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You searched for subject:(geometric lattice equivalence). Showing records 1 – 2 of 2 total matches.

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University of Alberta

1. Sebestyen, Mark D. Jacobi Theta and Dedekind Eta Function Identities Via Geometrical Lattice Equivalence.

Degree: MS, Department of Mathematical and Statistical Sciences, 2013, University of Alberta

Geometrical lattice equivalences are used to generate over 100 new quadratic identities involving classical modular forms, Jacobi theta functions, θ2, θ3, θ4, and the Dedekind eta function η. Generalizations are examined and a seemingly new observation on the nature of η is noted.

Subjects/Keywords: geometric lattice equivalence; Jacobi Theta function; Dedekind Eta Function

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Sebestyen, M. D. (2013). Jacobi Theta and Dedekind Eta Function Identities Via Geometrical Lattice Equivalence. (Masters Thesis). University of Alberta. Retrieved from https://era.library.ualberta.ca/files/vm40xs89b

Chicago Manual of Style (16th Edition):

Sebestyen, Mark D. “Jacobi Theta and Dedekind Eta Function Identities Via Geometrical Lattice Equivalence.” 2013. Masters Thesis, University of Alberta. Accessed August 05, 2020. https://era.library.ualberta.ca/files/vm40xs89b.

MLA Handbook (7th Edition):

Sebestyen, Mark D. “Jacobi Theta and Dedekind Eta Function Identities Via Geometrical Lattice Equivalence.” 2013. Web. 05 Aug 2020.

Vancouver:

Sebestyen MD. Jacobi Theta and Dedekind Eta Function Identities Via Geometrical Lattice Equivalence. [Internet] [Masters thesis]. University of Alberta; 2013. [cited 2020 Aug 05]. Available from: https://era.library.ualberta.ca/files/vm40xs89b.

Council of Science Editors:

Sebestyen MD. Jacobi Theta and Dedekind Eta Function Identities Via Geometrical Lattice Equivalence. [Masters Thesis]. University of Alberta; 2013. Available from: https://era.library.ualberta.ca/files/vm40xs89b


University of Illinois – Urbana-Champaign

2. Sharif, Behzad. Distortion-optimal parallel MRI with sparse sampling: from adaptive spatio-temporal acquisition to self-calibrating reconstruction.

Degree: PhD, 1200, 2010, University of Illinois – Urbana-Champaign

In this dissertation, we address several inverse problems associated with multi-channel sampling and reconstruction that pertain to parallel magnetic resonance imaging (pMRI). The first part of this dissertation addresses adaptive design of spatio-temporal acquisition and reconstruction in model-based pMRI wherein the signal model is a sparse support. We develop a highly-accelerated real-time dynamic MRI technique, dubbed PARADISE, which incorporates a physiologically-driven sparse support model in the joint spatial domain and temporal frequency dimension. The imaging scheme gains its acceleration from: (i) sparsity of the support model; and (ii) the redundancy in data acquired by the parallel receiver coils. The PARADISE adaptation procedure ensures that maximally compressed MR data is acquired by optimally exploiting the degrees of freedom in the joint k-t sampling space, thereby enabling high accelerations and quality in the cine reconstruction stage. We propose and verify the efficacy of a geometric multi-channel sampling design algorithm that does not require explicit knowledge of the channel characteristics. Accompanied by a customized pulse sequence, the fast semi-blind acquisition design technique enables streamlined implementation of the method in a clinical setting. Moreover, the unified multi-channel sampling framework explicitly accounts for speed limitations of gradient encoding, provides performance guarantees on achievable image quality both in terms of noise gain and aliasing distortion, and allows for analysis of the method's robustness to model mismatch. We present in-vivo results demonstrating the feasibility of the PARADISE scheme  – and its distinctive features and effectiveness  – for high resolution non-gated cardiac imaging during a short breath-hold. The second part of the dissertation addresses the problems of blind and nonblind perfect inversion of multi-channel multi-rate systems. Driven by applications in multi-sensor imaging systems such as pMRI, we focus on systems wherein each channel is subsampled relative to the Nyquist rate but the overall multi-channel system is oversampled. We address the feasibility of perfect reconstruction (PR) using short finite impulse response (FIR) synthesis filters given an oversampled but otherwise general FIR analysis filter bank (FB). We provide prescriptions for the shortest filter length of the synthesis bank that would guarantee PR and, in addition, study the requirements for achieving near-optimal noise performance. Next, we address the problem of multi-channel perfect interpolation (PI) by building upon the developed framework for the multi-channel PR problem. We present the theory and algorithms for identifying a FIR multi-input multi-output interpolation bank that achieves PI both with and without the knowledge of the channel characteristics. The theory developed for the latter case, called the blind PI problem, is in turn used to develop a self-calibrating algorithm, dubbed ACSIOM, for blind identification of the interpolation FB with… Advisors/Committee Members: Bresler, Yoram (advisor), Bresler, Yoram (Committee Chair), Liang, Zhi-Pei (committee member), Kamalabadi, Farzad (committee member), Sutton, Bradley P. (committee member), Do, Minh N. (committee member).

Subjects/Keywords: magnetic resonance; Magnetic resonance imaging (MRI); parallel magnetic resonance imaging (MRI); parallel imaging; dynamic magnetic resonance imaging (MRI); cardiac magnetic resonance imaging MRI; real-time magnetic resonance imaging MRI; image formation; nongated; model-based; patient-adaptive; k-t sampling; sensitivity encoding; multi-channel; filter banks; multi-rate systems; minimum filter length; frame theory; dual frame; oversampled; subsampled; generic; perfect reconstruction; perfect interpolation; multi-channel interpolation; distortion free; distortion optimal; aliasing free; self-calibrating; auto-calibrated; image reconstruction; blind identification; blind reconstruction; self calibration; Papoulis sampling; multi-channel sampling; aliasing error; geometric factor; equivalence class; lattice sampling; dual lattice; time-sequential; nonbandlimited; adaptive acquisition; sparse sampling; compressive sampling; spatio-temporal acquisition

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Sharif, B. (2010). Distortion-optimal parallel MRI with sparse sampling: from adaptive spatio-temporal acquisition to self-calibrating reconstruction. (Doctoral Dissertation). University of Illinois – Urbana-Champaign. Retrieved from http://hdl.handle.net/2142/17056

Chicago Manual of Style (16th Edition):

Sharif, Behzad. “Distortion-optimal parallel MRI with sparse sampling: from adaptive spatio-temporal acquisition to self-calibrating reconstruction.” 2010. Doctoral Dissertation, University of Illinois – Urbana-Champaign. Accessed August 05, 2020. http://hdl.handle.net/2142/17056.

MLA Handbook (7th Edition):

Sharif, Behzad. “Distortion-optimal parallel MRI with sparse sampling: from adaptive spatio-temporal acquisition to self-calibrating reconstruction.” 2010. Web. 05 Aug 2020.

Vancouver:

Sharif B. Distortion-optimal parallel MRI with sparse sampling: from adaptive spatio-temporal acquisition to self-calibrating reconstruction. [Internet] [Doctoral dissertation]. University of Illinois – Urbana-Champaign; 2010. [cited 2020 Aug 05]. Available from: http://hdl.handle.net/2142/17056.

Council of Science Editors:

Sharif B. Distortion-optimal parallel MRI with sparse sampling: from adaptive spatio-temporal acquisition to self-calibrating reconstruction. [Doctoral Dissertation]. University of Illinois – Urbana-Champaign; 2010. Available from: http://hdl.handle.net/2142/17056

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