## Local signal decomposition based methods for the calculation of three-dimensional scalar optical diffraction field due to a field given on a curved surface

##### Author

Şahin, Erdem

##### Advisor

Onural, Levent

##### Date

2013##### Publisher

Bilkent University

##### Language

English

##### Type

Thesis##### Item Usage Stats

74

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##### Abstract

A three-dimensional scene or object can be optically replicated via the threedimensional
imaging and display method holography. In computer-generated
holography, the scalar diffraction field due to a field given on an object (curved
surface) is calculated numerically. The source model approaches treat the building
elements of the object (such as points or planar polygons) independently to
simplify the calculation of diffraction field. However, as a tradeoff, the accuracies
of fields calculated by such methods are degraded. On the other hand,
field models provide exact field solutions but their computational complexities
make their application impractical for meaningful sizes of surfaces. By using
the practical setup of the integral imaging, we establish a space-frequency signal
decomposition based relation between the ray optics (more specifically the
light field representation) and the scalar wave optics. Then, by employing the
uncertainty principle inherent to this space-frequency decomposition, we derive an upper bound for the joint spatial and angular (spectral) resolution of a physically
realizable light field representation. We mainly propose two methods for
the problem of three-dimensional diffraction field calculation from fields given
on curved surfaces. In the first approach, we apply linear space-frequency signal
decomposition methods to the two-dimensional field given on the curved
surface and decompose it into a sum of local elementary functions. Then, we
write the diffraction field as a sum of local beams each of which corresponds to
such an elementary function on the curved surface. By this way, we increase
the accuracy provided by the source models while keeping the computational
complexity at comparable levels. In the second approach, we firstly decompose
the three-dimensional field into a sum of local beams, and then, we construct a
linear system of equations where we form the system matrix by calculating the
field patterns that the three-dimensional beams produce on the curved surface.
We find the coefficients of the beams by solving the linear system of equations
and thus specify the three-dimensional field. Since we use local beams in threedimensional
field decomposition, we end up with sparse system matrices. Hence,
by taking advantage of this sparsity, we achieve considerable reduction in computational
complexity and memory requirement compared to other field model
approaches that use global signal decompositions. The local Gaussian beams
used in both approaches actually correspond to physically realizable light rays.
Indeed, the upper joint resolution bound that we derive is obtained by such
Gaussian beams.

##### Keywords

Linear Space-Frequency Signal DecompositionGaussian Beam Decomposition

Curved Surfaces

Computer-Generated Holography

Scalar Optical Diffraction

Light Field Representation

Integral Imaging

Ray Optics

Scalar Wave Optics