THESIS
2016
xvi, 110 pages : illustrations ; 30 cm
Abstract
To process discrete digital images, a straightforward approach is to view them
as arrays of numbers and then directly operate on the arrays. An alternative
paradigm, however, is to regard them as samples from continuous signals. Hence,
we can apply a continuous model to analyze and resolve discrete image processing
problems. By virtue of this continuous-domain paradigm, more mathematical
tools become available and formal continuous analysis can be performed, providing
profound insights into the problem being considered. Moreover, for problems
involving entities that are inherently continuous, it is more accurate to do analysis
in the continuous domain.
In this thesis, we focus on two image processing problems: graph-based image
restoration and subpixel-based image scaling. Unl...[
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To process discrete digital images, a straightforward approach is to view them
as arrays of numbers and then directly operate on the arrays. An alternative
paradigm, however, is to regard them as samples from continuous signals. Hence,
we can apply a continuous model to analyze and resolve discrete image processing
problems. By virtue of this continuous-domain paradigm, more mathematical
tools become available and formal continuous analysis can be performed, providing
profound insights into the problem being considered. Moreover, for problems
involving entities that are inherently continuous, it is more accurate to do analysis
in the continuous domain.
In this thesis, we focus on two image processing problems: graph-based image
restoration and subpixel-based image scaling. Unlike existing works applying the
continuous-domain paradigm, it is non-trivial to analyze these two problems in
the continuous domain. However, we manage to do so and answer/resolve several
fundamental aspects of the two problems.
We first consider the problem of image restoration with graph Laplacian regularization.
A graph Laplacian regularizer is a recent, popular prior assuming the
target pixel patch to be smooth with respect to an appropriately chosen graph.
However, the mechanisms and implications of imposing the graph Laplacian regularizer
on the original inverse problem are not well understood. In this work,
we interpret neighborhood graphs of pixel patches as discrete counterparts of
Riemannian manifolds and perform analysis in the continuous domain, bringing
novel understandings to several key problems of graph Laplacian regularization. Specifically, we show the convergence of the graph Laplacian regularizer to a
continuous-domain functional and derive the optimal graph Laplacian regularizer
for the problem of image denoising. Then with the notion of anisotropic diffusion,
we explain the behavior of graph Laplacian regularization during iterations,
e.g., its tendency to promote piecewise smooth signals under certain settings. To
verify the analysis, an iterative image denoising algorithm is developed. Experimental
results show that our algorithm performs competitively with state-of-the-art
denoising methods for natural images, and outperforms them significantly for
piecewise smooth images.
In the second problem, we aim at re-scaling a given image for a color display
panel by controlling its subpixels individually, so as to improve the luminance resolution
of the displayed image. However, improved luminance resolution brings
chrominance distortion, making it crucial to suppress color error while maintaining
sharpness. Moreover, it is difficult to develop a scheme that is applicable
for various subpixel arrangements and for arbitrary scaling factors. By taking
into account the low-pass nature of the human visual system (HVS), we address
the aforementioned issues with a generalized continuous-domain analysis model.
Our developed continuous-domain algorithm is accurate. The error of the resulting
images is only introduced by the discrete implementation, and a higher
computational budget leads to results with higher accuracy. Experiments show
that the proposed method provides sharp images with negligible color distortions.
Our method is comparable to the state-of-the-art methods for the RGB stripe
arrangement, and outperforms them for other subpixel arrangements.
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