Shape matching is a fundamental problem in computer graphics with many applications. Functional maps translate the point-wise shape-matching problem into its functional counterpart and have inspired numerous solutions over the last decade. Nearly all the solutions based on functional maps rely on the eigenfunctions of the Laplace-Beltrami Operator (LB) to describe the functional spaces defined on the surfaces and then convert the functional correspondences into point-wise correspondences. However, this final step is often error-prone and inaccurate in tiny regions and protrusions, where the energy of LB does not uniformly cover the surface. We propose a new functional basis Principal Components of a Dictionary (PCD) to address such intrinsic limitation. PCD constructs an orthonormal basis from the Principal Component Analysis (PCA) of a dictionary of functions defined over the shape. These dictionaries can target specific properties of the final basis, such as achieving an even spreading of energy. Our experimental evaluation compares seven different dictionaries on established benchmarks, showing that PCD is suited to target different shape-matching scenarios, resulting in more accurate point-wise maps than the LB basis when used in the same pipeline. This evidence provides a promising alternative for improving correspondence estimation, confirming the power and flexibility of functional maps.

翻译：形状匹配是计算机图形学中的基本问题，具有众多应用。功能映射将逐点形状匹配问题转化为其功能对应问题，并在过去十年中催生了大量解决方案。几乎所有基于功能映射的解决方案都依赖拉普拉斯-贝尔特拉米算子（LB）的特征函数来描述定义在曲面上的函数空间，然后将功能对应关系转化为逐点对应关系。然而，在LB能量无法均匀覆盖曲面的微小区域和突起处，这最后一步常常容易出错且不精确。我们提出一种新的函数基——字典主成分（PCD），以解决这一固有局限性。PCD通过对定义在形状上的函数字典进行主成分分析（PCA）来构造一组标准正交基。这些字典可以针对最终基的特定性质（如实现能量的均匀分布）进行设计。我们的实验评估在公认基准上比较了七种不同字典，结果表明PCD适用于不同形状匹配场景，在相同流程下能比LB基生成更精确的逐点映射。这一发现为改进对应关系估计提供了有前景的替代方案，进一步证实了功能映射的强大性和灵活性。