Distributions of Euclidean distances between copies of the set of 2D points when one copy is randomly rotated or reflected

• Published in: Pattern recognition and image analysis Vol. 22; no. 3; pp. 433 - 445
• Main Authors: Zharkikh, A. A., Bychkova, S. M.
• Format: Journal Article
• Language: English
• Published: Moscow Nauka/Interperiodica 01.07.2012; Springer Nature B.V
• Subjects: Analysis; Computer Science; Decomposition; Density; Euclidean space; Image analysis; Image Processing and Computer Vision; Image processing systems; Pallets; Pattern Recognition; Probability distribution; Processing; Random variables; Reflection; Representation; Reproduction; Studies; Theorems; Transformations; Two dimensional; Understanding of Images; probability distribution density; reflection transformation; ordinary moment; Euclidean distance; rotation transformation
• ISSN: 1054-6618; 1555-6212
• DOI: 10.1134/S1054661812020241

Theorems are stated on the form of probabilistic distributions of Euclidean distances between ordered sets of 2D points with random transformations of their subsets. The theorems provide formulas for the probability distribution densities and ordinary moments. First, several variants of transformations are considered, including random rotation as a whole, random reflection as a whole, simultaneous independent random rotations of two ordered nonoverlapping subsets that form the initial ordered set, and simultaneous independent random reflections of two ordered nonoverlapping subsets that form the initial ordered set. Upon completion of the work, two theorems are stated on the form of probability distribution densities of Euclidean distances for the initial set decomposed into any number of nonoverlapping subsets.