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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].
Polytope-based tolerance analysis with consideration of form defects and surface deformations
Published in International Journal of Computer Integrated Manufacturing, 2021
Zhiqiang Zhang, Jianhua Liu, Laurent Pierre, Nabil Anwer
The tolerances applied to the corresponding features are summarized in Table 3. This subsection conducts the conventional polytope-based tolerance analysis, i.e. ideal surfaces. Based on the definitions of the polytope model, each polytope from Eq. (4) can be obtained. As we can see, there are three parallel chains: 2.0–2.3–1.3-1.0, 2.0–2.2-2.1–1.0, and 2.0–2.1–1.1-1.0. The polytope for each chain can be obtained by the minkowski sum of the corresponding polytopes, and the polytope in Eq. (4) can be obtained by the intersection of the three chains. This procedure is shown in Figure 13.