Hyperplane Sections, Groebner Bases, and Hough Transforms

TL;DR

This paper explores hyperplane sections of algebraic schemes, Groebner bases behavior, and Hough transforms, providing new methods for implicitization and insights into reconstructing 3D structures from 2D slices.

Contribution

It introduces techniques for lifting Groebner bases in non-homogeneous ideals and analyzes the dimension of Hough transforms, with applications to medical imaging reconstruction.

Findings

Conditions for Groebner bases to pass to quotients

Methods for efficient implicitization

Potential for reconstructing organ surfaces from tomography

Abstract

The purpose of this paper is twofold. In the first part we concentrate on hyperplane sections of algebraic schemes, and present results for determining when Gr\"obner bases pass to the quotient and when they can be lifted. The main difficulty to overcome is the fact that we deal with non-homogeneous ideals. As a by-product we hint at a promising technique for computing implicitization efficiently. In the second part of the paper we deal with families of algebraic schemes and the Hough transforms, in particular we compute their dimension, and show that in some interesting cases it is zero. Then we concentrate on their hyperplane sections. Some results and examples hint at the possibility of reconstructing external and internal surfaces of human organs from the parallel cross-sections obtained by tomography.