Introducción  a  la  Fotografia  3D  UBA/FCEN  Marzo  27  –  Abril  12  2013  

Clase  2  :  Miercoles  Abril  3  

Gabriel  Taubin  Brown  University  

 

Las  matemá)cas  de  la  triangulacion  3D  

•  TriangulaJon  por  interseccion  de  linea  y  plano  •  TriangulaJon  por  interseccion  de  linea  y  linea  •  Puntos,  vectores,  lineas,  rayos,  and  planos  •  Que  es  una  camara?  •  Que  es  una  image?  •  …  

3D  triangula)on:  ray-­‐plane  Intersec)on  

ray  

plane  

intersecJon  point  

projector  /    

coordinate  systems  world

coordinate system

Que  es  una  Camara?  

What is a Camera ?

What is a Camera ?

Modelo  Geometrico  General  de  una  Camara  

•  Funcion  que  a  cada  punto  de  la  imagen  le  hace  corresponder  un  rayo  •  El  dominio  esta  contenido  en  un  rectangulo  y  la  funcion  es  conJnua  •  En  muchos  casos  el  análisis  es  mas  simple  en  el  espacio  de  los  rayos  

W  

H  u

q(u)

v(u)

optics (black box)

pixel

ray

0 <= u1 <=W 0 <= u2 <= H

Representation of Lines and Rays

line

q

vvqp λ+=reference

point

vector

scale parameter

point

q

v

ray=“1/2  line” vqp λ+=

parameter is  positive

Representation of Planes

2211 vvqp λλ ++=

1v

2vqp

P

parametric

n

qp

0)( =− qpntP

implicit

2  scale parameters

reference point

2  vectors

point

reference point

normal vector

1  implicit equation

3D  Triangula)on  by  Line-­‐Plane  intersec)on  

object  being   scanned

pq n

projected  light  plane

p

illuminated  point  on  object

camera  ray

v

intersection of  light  plane with  object

world coordinate

system

image  plane

center  of  projection

ray  direction  vector

If  camera  and  projector  are  calibrated  

object  being   scanned

pq n

projected  light  plane

p

illuminated  point  on  object

camera  ray

vintersection of  light  plane with  object

common  world  

coordinate  system  

center  of  projection

ray  direction  vector  from  detected  pixel

Lq

3D  Triangula)on  by  Line-­‐Line  Intersec)on  

object  being   scanned

camera  ray 2v

projected  light  ray

p

lines  may  not  intersect  !

Implicit  equa)on  of  the  plane  

pq n

}0)(:{ =−= pt qpnpP

v

world coordinate

system Plane  normal  vector

(unit  length)

A  fixed  point  on  the  plane

p

Parametric  equa)on  of  the  ray  

pq n

L = {p = qL +λv :λ > 0}

camera  ray

Lq

v

world coordinate

system

ray  direction  vector

(unit  length)

p

v

Triangula)on  by  Line-­‐Plane  Intersec)on  

world coordinate

system }{ vqpL L λ+==

camera  ray

Lqv

1.  Compose  implicit  and  parametric  equations

2.  Eliminate  parameter  λ 3.  Replace  computed  λ  in  

parametric  equation  

p

pq n

}0)(:{ =−= pt qpnpP

projected  light  plane

Triangula)on  by  Line-­‐Line  Intersec)on  

object  being   scanned

}{ 2222 vqpL λ+==camera  ray

2q

2v

projected  light  ray

1q1v

}{ 1111 vqpL λ+==

p

lines  may  not  intersect  !

Triangula)on  by  Line-­‐Line  Intersec)on  

L2 = {p2 = q2 +λ2v2}

2q2v1q

1v

L1 = {p1 = q1 +λ1v1}

p1 = q1 +λ1v1

p2 = q2 +λ2v2

p1 − p2

v1t (p1 − p2 ) = 0v2t (p2 − p1) = 0

E(λ1,λ2 ) = dist(p2 − p1)2

Minimize

Necessary  conditions

p = (p1 + p2 ) / 2

p

Approximate Line-Line Intersection

1q1v

2q

2v

1111 vqp λ+=

2222 vqp λ+=

),( 2112 λλp

Midpoint  of  segment  joining arbitrary  points  in  the  two  lines

1q1v

2q

2v

),( 2112 λλp

Least-­‐‑squares  approach

Find  parameters  which  minimize

Approximate Line-Line Intersection

1111 vqp λ+=

2222 vqp λ+=

Perspective Projection (Pinhole Model)

center  of  projection

image point

image  plane

3D  point

light  direction for  a  projector

light  direction for  a  camera

Calibration: mapping from image points to rays

The Ideal Pinhole Camera

camera  coordinate  system  =  world  coordinate  system

1v3v2v

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

=

1

2

1

uu

u0=q

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

=3

2

1

ppp

p

1=f

The General Pinhole Model

world  coordinate  system camera  coordinate  system

WΧ

CΧ

1 2

31

2 3 up

TRXWC +=Χ

Ideal  assumptions •   Image  lengths  =  world  lengths •   Focal  length  =  1 •   Image  origin  =  optical  center •   Image  plane  spanned  by  two  basis  vectors

The General Pinhole Model

world  coordinate  system camera  coordinate  system

WΧ

CΧ

1 2

31

2 3 up

TRXWC +=Χ

Plane Defined by Image Line and Projection Center

center  of  projection q

n

image  plane

}0)(:{ =−= qpnpP t

}0:{ == uluL tImplicit  equation  of  line  in  image  coordinates

Structured  Ligh)ng  

Desktop  3D  Photography  

Es)ma)ng  the  equa)on  of  a  plane  

r3

r1

r2

T

Plane coordinate  system

Camera coordinate  system

pj vj

λ jv j = Rpj +T

What  is  the  minimum number  of  points necessary  to solve  the  problem?

j =1,...,N

vector point

We  want  to  estimate the  rotation  R and  the  translation  T

Given

How  to  solve this  problem?

R = r1r2r3[ ]

Es)ma)ng  the  equa)on  of  a  plane  

r3

r1

r2

T

pj vjCamera

coordinate  system

λ jv j = R pj +T

Plane coordinate  system

Implicit  equation  of  plane  in  the  camera coordinate  system

{q : r3t (q−T ) = 0}

Parametric  equation  of  plane  in  the  camera coordinate  system

{q = T + x r1 + y r2 + 0 r3} p = x, y, 0[ ]t

Es)ma)ng  the  equa)on  of  two  planes