By following [14, 20, 32, 33], we compute affine transformations
for patch [p.sub.i] to approximate various geometric changes and generate set ([p.sub.ia], [p.sub.ib], [??]) for learning.
IN V requires that the (physical) solution outcome be the same under positive affine transformations
of players' utilities.
Note that an affine transformation
is definitely not a conformal transformation: conformal transformations preserve the angles between two curves, whereas affine transformations
do not, because they skew the geometry.
(1) Computing affine transformations
, that is, [bar.y] = Ay + b, where A is a matrix and b is a vector
The group of affine transformations
on G, Aff(G) = G Aut(G), admits a natural left action on G:
Contractive similarities are affine transformations
of the form [F.sub.i] = [r.sub.i] [D.sub.i] x+[b.sub.i], where r is a scalar such that 0< [r.sub.i] < 1, [D.sub.i] is the matrix of rotations and [b.sub.i] is a vector in [R.sup.2] (Barnsley 1993).
Caption: Figure 2: Image transformations used for augmentation: (a) affine transformations
; (b) perspective transformations; (c) rotations.
In particular, features-based methods (based on SIFT or others feature descriptors) have been demonstrated so far to be effective against copy-move attack and constitute one of the most promising techniques addressing this issue because they are resistant to JPEG compression, scaling, rotation, and affine transformations
and also to digital/analog/digital conversion [10, 11].
Coloring the cells of the grid, stereo learners revisit arithmetic and algebra, meet tessellations, polyominoes, moires, affine transformations
, cellular automata, methods of computer graphics and animation.
It covers various topics including representing real numbers, vectors and points, linear transformations and matrices, affine transformations
, orientation representation, interpolation, viewing and projection, geometry and programmable shading, lighting, rasterization, random numbers, intersection testing, and rigid-body dynamics.
For this purpose, the affine transformations
The geometric identity of each 3D face is encoded as 3D affine transformations
between the face and the average human face.