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Two Decades of Multidimensional Systems Research and Future Trends
Published in Krzysztof Gałkowski, Jeff David Wood, Multidimensional Signals, Circuits and Systems, 2001
A polynomial or a rational function (matrix), characterizing a single-input single-output (multi-input multi-output) system, has the coefficients for parameters. The number of such free parameters defines the dimension of the space and a system with fixed parameters may be represented by a point in such a space. If the coefficients vary about the nominal values, a region in parameter space is generated. This region characterizes a family of systems instead of one fixed system. When the coefficients vary independently of each other within specified compact intervals, an intenal system is generated. A well explored case when the coefficients do not vary independently occurs when the region in parameter space is a bounded polyhedral set. A polyhedral set is formed from the intersection of a finite number of closed half-spaces and could be unbounded. A bounded polyhedral set is a convex polytope and vice versa. For an interval system, the polytope degenerates into a boxed domain or a hyper-rectangle. Extensive documentation of research results concerned with the extraction of information about the complete polytope from a very small subset of the polytope with respect to the property of stability for both continuous-time and discrete-time systems is available in several recent texts ((Kogan 1995). for example), since Kharitonov's trend-setting publications.
Image Unmixing and Segmentation
Published in Gerhard X. Ritter, Gonzalo Urcid, Introduction to Lattice Algebra, 2021
Gerhard X. Ritter, Gonzalo Urcid
There exist a multitude of different approaches to construct endmembers and endmember abundances (percentages) in mixed pixels [18, 31, 47, 48, 57, 126, 197, 317]. A large number of these approaches are based on the convex polytope model. This model is derived from the physical fact that radiance is a non-negative quantity. Therefore, the vectors formed by discrete radiance spectra are linear combinations of non-negative components and as such must lie in a convex region located in the non-negative orthant ℝ+n={x∈ℝn:x≥0}. The vertices of this convex region, which can be elements of X or vectors that are physically related to the elements of X, have proven to be excellent endmember candidates. The reason for this is based on the observation that endmembers exhibit maximal and minimal reflectances within certain bands and correspond to vertices of a high-dimensional polyhedron. The N-FINDR and MVT (minimal value transform) algorithms are two classic examples of the convex polytope based endmember selection approach [315, 57]. Since these earlier methods, researchers have developed various sophisticated endmember extraction and endmember generation techniques based on the convex polytope assumption [90, 91, 197, 317]. The WM-method described here differs from those described by Graña [101, 102] and Myers [194] as the endmembers we obtain have a physical relationship to the pixels of the hyperspectral image under consideration. The WM-method will always provide the same sets of candidate endmembers based on theoretical facts already exposed in Chapter 6. A brief comparison is made against two approaches based on convex optimization, namely vertex component analysis [197] and the minimal volume enclosing simplex [46].
Inner approximation algorithm for solving linear multiobjective optimization problems
Published in Optimization, 2021
Benson's algorithm works in stages by maintaining a ‘double description’ of an approximation of the final polytope . A convex polytope is uniquely determined as the convex hull of a finite set of points (for example, the set of vertices) and directions (in our case: ideal points) as well as the intersection of finitely many closed halfspaces (for example, halfspaces corresponding to the facets plus the ideal facet). The ‘double description’ refers to the technique keeping and maintaining both lists simultaneously. The iterative algorithm stops when the last approximation equals . At this stage, both the vertices and facets of are computed, thus the (MOLP) has been solved.
The method of codifferential descent for convex and global piecewise affine optimization
Published in Optimization Methods and Software, 2020
Suppose that is not a global minimizer of f. By definition belongs to the convex polytope (see Step 3 of the MGCD). Any convex polytope is equal to the disjoint union of the relative interiors of its faces, i.e. the relative interiors of all faces of a convex polytope are pairwise disjoint, and the polytope is equal to the union of these relative interiors (see [38], p. 61). Therefore, belongs to the relative interior of a face F of .