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In multilinear algebra, mode-m flattening [1] [2] [3], also known as matrixizing, matricizing, or unfolding, [4] is an operation that reshapes a multi-way array into a matrix denoted by [] (a two-way array). Matrixizing may be regarded as a generalization of the mathematical concept of vectorizing.
After flattening, arrays are represented as single-dimensional value vector V containing scalar elements, alongside auxiliary information recording the nested structure, typically in the form of a boolean flag vector F. The flag vector indicates, for the corresponding element in the value vector, whether it is the beginning of a new segment.
In Matlab/GNU Octave a matrix A can be vectorized by A(:). GNU Octave also allows vectorization and half-vectorization with vec(A) and vech(A) respectively. Julia has the vec(A) function as well. In Python NumPy arrays implement the flatten method, [note 1] while in R the desired effect can be achieved via the c() or as.vector() functions.
This reshaping is sometimes called matrixizing, matricizing, flattening or unfolding in the literature. A standard choice for the bijections μ 1 , μ 2 {\displaystyle \mu _{1},\ \mu _{2}} is the one that is consistent with the reshape function in Matlab and GNU Octave, namely
Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution respectively. Other terms used are ellipticity , or oblateness . The usual notation for flattening is f {\displaystyle f} and its definition in terms of the semi-axes a {\displaystyle a} and b {\displaystyle b} of ...
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 ...
An increase of Laravel's userbase and popularity lined up with the release of Laravel 3. [1] Laravel 4, codenamed Illuminate, was released in May 2013. It was made as a complete rewrite of the Laravel framework, migrating its layout into a set of separate packages distributed through Composer, which serves as an application-level package manager.
In image processing, a kernel, convolution matrix, or mask is a small matrix used for blurring, sharpening, embossing, edge detection, and more. This is accomplished by doing a convolution between the kernel and an image. Or more simply, when each pixel in the output image is a function of the nearby pixels (including itself) in the input image ...