Affine-Permutation Invariance of 2-D Shapes

Shapes provide a rich set of clues on the identity and topological properties of an object. In many imaging environments, however, the same object appears to have different shapes due to distortions such as translation, rotation, reflection, scaling, or skewing. Further, the order by which the object’s feature points are scanned changes, i.e., the order of the pixels may be permuted. Relating two-dimensional shapes of the same object distorted by different affine and permutation transformations is a challenge.We introduce a shape invariant that we refer to as the intrinsic shape of an object and describe an algorithm, BLAISER, to recover it. The intrinsic shape is invariant to affine-permutation distortions. It is a uniquely defined representative of the equivalence class of all affine-permutation distortions of the same object. BLAISER computes the intrinsic shape from any arbitrarily affine-permutation distorted image of the object, without prior knowledge regarding the distortions or the undistorted shape of the object. The critical step of BLAISER is the determination of the shape orientation and we provide a detailed discussion on this topic. The operations of BLAISER are based on low-order moments of the input shape and, thus, robust to error and noise. Examples illustrate the performance of the algorithm.