This note presents a theoretical discussion of two structure tensor constructions: one proposed by Bigun and Granlund 1987, and the other by Granlund and Knutsson 1995. At first glance, these approaches may appear quite different--the former is implemented by averaging outer products of gradient filter responses, while the latter constructs the tensor from weighted outer products of tune-in frequency vectors of quadrature filters. We argue that when both constructions are viewed through the common lens of Total Least Squares (TLS) line fitting to the power spectrum, they can be reconciled to a large extent, and additional benefits emerge. From this perspective, the correction term introduced in Granlund and Knutsson 1995 becomes unnecessary. Omitting it ensures that the resulting tensor remains positive semi-definite, thereby simplifying the interpretation of its eigenvalues. Furthermore, this interpretation allows fitting more than a single 0rientation to the input by reinterpreting quadrature filter responses without relying on a structure tensor. It also removes the constraint that responses must originate strictly from quadrature filters, allowing the use of alternative filter types and non-angular tessellations. These alternatives include Gabor filters--which, although not strictly quadrature, are still suitable for structure tensor construction--even when they tessellate the spectrum in a Cartesian fashion, provided they are sufficiently concentrated.

翻译：本文从理论角度探讨了两种结构张量构造方法：Bigun与Granlund于1987年提出的方法，以及Granlund与Knutsson于1995年提出的方法。初看之下，这两种方法似乎差异显著——前者通过对梯度滤波器响应的外积进行平均实现，后者则通过正交滤波器的调谐频率向量的加权外积构建张量。我们认为，当通过总体最小二乘法对功率谱进行直线拟合这一共同视角审视这两种构造时，它们在很大程度上可以得到统一，并展现出额外的优势。基于此视角，Granlund与Knutsson在1995年引入的修正项变得不再必要。省略该修正项可确保所得张量保持半正定性，从而简化其特征值的解释。此外，这种解释方式允许通过重新诠释正交滤波器响应来为输入拟合多个方向，而无需依赖结构张量。该方法还消除了响应必须严格源自正交滤波器的约束，允许使用其他滤波器类型及非角度镶嵌方案。这些替代方案包括Gabor滤波器——虽然并非严格正交，但只要其频谱足够集中，即使以笛卡尔方式镶嵌频谱，仍适用于结构张量构造。