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 are sufficient.

The geometric identity of each 3D face is encoded as 3D

affine transformations between the face and the average human face.