Two-dimensional (2D) image processing systems are concerned with the processing of the images represented as 2D arrays and are widely used in medicine, transportation and many other autonomous systems. The dynamics of these systems are generally modeled using 2D difference equations, which are mathematically analyzed using the 2D z-transform. It mainly involves a transformation of the difference equations-based models of these systems to their corresponding algebraic equations, mapping the 2D arrays (2D discrete-time signals) over the (\(z_1\),\(z_2\))-domain. Finally, these (\(z_1\),\(z_2\))-domain representations are used to analyze various properties of these systems, such as transfer function and stability. Conventional techniques, such as paper-and-pencil proof methods, and computer-based simulation techniques for analyzing these filters cannot assert the accuracy of the analysis due to their inherent limitations like human error proneness, limited computational resources and approximations of the mathematical expressions and results. In this paper, as a complimentary technique, we propose to use formal methods, higher-order logic (HOL) theorem proving, for formally analyzing the image processing filters. These methods can overcome the limitations of the conventional techniques and thus ascertain the accuracy of the analysis. In particular, we formalize the 2D z-transform based on the multivariate theories of calculus using the HOL Light theorem prover. Moreover, we formally analyze a generic (\(L_1,L_2\))-order 2D infinite impulse response image processing filter. We illustrate the practical effectiveness of our proposed approach by formally analyzing a second-order image processing filter.

二维 (2D) 图像处理系统涉及对表示为 2D 阵列的图像的处理，并广泛用于医学、交通和许多其他自主系统。这些系统的动力学通常使用 2D 差分方程建模，使用 2D z变换对其进行数学分析。它主要涉及将这些系统的基于差分方程的模型转换为相应的代数方程，将二维数组（二维离散时间信号）映射到（\(z_1\)，\(z_2\)）域上。最后，这些 ( \(z_1\)，\(z_2\))-域表示用于分析这些系统的各种属性，例如传递函数和稳定性。用于分析这些滤波器的传统技术（例如纸笔证明方法）和基于计算机的模拟技术无法断言分析的准确性，因为它们存在人为错误倾向、有限的计算资源和数学表达式的近似等固有限制，以及结果。在本文中，作为一种补充技术，我们建议使用形式化方法、高阶逻辑 (HOL) 定理证明来形式化分析图像处理滤波器。这些方法可以克服传统技术的局限性，从而确定分析的准确性。特别是，我们将 2D z- 使用HOL Light 定理证明器基于微积分的多元理论进行变换。此外，我们正式分析了一个通用的 ( \(L_1,L_2\) )-阶 2D 无限脉冲响应图像处理滤波器。我们通过正式分析二阶图像处理滤波器来说明我们提出的方法的实际有效性。