Jacobi sets are an important tool to study the relationship between functions. Defined as the set of all points where the function's gradients are linearly dependent, Jacobi sets extend the notion of critical point to multifields. In practice, Jacobi sets for piecewise-linear approximations of smooth functions can become very complex and large due to noise and numerical errors. Existing methods that simplify Jacobi sets exist, but either do not address how the functions' values have to change in order to have simpler Jacobi sets or remain purely theoretical. In this paper, we present a method that modifies 2D bivariate scalar fields such that Jacobi set components that are due to noise are removed, while preserving the essential structures of the fields. The method uses the Jacobi set to decompose the domain, stores the and weighs the resulting regions in a neighborhood graph, which is then used to determine which regions to join by collapsing the image of the region's cells. We investigate the influence of different tie-breaks when building the neighborhood graphs and the treatment of collapsed cells. We apply our algorithm to a range of datasets, both analytical and real-world and compare its performance to simple data smoothing.

翻译：雅可比集是研究函数间关系的重要工具。其定义为函数梯度线性相关的所有点集，将临界点的概念推广至多场情形。在实际应用中，由于噪声和数值误差，光滑函数的分段线性逼近所对应的雅可比集可能变得极其复杂且规模庞大。现有雅可比集简化方法存在局限性：要么未涉及如何调整函数值以获得更简化的雅可比集，要么仅停留在理论层面。本文提出一种二维二元标量场修正方法，该方法在保持场域本质结构的前提下，可消除由噪声引起的雅可比集分量。本方法利用雅可比集对定义域进行分解，将生成区域存储于邻域图中并赋予权重，继而通过坍缩区域单元像集来确定待合并区域。我们研究了构建邻域图时不同平局决胜策略的影响，以及坍缩单元的处理方式。将本算法应用于解析数据集和真实数据集，并与简单数据平滑方法进行性能对比。