Improving the Stability of the Recovery of Algebraic Curves via Bernstein Basis Polynomials and Neural Networks

dc.contributor.advisorLabate, Demetrio
dc.contributor.committeeMemberMang, Andreas
dc.contributor.committeeMemberPapadakis, Emanuel I.
dc.contributor.committeeMemberGuillén-Rondón, Pablo
dc.creatorMolina, Wilfredo
dc.creator.orcid0000-0001-5487-7906
dc.date.accessioned2021-08-04T02:12:41Z
dc.date.available2021-08-04T02:12:41Z
dc.date.createdAugust 2020
dc.date.issued2020-08
dc.date.submittedAugust 2020
dc.date.updated2021-08-04T02:12:42Z
dc.description.abstractWe present new methods for the stable reconstruction of a class of binary images from sparse measurements. The images that we consider are characteristic functions of algebraic shapes, that is, interiors of zero sets of bivariate polynomials, and we assume that we only know a finite set of samples of these images. A solution to this problem can be formulated in terms of a system of linear equations of moments. Although it was shown in the literature that one can improve the stability of the reconstruction by increasing the number of moments, the recovery of an algebraic shape remains unstable in the sense that small errors in the computation of the moments may have a catastrophic impact on the recovery algorithm. To address this numerical and theoretical instability, we introduce a novel approach where we represent bivariate polynomials and moments in terms of Bernstein basis polynomials and use them in combination with a polynomial-reproducing, refinable sampling kernel. We show that this is approach is very robust, straightforward to implement, and fast to compute. We also address the same reconstruction problem using an alternative approach that combines a convolutional neural network with a model-based constraint supported by our prior theoretical study. This approach also yields very competitive results and is even more robust to noise. We illustrate the performance of our algorithms on noisy samples through extensive experiments. Our code is publicly accessible on GitHub at github.com/wjmolina/AlgebraicCurves.
dc.description.departmentMathematics, Department of
dc.format.digitalOriginborn digital
dc.format.mimetypeapplication/pdf
dc.identifier.urihttps://hdl.handle.net/10657/7973
dc.language.isoeng
dc.rightsThe author of this work is the copyright owner. UH Libraries and the Texas Digital Library have their permission to store and provide access to this work. Further transmission, reproduction, or presentation of this work is prohibited except with permission of the author(s).
dc.subjectalgebraic
dc.subjectcurve
dc.subjectshape
dc.subjectpolynomial
dc.subjectbivariate
dc.subjectkernel
dc.subjectneural
dc.subjectnetwork
dc.subjectspline
dc.subjectBernstein
dc.subjectmoment
dc.titleImproving the Stability of the Recovery of Algebraic Curves via Bernstein Basis Polynomials and Neural Networks
dc.type.dcmiText
dc.type.genreThesis
thesis.degree.collegeCollege of Natural Sciences and Mathematics
thesis.degree.departmentMathematics, Department of
thesis.degree.disciplineMathematics
thesis.degree.grantorUniversity of Houston
thesis.degree.levelDoctoral
thesis.degree.nameDoctor of Philosophy

Files

Original bundle

Now showing 1 - 4 of 4
Loading...
Thumbnail Image
Name:
MOLINA-DISSERTATION-2020.pdf
Size:
2.7 MB
Format:
Adobe Portable Document Format
No Thumbnail Available
Name:
wjm_dissertation_1.zip
Size:
2.67 MB
Format:
Unknown data format
No Thumbnail Available
Name:
wjm_dissertation.zip
Size:
2.67 MB
Format:
Unknown data format
No Thumbnail Available
Name:
MOLINA-SOURCE-FILES-2020.zip
Size:
2.67 MB
Format:
Unknown data format

License bundle

Now showing 1 - 2 of 2
No Thumbnail Available
Name:
PROQUEST_LICENSE.txt
Size:
4.43 KB
Format:
Plain Text
Description:
No Thumbnail Available
Name:
LICENSE.txt
Size:
1.82 KB
Format:
Plain Text
Description: