# 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.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## 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 . 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 ?
(8) It is without loss of generality to assume that [v.sub.2]([c.sub.1]) = [u.sub.2]([c.sub.1]) and [v.sub.2]([c.sub.2]) = [u.sub.2]([c.sub.2]) (otherwise, we can apply a positive affine transformation to [v.sub.2] such that the two equalities hold).
Transform [P.sub.CCT] using an affine transformation with parameters [R.sub.int] and [T.sub.int].
However, (11) can be transformed into Laplace's equation by an affine transformation [summation](x, y) [right arrow] [summation]'(x', y').
Affine transformation can offer better performance than linear method in [17], but the projection of the objects from image B to image A is still not precisely correct.
(1) Compute the first affine transformation [L.sub.1] via invoking matrix-vector multiplication and vector addition.
Finally, an affine transformation is performed on the moving image sequences (the segmented results of PET image sequences) after calculating the transformation matrixes for all of the slices.
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,
Proof of Proposition 2: First note that a positive affine transformation does not change the optimal level of self-protection in our model.
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 fractal video coding, the best matched domain block is decided after applying the proper affine transformation which gives least mean square error.
Equation 3 is often discussed as the tissue motion model in ultrasound-based strain estimation [16, 33, 35], and it is also an affine transformation. Coefficients in the affine transformation have been estimated using optical flow method [16], least square method [34], and numerical optimization method [35], However, Pan [16] proposed direct a cost function which minimizes the sum of squared differences between pre- and post-deformation sub-images, and this method has higher time cost.

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