Complexity of Shapes Embedded in Zn with a Bias Towards Squares.

Shape complexity is a hard-to-quantify quality, mainly due to its relative nature. Biased by Euclidean thinking, circles are commonly considered as the simplest. However, their constructions as digital images are only approximations to the ideal form. Consequently, complexity orders computed in reference to circle are unstable. Unlike circles which lose their circleness in digital images, squares retain their qualities. Hence, we consider squares (hypercubes in Zn) to be the simplest shapes relative to which complexity orders are constructed. Using the connection between L∞ norm and squares we effectively encode squareness-adapted simplification through which we obtain multi-scale complexity measure, where scale determines the level of interest to the boundary. The emergent scale above which the effect of a boundary feature (appendage) disappears is related to the ratio of the contacting width of the appendage to that of the main body. We discuss what zero complexity implies in terms of information repetition and constructibility and what kind of shapes in addition to squares have zero complexity.

Disconnected skeleton: shape at its absolute scale.

We present a new skeletal representation along with a matching framework to address the deformable shape recognition problem. The disconnectedness arises as a result of excessive regularization that we use to describe a shape at an attainably coarse scale. Our motivation is to rely on stable properties the shape instead of inaccurately measured secondary details. The new representation does not suffer from the common instability problems of the traditional connected skeletons, and the matching process gives quite successful results on a diverse database of 2D shapes. An important difference of our approach from the conventional use of skeleton is that we replace the local coordinate frame with a global Euclidean frame supported by additional mechanisms to handle articulations and local boundary deformations. As a result, we can produce descriptions that are sensitive to any combination of changes in scale, position, orientation and articulation, as well as invariant ones.