Rescaled

Rescaled is the past participle of rescale and is used to describe an object, dataset, or signal that has undergone a scale transformation. This can involve changing the units, magnitudes, or spatial dimensions by applying scale factors or affine transformations.

In mathematics and geometry, rescaling typically means modifying coordinates or values by a scale factor. If x is a vector in R^n and D is a diagonal matrix with positive entries d_i, a rescaled version is x' = D x. More generally, a rescaled object may be expressed as x' = A x + b, where A is a linear transformation and b is a translation. Rescaling can preserve certain properties, such as ratios or relative structure, while altering absolute magnitudes.

In data processing and statistics, rescaling describes transforming data to a common scale. Common methods include min–max normalization, x' = (x − min) / (max − min), which maps values to [0, 1], and standardization, x' = (x − μ) / σ, which centers data around zero with unit variance. These techniques improve comparability across variables with different units or ranges and can enhance the performance of learning algorithms. Rescaled data should be interpreted in the context of the chosen scale and, if needed, the original scale can be recovered via inverse transformations.

In computer graphics and image processing, rescaling refers to resizing images or 3D models, often using interpolation (nearest-neighbor, bilinear, bicubic) to estimate new pixel or vertex values. This operation changes spatial resolution and may affect detail and sharpness.

Overall, rescaled objects have been transformed to a different scale for analysis, comparison, or practical purposes, with attention paid to preserve meaningful relationships and, when possible, to allow inverse recovery.