L1 Generalized Procrustes 2D Shape Alignment 
Rasmus Larsen, Hrafnkell Eiriksson

Abstract  We describe a new method for resistant and robust alignment of sets of
2D shapes wrt. position, rotation, and isotropical scaling based on minimization of absolute distances. The shapes are represented by $k$ landmarks in two dimensions. It is formulated as a linear programming (LP) problem, thus enabling the use of wellknown and thoroughly tested standard numerical software. This is achieved by using the city block distance between points in the plane. However, the city block distance is dependent on the orientation of the coordinate system. By simultaneously minimising the city block distances in a series of rotated coordinate systems we are able to approximate the circular equidistance curves of Euclidean distances with a regular polygonal equidistance curve to the precision needed. Using 3 coordinate systems rotated $30^\circ$ we get a 12 sided regular polygon, with which we achieve
deviations from Euclidean distances less than 2 \% over all directions.
This new formulation allows for minimization in the $L_1$norm using LP. We demonstrate that the use of the $L_1$norm results in resistance towards object as well as landmark outliers. Examples that illustrate the properties of the robust norm are given on simulated as well as medical datasets. 
Keywords  shape alignment, Procrustes methods, robust, LP 
Type  Conference paper [With referee] 
Conference  Eleventh International Workshop on Matrices and Statistics 
Editors  
Year  2002 Month August 
Publisher  Informatics and Mathematical Modelling, Technical University of Denmark, DTU 
Address  Richard Petersens Plads, Building 321, DK2800 Kgs. Lyngby 
BibTeX data  [bibtex] 
IMM Group(s)  Image Analysis & Computer Graphics 