Related papers: Reconstruction of Convex Sets from One or Two X-ra…
Tomography is the area of reconstructing objects from projections. Here we wish to reconstruct a set of cells in a two dimensional grid, given the number of cells in every row and column. The set is required to be an hv-convex polyomino,…
In the usual aim of discrete tomography, the reconstruction of an unknown discrete set is considered, by means of projection data collected along a set $U$ of discrete directions. Possible ambiguous reconstructions can arise if and only if…
Any solid object can be decomposed into a collection of convex polytopes (in short, convexes). When a small number of convexes are used, such a decomposition can be thought of as a piece-wise approximation of the geometry. This…
Reconstructing a composition (union) of convex polytopes that perfectly fits the corresponding input point-cloud is a hard optimization problem with interesting applications in reverse engineering and rigid body dynamics simulations. We…
We study a combinatorial notion where given a set of lattice points one takes the set of all sums of subsets of a fixed size, and we ask if the given set comes from a convex lattice polytope whether the resulting set also comes from a…
In pure integer linear programming it is often desirable to work with polyhedra that are full-dimensional, and it is well known that it is possible to reduce any polyhedron to a full-dimensional one in polynomial time. More precisely, using…
Convex approximation sets for multiobjective optimization problems are a well-studied relaxation of the common notion of approximation sets. Instead of approximating each image of a feasible solution by the image of some solution in the…
Convolutional Neural Networks (CNN) based image reconstruction methods have been intensely used for X-ray computed tomography (CT) reconstruction applications. Despite great success, good performance of this data-based approach critically…
We investigate how much information about a convex body can be retrieved from a finite number of its geometric moments. We give a sufficient condition for a convex body to be uniquely determined by a finite number of its geometric moments,…
A variational model for learning convolutional image atoms from corrupted and/or incomplete data is introduced and analyzed both in function space and numerically. Building on lifting and relaxation strategies, the proposed approach is…
In two dimensions, we consider the problem of reconstructing a vector field from partial knowledge of its zeroth and first moment ray transforms. Different from existing works the data is known on a subset of lines, namely the ones…
This paper studies the problem of reconstructing binary matrices that are only accessible through few evaluations of their discrete X-rays. Such question is prominently motivated by the demand in material science for developing a tool for…
We discuss how well a given convex body B in a real d-dimensional vector space V can be approximated by a set X for which the membership question: ``given an x in V, does x belong to X?'' can be answered efficiently (in time polynomial in…
We study the problem of decomposing (i.e. partitioning and covering) polygons into components that are $\alpha$-fat, which means that the aspect ratio of each subpolygon is at most $\alpha$. We consider decompositions without Steiner…
This work proposes a new formulation to the long-standing problem of convex decomposition through learning feature fields, enabling the first feed-forward model for open-world convex decomposition. Our method produces high-quality…
Binary tomography is concerned with the recovery of binary images from a few of their projections (i.e., sums of the pixel values along various directions). To reconstruct an image from noisy projection data, one can pose it as a…
The emergence of deep-learning-based methods to solve image-reconstruction problems has enabled a significant increase in reconstruction quality. Unfortunately, these new methods often lack reliability and explainability, and there is a…
Given a finite set $V$, a convexity $\mathscr{C}$, is a collection of subsets of $V$ that contains both the empty set and the set $V$ and is closed under intersections. The elements of $\mathscr{C}$ are called convex sets. The digital…
A convex polyhedron, that is, a compact convex subset of $\mathbb{R}^3$ which is the intersection of finitely many closed half-spaces, can be rectified by taking the convex hull of the midpoints of the edges of the polyhedron. We derive…
3D teeth reconstruction from X-ray is important for dental diagnosis and many clinical operations. However, no existing work has explored the reconstruction of teeth for a whole cavity from a single panoramic radiograph. Different from…