Note on the Construction of Structure Tensor

TL;DR

This paper unifies two structure tensor constructions through Total Least Squares, simplifying interpretation, removing unnecessary correction terms, and broadening the scope to include various filter types like Gabor filters.

Contribution

It demonstrates that two seemingly different structure tensor methods can be reconciled via TLS, leading to simplified interpretation and greater flexibility in filter choice.

Findings

The correction term in Granlund and Knutsson 1995 is unnecessary when viewed through TLS.

The resulting tensor remains positive semi-definite without the correction.

Alternative filters like Gabor can be used for structure tensor construction.

Abstract

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.