Structure tensor: Difference between revisions

Revision as of 19:13, 14 February 2021

Revision as of 19:14, 14 February 2021

Line 81:

</ref> This is a reason for why <math>(\nabla I)(\nabla I)^\text{T}</math> is used in the averaging of the structure tensor to optimize the direction instead of <math>\nabla I</math>.

By expanding the effective radius of the window function <math>w</math> (that is, increasing its variance), one can make the structure tensor more robust in the face of noise, at the cost of diminished spatial resolution.<ref name=MedioniEA /><ref name=lin94book>T. Lindeberg (19941993), ''[http://www.nadacsc.kth.se/~tony/book.html Scale-Space Theory in Computer Vision]''. Kluwer Academic Publishers, (see sections 14.4.1 and 14.2.3 on pages 359–360 and 355–356 for detailed statements about how the multi-scale second-moment matrix/structure tensor defines a true and uniquely determined multi-scale representation of directional data).

</ref> The formal basis for this property is described in more detail below, where it is shown that a multi-scale formulation of the structure tensor, referred to as the [[Structure tensor#The multi-scale structure tensor|multi-scale structure tensor]], constitutes a ''true multi-scale representation of directional data under variations of the spatial extent of the window function''.