lEEE TRANSACTIONS ON PAmERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 16, NO. 6, JUNE 1994 589 Optimal Structuring Elements for the Morphological Pattern Restoration of Binary Images Dan Schonfeld, Member. IEEE Abstruct- In this paper, we derive the optimal structuring elements of morphological filters in image restoration. The ex- pected pattern transformation of random sets is presented. An estimation theory framework for random sets is subsequently proposed. This framework is based on the least mean difference (LMD) estimator. The least mean difference (LMD) estimator is defined to minimize the cardinality of the expected pattern transformation of the set-difference of the parameter and the estimate. Several important results for the determination of the least mean difference (LMD) estimator are derived. The least mean difference (LMD) structuring elements of morphological filters in image restoration are finally derived. Index Terms- Mathematical morphology, nonlinear filtering, pattern restoration, random set theory. I. INTRODUCTION ATTERN restoration is an important problem in image P processing and analysis applications. Various methods have been proposed for pattem restoration. Among the most important techniques is the median filter (and its repeated iter- ations) [ 11-[2]. The median filter has been proven to minimize the absolute value of the error [I]. The ubiquity of the median filter (and its repeated iterations) is further attributed to its superior performance in pattem restoration applications [2]. The implementation of this technique, however, results in a high computational complexity due to the large number of iterations required. A more recent approach to pattem restoration is based on morphological filters (i.e., increasing and idempotent opera- tors) [3]-[9]. In [ lo]-[ 131, morphological filters have been demostrated to provide an approximation of median filters. The popularity of morphological filters in pattem restora- tion applications is ultimately attributed to their performance [14]-[19]. The main advantage of morphological filters is due to their idempotence (i.e., single iteration filters) [3]-[9]. Fast algortithms for the implementation of morphological filters have also been proposed [20]. As a result, a very low computational complexity is required for the implementation of morphological filters. In the theory of mathematical morphology, an image is probed by a structuring element which interacts with the image Manuscript received January 13, 1992; revised November 15, 1993. This work was supported by the Office of Naval Research under Award N00014- 91-J-1725. Recommended for acceptance by Associate Editor E. Delp. The author is with the Signal and Image Research Laboratory, Department of Electrical Engineering and Computer Science, The University of Illinois at Chicago, Chicago, IL 60680-4348 USA. IEEE Log Number 9400022. in order to extract useful information about the geometrical structure of the image [3]-[9]. A fundamental problem in the application of mathematical morphology is the determination of the optimal structuring element. The performance of the morphological filters in pattem restoration is critically depen- dent on the choice of the structuring element [14]-[19]. The optimal “size” of a given structure element of morphological filters in pattem restoration has been derived in [I41 and [15]. The determination of the optimal structuring element of morphological filters in pattem restoration, however, has remained an open problem. Our approach to the determination of the optimal structuring elements of morphological filters in image restoration is to formulate the problem as the solution of a parameter estimation problem in terms of random set theory [21]-[25]. The theory of random sets characterizes the distribution of a collection of sets in a probability space. Its utility, however, is severely limited due to the absence of an estimation theory framework for random sets. The success of our approach, therefore, relies on the development of an estimation theory for random sets. In this paper, we derive the optimal structuring elements of morphological filters in image restoration. In Section 11, we provide some basic notation used throughout this presentation. In Section 111, we present the expected pattern transformation of random sets. Some basic properties of expected pattem transformations are derived. In Section IV, we propose an estimation theory for random sets. This framework is based on the least mean difference (LMD) estimator. The least mean difference (LMD) estimator is defined to minimize the cardinality of the expected pattem transformation of the set- difference of the parameter and the estimate. Several important results for the determination of the least mean difference (LMD) estimator are derived. In Section V, we present the noise model used in the characterization of the degradation process. In this presentation, the noise model is assumed to be the germ-grain model [3], [ 141-[ 151 [21]-[23]. This model consists of a collection of random sets (primary grains) located at sites determined by a point process (germs). In Section VI, we derive the least mean difference (LMD) structuring elements of the morphological opening for the restoration of binary images. The degradation process in this section is assumed to be the addition (union) of the noise model to the original pattem. In Section VII, we derive the least mean difference (LMD) structuring elements of the morphological closing for the restoration of binary images. The degradation process in this section is assumed to be the subtraction (set- subtraction) of the noise model from the original pattem. 0162-8828/94$04.00 0 1994 IEEE