SkalierungAddition

SkalierungAddition is a term used in German-language texts to describe a simple affine transformation consisting of two operations: scaling (Skalierung) and addition (Addition). The basic form maps an input value x to a new value y = s · x + o, where s is the scaling factor and o is the offset. This combination is widely used to adjust the magnitude and the baseline of data or signals.

In one dimension, the transformation is invertible when s ≠ 0, with the inverse given by x = (y − o) / s. The inverse has parameters s' = 1/s and o' = −o/s. In higher dimensions, SkalierungAddition generalizes to y = Sx + o, where S is a scaling (often diagonal) matrix and o is an offset vector. The order of operations matters: scaling before addition yields y = Sx + o, whereas addition before scaling gives y = S(x + o) = Sx + So, which has a different offset.

Composition of two SkalierungAddition steps produces another affine transformation: y = S1(S2x + o2) + o1 = (S1S2)x + (S1o2 + o1). This closure under composition makes the concept central in linear algebra and graphics.

Applications include data normalization and standardization, sensor calibration, image processing for contrast and bias adjustments, and as the linear component of neural network layers (weights and biases implement an affine transformation). Related concepts are affine transformations, linear transformations, and normalization procedures. Variants and special cases arise from choosing different orders of operations or restricting S to diagonal matrices for per-coordinate scaling.