## Complexity of Shapes Embedded in Z with a Bias towards Squares

IEEE Transactions on Image Processing, cilt.29, ss.8870-8879, 2020 (SCI Expanded İndekslerine Giren Dergi)

• Yayın Türü: Makale / Tam Makale
• Cilt numarası: 29
• Basım Tarihi: 2020
• Doi Numarası: 10.1109/tip.2020.3021316
• Dergi Adı: IEEE Transactions on Image Processing
• Sayfa Sayıları: ss.8870-8879

#### Özet

© 1992-2012 IEEE.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.