## Section: Scientific Foundations

### 3D scene modelling based on projective geometry

3d reconstruction is the process of estimating the shape and position of 3d objects
from views of these objects.
TEMICS deals more specifically with the modelling of large
scenes from monocular video
sequences.
3d reconstruction using projective geometry is by definition
an inverse problem.
Some key issues which do not have yet satisfactory solutions are
the estimation of camera parameters,
especially in the case of a moving camera. Specific problems
to be addressed are e.g. the matching of features between images,
and the modelling of hidden areas and depth discontinuities.
3d reconstruction uses theory and methods from the areas of
computer vision and projective geometry.
When the camera is modelled as a *perspective projection* ,
the *projection
equations* are :

where is a 3d point with homogeneous coordinates
in the scene reference frame ,
and where are the coordinates of its projection
on the image plane I_{i} .
The *projection matrix* P_{i} associated to the camera is defined as P_{i} = K(R_{i}|t_{i}) . It is
function of both the *intrinsic parameters* K of the camera,
and of transformations (rotation R_{i} and translation t_{i} )
called the *extrinsic parameters* and characterizing the position of
the camera reference frame with respect to the scene
reference frame .
Intrinsic and extrinsic parameters
are obtained through calibration or self-calibration procedures.
The *calibration* is the estimation of camera parameters
using a calibration pattern (objects
providing known 3d points), and images of this calibration pattern.
The *self-calibration* is the estimation of camera parameters using only
image data. These data
must have previously been matched by identifying and grouping
all the image 2d points resulting from
projections of the same 3d point.
Solving the 3d reconstruction problem is then equivalent to
searching for , given
, i.e. to solve Eqn. (1 )
with respect to coordinates
.
Like any inverse problem, 3d reconstruction is very sensitive
to uncertainty.
Its resolution requires a good accuracy for the image measurements,
and the choice of adapted
numerical optimization techniques.