In evolving access structures, the number of participants is countably infinite with no predetermined upper bound. While such structures have been realized in secret sharing, research in secret image sharing has primarily focused on visual cryptography schemes (VCS). However, there exists no construction for $(k,\infty)$ VCS that applies to arbitrary $k$ values without pixel expansion currently, and the contrast requires enhancement. In this paper, we first present a formal mathematical definition of $(k,\infty)$ VCS. Then, propose a $(k,\infty)$ VCS based on random grids that works for arbitrary $k$. In addition, to further improve contrast, we develop optimized $(k,\infty)$ VCS for $k=2$ and $3$, along with contrast enhancement strategies for $k\geq 4$. Theoretical analysis and experimental results demonstrate the superiority of our proposed schemes.

翻译：在演化型访问结构中，参与者数量为可数无限且无预设上限。尽管此类结构已在秘密共享中实现，但秘密图像共享领域的研究主要集中于视觉密码方案（VCS）。然而，目前尚不存在适用于任意k值且无需像素扩展的$(k,\infty)$ VCS构造方法，且其对比度有待提升。本文首先给出$(k,\infty)$ VCS的正式数学定义，随后提出一种基于随机网格的$(k,\infty)$ VCS方案，该方案适用于任意k值。此外，为进一步提升对比度，我们针对k=2和3的情形开发了优化$(k,\infty)$ VCS方案，并提出了适用于k≥4的对比度增强策略。理论分析与实验结果证明了所提方案的优越性。