Histogram of Directions by the Structure Tensor Josef Bigun Stefan M. Karlsson Halmstad University Halmstad University IDE SE-30118 IDE SE-30118 Halmstad, Sweden Halmstad, Sweden
[email protected] [email protected] ABSTRACT entity). Also, by using the approach of trying to reduce di- Many low-level features, as well as varying methods of ex- rectionality measures to the structure tensor, insights are to traction and interpretation rely on directionality analysis be gained. This is especially true for the study of the his- (for example the Hough transform, Gabor filters, SIFT de- togram of oriented gradient (HOGs) features (the descriptor scriptors and the structure tensor). The theory of the gra- of the SIFT algorithm[12]). We will present both how these dient based structure tensor (a.k.a. the second moment ma- are very similar to the structure tensor, but also detail how trix) is a very well suited theoretical platform in which to they differ, and in the process present a different algorithm analyze and explain the similarities and connections (indeed for computing them without binning. In this paper, we will often equivalence) of supposedly different methods and fea- limit ourselves to the study of 3 kinds of definitions of di- tures that deal with image directionality. Of special inter- rectionality, and their associated features: 1) the structure est to this study is the SIFT descriptors (histogram of ori- tensor, 2) HOGs , and 3) Gabor filters. The results of relat- ented gradients, HOGs). Our analysis of interrelationships ing the Gabor filters to the tensor have been studied earlier of prominent directionality analysis tools offers the possibil- [3], [9], and so for brevity, more attention will be given to ity of computation of HOGs without binning, in an algo- the HOGs.