Related papers: Reconstruction of Convex Bodies from Projection Cu…
We present a novel, log-radius profile representation for convex curves and define a new operation for combining the shape features of curves. Unlike the standard, angle profile-based methods, this operation accurately combines the shape…
For a compact convex set F in R^n, with the origin in its interior, we present a formula to compute the curvature at a fixed point on its boundary, in the direction of any tangent vector. This formula is equivalent to the existing ones, but…
In this paper, we derive uniqueness and stability results for surface tensors. Further, we develop two algorithms that reconstruct shape of $n$-dimensional convex bodies. One algorithm requires knowledge of a finite number of surface…
For a convex body $K\subset\R^n$ and $i\in\{1,...,n-1\}$, the function assigning to any $i$-dimensional subspace $L$ of $\R^n$, the $i$-dimensional volume of the orthogonal projection of $K$ to $L$, is called the $i$-th projection function…
Given a set of radii measured from a fixed point, the existence of a convex configuration with respect to the set of distinct radii in the two-dimensional case is proved when radii are distinct or repeated at most four points. However, we…
We propose a generalization of the method of cyclic projections, which uses the lengths of projection steps carried out in the past to learn about the geometry of the problem and decides on this basis which projections to carry out in the…
Reconstructing a complete object from its parts is a fundamental problem in many scientific domains. The purpose of this article is to provide a systematic survey on this topic. The reassembly problem requires understanding the attributes…
Tomographic image reconstruction can be mapped to a problem of finding solutions to a large system of linear equations which maximize a function that includes \textit{a priori} knowledge regarding features of typical images such as…
We construct examples of two convex bodies $K,L$ in $\mathbb{R}^n$, such that every projection of $K$ onto a $(n-1)$-dimensional subspace can be rotated to be contained in the corresponding projection of $L$, but $K$ itself cannot be…
We propose a novel method to accurately reconstruct a set of images representing a single scene from few linear multi-view measurements. Each observed image is modeled as the sum of a background image and a foreground one. The background…
A set of orthonormal polynomials is proposed for image reconstruction from projection data. The relationship between the projection moments and image moments is discussed in detail, and some interesting properties are demonstrated.…
We propose a neural parameterization of convex sets by learning sublinear (positively homogeneous and convex) functions. Our networks implicitly represent both the support and gauge functions of a convex body. We prove a universal…
The reconstruction problem of voxels with individual weightings can be modeled a position- and angle- dependent function in the forward-projection. This changes the system matrix and prohibits to use standard filtered backprojection. In…
Computed tomography is a method for synthesizing volumetric or cross-sectional images of an object from a collection of projections. Popular reconstruction methods for computed tomography are based on idealized models and assumptions that…
We have derived long series expansions for the perimeter generating functions of the radius of gyration of various polygons with a convexity constraint. Using the series we numerically find simple (algebraic) exact solutions for the…
In this paper we prove different functional inequalities extending the classical Rogers-Shephard inequalities for convex bodies. The original inequalities provide an optimal relation between the volume of a convex body and the volume of…
Industrial cone-beam X-ray computed tomography (CT) scans of additively manufactured components produce a 3D reconstruction from projection measurements acquired at multiple predetermined rotation angles of the component about a single…
We introduce combinatorial types of arrangements of convex bodies, extending order types of point sets to arrangements of convex bodies, and study their realization spaces. Our main results witness a trade-off between the combinatorial…
We give an explicit construction of a closed curve with constant torsion and everywhere positive curvature. We also discuss the restrictions on closed curves of constant torsion when they are constrained to lie on convex surfaces.
Self-concordance is the most important property required for barriers in convex programming. It is intrinsically linked to the affine structure of the underlying space. Here we introduce an alternative notion of self-concordance which is…