoints, double* modelPoints, double* imagePoints, double* H){ int k; double* L=(double*)malloc(2*nPoints*9*sizeof(double)); /* L is a 2nx9 matrix where Lij is in L[9*i+j] */

/* Assembles coeficiente matrix L */ for(k=0; k<nPoints; k++) { double X=modelPoints[3*k+0]; /* X coord of model point k */ double Y=modelPoints[3*k+1]; /* Y coord of model point k */ double W=modelPoints[3*k+2]; /* W coord of model point k */ double x=imagePoints[2*k+0]; /* x coord of image point k */ double y=imagePoints[2*k+1]; /* y coord of image point k */ int i=2*k; /* line number in matrix L */

L[9*i+0] = X; L[9*i+1] = Y; L[9*i+2] = W; L[9*i+3] = 0; L[9*i+4] = 0; L[9*i+5] = 0; L[9*i+6] = -x*X; L[9*i+7] = -x*Y; L[9*i+8] = -x*W; i=2*k+1; L[9*i+0] = 0; L[9*i+1] = 0; L[9*i+2] = 0; L[9*i+3] = X; L[9*i+4] = Y; L[9*i+5] = W; L[9*i+6] = -y*X; L[9*i+7] = -y*Y; L[9*i+8] = -y*W; } solveAx0(2*nPoints,9,L,H); /* solves the system [L]{h}={0} */

if (OPTIMIZE) { double lmH[9]; lmHomography(imagePoints,modelPoints,H,nPoints,lmH); mtxMatCopy(lmH,3,3,H); } free(L); return 0;}

Restrições na matriz de parâmetros intrínsicos

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~~ s tRAH

trrAhhh 21321

21

2

11

1

1

1

hAr

hAr

1

0

2211

21

rrrr

rr

0

0

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21

rrrr

rrTT

T

0

0

21

211

1

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1

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Das homografias para parâmetros intrínsecos

0

0

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1

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1

hAAhhAAh

hAAhTTTT

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0

0

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0)()()( 576446733167024311340010 bhhbhhhhbhhhhbhhbhhhhbhh

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2 equações por homografia. 6 incógnitas

void calcB(int nH, double* H, double* B){ int m = 2*nH; int n = 6; double* V =(double*) calloc(sizeof(double),m*n); double* b =(double*) malloc(sizeof(double)*n); int i;

for(i=0;i<nH;i++){ double* h = &H[9*i]; int line1 = 2*n*i; int line2 = line1+6;

V[line1+0]=h[0]*h[1]; V[line1+1]=h[0]*h[4] + h[3]*h[1]; V[line1+2]=h[3]*h[4]; V[line1+3]=h[0]*h[7]+h[6]*h[1]; V[line1+4]=h[3]*h[7]+h[6]*h[4]; V[line1+5]=h[6]*h[7];

V[line2+0]=h[0]*h[0] - h[1]*h[1]; V[line2+1]=2*(h[0]*h[3] - h[1]*h[4]); V[line2+2]=h[3]*h[3] - h[4]*h[4]; V[line2+3]=2*(h[0]*h[6] - h[1]*h[7]); V[line2+4]=2*(h[3]*h[6] - h[4]*h[7]); V[line2+5]=h[6]*h[6] - h[7]*h[7]; if (NORMALIZE) { mtxNormalizeVector(6,&V[line1]); mtxNormalizeVector(6,&V[line2]); } }

solveAx0(m,n,V,b); /* solves the system [V]{x}={0} */

i=0; B[i++]=b[0]; B[i++]=b[1]; B[i++]=b[3]; B[i++]=b[1]; B[i++]=b[2]; B[i++]=b[4]; B[i++]=b[3]; B[i++]=b[4]; B[i++]=b[5];

free(b); free(V);}

Cálculo da matrix A

/* Get intrinsic parameters from matrix B*/int calcA(double* B, double* A){ double alpha,betha,gamma,u0,v0,lambda;

double den=B[0]*B[4]-B[1]*B[1]; if (fabs(den)< TOL ) return 0;

v0 = (B[1]*B[2]-B[0]*B[5])/den;

if (fabs(B[0])<TOL) return 0; lambda = B[8]-(B[2]*B[2]+v0*(B[1]*B[2]-B[0]*B[5]))/B[0];

if (lambda/B[0]<0) return 0; alpha=sqrt(lambda/B[0]);

if ((lambda*B[0]/den)<0) return 0; betha = sqrt(lambda*B[0]/den);

gamma = - B[1]*alpha*alpha*betha/lambda; u0=gamma*v0/betha-B[2]*alpha*alpha/lambda;

A[0]=alpha; A[1]=gamma; A[2]=u0; A[3]=0; A[4]=betha; A[5]=v0; A[6]=0; A[7]=0; A[8]=1;

return 1;}

Cálculo de R e t

Calibração de Zhang para uma câmera “boa”

yim

f xim

yc

vista lateral

oczc

f

fovy oy

xim

yim

h pi

xels

ox

oc

eixo óptico

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y0

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zc

Z

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x'

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Notação do artigo de Zhang:

1

0100

00

00

1321

Y

X

f

f

v

u

s x

x

trrr

MtRAm~~ s

1100

00

00

121

Y

X

f

f

v

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x

trr

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x

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1

0100

0

13210

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X

v

u

v

u

s trrr

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)(

00

vu

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xx

Determinação de uma Homografia

MtRAm~~ s MHm

~~ s

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543

210

i

i

i

i

i

Y

X

hhh

hhh

hhh

s

v

u

tRAH

210 hYhXhu iii

543 hYhXhv iii

876 hYhXhs iii

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210

hYhXh

hYhXhx

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iii

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543

hYhXh

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iii

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X

Y

x

y

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0100 876543210 hxhYxhXxhhYhXhhh iiiiiii

210876 hYhXhxhYhXh iiiii

543876 hYhXhyhYhXh iiiii

2 equações por ponto. 9 (8) incógnitas

int homography(int nPoints, double* modelPoints, double* imagePoints, double* H){ int k; double* L=(double*)malloc(2*nPoints*9*sizeof(double)); /* L is a 2nx9 matrix where Lij is in L[9*i+j] */

/* Assembles coeficiente matrix L */ for(k=0; k<nPoints; k++) { double X=modelPoints[3*k+0]; /* X coord of model point k */ double Y=modelPoints[3*k+1]; /* Y coord of model point k */ double W=modelPoints[3*k+2]; /* W coord of model point k */ double x=imagePoints[2*k+0]; /* x coord of image point k */ double y=imagePoints[2*k+1]; /* y coord of image point k */

int i=2*k; /* line number in matrix L */ L[9*i+0] = X; L[9*i+1] = Y; L[9*i+2] = W; L[9*i+3] = 0; L[9*i+4] = 0; L[9*i+5] = 0; L[9*i+6] = -x*X; L[9*i+7] = -x*Y; L[9*i+8] = -x*W; i=2*k+1; L[9*i+0] = 0; L[9*i+1] = 0; L[9*i+2] = 0; L[9*i+3] = X; L[9*i+4] = Y; L[9*i+5] = W; L[9*i+6] = -y*X; L[9*i+7] = -y*Y; L[9*i+8] = -y*W; } solveAx0(2*nPoints,9,L,H); /* solves the system [L]{h}={0} */

free(L); return 0;}

Minimização (Levenberg-Marquardt)

N

i i

ii

i

ii s

vy

s

ux

0

22

1876

543

210

i

i

i

i

i

Y

X

hhh

hhh

hhh

s

v

uonde:

Tome H como solução inicial

Minimize:

int homography(int nPoints, double* modelPoints, double* imagePoints, double* H){ int k; double* L=(double*)malloc(2*nPoints*9*sizeof(double)); /* L is a 2nx9 matrix where Lij is in L[9*i+j] */

/* Assembles coeficiente matrix L */ for(k=0; k<nPoints; k++) { double X=modelPoints[3*k+0]; /* X coord of model point k */ double Y=modelPoints[3*k+1]; /* Y coord of model point k */ double W=modelPoints[3*k+2]; /* W coord of model point k */ double x=imagePoints[2*k+0]; /* x coord of image point k */ double y=imagePoints[2*k+1]; /* y coord of image point k */ int i=2*k; /* line number in matrix L */

L[9*i+0] = X; L[9*i+1] = Y; L[9*i+2] = W; L[9*i+3] = 0; L[9*i+4] = 0; L[9*i+5] = 0; L[9*i+6] = -x*X; L[9*i+7] = -x*Y; L[9*i+8] = -x*W; i=2*k+1; L[9*i+0] = 0; L[9*i+1] = 0; L[9*i+2] = 0; L[9*i+3] = X; L[9*i+4] = Y; L[9*i+5] = W; L[9*i+6] = -y*X; L[9*i+7] = -y*Y; L[9*i+8] = -y*W; } solveAx0(2*nPoints,9,L,H); /* solves the system [L]{h}={0} */

if (OPTIMIZE) { double lmH[9]; lmHomography(imagePoints,modelPoints,H,nPoints,lmH); mtxMatCopy(lmH,3,3,H); } free(L); return 0;}

Restrições na matriz de parâmetros intrínsicos

MtRAm~~ s MHm

~~ s tRAH

trrAhhh 21321

21

2

11

1

1

1

hAr

hAr

1

0

2211

21

rrrr

rr

0

0

2211

21

rrrr

rrTT

T

0

0

21

211

1

21

1

hAAhhAAh

hAAhTTTT

TT

Das homografias para parâmetros intrínsecos

0

0

21

211

1

21

1

hAAhhAAh

hAAhTTTT

TT

0

0

2211

21

hBhhBh

hBhTT

T

100

01

0

001

2

2

543

421

3101

x

x

T

f

f

bbb

bbb

bbb

AABonde:

876

543

210

hhh

hhh

hhh

He

0)()()( 576446733167024311340010 bhhbhhhhbhhhhbhhbhhhhbhh

0)()(3)(2)()(2)( 52

72

6474633716022

42

31413002

12

0 bhhbhhhhbhhhhbhhbhhhhbhh 0)()(3)(2)()(2)( 52

72

6474633716022

42

31413002

12

0 bhhbhhhhbhhhhbhhbhhhhbhh

076243

210 hh

f

hh

f

hh

xx

0)( 27

262

24

23

2

21

20

hh

f

hh

f

hh

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76

4310

hh

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27

26

24

23

21

20 )()(

hh

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Cálculo de R e t

Proceedings of SIBGRAPI'98, Rio de Janeiro, Brazil, 1998, pp. 388-399.

Equações de projeção

c

c

c

yy

xx

Z

Y

X

of

of

w

v

u

100

0

0

wv

wu

y

x

im

im

yc

c

yim

xc

c

xim

oZ

Yfy

oZ

Xfx

z

y

x

w

w

w

c

c

c

T

T

T

Z

Y

X

rrr

rrr

rrr

Z

Y

X

333231

332221

131211

1333231

232221

131211

w

w

w

z

y

x

c

c

c

Z

Y

X

Trrr

Trrr

Trrr

Z

Y

X

1100

0

0

333231

232221

131211

w

w

w

z

y

x

yy

xx

Z

Y

X

Trrr

Trrr

Trrr

of

of

w

v

u

Equações de projeção

1100

0

0

333231

232221

131211

w

w

w

z

y

x

yy

xx

Z

Y

X

Trrr

Trrr

Trrr

of

of

w

v

u

134333231

24232221

14131211

w

w

w

Z

Y

X

mmmm

mmmm

mmmm

w

v

u

1333231

332332223121

331332123111

w

w

w

z

zxyyxyxyxy

zxxxxxxxxx

Z

Y

X

Trrr

ToTfrorfrorfrorf

ToTfrorfrorfrorf

w

v

u

Calibração a partir da matriz

134333231

24232221

14131211

wi

wi

wi

i

i

i

Z

Y

X

mmmm

mmmm

mmmm

w

v

u

34333231

24232221

34333231

14131211

mZmYmXm

mZmYmXmy

mZmYmXm

mZmYmXmx

www

www

i

www

www

i

iii

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0

0

3433323124232221

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mZmYmXmxmZmYmXmwww

iwww

wwwi

www

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iiiiii

Para cada ponto:

Calibração a partir da matriz

0

0

3433323124232221

3433323114131211

mZmYmXmymZmYmXm

mZmYmXmxmZmYmXmwww

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03

0

3433323124232221

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myZmymYymXymmZmYmX

mxZmxmYxmXxmmZmYmX

iw

iw

iw

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iw

iw

iw

iwww

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Para cada ponto:

34

33

12

11

m

m

m

m

m

0Am 0m

Calibração a partir da matriz

34333231

24232221

14131211

mmmm

mmmm

mmmm

34333231

24232221

14131211

mmmm

mmmm

mmmm1q

2q

3q

4q

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Trrr

ToTfrorfrorfrorf

ToTfrorfrorfrorf

333231

332332223121

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233

232

231

233

232

2313 rrrmmmq

01 zTquetalescolha 34mTz

333332323131 ,, mrmrmr

Calibração a partir da matriz

34333231

24232221

14131211

mmmm

mmmm

mmmm

34333231

24232221

14131211

mmmm

mmmm

mmmm1q

2q

3q

4q

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Trrr

ToTfrorfrorfrorf

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333231

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cycy

cxcx

c

of

of

kjq

kiq

kq

2

1

3

yy

xx

of

of

22

11

qq

qq

32

31

qq

qq

y

x

o

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Calibração a partir da matriz

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24232221

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mmmm

mmmm

mmmm

34333231

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mmmm

mmmm

mmmm1q

2q

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3,2,1/)(

3,2,1/)(

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1433

232

131

TT UIVRUDVR

Calibração no Juiz Virtual

ts

vs

us

w

y

x

hhh

hhh

hhh

w

y

x

H

333231

232221

131211

0

0

0

333231

232221

131211

kkkkk

kkkkk

kkkkk

sthwhyhx

svhwhyhx

suhwhyhxk = 1..n-1

1333231

232221

131211

hwhyhx

vhwhyhx

uhwhyhx

nnn

nnnn

nnnn

1a Opção

Calibração no Juiz Virtual

1

0

0

0

0

0

0

0

0

0

000000000

000000000

000000000

00000000

00000000

00000000

00000000

00000000

00000000

00000000

00000000

00000000

1

2

1

33

32

31

23

22

21

13

12

11

1111

1111

1111

2222

2222

2222

1111

1111

1111

n

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nnnn

nnn

nnn

nnnn

nnnn

nnnn

v

u

s

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s

h

h

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h

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h

h

h

wyx

wyx

wyx

twyx

vwyx

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twyx

vwyx

uwyx

twyx

vwyx

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Calibração no Juiz Virtual

ts

vs

us

w

y

x

hhh

hhh

hhh

w

y

x

H

333231

232221

131211

2a Opção

kkk

kkkk whyhxh

whyhxhu

333231

131211

kkk

kkkk whyhxh

whyhxhv

333231

232212

Mimimiza-se , onde é o resultado obtido pelatransformação encontrada.

n

kkkkk vvuu

1

22 kk vu ,

Problema não linear

Onde é o centróide dos pontos próprios Pk.

Calibração no Juiz Virtual

kkk

kkkk whyhxh

whyhxhu

333231

131211

kkk

kkkk whyhxh

whyhxhv

333231

232212

0

0

333231232221

333231131211

kkkkkkk

kkkkkkk

whyhxhvwhyhxh

whyhxhuwhyhxhk=1..n

1333231 hyhxh

Para eliminar a solução trivial H = 0,

1,0,, yx

Calibração no Juiz Virtual

A solução desse problema se dá por resolver

Minimizar || Mt ||sujeito a mTt = 1,

onde

333231232221131211 hhhhhhhhhtT

1000000 yxm e M é a representação matricial da transformação,

dada por:

vnnnnnnn

nnnnnnnn

vyvxvwyx

vyvxvwyx

vyvxvwyx

uyuxuwyx

uyuxuwyx

uyuxuwyx

M

000

000

000

000

000

000

22222222

11111111

22222222

11111111

É usado o método de multiplicadores de Lagrange, conduzindo ao seguinte sistema linear:

1

0

tm

mMtMT

T

(2n x 9)