  In image analysis, many tasks require representing two-dimensional (2D)
shape, often specified by a set of 2D points, for comparison purposes. The
challenge of the representation is that it must not only capture the
characteristics of the shape but also be invariant to relevant transformations.
Invariance to geometric transformations, such as translation, rotation, and
scale, has received attention in the past, usually under the assumption that
the points are previously labeled, i.e., that the shape is characterized by an
ordered set of landmarks. However, in many practical scenarios, the points
describing the shape are obtained from automatic processes, e.g., edge or
corner detection, thus without labels or natural ordering. Obviously, the
combinatorial problem of computing the correspondences between the points of
two shapes in the presence of the aforementioned geometrical distortions
becomes a quagmire when the number of points is large. We circumvent this
problem by representing shapes in a way that is invariant to the permutation of
the landmarks, i.e., we represent bags of unlabeled 2D points. Within our
framework, a shape is mapped to an analytic function on the complex plane,
leading to what we call its analytic signature (ANSIG). To store an ANSIG, it
suffices to sample it along a closed contour in the complex plane. We show that
the ANSIG is a maximal invariant with respect to the permutation group, i.e.,
that different shapes have different ANSIGs and shapes that differ by a
permutation (or re-labeling) of the landmarks have the same ANSIG. We further
show how easy it is to factor out geometric transformations when comparing
shapes using the ANSIG representation. Finally, we illustrate these
capabilities with shape-based image classification experiments.
