Universität Des Saarlandes Fachrichtung 6.1 – Mathematik Nonlinear Structure Tensors Nonlinear Structure Tensors
Universität Des Saarlandes Fachrichtung 6.1 – Mathematik Nonlinear Structure Tensors Nonlinear Structure Tensors
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通讯作者：
T. Brox;J. Weickert;B. Burgeth;P. Mrázek
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作者：
T. Brox;J. Weickert;B. Burgeth;P. Mrázek
在这篇文章中，我们介绍了流行的结构张量，也被称为二阶矩矩阵的非线性版本。这些非线性结构张量取代高斯光滑的经典结构张量的不连续性保持非线性扩散。虽然非线性扩散是标量和向量值数据的成熟工具，但到目前为止，它还不常用于张量图像。研究了两类张量数据的非线性扩散过程：一类是具有标量扩散系数的各向同性扩散过程，另一类是具有扩散张量的各向异性扩散过程。我们证明了这些方案保持了矩阵场的正半定性，因此适用于光滑结构张量场。使用的扩散函数的总变差（TV）类型允许我们构建非线性结构张量，而无需指定额外的参数相比，传统的结构张量。非线性结构张量的性能表现在三个领域中，经典的结构张量是经常使用的：方向估计，光流计算，和角点检测。在所有这些情况下，非线性结构张量显示出其优于经典的线性。我们的实验还表明，对于基于非线性结构张量的角点检测，各向异性非线性张量给出了最精确的定位。
In this article we introduce nonlinear versions of the popular structure tensor, also known as second moment matrix. These nonlinear structure tensors replace the Gaussian smoothing of the classical structure tensor by discontinuity-preserving nonlinear diffusions. While nonlinear diffusion is a well-established tool for scalar and vector-valued data, it has not often been used for tensor images so far. Two types of nonlinear diffusion processes for tensor data are studied: an isotropic one with a scalar-valued diffusivity, and its anisotropic counterpart with a diffusion tensor. We prove that these schemes preserve the positive semidefiniteness of a matrix field and are therefore appropriate for smoothing structure tensor fields. The use of diffusivity functions of total variation (TV) type allows us to construct nonlinear structure tensors without specifying additional parameters compared to the conventional structure tensor. The performance of nonlinear structure tensors is demonstrated in three fields where the classic structure tensor is frequently used: orientation estimation, optic flow computation, and corner detection. In all these cases the nonlinear structure tensors demonstrate their superiority over the classical linear one. Our experiments also show that for corner detection based on nonlinear structure tensors, anisotropic nonlinear tensors give the most precise localisation.