Image Processing Reference
In-Depth Information
accurate version of this model by assuming a different affine transform, and thus a different d , for
each facet of the mesh. This approximation neglects depth variation across individual facets, rather
than across the whole surface. Under this assumption, the projection of a 3D point q i lying on facet
f of the mesh can be expressed as
d f u i
v i
A I 2 × 2
0 q i +
0 ,
where d f accounts for the average depth of facet f . Here, without loss of generality, we expressed
the 3D point in the camera referential, and therefore replaced R with the first two rows of the
3 identity matrix and the translation with a zero vector. Note that this does not prevent us from
accounting for camera motion. It simply means that it will be interpreted as a rigid motion of the
object of interest.
Under the full perspective model, the projection of a 3D point q i
is written as
u i
v i
A I 3 × 3 q i +
0 ,
d i
where the matrix of internal camera parameters A is now a 3
3 matrix, and each point i has a
different depth factor d i .
Detecting feature points in images has received enormous attention in the Computer Vision commu-
nity. For most template-based approaches, feature points are typically detected with either the SIFT
keypoints detector Lowe [ 2004 ] or Harris's corners detector Harris and Stephens [ 1988 ]. Once fea-
ture points have been detected in two images, they need to be matched to produce correspondences.
When using SIFT, this can be done by a simple dot-product between specific vector representa-
tions of the feature points. For Harris's corners, methods based on Randomized Trees have proved
efficient Lepetit and Fua [ 2006 ]. From a large set of views obtained by applying random affine trans-
formations to a reference image, a tree that models the relationships between neighboring keypoints
is built. Each leaf-node of the tree then corresponds to a specific keypoint, and matching can be
done by dropping the feature points of a new image down the tree.
Another way to establish correspondences is to first tackle the 2D non-rigid image regis-
tration problem. Non-rigid image registration aims at finding a transformation between two im-
ages of the same surface undergoing different deformations. Different parameterizations have been
proposed to represent the transformation, such as RBFs Bartoli and Zisserman [ 2004 ], thin-plate
splines Bookstein [ 1989 ], or 2D meshes Pilet et al. [ 2008 ]. Since the resulting warp is defined over
the entire image, discrete correespondences can then be obtained by sampling it. Note that, while
2D non-rigid registration can be thought of as related to 3D non-rigid reconstruction, we believe
Search WWH ::

Custom Search