Pattern Recognition Letters 10 (1989) 25 31 July 1989 North-Holland More on path generated digital metrics P.P. DAS* D~Tartment of Eh, ctronics and Electrical Communication Engineering, Indian Instilute of Technology, Kharagpur-7 21302, India Received 19 August 1988 Revised 20 December 1988 Abstract: Melter and Tomescu [7] have introduced the concept of path generated digital metrics (PGDM) in the digital plane. In this note we present several properties of these metrics including the closed form analytical expressions, the minimal paths and path-tracing algorithms, the circles and the error estimates with the Euclidean distance. Key words: Digital geometry, metric, path, distance transform, circle. 1. Introduction 2. Path generated digital metrics In [7] Melter and Tomescu have discussed the path generated digital metrics (PGDM) in the two- dimensional digital plane for 3 x 3 symmetric neighbourhoods. They show that out of 15 possible symmetric neighbourhoods only 10 define total dis- tance functions (where a path exists between every pair of points in the plane) which can be further grouped into 5 distinct classes according to the isomorphisms of the traced paths. In this note we derive closed form analytic func- tions for these PGDMs. Given a neighbourhood a PGDM function is an expression involving the co- ordinates of the end points between which the met- ric has been used. Then the isomorphic classes are proved from suitable bijective mappings. We also establish expressions for the perimeter and the area of the circles defined by a PGDM in terms of the radius of the circle. Finally we estimate the absolute and relative errors between a PGDM and the Eu- clidean norm. * Present address: Department of Computer Sc. & Engg. Indian Institute of Technology, Kharagpur-721302, India. Let Z 2=~ x Z={x=(xx,x2)lx1,x2EZ} be the digital plane where 7/ is the set of integers. If N c Y 2 is the neighbourhood relation, then a se- quence ~s(x,y): uo (= x), ul,u2 ..... uu (=y) of points of 7/2 is an N-path of length InN(x,y)l = M from x to y, provided (ul + 1 - u3 • N, 0 < i < M - 1, i.e., u~ + 1 and u~ are neighbours. A shortest (mini- mal) path between x and y is denoted by =}(x,y). In this note we assume neighbourhood relations to be 3 x 3 in size and symmetric in shape in the sense that a neighbourhood N satisfies the following con- ditions: (1) if x• N, then x # (0,0) (no point can be its own neighbour), (2) ifx•J, then tXl[. Ix l -< 1 (3 x 3 size), (3) if x • N, then - x • N (symmetry). A total function d: 77 2 x 7/2 __. N +, where N + is the set of non-negative real numbers, is called a metric if (i) d(x,y) = 0 iffx = y (definiteness), (2) d(x,y) = d(y,x) (symmetry), (3) d(x,y) + d(y,z) > d(x, z) (triangularity), x, y, ~,622. Two metrics d~ and d 2 are called isomorphic to each other if there exists a bijection f12: 7/2._, 7/2 0167-8655/89/$3.50 O 1989, Elsevier Science Publishers B.V. (North-Holland) 25