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Glossary of Computer Vision Terms
Published in Edward R. Dougherty, Digital Image Processing Methods, 2020
Robert M. Haralick, Linda G. Shapiro
An octree is a data in a tree data structure which represents a function defined as a three-dimensional space volume. Each node of the octree represents a cube subset of the volume. The root node of the quadtree represents the entire volume. If all the voxels represented by a node have the same function values, then the value of the node is their function value. Such a node is called a pure node. If the node is a mixed or impure node, then the cube represented by the node is partitioned into eight volume octants and the node has eight children, one child for each octant. If the function is binary, then the corresponding octree is called a binary octree. The binary octree is useful for representing three- dimensional volumes. If the function being represented is a non-binary, such as a real or integer valued function, then the corresponding octree is called a gray scale octree.
Characteristic classes
Published in Peter B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, 2018
be the corresponding eigenbundles; Π±q(x)=ℜ(π±(x)). Since Tr(q(x)) is a continuous integer valued function, it is constant. It vanishes since q(−x) = −q(x). Thus dim(Π±q)=12k is constant so the Π±q define smooth complementary sub-bundles; () Sm×Ck=Π+q⊕Π−q.
On the Extraction of Local Material Parameters of Metamaterials from Experimental or Simulated Data
Published in Filippo Capolino, Theory and Phenomena of Metamaterials, 2017
In these formulas k0=øε0μ0 is the free space wave number, and η0=μ0/ε0andη=μ0/ε0εm are wave impedances of free space and of the slab host medium, respectively. Parameter m in Equation 11.2 is the integer valued function of frequency. In the region ∣Re(qd)∣ < π/2, it must be m = 0. Also, parameter m can be found from the requirement of smoothness for the frequency dependence of extracted parameters [4] or from the requirement of the correct sign for the imaginary part of εeff and μeff, as was suggested in [31]. Notice, however, that the requirement of the “correct sign” [31] is senseless when applied to EMP, which are by definition nonlocal.
A survey of fundamental operations on discrete convex functions of various kinds
Published in Optimization Methods and Software, 2021
In this section we deal with the operations related to integral conjugacy for integer-valued discrete convex functions. For an integer-valued function , we define a function on by where is the inner product of and . This function is referred to as the integral conjugate of f, or the integral Legendre–Fenchel transform of f. The function takes values in , since and are integers (or ) for all and and is finite for some x by the assumption of . That is, we have . This allows us to apply the transformation (82) to to obtain . This function is called the integral biconjugate of f.
Relative Morse index and multiple homoclinic orbits for a nonperiodic Hamiltonian system
Published in Applicable Analysis, 2023
From (22), we have Thus, it is suffices to prove Let . Since is a finite integer-valued function, by (22), we deduce that there exists such that Arguing as in Lemma 2.2, for each p, we know that there is a basis of , and a sequence in as such that By the definition of index function and (24), we obtain Fix . Since and are bounded, there exist with such that Note that is bounded and is a compact operator. From (23), we deduce that Taking the limit in (25) as , we have This means, by Lemma 2.2 and Definition 2.1,
Index theory and multiple solutions for asymptotically linear wave equation
Published in Applicable Analysis, 2023
By Step 1, we have . Thus, it is suffices to prove . Since is a finite integer valued function, there exists such that Let . Assume that are the negative eigenvalues of . and are the corresponding eigenfunctions. That is, Fix , since are bounded in , there exist such that In particular, . By the definitions of and (23), we obtain Assume , . Then in , in . Recall that is finite dimensional and is a compact operator, we have It follows from that . Then . Thus, . Taking the limit in (23) as , we obtain Thus, . We finish this step.