A discrete projection analogue to Pick’s theorem

Abstract

Pick’s theorem expresses the area of a polygon on a grid in terms of the number of boundary and interior integer lattice points. Here, we present an analogous theorem for the area of a symmetric, convex polygon in terms of the number of polygon edges and total projection bins. These polygons arise naturally through discrete projection ghosts. Ghosts are geometric objects that define locations in discrete tomographic systems which are not uniquely determinable. In this work, we show that the area A of a ghost’s convex hull is related to the number of non-trivial discrete projection bins B over the ghost image for any set of N 2D discrete projections by $A=B/2-N/2$. The ratio B/A has a strong upper bound of exactly 2. This relation is analogous to Pick’s theorem for polygons with lattice point vertices.

Introduction

Discrete projection preserves several properties of discrete images such as moments, correlations and oriented slices of the Fourier Transform (via the central slice theorem). We establish here a projective parallel to the well-known Pick’s theorem. We relate the total number of projections bins over a given set of discrete projection angles to a simple polygonal area.

The problem of counting the number of discrete projection bins is usually concerned with a rectangular region, which has been studied in great detail by Verbert [1] for n-dimensional projections. In this work we find the total number of non-trivial discrete (parallel ray) projection bins B over the null-set or ghost image for any set of 2D discrete projections comprised of N discrete angles, (p_i, q_i) for $i=1,2,\cdots ,N$. We prove $B=2A+N,$ where A is the area of the convex hull of the ghost.

Though we state the result for ghosts, it may be applied to any polygonal region of interest defined by the discrete projection directions. This allows for quick calculation of the number of bins required to store information contained within an area of interest in tomographic reconstructions, also of relevance to digital security applications where data is encoded in discrete projections.

Pick’s theorem evaluates the exact area of a simple polygon whose vertices are fixed to any set of 2D integer lattice points. The polygon area A is evaluated by counting the number of interior i and boundary points b through $A=i+b/2-1$ [2]. Generalisations of Pick’s theorem to higher dimensions do not exist, as demonstrated through the Reeve tetrahedron [3]. The number of projection bins over a discrete ghost in 3D produces similar difficulties.

The Mojette transform is an example of a discrete Radon transform that uses discrete projections for reconstruction. This can allow for exact tomographic inversion in the absence of noise. The word “Mojette” comes from a type of bean used to teach children counting and arithmetic. Projections under the Mojette transform are discrete, and sum pixel values at sites separated by p steps in the x direction, and q steps in the y direction. Each projection sum is stored in a projection bin. A Mojette transform is demonstrated in Fig. 1.

For a set of non-degenerate projections {(p_i, q_i)} on a P × Q array, unique reconstruction is possible if and only if the Katz critereon is met [4].

$$\sum |p_i|\ge P\text{or}\sum |q_i|\ge Q$$

When this condition is not met, some values in the reconstructed array do not have a unique solution. The locations of these indeterminate values are fixed by the pattern of non-zero entries in the discrete projection ghost for that set of angles.

Ghosts are arrays that have zero sum projections along a set of prescribed directions. If a ghost can be embedded into an image, it will not change the value of projections and therefore the values of the image where the ghost array has a non-zero value can not be recovered uniquely. An elementary ghost is the simplest possible ghost for a single projection direction, consisting of a $+1$ value at the point (0,0), and a $-1$ value at the point (p, q). A ghost for a set of directions can be constructed through convolution of elementary ghosts for each projection direction. In this work we are interested in the size of these ghosts, as viewed by its projections. The following is an example of a ghost for the directions {(0, 1), (1,2), (2,1), (0, 1)}.

$$\left[\begin{array}{ccccc}0& 0& 0& -1& +1\\ 0& +1& -1& +1& -1\\ 0& -1& +2& -1& 0\\ -1& +1& -1& +1& 0\\ +1& -1& 0& 0& 0\end{array}\right]$$