Browsing by Author "Levin, D"

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Open Access
Approximation of Functions over Manifolds: A Moving Least-Squares Approach
Sober, B; Aizenbud, Y; Levin, D
We present an algorithm for approximating a function defined over a $d$-dimensional manifold utilizing only noisy function values at locations sampled from the manifold with noise. To produce the approximation we do not require any knowledge regarding the manifold other than its dimension $d$. We use the Manifold Moving Least-Squares approach of (Sober and Levin 2016) to reconstruct the atlas of charts and the approximation is built on-top of those charts. The resulting approximant is shown to be a function defined over a neighborhood of a manifold, approximating the originally sampled manifold. In other words, given a new point, located near the manifold, the approximation can be evaluated directly on that point. We prove that our construction yields a smooth function, and in case of noiseless samples the approximation order is $\mathcal{O}(h^{m+1})$, where $h$ is a local density of sample parameter (i.e., the fill distance) and $m$ is the degree of a local polynomial approximation, used in our algorithm. In addition, the proposed algorithm has linear time complexity with respect to the ambient-space's dimension. Thus, we are able to avoid the computational complexity, commonly encountered in high dimensional approximations, without having to perform non-linear dimension reduction, which inevitably introduces distortions to the geometry of the data. Additionaly, we show numerical experiments that the proposed approach compares favorably to statistical approaches for regression over manifolds and show its potential.
Open Access
Computer aided restoration of handwritten character strokes
(CAD Computer Aided Design, 2017-08-01) Sober, B; Levin, D
© 2017 Elsevier Ltd This work suggests a new variational approach to the task of computer aided segmentation and restoration of incomplete characters, residing in a highly noisy document image. We model character strokes as the movement of a pen with a varying radius. Following this model, in order to fit the digital image, a cubic spline representation is being utilized to perform gradient descent steps, while maintaining interpolation at some initial (manually sampled) points. The proposed algorithm was used in the process of restoring approximately 1000 ancient Hebrew characters (dating to ca. 8th–7th century BCE), some of which are presented herein and show that the algorithm yields plausible results when applied on deteriorated documents.