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Level Set Methods in Segmentation of SDOCT Retinal Images
Published in Ayman El-Baz, Jasjit S. Suri, Level Set Method in Medical Imaging Segmentation, 2019
N Padmasini, R Umamaheswari, Yacin Sikkandar Mohamed, Manavi D Sindal
Convex hull or convex envelope of a set X of points in the Euclidean plane or in a Euclidean space (or, more generally, in an affine space over the reals) is the smallest convex set that contains X. For instance, when X is a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around X [71]. Formally, the convex hull may be defined as the intersection of all convex sets containing X or as the set of all convex combinations of points in X. With the latter definition, convex hulls may be extended from Euclidean spaces to arbitrary real vector spaces; they may also be generalized further, to oriented matroids [72], the algorithmic problem of finding the convex hull of a finite set of points in the plane or other low dimensional Euclidean spaces is one of the fundamental problems of computational geometry.
QP Subject to Linear Equality Constraints
Published in Michael J. Best, Quadratic Programming with Computer Programs, 2017
Let x1 and x2 be any two points. For σ between 0 and 1, σx1 + (1 − σ)x2 is on the line segment joining x1 and x2. Such a point is called a convex combination of x1 and x2. The following characterization of a convex function relates the function value at a point to the corresponding convex combination of function values (see Figure 3.2).
Comparison of optimal linear, affine and convex combinations of metamodels
Published in Engineering Optimization, 2021
In this article, a general framework for generating optimal linear, affine and convex combinations of metamodels by minimizing the PRESS using the taxicab, Euclidean or infinity norm are presented and studied. Thus, the formulation proposed by Viana, Haftka, and Steffen (2009) is not the starting point in this article, but instead derived symbolically as a special case for affine combinations using the Euclidean norm. It is concluded from the study that convex combinations are preferable over linear as well as the established affine combinations. The risk of overfitting is less and automatic pruning of metamodels is obtained. In conclusion, it is suggested that the optimal weights of the convex combinations of metamodels be established by minimizing the Euclidean norm or the taxicab norm of the residual vector of the leave-one-out cross-validation errors, where the former formulation implies a QP-problem and the latter one is an LP-problem to be solved. For future work, it is suggested to study the performance of these two formulations for large problems, in order to establish if the QP-problem is preferable over the LP-problem or vice versa.
On new refinement of the Jensen inequality using uniformly convex functions with applications
Published in Applicable Analysis, 2023
Yamin Sayyari, Hasan Barsam, Ali Reza Sattarzadeh
Let be points, and let be coefficients such that . The sum is called the convex combination of points (with coefficients ).
A Comparative Survey of Convex Combination of Adaptive Filters
Published in IETE Journal of Research, 2023
Ángel A. Vázquez, J. Gerardo Avalos, Giovanny Sánchez, Juan C. Sánchez, Héctor Pérez
In the past five decades, several authors have proposed diverse adaptive algorithms, which range from the simple least mean square (LMS) to complex alternatives, such as the affine projection (AP) algorithm and the recursive least mean square (RLS) algorithm [1], with the aim of improving the performance of advanced signal processing applications [2,3]. Some of their properties include convergence speed, steady-state MSE, tracking ability, robustness, misadjustment and computational complexity. For example, the LMS algorithm exhibits low computational complexity and robustness resulting in an attractive solution for some applications. However, its low convergence speed limits its use in some real-time applications. To overcome this limitation, several authors have proposed variants to improve its performance. One of these variants is based on the affine projection (AP) algorithm which improves convergence speed when compared with the LMS algorithm at the cost of increasing its computational complexity since this algorithm requires a large number of input vectors (called projection order – L) to update the weight vector. In addition, the steady-state mean square error becomes worse when the projection order is increased significantly. Another alternative can be found in the approaches based on RLS algorithms which exhibit the fastest convergence speed. However, its computational complexity is also the highest and it presents critical stability problems. Recently, convex combinations have attracted much interest in the development of new alternative adaptive filtering schemes. In particular, these solutions combine two filters with the aim of improving filter behaviour with regard to well-known tradeoffs, such as convergence speed and steady-state mean square error versus computational complexity. Currently, some of these solutions combine two filters in a simple way, i.e. without employing specific mixing parameters [4–7]. As a consequence, the performance of their proposals is still limited. To overcome this aspect, optimal convex combinations have been proposed. Some of them are described as follows: Combination of adaptive filters based on LMS algorithms