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For such
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o obtain a
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es crucial.
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do so, the
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Fig. 1. The adopted camera model.
2. THE SYSTEM SETUP
2.1. Camera Model
What it is normally meant with camera model is the
mathematical relationship between the position of a point
in the three-dimensional space imaged by the camera,
and the corresponding position of that point on the image
plane. The scheme of the camera model that we adopted
for our 3D reconstruction system is shown in Fig. 1,
where three reference frames are visible:
e World reference frame, attached to the imaged
scene;
e Camera reference frame, attached to the camera
system. Notice that the Z axis is the optical axis,
while the X and Y axes are parallel to the image
plane.
e Image reference frame, where the x, and y; axes
respectively define the horizontal and vertical
directions in the digital image.
The camera model consists of a set of equations that
map the 3D world-coordinates (Xy, Yu, Zu) of a generic
point P into the 2D coordinates (x; y; of its projection
onto the image plane:
e Change of reference frame from world-coordinates to
camera-coordinates:
X
cam
PaT7|Yas|-R: B,*T;
cam
Z
cam
R and T being the rotation matrix and the translation
vector, respectively.
e Perspective projection of a scene point to the image
plane (the center of projection is the center of the lens
and the projection plane is the camera CCD sensor):
X
P^ end
Yu z
cam
e Lens distortion shift of the point position p,, predicted
via perspective projection, to the actual position py on
the image plane. When standard-resolution CCD
cameras are being used, only radial distortion is
507
normally considered. In this case, in fact, the
tangential distortion can be neglected. Radial
distortion is usually approximated by a power series:
K, = (Lkr hor +.)
This series can be truncated at the fifth order (only
the first two coefficients are used) as the residual
error results as being far below 1 pixel [4].
e Change of coordinate frame from camera-coordinates
Pd(XaYa). to image-coordinates p=(x,y;). This
operation simply consists of a 2-D translation and
scale change (see Fig. 1):
X
x, =C, +++;
y A de
Ya
Gk
Yr Y d,
d, and d, being the horizontal and vertical size of an
image pixel, respectively, and (C,,C,) the image-
coordinates of the optical center OC.
As we can see from the above description, a camera
model is completely specified by a limited set of
parameters. In particular, the intrinsic parameters (f, ks,
ks, C,, Cy) incorporate the physical characteristics of the
camera, while the extrinsic parameters (R and T) define
the projective geometry, and they all are estimated
through an appropriate calibration procedure.
2.2. Multi-View Geometry
The multi-camera acquisition system is designed in such
a way to guarantee a satisfactory camera geometry for
back-projection. More precisely, the cameras are placed
far enough from each other to guarantee an accurate 3D
triangulation, and they are approximately pointed to a
common center in the scene. By doing so, in fact, we
maximize the 3D volume which is being simultaneously
imaged by all cameras.
It is well-known that the minimum number of cameras
that is required for obtaining a 3D description of the
scene is two. Increasing the number of cameras,
however, improves the precision and the reliability of the
3D reconstruction dramatically [6]. In particular, the
introduction of a third camera has been proven to provide
the most significant improvement with respect to the
binocular setup, with a minimum cost increment. The
results presented in this paper are thus obtained by using
a trinocular system.
Once the cameras are properly placed, camera
calibration is performed in order to determine the intrinsic
and extrinsic parameters of the acquisition system. In
order to do so, a set of calibration target points, whose
3D world-coordinates are known with good precision, is
used. It is worth noticing that the camera model
introduced above is used throughout the whole 3D
reconstruction chain (matching and back-projection
algorithms). This means that there is no need of image
rectification or any constraint on the camera geometry.
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B5. Vienna 1996
