Modelos de deconvolución ciega fraccionaria. Aplicaciones a la restauración de obras pictóricas.
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2009
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2009-03-30
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Abstract
The goal of this work is to get mathematical models for restoration of
artistic paintings, which may be used in real-life projects. In particular, we
validate our models in a special case.
Motivation of this work dates back in 2004, when the artistic restorers in
Grupo Absidebegan their work related to the preservation of the altarpiece
of San Bartolomé Church, in Bienservida (Spain) . While carrying these
tasks, the artistic team suggested the need of a harmless intervention of the
work, via digitalization, adapted to their criteria in reintegration and their
principles of art and restoration. Most models in literature are general, but
loosely related to this scenario, and they do not provide some (or many) of
the requirements needed in actual restoration.
Among all the dierent aspects of digital restoration, this thesis deals
mainly with two of them: deconvolution and inpainting. Due to the nature of
the work, all the obtained models were devised to be useful for color images,
and, particularly, for Spanish Baroque paintings, some of their features have
been included in the models. Color analysis is, then, an important part of
this research.
The central part of the thesis is the emph deconvolution. A bad decon-
volution does not adequately eliminates diusion in the image, leading to a
poor digital inpainting, because the inpainting process will use as contour
conditions those obtained by deconvolution. As we do not know the exact
natural deterioration process, and we can only guess their reasons, we are
set in the case of blind deconvolution problems, presenting some additional
diculties to those in the case were we have information about the blur, or
smoothing of the painting.
Following the directives of the artistic team, we can assume blur is global
and mostly regular, leading to a quasi-Gaussian type of convolution. Thus,
the model we propose is based on fractional powers of the Laplacian, which
allows us to process ner scales than the usual ones in general Fourier met-
hods, in a direct (and fast) way.
In a second stage of our work, some inpainting algorithms were devised.
Though adapted from dierent published models, they were improved by including the geometry of the images as an essential part of them, together
with a multiresolution analysis, in order to process large holes and dierent
levels of texture in the images.
Finally, we validate the models and algorithms, showing the obtained
results in the altarpiece, and we compare their eciency in some other more
widely known images.