Invariant Level Line Encoding

Chapters 2 and 3 described the level lines extraction, selection and smoothing procedures, as well as the selection of a few stable, local directions on these curves. These procedures yield shape elements which cannot be directly compared or recognized since they have undergone an unknown affine transformation. The classical way to address this problem is normalization. We call affine invariant normalization a method to build shape representations that are invariant to any planar affine transformation T(x) = Ax + b, such that det(A) > 0. In other words, an affine invariant normalization transforms a planar shape F into a normalized shape such that any deformation of F by a planar affine transformation will give back the same normalized shape. Notice that shapes related by axial symmetry are not considered to be equivalent in this framework and will not yield the same normalized shape. Similarity invariant normalization is simpler and will be defined in the same way. Section 4.1 first presents the most classical moment method for affine normalization. We will show that this method is not efficient. In Sect. 4.1.3, a much more accurate normalization method is proposed, involving local and robust features of a level line such as bitangent lines and flat parts. This method is applied first to global level lines and then adapted in Sect. 4.2 to pieces of level lines, thus making shape recognition robust to occlusions. These normalization techniques will be used to describe, first, the MSER moment normalization method. The more sophisticated geometric affine normalization methods will be applied throughout the book to the recognition of LLDs (level line descriptors).