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Reconstruction of Convex Sets from One or Two X-rays

Yan Gerard

TL;DR

This work tackles reconstructing convex lattice sets from horizontal and vertical X-rays in discrete tomography, focusing on HV-convex polyominoes and digital convex sets. It presents three core contributions: (i) a counterexample, the Bad Guy, showing that the standard 2-SAT aggregation step can fail, refuting a long-standing conjecture; (ii) a polynomial-time method DAGTomo1 to reconstruct digital convex sets from a single X-ray by encoding the problem as a path in a DAG; and (iii) a polynomial-time method DAGTomo2 to reconstruct fat digital convex sets from two X-rays via a DAG-based aggregation using the fatness property. These results illuminate both the limitations and the algorithmic possibilities in discrete tomography, providing explicit time bounds and a unified DAG-based framework, while leaving open questions for non-fat digital convex sets and more general configurations. $DT_{\mathcal{C}}(v)$ and $DT_{\mathcal{C}\cap\mathcal{F}}(h,v)$ are central to the methods, with polynomial-time reconstruction achieved for the fat subclass via $O(m^7n^7)$ and $O\left(m^{11}+m\big(\sum_i v_i\big)^5\right)$ bounds in the respective cases.

Abstract

We consider a class of problems of Discrete Tomography which has been deeply investigated in the past: the reconstruction of convex lattice sets from their horizontal and/or vertical X-rays, i.e. from the number of points in a sequence of consecutive horizontal and vertical lines. The reconstruction of the HV-convex polyominoes works usually in two steps, first the filling step consisting in filling operations, second the convex aggregation of the switching components. We prove three results about the convex aggregation step: (1) The convex aggregation step used for the reconstruction of HV-convex polyominoes does not always provide a solution. The example yielding to this result is called \textit{the bad guy} and disproves a conjecture of the domain. (2) The reconstruction of a digital convex lattice set from only one X-ray can be performed in polynomial time. We prove it by encoding the convex aggregation problem in a Directed Acyclic Graph. (3) With the same strategy, we prove that the reconstruction of fat digital convex sets from their horizontal and vertical X-rays can be solved in polynomial time. Fatness is a property of the digital convex sets regarding the relative position of the left, right, top and bottom points of the set. The complexity of the reconstruction of the lattice sets which are not fat remains an open question.

Reconstruction of Convex Sets from One or Two X-rays

TL;DR

This work tackles reconstructing convex lattice sets from horizontal and vertical X-rays in discrete tomography, focusing on HV-convex polyominoes and digital convex sets. It presents three core contributions: (i) a counterexample, the Bad Guy, showing that the standard 2-SAT aggregation step can fail, refuting a long-standing conjecture; (ii) a polynomial-time method DAGTomo1 to reconstruct digital convex sets from a single X-ray by encoding the problem as a path in a DAG; and (iii) a polynomial-time method DAGTomo2 to reconstruct fat digital convex sets from two X-rays via a DAG-based aggregation using the fatness property. These results illuminate both the limitations and the algorithmic possibilities in discrete tomography, providing explicit time bounds and a unified DAG-based framework, while leaving open questions for non-fat digital convex sets and more general configurations. and are central to the methods, with polynomial-time reconstruction achieved for the fat subclass via and bounds in the respective cases.

Abstract

We consider a class of problems of Discrete Tomography which has been deeply investigated in the past: the reconstruction of convex lattice sets from their horizontal and/or vertical X-rays, i.e. from the number of points in a sequence of consecutive horizontal and vertical lines. The reconstruction of the HV-convex polyominoes works usually in two steps, first the filling step consisting in filling operations, second the convex aggregation of the switching components. We prove three results about the convex aggregation step: (1) The convex aggregation step used for the reconstruction of HV-convex polyominoes does not always provide a solution. The example yielding to this result is called \textit{the bad guy} and disproves a conjecture of the domain. (2) The reconstruction of a digital convex lattice set from only one X-ray can be performed in polynomial time. We prove it by encoding the convex aggregation problem in a Directed Acyclic Graph. (3) With the same strategy, we prove that the reconstruction of fat digital convex sets from their horizontal and vertical X-rays can be solved in polynomial time. Fatness is a property of the digital convex sets regarding the relative position of the left, right, top and bottom points of the set. The complexity of the reconstruction of the lattice sets which are not fat remains an open question.
Paper Structure (39 sections, 4 theorems, 2 equations, 21 figures, 1 table)

This paper contains 39 sections, 4 theorems, 2 equations, 21 figures, 1 table.

Table of Contents

  1. Introduction
  2. About discrete tomography
  3. Problem statement
  4. State of the art
  5. Three results
  6. 1 - The Bad Guy.
  7. 2 - Only One X-ray.
  8. 3 - With Two X-rays and a Fatness Property.
  9. Plan
  10. The structure of the bad guy counter-example
  11. Episode 1: the reconstruction procedure with its filling/aggregation strategy
  12. Choosing the Feet.
  13. Kernel, Excluded and Undetermined Points.
  14. Filling Step.
  15. The Switching Components.
  16. ...and 24 more sections

Key Result

Theorem 1.2

The algorithm DAGTomo1 solves $DT_{\mathcal{C}} (v)$ with a worst case time complexity in $O(m^{11} + m (\sum _{i=0}^{m-1} v_i)^5 )$ where $m$ is the number of columns of the X-ray and $\sum _{i=0}^{m-1} v_i$ the number of points to determine.

Figures (21)

  • Figure 1: The horizontal and vertical X-rays of the lattice set $S$ are the vectors $V(S)=(1,2,4,5,3,1)$ and $H(S)=(2,4,4,5,1)$.
  • Figure 2: Main classes of convex lattice sets. The set $S_1$ is not a polyomino since not connected. The set $S_2$ is not horizontally convex, $S_3$ is not vertically convex. The set $S_4$ is connected, horizontally and vertically convex. It is an HV-convex polyomino and then belongs to the class $\mathcal{H} \cap \mathcal{V} \cap \mathcal{P}$. The set $S_5$ is not digital convex while $S_6$ is digital convex ($S_6 \in \mathcal{C}$).
  • Figure 3: Reduction of an instance of $DT_\mathcal{W} (h,v)$ to a problem of flow.
  • Figure 4: The four feet of a lattice set $S$ are the subsets denoted $\mathrm{South}$, $\mathrm{West}$, $\mathrm{North}$ and $\mathrm{East}$.
  • Figure 5: Thin VS fat lattice sets. The fatness/thinness of an HV-convex lattice set depends on the relative positions of its feet. A lattice set is thin if there exists an integer point $(X,Y)$ (represented by the green cross) strictly separating the pairs of feet in diagonally opposite quadrants. Otherwise it is fat.
  • ...and 16 more figures

Theorems & Definitions (7)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Definition 2.1
  • Proposition 3.1
  • Proposition 4.1