J Math Imaging Vis DOI 10.1007/s10851-013-0444-5 Fast Circular Arc Segmentation Based on Approximate Circularity and Cuboid Graph Partha Bhowmick · Shyamosree Pal © Springer Science+Business Media New York 2013 Abstract A fast and efficient algorithm for circular arc seg- mentation is presented. The algorithm is marked by several novel features including approximate circularity for arc de- tection, cuboid graph defined by the detected arcs in the 3D parameter space, and resolving all delimited cliques in the cuboid graph to form larger arcs. As circular arcs present in a digitized document often deviate from the ideal conditions of digital circularity, we have loosened their radius intervals and center locations depending on an adaptive tolerance so as to detect the arcs by approximate circularity. The notion of approximate circularity is realized by modifying certain number-theoretic properties of digital circularity, which en- sures that the isothetic deviation of each point in an input curve segment from the reported circle does not exceed the specified tolerance. Owing to integer computation and ju- diciousness of delimited cliques, the algorithm runs signif- icantly fast even for very large images. Exhaustive experi- mentation with benchmark datasets demonstrate its speed, efficiency, and robustness. Keywords Arc segmentation · Circle detection · Digital circularity · Digital geometry · Hough transform 1 Introduction Theorization and experimentation with different techniques on circular arc segmentation in the early research stage [17, 18, 27, 38, 42] have gained new directions in recent times with the advent of new paradigms like digital geometry [34, P. Bhowmick ( ) · S. Pal Indian Institute of Technology, Kharagpur, Kharagpur, West Bengal, India e-mail: bhowmick@gmail.com 36], computational imaging [3, 4, 47], and theory of words and numbers [9, 35]. These techniques are of higher rele- vance today with the advent of digitization in general, and vectorization in particular [28, 51]. Most of the early works were based on notions inherited from real/Euclidean geom- etry and did not consider the digital-geometric properties of digital discs/circles, which indeed are inherently com- plex [12]. These digital-geometric properties of circular arcs have been manifested recently in several interesting forms [6, 14, 15, 25, 44, 46]. The latest of these approaches is based on number theory and forms the cornerstone of the proposed algorithm. Out of all the techniques for circular arc detection, the most widely used is Hough transform (HT) [5, 22, 29, 30, 33, 40]. For a large image, however, an HT-based ap- proach is not practically attractive for its O(n 3 ) compu- tational complexity. Hence, improvements have been pro- posed over the years with various alternatives, e.g., gradient information [17], decomposition of parameter space [55], randomized HT [13, 54], domain reconstruction [12, 48], Hough space voting [21], radius histogram [31, 32], and chord pair voting [14, 32]. As a computational-geometric problem, the arc segmen- tation has been addressed as the problem of arc separabil- ity using Voronoi diagrams in the Euclidean plane [20, 37]. As shown in [15], the time complexity of such algorithms is O(n 2 log n). The algorithm in [49] uses the idea of mini- mum covering circle, which is computed by linear program- ming proposed in [41], and can recognize only full circles and not circular arcs. The work in [16] is also based on the linear programming approach from [41], but it does not con- tain any experimental result. A recent algorithm in [43] uses the notion of tangent space [39] defined by the sequence of equi-length chords obtained by a polygonal approximation of the input curve. Although the time complexity of the al-