arXiv subjects
Publications and source records attributed to Thomas J. Asaki.
In this paper, we propose a new fractional derivative, which is based on a Caputo-type derivative with a smooth kernel. We show that the proposed fractional derivative reduces to the classical derivative and has a smoothing effect which is compatible with $\ell_{1}$ regularization. Moreover, it satisfies some classical properties.
In this work, we study the problem of reconstructing shapes from simple nonasymptotic densities measured only along shape boundaries. The particular density we study is also known as the integral area invariant and corresponds to the area of a disk centered on the boundary that is also inside the shape. It is easy to show uniqueness when these densities are known for all radii in a neighborhood of r = 0, but much less straightforward when we assume that we only know the area invariant and its derivatives for only one r > 0. We present variations of uniqueness results for reconstruction (modulo translation and rotation) of polygons and (a dense set of) smooth curves under certain regularity conditions.
In search of a meaningful 2-dimensional analog to mono- tonicity, we introduce two new definitions and give examples of and dis- cuss the relationship between these definitions and others that we found in the literature. Note: After we published the article in Abstract and Applied Analysis and after we searched multiple times for previous work, we discovered that Clarke at al. had introduced the definition of cone monotonicity and given a characterization. See the addendum at the end of this paper for full reference information.