Complexity of Shapes Embedded in ${\mathbb Z^n}$ with a Bias Towards Squares
This work addresses the challenge of defining stable complexity measures for shapes in digital imaging, which is important for fields like computer vision and image processing, though it appears incremental as it builds on existing norms and scale-based approaches.
The paper tackles the problem of quantifying shape complexity in digital images by proposing squares as the simplest reference shapes instead of circles, which are unstable due to approximation issues, and develops a multi-scale complexity measure based on squareness-adapted simplification, showing that the scale at which boundary features disappear relates to the ratio of appendage width to main body width.
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 $\mathbb Z^n$) to be the simplest shapes relative to which complexity orders are constructed. Using the connection between $L^\infty$ 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.