affine transformation

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affine transformation

[ə′fīn ‚tranz·fər′mā·shən]
(mathematics)
A function on a linear space to itself, which is the sum of a linear transformation and a fixed vector.

Affine Transformation

 

a pointwise mutually single-valued mapping of a plane (space) onto itself in which straight lines are transformed into straight lines. If a Cartesian coordinate system is given in a plane, then any affine transformation of this plane can be defined by means of a so-called nonsingu-lar linear transformation of the coordinates x and y of the points of this plane. Such a transformation is given by the formulas x’ = ax + by + p and y’ = cx + dy + q with the additional requirement that Affine Transformation. Analogously, any affine transformation of a space can be defined by means of nonsingular linear transformations of the coordinates of points in space. The set of all affine transformations of a plane (space) into itself forms a group of affine transformations. This denotes, in particular, that the successive execution of two affine transformations is equivalent to some single affine transformation.

Examples of affine transformations are the orthogonal transformation—a motion of a plane or space or motion with a reflection; the transformation of similitude; and uniform “compression.” A uniform “compression” with coefficient k of the plane π toward a straight line a located in it is a transformation in which the points of a remain stationary and every point M of the plane π which does not lie on a is displayed along a ray passing through M perpendicularly to a to a point M’ such that the ratio of the distances from M and M’ to a is equal to k. Analogously, one defines a uniform “compression” of space to a plane. Each affine transformation of the plane can be obtained by performing a certain orthogonal transformation and a successive “compression” on some two perpendicular lines. Any affine transformation of space can be accomplished by means of a certain orthogonal transformation and successive “compressions” on some three mutually perpendicular lines. In an affine transformation, parallel lines and planes are transformed into parallel lines and planes. The properties of the affine transformation are widely used in various branches of mathematics, mechanics, and theoretical physics. Thus, in geometry the affine transformation is used for the so-called affine classification of figures. In mechanics, it is used in the study of small deformations of continuous media; in such deformations, small elements of the medium in the first approximation undergo affine transformations.

REFERENCES

Muskhelishvili, N. I. Kurs analiticheskoi geometrii, 4th ed. Moscow, 1967.
Aleksandrov, P. S. Lektsii po analiticheskoi geometrii. Moscow, 1968.
Efimov, N. V. Vysshaia geometriia, 4th ed. Moscow, 1961.
E. G. POZNIAK

affine transformation

(mathematics)
A linear transformation followed by a translation. Given a matrix M and a vector v,

A(x) = Mx + v

is a typical affine transformation.
References in periodicals archive ?
For any asymmetric claims problem, we can always find a positive affine transformation to transform it into a new claims problem, in which the two players' claims in the new problem are symmetric.
Section 3 briefly repeats the affine transformation of van der Pauw and specifies it for (100)-planes of cubic crystals, like silicon.
The work in [42] uses differential power analysis to attack SHA-1 module; thus we take a method to protect affine transformations. However, the countermeasure mentioned above is theoretical; we should be able to implement and verify it on hardware.
Finally, let's call G the composition of [F.sub.k] and [W.sub.j], G=[F.sub.k] [omicron] [W.sub.j], assuring that G is also an affine transformation, as it is a composition of two affine transformations, thus getting,
Transformations applied in augmentation process are illustrated in Figure 2, where the first row represents resulting images obtained by applying affine transformation on the single image; the second row represents images obtained from perspective transformation against the input image and the last row visualizes the simple rotation of the input image.
In particular, affine transformations have been considered in order to model geometric distortions such as scaling, rotation, and shearing between the original and the copied patches.
Given an initial solution of the ROI ([S.sup.0]) belonging to the target image, {I, [S.sup.0]} are aligned into the selected atlas as a reference using the same affine transformation method that was used to construct the probabilistic atlas.
where T is the linear operator of the average face surface, [[??].sub.c] is the affine transformations (from [[??].sub.c]) from the IE algorithm, and x is the vector containing the vertex positions of the resultant identity-embodied artistic face.
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Define an affine transformation of the plane and apply it to the set.
The traditional image registration is based on classic features such as the Harris corner and the scaled-invariant feature transform (SIFT) corner, which are both weak to affine transformations. In our research, we introduce affine invariant features to improve subpixel image registration, which considerably reduces the number of mismatched points and hence makes traditional image registration more efficient and more accurate.