# Stereo Vision by lJt41jZz

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```									Computer Vision

Stereo Vision
Pinhole Camera

Perspective Projection

x' y' f '
 
x  y  z

Stereo Vision

    Two cameras.
    Known camera positions.
    Recover depth.

scene point

p                     p’
image plane

optical center

Correspondences

p             p’

Matrix form of cross product
a=axi+ayj+azk
a×b=|a||b|sin(η)u
b=bxi+byj+bzk

 a y bz  az by   0         az    a2 
                                      
a  b   az bx  ax bz    az        0    ax  b     a  b
 ax by  a y bz   a y      ax     0 
                                      

a  (a  b)  0
b  (a  b)  0

The Essential Matrix

pT Ep '  0

Essential matrix     p  (u, v,1)T
p '  (u ', v ',1)T
Stereo Constraints
M
Image plane             Epipolar Line

Y1            p
p’
Y2
X2
O1
Z1                            X1

O2         Z2
Focal plane
Epipole

A Simple Stereo System

LEFT CAMERA                     RIGHT CAMERA
baseline

Left image:                              Right image:
reference                                target
disparity

Depth Z

Elevation Zw
Zw=0 K. Gunturk
Stereo View

Left View                 Right View

Stereo Disparity
    The separation between two matching objects
is called the stereo disparity.

Parallel Cameras
P
T x x T
r
      l

Z f   Z
Z                               T
xl                     xr
Z f
x x
f              pl              pr
l       r

Ol                        Or
T
Disparity:       d  x x
l        r

T is the stereo baseline

Finding Correspondences

Correlation
LEFT IMAGE               Approach
(x , y )
l   l

   For Each point (xl, yl) in the left image, define a window
centered at the point
Correlation
RIGHT IMAGE             Approach
(x , y )
l   l

   … search its corresponding point within a search region in
the right image
Correlation
RIGHT IMAGE      (xr, yr)   Approach
dx (x , y )
l   l

   … the disparity (dx, dy) is the displacement when the
correlation is maximum
?
Comparing Windows        =
f         g

Most
popular

Comparing Windows

Minimize          Sum of Squared
Differences

Maximize          Cross correlation

Correspondence Difficulties
    Why is the correspondence problem difficult?
      Some points in each image will have no
corresponding points in the other image.
(1) the cameras might have different fields of view.
(2) due to occlusion.
    A stereo system must be able to determine
the image parts that should not be matched.

Structured Light
   Structured lighting
    Feature-based methods are not applicable when the
objects have smooth surfaces (i.e., sparse disparity
maps make surface reconstruction difficult).
    Patterns of light are projected onto the surface of
objects, creating interesting points even in regions
which would be otherwise smooth.
   Finding and matching such
points is simplified by
knowing the geometry of the
projected patterns.

Stereo results
    Data from University of Tsukuba

Scene                        Ground truth

(Seitz)                      21
Results with window correlation

Estimated depth of field             Ground truth
(a fixed-size window)
(Seitz)