Graphics Reference
In-Depth Information
Instead, we look for the optimal value of
P
that lies closest to the rays
v
i
. Mathematically, if
K
i
,
3 matrix that contains the
intrinsic parameters of the camera and
R
i
and
t
i
are the pose of the
i
th camera with respect to
the world coordinate system, the rays
v
i
originating at
C
i
and passing through
p
i
are in the
direction of
R
−
i
K
−
i
p
i
. The optimal value of
P
that lies closest to all the rays
R
i
,
t
i
are the parameters of the camera
C
i
, where
K
i
is the 3
×
v
i
,
p
minimizes
the distance:
2
c
j
+
d
j
v
j
−
p
(1.3)
Methods based on the
optical triangulation
need to solve two problems:
(i)
the matching
problem, and
(ii)
the reconstruction problem. The correspondence problem consists of finding
matching points across the different cameras. Given the corresponding points, the reconstruc-
tion problem consists of computing a 3D disparity map of the scene, which is equivalent
to the depth map (
z
-coordinate on each pixel). Consequently, the quality of the reconstruc-
tion depends crucially on the solution to the correspondence problem. For further reading on
stereo vision
(cameras calibration, stereo matching algorithms, reconstruction, etc.), we refer
the reader to download the PDF of the Richard Szeliski's
Computer Vision: Algorithms and
Applications
available at http://szeliski.org.
1
Existing
optical triangulation
-based 3D reconstruction techniques, such as multi-view
stereo, structured-light techniques, and laser-based scanners, differ in the way the corre-
spondence problem is solved. Multiview stereo reconstruction uses the triangulation principle
to recover the depth map of a scene from two or more projections. The same mechanism
is unconsciously used by our brain to work out how far an object is. The correspondence
problem in stereo vision is solved by looking for pixels that have the same appearance in the
set of images. This is known as
stereo matching
. Structured-light techniques use, in addition to
camera(s), a light source to project on the scene a set of light patterns that are used as
codes
for
finding correspondences between stereo images. Laser scanners use the triangulation principle
by replacing one camera with a laser source that emits a laser ray in the direction of the object
to scan. A camera from a different viewpoint captures the projected pattern.
1.2.2 Shape from Shading
Artists have reproduced, in paintings, illusions of depth using lighting and shading. Shape From
Shading (SFS) addresses the shape recovery problem from a gradual variation of shading in the
image. Image formation is a key ingredient to solve the SFS problem. In the early 1970s, Horn
was the first to formulate the SFS problem as that of finding the solution of a nonlinear first-
order Partial Differential Equation (PDE) also called the brightness equation. In the 1980s, the
authors address the computational part of the problem, directly computing numerical solutions.
Bruss and Brooks asked questions about the existence and uniqueness of solutions. According
to the Lambertian model of image formation, the gray level at an image pixel depends on the
light source direction and surface normal. Thus, the aim is to recover the illumination source
1
http://szeliski.org/Book/
