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A scaling can be represented by a scaling matrix. To scale an object by a vector v = (v x, v y, v z), each point p = (p x, p y, p z) would need to be multiplied with this scaling matrix: = []. As shown below, the multiplication will give the expected result:
In other words, the matrix of the combined transformation A followed by B is simply the product of the individual matrices. When A is an invertible matrix there is a matrix A −1 that represents a transformation that "undoes" A since its composition with A is the identity matrix. In some practical applications, inversion can be computed using ...
2D computer graphics is the computer-based generation of digital images—mostly from two-dimensional models ... A scaling can be represented by a scaling matrix.
An image scaled with nearest-neighbor scaling (left) and 2×SaI scaling (right) In computer graphics and digital imaging, image scaling refers to the resizing of a digital image. In video technology, the magnification of digital material is known as upscaling or resolution enhancement.
Let X be an affine space over a field k, and V be its associated vector space. An affine transformation is a bijection f from X onto itself that is an affine map; this means that a linear map g from V to V is well defined by the equation () = (); here, as usual, the subtraction of two points denotes the free vector from the second point to the first one, and "well-defined" means that ...
For magnifying computer graphics with low resolution and few colors (usually from 2 to 256 colors), better results will be achieved by pixel art scaling algorithms such as hqx or xbr. These produce sharp edges and maintain high level of detail.
Homogeneous coordinates are ubiquitous in computer graphics because they allow common vector operations such as translation, rotation, scaling and perspective projection to be represented as a matrix by which the vector is multiplied. By the chain rule, any sequence of such operations can be multiplied out into a single matrix, allowing simple ...
It is also known as Principal Coordinates Analysis (PCoA), Torgerson Scaling or Torgerson–Gower scaling. It takes an input matrix giving dissimilarities between pairs of items and outputs a coordinate matrix whose configuration minimizes a loss function called strain, [2] which is given by (,,...,) = (, (),) /, where denote vectors in N-dimensional space, denotes the scalar product between ...