Error quadrics are a fundamental and powerful building block in many geometry processing algorithms. However, finding the minimizer of a given quadric is in many cases not robust and requires a singular value decomposition or some ad‐hoc regularization. While classical error quadrics measure the squared deviation from a set of ground truth planes or polygons, we treat the input data as genuinely uncertain information and embed error quadrics in a probabilistic setting (“probabilistic quadrics”) where the optimal point minimizes the expected squared error. We derive closed form solutions for the popular plane and triangle quadrics subject to (spatially varying, anisotropic) Gaussian noise. Probabilistic quadrics can be minimized robustly by solving a simple linear system — 50× faster than SVD. We show that probabilistic quadrics have superior properties in tasks like decimation and isosurface extraction since they favor more uniform triangulations and are more tolerant to noise while still maintaining feature sensitivity. A broad spectrum of applications can directly benefit from our new quadrics as a drop‐in replacement which we demonstrate with mesh smoothing via filtered quadrics and non‐linear subdivision surfaces.

中文翻译：

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使用概率二次方程快速且稳健的 QEF 最小化

误差二次曲面是许多几何处理算法中的基本且强大的构建块。然而，找到给定二次方程的最小值在许多情况下并不稳健，需要奇异值分解或一些特殊的正则化。虽然经典误差二次曲线测量与一组地面实况平面或多边形的平方偏差，但我们将输入数据视为真正不确定的信息，并将误差二次曲线嵌入概率设置（“概率二次曲线”）中，其中最佳点最小化预期平方误差. 我们为受（空间变化、各向异性）高斯噪声影响的流行平面和三角形二次曲面推导出封闭形式的解决方案。通过求解一个简单的线性系统，可以稳健地最小化概率二次方程——比 SVD 快 50 倍。我们表明概率二次方程在抽取和等值面提取等任务中具有优越的特性，因为它们有利于更均匀的三角剖分，并且在保持特征灵敏度的同时更能容忍噪声。广泛的应用程序可以直接受益于我们的新二次曲面作为直接替代品，我们通过过滤二次曲面和非线性细分曲面进行网格平滑演示。