Single  View  Metrology  

Slides  taken  from  Steve  Seitz,  A.  Efros  

Applica=ons  of  projec=ve  geometry    

–  Criminisi  et  al.,  “Single  View  Metrology”,  ICCV  1999  –  Other  methods  

•  Horry  et  al.,  “Tour  Into  the  Picture”,  SIGGRAPH  96  •  Shum  et  al.,  CVPR  98  •  ...  

Vermeer’s Music Lesson

1   2   3   4  

1  

2  

3  

4  

Measurements  on  planes  

Approach: unwarp then measure

What kind of warp is this?

•  A Homography

Image  rec=fica=on,  and  measurement  

Vanishing  points  

•  Vanishing  point  – projec=on  of  a  point  at  infinity  

image plane

camera center

ground plane  

vanishing point

Vanishing  points  (2D)  image  plane  

camera  center  

line  on  ground  plane  

vanishing  point  

Vanishing  points  

•  Proper=es  – Any  two  parallel  lines  have  the  same  vanishing  point  –  The  ray  from  C  through  v  point  is  parallel  to  the  lines  – An  image  may  have  more  than  one  vanishing  point  

image  plane  

camera  center  C  

line  on  ground  plane  

vanishing  point  V  

line  on  ground  plane  

Vanishing  lines  

•  Mul=ple  Vanishing  Points  –  Any  set  of  parallel  lines  on  the  plane  define  a  vanishing  point  –  The  union  of  all  of  these  vanishing  points  is  the  horizon  line  

•  also  called  vanishing  line  –  Note  that  different  planes  define  different  vanishing  lines  

v1   v2  

Compu=ng  vanishing  lines  

•  Proper=es  –  l  is  intersec=on  of  horizontal  plane  through  C  with  image  plane  –  Compute  l  from  two  sets  of  parallel  lines  on  ground  plane  –  All  points  at  same  height  as  C  project  to  l  –  Provides  way  of  comparing  height  of  objects  in  the  scene  

ground  plane  

l C  

Fun  with  vanishing  points  

Perspec=ve  cues  

Perspec=ve  cues  

Perspec=ve  cues  

Comparing  heights  Vanishing

Point

Measuring  height  

1  

2  

3  

4  

5  5.4  

2.8  

3.3  

Camera height

q1

Compu=ng  vanishing  points  (from  lines)  

•  Intersect  p1q1  with  p2q2    

v  

p1

p2

q2

Least squares version

•  Better to use more than two lines and compute the “closest” point of intersection

•  See notes by Bob Collins for one good way of doing this: –  http://www-2.cs.cmu.edu/~ph/869/www/notes/vanishing.txt

Vanishingpoint

Vanishingline

Vanishingpoint

Vertical vanishingpoint

(at infinity)

Criminisi ’99

C  

Measuring  height  without  a  ruler  

ground  plane  

Compute Y from image measurements •  Need more than vanishing points to do this

Y  

Measuring height

RH

vz

r

b

t

H

b0

t0

vvx vy

vanishing line (horizon)

Measuring height vz

r

b

t0

vx vy

vanishing line (horizon)

v

t0

m0

What if the point on the ground plane b0 is not known?

• Here the guy is standing on the box

• Use one side of the box to help find b0 as shown above

b0

t1

b1

What if vz is not infinity?

The  cross  ra=o  •  A  Projec=ve  Invariant  

–  Something  that  does  not  change  under  projec=ve  transforma=ons  (including  perspec=ve  projec=on)  

P1  P2  

P3  P4  

1423

2413

PPPPPPPP

−−

−−

The cross-ratio of 4 collinear points

Can permute the point ordering •  4! = 24 different orders (but only 6 distinct values)

This is the fundamental invariant of projective geometry

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

1i

i

i

i ZYX

P

3421

2431

PPPPPPPP

−−

−−

vZ  

   r  t  

b  

tvbrrvbt−−

−−

Z

Z

image cross ratio

Measuring  height  

B   (bottom of object)

T    (top of object)

R    (reference point)

ground plane

H  C  

TBRRBT

−∞−

−∞−

scene cross ratio

∞  

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

1ZYX

P⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

1yx

pscene points represented as image points as

RH

=

RH

=

R  

Measuring height

RH

vz

r

b

t

R

H

Z

Z=

−−

−−

tvbr

rvbt

image cross ratio

H

b0

t0

vvx vy

vanishing line (horizon)

Measuring heights of people

Here we go !

reference185.3 cm

Forensic Science: measuring heights of suspects

Vanishing line

Refe

ren

ce h

eig

ht

Reference height

Assessing geometric accuracy

Flagellation, Piero della Francesca

Estimated relative heights

Are the heights of the 2 groups of people consistent with each other?