Towards a unified treatment of 3D display using partially coherent light

Thesis (S.M.)--Massachusetts Institute of Technology, School of Architecture and Planning, Program in Media Arts and Sciences, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 111-120).This thesis develops a novel method of decomposing a 3D phase space description of light into multiple partially coherent modes, and applies this decomposition to the creation of a more flexible 3D display format. Any type of light, whether it is completely coherent, partially coherent or incoherent, can be modeled either as a sum of coherent waves or as rays. A set of functions, known as phase space functions, provide an intuitive model for these waves or rays as they pass through a 3D volume to a display viewer's eyes. First, this thesis uses phase space functions to mathematically demonstrate the limitations of two popular 3D display setups: parallax barriers and coherent holograms. Second, this thesis develops a 3D image design algorithm based in phase space. The "mode-selection" algorithm can find an optimal holographic display setup to create any desired 3D image. It is based on an iterative algebraic-rank restriction process, and can be extended to model light with an arbitrary degree of partial coherence. Third, insights gained from partially coherent phase space representations lead to the suggestion of a new form of 3D display, implemented with multiple time-sequential diffracting screens. The mode-selection algorithm determines an optimal set of diffracting screens to display within the flicker-fusion rate of a viewer's eye. It is demonstrated both through simulation and experiment that this time-sequential display offers improved performance over a fixed holographic display, creating 3D images with increased intensity variation along depth. Finally, this thesis investigates the tradeoffs involved with multiplexing a holographic display over time with well-known strategies of multiplexing over space, illumination angle and wavelength. The examination of multiplexing tradeoffs is extended into the incoherent realm, where comparisons to ray-based 3D displays can hopefully offer a more unified summary of the limitations of controlling light within a volume.by Roarke Horstmeyer.S.M

Efficient alternating gradient-type algorithms for the approximate non-negative matrix factorization problem

The approximate non-negative matrix factorization problem (ANMF) seeks to interpret a given set of non-negative data vectors, where the non-negativity is physically meaningful, through the extraction of a sufficiently small set of non-negative "features". Each data vector is then approximated by a non-negative composition of the extracted features. Mathematically, this process is equivalent to approximating a non-negative matrix A by the product of two non-negative matrices, i.e. A ≈ WH, where W ∈ realsm xp and H ∈ reals pxn, and where p is typically chosen to be significantly smaller than m and n. A standard approach for solving the ANMF problem is to use a alternating gradient-type algorithm that keeps the iterates strictly positive. Lee and Seung have proposed what is arguably the most popular of these alternating gradient approaches. In this study we propose several variations of this particular Lee and Seung algorithm with notably improved performance. One of the proposed algorithms optimizes the step-length parameter to achieve greater reduction in the objective function, while another uses different weights derived from the first order necessary conditions of the ANMF problem. The latter has exhibited better empirical performance than the former, while both perform better in practice than the original Lee and Seung algorithm. Theoretical questions concerning the convergence of these feasible alternating gradient-type algorithms have remained unresolved. Hence, from an optimization point of view, these approaches may be seen as unsatisfactory. Our research has contributed to solving these theoretical questions by proving that a class of these algorithms will converge to a continuum, and in specific cases, to a KKT point. These convergence results are applicable to multiple variations of the popular Lee and Seung algorithm. A low-rank Singular Value Decomposition (SVD) of a given matrix A is an optimal approximation under the Frobenius norm whose fundamental subspaces are most relevant within the given measure. In this study we utilize these subspaces to produce a compact reformulation of the ANMF problem to a lower dimensional optimization problem. We introduce an alternating algorithmic approach for this reformulation that is extremely efficient and competitive when producing a low rank ANMF