Pattern Recognition, Vol. 26, No. 9, pp. 1363 1372, 1993 Printed in Great Britain 0031 3203/93 $6.00+.00 Pergamon Press Ltd Pattern Recognition Society IMPLEMENTING CONTINUOUS-SCALE MORPHOLOGY VIA CURVE EVOLUTION GUILLERMOSAPIRO,'J"RON KIMMEL,~DORON SHAKED,~BENJAMINB. KIMIA{ and ALFRED M. BRUCKSTEIN§ ]"Department of Electrical Engineering,Technion-I.I.T., Haifa 32000, Israel + Laboratory for Engineering Man-Machine Systems, Brown University, RI 02912, U.S.A. §Department of Computer Science, Technion-l.I.T., Haifa 32000, Israel (Received 19 June 1992; in revisedJorm 19 January 1993 received ]br publication 8 February 1993) Abstraet--A new approach to digital implementation of continuous-scale mathematical morphology is presented. The approach is based on discretization of evolution equations associated with continuous mul- tiscale morphological operations. Those equations, and their corresponding numerical implementation, can be derived either directly from mathematical morphology definitions or from curve evolution theory. The advantages of the proposed approach over the classical discrete morphology are demonstrated. Mathematical morphology Scale-space Curve evolution Digital implementation Numerical algorithms Partial differentialequations I. INTRODUCTION A new definition of discrete mathematical morphology is presented. First, continuous mathematical morphol- ogy is given as a dynamic process, where the basic mor- phological operations are obtained as solutions of par- tial differential equations. Then, discrete mathematical morphology is defined via an efficient numerical im- plementation of this continuous process. The result is that this new discrete morphology approximates con- tinuous morphology much better than the classical dis- crete one. Traditionally, mathematical morphology is intro- duced in a set-theoretical setting." 3) Morphological operators are defined as operators on sets in RN(R2 in case of shapes or binary images). The dilation 8~: R N R v and the erosion e~:R N~R N of a set X ~ R '~' by a structuring element B ~ R N are defined as the sets 3B( X) A= U ~) x+b={x+b:xeX, baB} (1) bEB xeX ~:B(X) A=(~ U x - b . (2) beB xeX It is well known that erosion can be derived from dilation sincetl ~:8(x) = (6~(xc)) c where X ~is the complement of X, and/} is the "'transpose" of B,/3 a= {b:-beB}. Then, dilation is obtained via vector addition of all elements of the set X and the structuring element B, and erosion is the dual operation ("dilation of the background"). The second pair of dual morphological operations is obtained via the concatenation of erosion and dila- tion. Opening is defined by C L 6B(~(X)) and closing by ~ L ~B(6B(X)). Figure 1 shows an example of these four operations on the plane (R2). Note that opening smoothes the figure, and closing smoothes the background. From the definitions above we see that all the basic operations of mathematical morphology are derived from the dilation operator. In the sequel, we shall there- fore refer to dilation only. Function, or multi-level, morphology is usually de- rived from set morphology via a homeomorphism between the space of functions f: RN~ fi where R = R~ {zc, - ~}, and the subspace of umbra sets in R N+ t An umbra set S is a set for which (xt,x2 ..... XN,XN+t)~S ::~(XI,X 2 ..... XN, y)ES , Vy ~__~ XN+ t" The dilation of functions can also be formulated in function terminology") 6.(f)(x)= sup {f(x-y)+,qO')} (3) y~R N where g : R N~ R is a function or multi-level structuring element. Usually the support of the structuring element of the morphological operation is finite. In set mor- phology, finite support simply means that the structur- ing element B has finite extent. In function morphology, finite support of the structuring function g means, that the support set G = {y:,q(y)> -~} is finite. In those 1363