L1 Generalized Procrustes 2D Shape Alignment

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\^\$\backslash\$circ\$ we get a 12 sided regular polygon, with which we achieve deviations from Euclidean distances less than 2 \$\backslash\$% 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.