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”

Notação

Z

Yfy

Z

Xfx ,

xc

yc

zcp’

Z

Y

X

P

y'

x'

Z

Yfy

Z

Xfx ,

sinal !

OpenGLVisão

(X,Y,Z)T (x’,y’)T(xim,yim)T

yim

f xim

yc

vista lateral

oczc

f

fovy oy

xim

yim

h pi

xels

ox

oc

eixo óptico

x0

y0

yc

xc

zc

Z

Y

X

P

y'

x'

w pixels

x'

y'

(xim,yim)T (x’,y’)T

xim

yim

x’

y’

01 3 4

1

2

0

3

2

sx

sy

p'

ooy

ox

5

ximx

oxs

x

'yim

y

oys

y

'

)(

)(

yimy

ximx

oys

oxs

y

xp

(X,Y,Z)T (xim,yim)T

Z

Y

X

osf

osf

w

y

x

yy

xx

h

h

100

0

0

yy

im

xx

im

oZ

Y

s

fy

oZ

X

s

fx

wy

wx

y

x

h

h

im

im

)(

)(

yimy

ximx

oys

oxs

y

xp

ZY

ZX

f

Parâmetros intrínsecos com cisalhamento

10100

0010

0001

100

0Z

Y

X

of

osf

w

y

x

yy

xhx

h

h

P0IKp 3

yy

xx

sff

sff

][ yxhyx oosff

Parâmetros intrínsecos com distorção radial

][ 1koosff yxhyx

22

42

21

42

21

)1('

)1('

ddd

ddd

ddd

yxr

rkrkyy

rkrkxx

02

12

k

kk

Parâmetros extrínsecos

• Translação –Tx, Ty, Tz (3 g.l.)

• Rotação (3 g.l.)

333231

332221

131211

rrr

rrr

rrr

w

w

w

k

j

i

1

0

wwwwww

wwwwww

kkjjii

jkkiji

Parâmetros extrínsecosxc

yc

zc

yw

xw

zw

w

w

w

cwcwcw

cwcwcw

cwcwcw

c

c

c

cw

Z

Y

X

Z

Y

X

kkkjki

jkjjji

ikijii

P

Pw

Pc T

TPP wc

1w

w

w

z

y

x

cwcwcw

cwcwcw

cwcwcw

c

c

c

cZ

Y

X

t

t

t

Z

Y

X

kkkjki

jkjjji

ikijii

P

wc PTRP

wPTRKp

cKPp

Compondo as transformações

PTRKp ][

parâmetros extrínsecos

parâmetros intrínsecos

1100

0w

w

w

z

y

x

cwcwcw

cwcwcw

cwcwcw

yy

xhx

h

h

Z

Y

X

t

t

t

of

osf

w

y

x

kkkjki

jkjjji

ikijii

wy

wx

y

x

h

h

im

im

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

2,,

,,|)(|min iTRKi

TRKPfp

pontos da cena

pontos da imagem

projeção (função não linear)

Método de Tsai(compativel com notação do OpenGL)

X c

Y c

Z c p’

c

c

c

c

Z

Y

X

P

'

'

y

xp

c

c

c

c

ZY

ZX

f

y'

x'

Câmera para imagem

c

c

yyim

c

c

xxim

Z

Y

s

foy

Z

X

s

fox

)(

)(

yimy

ximx

oys

oxs

y

xp

c

c

c

c

ZY

ZX

f

Concatenando

c

c

yyim

c

c

xxim

Z

Y

s

foy

Z

X

s

fox

zwww

ywww

yyim

zwww

xwww

xxim

TZrYrXr

TZrYrXrfoy

TZrYrXr

TZrYrXrfox

333231

232221

333231

131211

cameraz

y

x

w

w

w

c

c

c

T

T

T

Z

Y

X

rrr

rrr

rrr

Z

Y

X

333231

332221

131211

Os passos do Método de TsaiPasso 1:

),( xx oo conhecidos

Distorção radial insignificante

Assuma:

zwww

ywww

yi

zwww

xwww

xi

TZrYrXr

TZrYrXrfy

TZrYrXr

TZrYrXrfx

iii

iii

iii

iii

333231

232221

333231

131211

yimiximi oyyoxx

Método de Tsai

zwww

ywww

yi

zwww

xwww

xi

TZrYrXr

TZrYrXrfy

TZrYrXr

TZrYrXrfx

iii

iii

iii

iii

333231

232221

333231

131211

ywww

i

yz

www

xwww

i

xz

www

TZrYrXry

fTZrYrXr

TZrYrXrx

fTZrYrXr

iiiiii

iiiiii

232221333231

131211333231

Método de TsaiPasso 1:

ywww

i

yz

www

xwww

i

xz

www

TZrYrXry

fTZrYrXr

TZrYrXrx

fTZrYrXr

iiiiii

iiiiii

232221333231

131211333231

xwww

xiywww

yi TZrYrXrfyTZrYrXrfx iiiiii 131211232221

xiw

iw

iw

iyiw

iw

iw

i TyrZyrYyrXyTxrZxrYxrXx iiiiii 131211232221

x

yy

xy

x

s

s

f

s

s

f

f

f


xiw

iw

iw

iyiw

iw

iw

i TyrZyrYyrXyTxrZxrYxrXx iiiiii 131211232221

87654321 vyvZyvYyvXyvxvZxvYxvXx iw

iw

iw

iiw

iw

iw

i iiiiii

087654321 vyvZyvYyvXyvxvZxvYxvXx iw

iw

iw

iiw

iw

iw

i iiiiii

xy TvTv

rvrv

rvrv

rvrv

84

137233

126222

115211

Correspondência

11, yx

22 , yx

33 , yx

NN yx ,

www ZYX111

,,

www ZYX222

,,

www ZYX333

,,

www

NNNZYX ,,

Sistema Ax=0

0

0

0

0

0

0

0

0

8

7

6

5

4

3

2

1

2222222222222

111111111111

2

1

v

v

v

v

v

v

v

v

yZyYyXyxZxYxXx

yZyYyXyxZxYxXx

yZyYyXyxZxYxXx

NwNN

wNN

wNNN

wNN

wNN

wNN

wwwwww

wwwwwwi

0Av 7)( Arank

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

Fator de escala

Txy TrrrTrrrγv

trivialnãosolução umav

131211232221

, Seja

2

232

222

2122

322

21

223

222

221 ,1 Como

rrrvvv

rrr

Sinal do fator de escala

Sinal de

0131211 xww

iw

i TZrYrXrx ii

zwww

xwww

xi TZrYrXr

TZrYrXrfx

iii

iii

333231

131211

0333231 zwwwc

i TZrYrXrZ iiiComo:

Temos

Caso isto não seja verdade troque o sinal de v

Estimativa do fator

2

232

222

21222

726

25

223

222

221 have we,1 Since

rrrvvv

rrr

27

26

25

1vvv

Última linha da matriz de rotação

332221

131211

23

22

21

13

12

11

33

32

31

det

rrr

rrr

r

r

r

r

r

r

r

r

r kji

333231

332221

131211

rrr

rrr

rrr

Reortogonalize:

TT UIVRUDVR

Cáculo de fx fy e Tz

wwwii iii

ZYXyx ,, and ,pair ingcorrespondeach For

bf

TA

x

Z

Solve

xwww

xzwww

i TZrYrXrfTZrYrXrx iiiiii 131211333231

wwwixx

wwwzi iiiiii ZrYrXrxfTZrYrXrTx 333231131211


xwN

wN

wNN

xwww

xwww

TZrYrXrx

TZrYrXrx

TZrYrXrx

131211

2132122112

1131121111

A

wN

wN

wNN

www

www

ZrYrXrx

ZrYrXrx

ZrYrXrx

333231

2332322312

1331321311

b

bAx bxUDV T bUVDx T1

bAx bAAxA TT bAAAx TT 1

Ponto de fuga

Cálculo do centro ótico pelos pontos de fuga

Passo 2 do Tsai

Computing Image Center yx oo ,

vv11

vv22

vv33

Pontos de fuga do padrão 3D

Tsai 2D

0wiZ

Método de Tsai planoPasso 1:

zwww

ywww

yi

zwww

xwww

xi

TZrYrXr

TZrYrXrfy

TZrYrXr

TZrYrXrfx

iii

iii

iii

iii

333231

232221

333231

131211

zww

yww

yi

zww

xww

xi

TYrXr

TYrXrfy

TYrXr

TYrXrfx

ii

ii

ii

ii

3231

2221

3231

1211

=0

Método de Tsai plano

yww

i

yz

ww

xww

i

xz

ww

TYrXry

fTYrXr

TYrXrx

fTYrXr

iiii

iiii

22213231

12113231

xww

xiyww

yi TYrXrfyTYrXrfx iiii 12112221

xiw

iw

iyiw

iw

i TyrYyrXyTxrYxrXx iiii 12112221

xiw

iw

iyiw

iw

i TyrYyrXyTxrYxrXx iiii 12112221

1


0654321 vyvYyvXyvxvYxvXx iw

iw

iiw

iw

i iiii

xy TvTv

rvrv

rvrv

63

125222

114211

0v0Av

3

6

222

121

512

411

vT

vT

vr

vr

vr

vr

y

x

?

?

23

13

r

r

Método de Tsai

Tc rrr 131211

ˆ i

Tc rrr 232221

ˆ j

3

6

222

121

512

411

vT

vT

vr

vr

vr

vr

y

x

051422

54214 vvvvvvvv

023132

2514

223

222

21

213

225

24

rrvvvv

rvv

rvv

5142

254215421

2 42

1vvvvvvvvvvvv

Sinal de

01211 xw

iw

i TYrXrx i

zww

xww

xi TYrXr

TYrXrfx

ii

ii

3231

1211

0ciZ

Logo

Caso isto não seja verdade troque o sinal de

2

3

6

222

121

512

411

vT

vT

vr

vr

vr

vr

y

x

escolha um sinal

Fator de escala

3

6

222

121

512

411

vT

vT

vr

vr

vr

vr

y

x

212

21113 1 rrr

222

22123 1 rrr

221221112313 rrrrrr

escolha um sinal

escolha um sinal

corrija a escolha

23

22

21

13

12

11

33

32

31

r

r

r

r

r

r

r

r

r

?

?

?

?

23

13

zT

f

r

r


wwwii iii

ZYXyx ,, and ,pair ingcorrespondeach For

bf

TA

x

Z

Solve

xww

xzww

i TYrXrfTYrXrx iiii 12113231

wwixx

wwzi iiii YrXrxfTYrXrTx 32311211

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

210 hYhXhu iii

543 hYhXhv iii

876 hYhXhs iii

876

210

hYhXh

hYhXhx

ii

iii

876

543

hYhXh

hYhXhy

ii

iii

210876 hYhXhxhYhXh iiiii

543876 hYhXhyhYhXh iiiii

X

Y

x

y

00001 876543210 hxhYxhXxhhhhhYhX iiiiiii

0100 876543210 hxhYxhXxhhYhXhhh iiiiiii



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:


/* 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

543

421

3101

bbb

bbb

bbbT AABonde:

876

543

210

hhh

hhh

hhh

He

0)()()( 576446733167024311340010 bhhbhhhhbhhhhbhhbhhhhbhh

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

72

6474633716022

42

31413002

12

0 bhhbhhhhbhhhhbhhbhhhhbhh

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

x0

y0

yc

xc

zc

Z

Y

X

P

y'

x'

w pixels

x'

y'

2

2h

o

wo

y

x

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

u

s x

x

trr

X

Y

x

y

s

vy

s

ux

1

0100

0

13210

0 Y

X

v

u

v

u

s trrr

0

)(

00

vu

pixelemfpf

xx

Determinação de uma Homografia

MtRAm~~ s MHm

~~ s

1876

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

876

210

hYhXh

hYhXhx

ii

iii

876

543

hYhXh

hYhXhy

ii

iii



X

Y

x

y

00001 876543210 hxhYxhXxhhhhhYhX iiiiiii

0100 876543210 hxhYxhXxhhYhXhhh iiiiiii



2 equações por ponto. 9 (8) incógnitas


/* 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:


/* 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

xx

76

4310

hh

hhhhf x

27

26

24

23

21

20 )()(

hh

hhhhf x

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

iii

iii

iii

0

0

3433323124232221

3433323114131211

mZmYmXmymZmYmXm

mZmYmXmxmZmYmXmwww

iwww

wwwi

www

iiiiii

iiiiii

Para cada ponto:


0

0

3433323124232221

3433323114131211

mZmYmXmymZmYmXm

mZmYmXmxmZmYmXmwww

iwww

wwwi

www

iiiiii

iiiiii

03

0

3433323124232221

3433323114131211

myZmymYymXymmZmYmX

mxZmxmYxmXxmmZmYmX

iw

iw

iw

iwww

iw

iw

iw

iwww

iiiiii

iiiiii

Para cada ponto:

34

33

12

11

m

m

m

m

m

0Am 0m


34333231

24232221

14131211

mmmm

mmmm

mmmm

34333231

24232221

14131211

mmmm

mmmm

mmmm1q

2q

3q

4q

z

zxyyxyxyxy

zxxxxxxxxx

Trrr

ToTfrorfrorfrorf

ToTfrorfrorfrorf

333231

332332223121

331332123111

233

232

231

233

232

2313 rrrmmmq

01 zTquetalescolha 34mTz

333332323131 ,, mrmrmr


34333231

24232221

14131211

mmmm

mmmm

mmmm

34333231

24232221

14131211

mmmm

mmmm

mmmm1q

2q

3q

4q

z

zxyyxyxyxy

zxxxxxxxxx

Trrr

ToTfrorfrorfrorf

ToTfrorfrorfrorf

333231

332332223121

331332123111

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

o


34333231

24232221

14131211

mmmm

mmmm

mmmm

34333231

24232221

14131211

mmmm

mmmm

mmmm1q

2q

3q

4q

z

zxyyxyxyxy

zxxxxxxxxx

Trrr

ToTfrorfrorfrorf

ToTfrorfrorfrorf

333231

332332223121

331332123111

yyy

xxx

yiiyi

xiixi

fmmoT

fmmoT

ifmmor

ifmmor

/)(

/)(

3,2,1/)(

3,2,1/)(

2433

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


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

n

nnnn

nnn

nnn

nnnn

nnnn

nnnn

v

u

s

s

s

h

h

h

h

h

h

h

h

h

wyx

wyx

wyx

twyx

vwyx

uwyx

twyx

vwyx

uwyx

twyx

vwyx

uwyx


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.


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


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)

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

Documents