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The images below present a simple opening-by-reconstruction example which extracts the vertical strokes from an input text image. Since the original image is converted from grayscale to binary image, it has a few distortions in some characters so that same characters might have different vertical lengths.
Mathematical Morphology was developed in 1964 by the collaborative work of Georges Matheron and Jean Serra, at the École des Mines de Paris, France.Matheron supervised the PhD thesis of Serra, devoted to the quantification of mineral characteristics from thin cross sections, and this work resulted in a novel practical approach, as well as theoretical advancements in integral geometry and ...
A category C consists of two classes, one of objects and the other of morphisms.There are two objects that are associated to every morphism, the source and the target.A morphism f from X to Y is a morphism with source X and target Y; it is commonly written as f : X → Y or X Y the latter form being better suited for commutative diagrams.
In mathematics, many types of algebraic structures are studied. Abstract algebra is primarily the study of specific algebraic structures and their properties. Algebraic structures may be viewed in different ways, however the common starting point of algebra texts is that an algebraic object incorporates one or more sets with one or more binary operations or unary operations satisfying a ...
The composition of regular maps is again regular; thus, algebraic varieties form the category of algebraic varieties where the morphisms are the regular maps. Regular maps between affine varieties correspond contravariantly in one-to-one to algebra homomorphisms between the coordinate rings: if f : X → Y is a morphism of affine varieties ...
See algebraic curves for examples. The classification of smooth curves can be refined by the degree for projectively embedded curves, in particular when restricted to plane curves. Note that all complete smooth curves are projective in the sense that they admit embeddings into projective space, but for the degree to be well-defined a choice of ...
For example, the unit sphere is an algebraic surface, as it may be defined by the implicit equation x 2 + y 2 + z 2 − 1 = 0. {\displaystyle x^{2}+y^{2}+z^{2}-1=0.} A surface may also be defined as the image , in some space of dimension at least 3, of a continuous function of two variables (some further conditions are required to ensure that ...
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems. Classically, it studies zeros of multivariate polynomials ; the modern approach generalizes this in a few different aspects.