A New Geometric Metric in the Shape and Size Space of Curves in R n

Irene Epifanio, Vicent Gimeno, Ximo Gual-Arnau and M. Victoria Ibáñez-Gual
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Irene Epifanio: Department of Mathematics-IF, Universitat Jaume I, 12071 Castelló, Spain
Vicent Gimeno: Department of Mathematics-IMAC, Universitat Jaume I, 12071 Castelló, Spain
Ximo Gual-Arnau: Department of Mathematics-INIT, Universitat Jaume I, 12071 Castelló, Spain
M. Victoria Ibáñez-Gual: Department of Mathematics-IMAC, Universitat Jaume I, 12071 Castelló, Spain

Mathematics, 2020, vol. 8, issue 10, 1-19

Abstract: Shape analysis of curves in R n is an active research topic in computer vision. While shape itself is important in many applications, there is also a need to study shape in conjunction with other features, such as scale and orientation. The combination of these features, shape, orientation and scale (size), gives different geometrical spaces. In this work, we define a new metric in the shape and size space, S 2 , which allows us to decompose S 2 into a product space consisting of two components: S 4 × R , where S 4 is the shape space. This new metric will be associated with a distance function, which will clearly distinguish the contribution that the difference in shape and the difference in size of the elements considered makes to the distance in S 2 , unlike the previous proposals. The performance of this metric is checked on a simulated data set, where our proposal performs better than other alternatives and shows its advantages, such as its invariance to changes of scale. Finally, we propose a procedure to detect outlier contours in S 2 considering the square-root velocity function (SRVF) representation. For the first time, this problem has been addressed with nearest-neighbor techniques. Our proposal is applied to a novel data set of foot contours. Foot outliers can help shoe designers improve their designs.

Keywords: shape space; square-root velocity function (SRVF); outliers (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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