Topic
Biorthogonal system
About: Biorthogonal system is a research topic. Over the lifetime, 2190 publications have been published within this topic receiving 32209 citations.
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TL;DR: A procedure is proposed which efficiently computes so-called connectivity matrices for biorthogonal basis functions and the applicability of the theoretical findings is demonstrated with several numerical experiments.
Abstract: In this paper, the discretization of a non-symmetric elliptic obstacle problem with h p -adaptive H 1 ( Ω ) -conforming finite elements is discussed. For this purpose, a higher-order mixed finite element discretization is introduced where the dual space is discretized via biorthogonal basis functions. The h p -adaptivity is realized via automatic adaptive mesh refinement based on a residual a posteriori error estimation which is also derived in this paper. The use of biorthogonal basis functions leads to unilateral box constraints and componentwise complementarity conditions enabling the highly efficient application of a quadratically converging semi-smooth Newton scheme, which can be modified to ensure global convergence. h p -adaptivity usually implies meshes with hanging nodes and varying polynomial degrees which have to be handled appropriately within the H 1 ( Ω ) -conforming finite element discretization. This is typically done by using so-called connectivity matrices. In this paper, a procedure is proposed which efficiently computes these matrices for biorthogonal basis functions. Finally, the applicability of the theoretical findings is demonstrated with several numerical experiments.
31 citations
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TL;DR: This paper proposes to combine the HT framework with the decimated biorthogonal Haar (Bi-Haar) transform instead of the classical Haar, and benefits from the regular Bi- Haar filter bank to gain a smooth estimate while always maintaining a low computational complexity.
Abstract: Methods based on hypothesis tests (HTs) in the Haar domain are widely used to denoise Poisson count data. Facing large datasets or real-time applications, Haar-based denoisers have to use the decimated transform to meet limited-memory or computation-time constraints. Unfortunately, for regular underlying intensities, decimation yields discontinuous estimates and strong "staircase" artifacts. In this paper, we propose to combine the HT framework with the decimated biorthogonal Haar (Bi-Haar) transform instead of the classical Haar. The Bi-Haar filter bank is normalized such that the p-values of Bi-Haar coefficients (pBH) provide good approximation to those of Haar (pH) for high-intensity settings or large scales; for low-intensity settings and small scales, we show that pBH are essentially upper-bounded by pH. Thus, we may apply the Haar-based HTs to Bi-Haar coefficients to control a prefixed false positive rate. By doing so, we benefit from the regular Bi-Haar filter bank to gain a smooth estimate while always maintaining a low computational complexity. A Fisher-approximation-based threshold imple- menting the HTs is also established. The efficiency of this method is illustrated on an example of hyperspectral-source-flux estimation.
31 citations
TL;DR: In this article, an approximate method for solving a system of nonlinear Volterra integro-differential equations is proposed, which involves the use of biorthogonal systems in adequate spaces of continuous functions associated with such a system.
31 citations
TL;DR: In this paper, a large deviations principle for the empirical measures of a class of biorthogonal and multiple orthogonal polynomial ensembles was proved for disordered bosons.
Abstract: We prove a large deviations principle for the empirical measures of a class of biorthogonal and multiple orthogonal polynomial ensembles that includes biorthogonal Laguerre, Jacobi, and Hermite ensembles, the matrix model of Lueck, Sommers, and Zirnbauer for disordered bosons, the Stieltjes-Wigert matrix model of Chern-Simons theory, and Angelesco ensembles.
31 citations
TL;DR: The various performance metrics like Ratio of Edge pixels to size of image (REPS), peak signal to noise ratio (PSNR) and computation time are compared for various wavelets for edge detection and biorthogonal wavelet bior1.3 performs well in detecting the edges with better quality.
31 citations