Inflation of 2D boundary ghosts and digital watermarking

TL;DR

This paper develops a framework for 2D projection ghosts in discrete tomography, introducing minimal binary ghosts and their boundary variants that vanish in a fixed set of $n$ projection directions. It then shows how boundary ghosts can be inflated through plane tiling to form larger, differently shaped ghosts $W_n(m)$ without altering the zero-projection property, and analyzes the perimeter and area relationships via segment-length recurrences. The authors demonstrate practical watermarking applications by embedding inflated boundary ghosts into digital images using MT and FRT projections, and provide examples of shape variation and robustness against tampering. The work offers a flexible toolkit for secure watermarking and for guiding image reconstruction in domains constrained by curved boundaries, with a rich combinatorial structure for generating diverse yet projection-stable ghost shapes.

Abstract

Projection ghosts are discrete arrays of signed values positioned so that their discrete projections vanish for some chosen set of n projection angles. Minimal ghosts are designed to be compact, with no internal pixels having value zero. Here we control the shape, number of boundary pixels and area that each minimal ghost encloses. Binary minimal ghosts and their boundaries can themselves be inflated by tiling copies of themselves to make ghosts with larger sizes and different shapes, whilst still retaining the same set of n zero projection angles. The intricate perimeters of minimal ghosts are formed by three strings of connected pixels that are defined by the minimal projection angles. We show that large changes to the ghost areas can be made whilst keeping the length of their segmented perimeters fixed. These inflated boundary ghosts may prove useful as secure watermarks to embed into digital image data. Boundary ghosts may also help guide the selection of angles used to reconstruct images where the object domain is confined to oval shaped arcs.

Figures (14)

• Figure 1: Image of binary ghost, $S_{10}$. The 20 pixels with value $+1$ and the 20 pixels with value $-1$ together make projections that are everywhere zero for the 10 directions of Example 2.1.
• Figure 2: Tiling pattern of the array T from Table \ref{['tab:tileT_5x3example']}, centred at location $(0, 0)$, shifted by vectors ($i, j$).
• Figure 3: (a) Tiling of the plane by copies of the $12$ pixel starting tile $T$ from Table \ref{['tab:tile_T_signed_3x6']}, with alternating signs, shifted by $v_n$. The array has zero projections in $5$ directions. The image size is $24 \times 27$. (b) Boundary ghost derived from (a) by applying boundary vector $(0,1)$. The image size is $24 \times 28$. The boundary ghost (b) has zero projections in $6$ directions and consists of $52~ (= 2 \times 24 + 4)$ pixels, covering an area of $410~ (=12 \times 2^5 + 52:2)$ pixels.
• Figure 4: (a) (b) (a) Minimal ghost $U_7^a$ has image size $20 \times 13$ with a total of $2^7 = 128$ signed pixels, arranged in alternating columns of $\pm1$. (b) $V_8^a$, with boundary direction $(0,1)$, has image size $20 \times 14$ with $48$ signed pixels that define the perimeter of the shape in (a). The vertically-defined edges enclose a total area of $152$ pixels. The boundary ghost is an 8-connected curve where each adjacent pixel has the opposite sign.
• Figure 5: (a) (b) (a) Minimal ghost $U_7^{a'}$ has image size $22 \times 13$ with a total of $2^7 = 128$ signed pixels, arranged in alternating columns of $\pm1$. (b) $V_8^{a'}$, with boundary direction $(0,1)$, has image size $22 \times 14$ with $52$ signed pixels that define the perimeter of the shape in (a). The vertically-defined edges enclose a total area of $154$ pixels. The boundary ghost is an 8-connected curve where each adjacent pixel has the opposite sign.