Thinning is the transformation of a digital image into a simplified, but topologically equivalent image. It is a type of topological skeleton, but computed using mathematical morphology operators.
Let , and consider the eight composite structuring elements, composed by:
and the three rotations of each by , , and . The corresponding composite structuring elements are denoted .
For any i between 1 and 8, and any binary image X, define
where denotes the set-theoretical difference and denotes the hit-or-miss transform.
The thinning of an image A is obtained by cyclically iterating until convergence:
Thickening is the dual of thinning that is used to grow selected regions of foreground pixels. In most cases in image processing thickening is performed by thinning the background [1]
where denotes the set-theoretical difference and denotes the hit-or-miss transform, and is the structural element and is the image being operated on.
In computer graphics, alpha compositing is the process of combining one image with a background to create the appearance of partial or full transparency. It is often useful to render picture elements (pixels) in separate passes or layers and then combine the resulting 2D images into a single, final image called the composite. Compositing is used extensively in film when combining computer-rendered image elements with live footage. Alpha blending is also used in 2D computer graphics to put rasterized foreground elements over a background.
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue, although according to the Bourbaki group they were first introduced by Frigyes Riesz. Lp spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, finance, engineering, and other disciplines.
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of the next. Associated to a chain complex is its homology, which describes how the images are included in the kernels.
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In mathematics, there are usually many different ways to construct a topological tensor product of two topological vector spaces. For Hilbert spaces or nuclear spaces there is a simple well-behaved theory of tensor products, but for general Banach spaces or locally convex topological vector spaces the theory is notoriously subtle.
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In mathematical morphology, hit-or-miss transform is an operation that detects a given configuration in a binary image, using the morphological erosion operator and a pair of disjoint structuring elements. The result of the hit-or-miss transform is the set of positions where the first structuring element fits in the foreground of the input image, and the second structuring element misses it completely.
In multilinear algebra, the tensor rank decomposition or canonical polyadic decomposition (CPD) is one generalization of the matrix singular value decomposition (SVD) to tensors, which have found application in statistics, signal processing, computer vision, computer graphics, psychometrics, linguistics and chemometrics. The tensor rank decomposition was introduced by Hitchcock in 1927 and later rediscovered several times, notably in psychometrics. For this reason, the tensor rank decomposition is sometimes historically referred to as PARAFAC or CANDECOMP.
In algebraic geometry, a derived scheme is a pair consisting of a topological space X and a sheaf of commutative ring spectra on X such that (1) the pair is a scheme and (2) is a quasi-coherent -module. The notion gives a homotopy-theoretic generalization of a scheme.
In mathematics, the base change theorems relate the direct image and the pull-back of sheaves. More precisely, they are about the base change map, given by the following natural transformation of sheaves:
Anyon fusion is the process by which multiple anyons behave as one larger composite anyon. Anyon fusion is essential to understanding the physics of non-abelian anyons and how they can be used in quantum information.
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In mathematics, the injective tensor product of two topological vector spaces was introduced by Alexander Grothendieck and was used by him to define nuclear spaces.
