Gray-scale Structuring Element Research Articles

Decomposition of arbitrary gray-scale morphological structuring elements

Mathematical morphology has been widely used for many applications in image processing and analysis. Most image processing architectures adapted to morphological operations use structuring elements of a limited size. Therefore, difficulties arise when we deal with a large-sized structuring element. In this paper, we present algorithms for the decomposition of arbitrary gray-scale structuring elements into combined dilations or maximum operators of smaller structuring components. Our method does not need to perform additional pre-processes, such as checking the type of structuring elements and the decomposition rules. Furthermore, it is suited for a parallel pipelined architecture.

A Constant-time Algorithm for Erosions/Dilations with Applications to Morphological Texture Feature Computation

A computationally efficient algorithm for computing erosions and dilations by one- dimensional grayscale structuring elements with constant slope is proposed. The computational complexity of this algorithm is independent of the size of the support of the structuring function. This is a generalization of the method proposed by Van Herk for the case of erosion and dilation by flat one-dimensional structuring elements. By appropriate combinations of these structuring elements, it is possible to approximate many useful structuring elements. This enables efficient computation of granulometries where the number of operations depends linearly on the number of openings. Theoretical and experimental results comparing the complexity of this algorithm with other standard techniques is presented. Two memory efficient algorithms are then presented. Several implementation issues in computing a granulometry and moments of the associated morphological pattern spectrum are then addressed. An efficient implementation of granulometries for large images on machines with limited memory, by dividing the image into smaller rectangular patches is then discussed. The optimum size of these patches is a function of the specific hardware and has been obtained experimentally for three different hardware platforms. Finally, parallel implementation of the different algorithms on two multi-processor machines is discussed.

Basis Matrix Representation of Morphological Filters with N-Dimensional Structuring Elements

In this paper, we present a basis matrix representation of grayscale morphological filters in N-dimensions. A procedure is proposed to derive the basis matrix and the block basis matrix (BBM) from an N-dimensional grayscale structuring element (GSE). It is shown that both opening and closing with arbitrary N-dimensional GSE can be accomplished by a local matrix operation using the basis matrix. Furthermore, these basis matrix representations are extended to the efficient implementation of open-closing (OC) and close-opening (CO) using the BBM.

Learning optimization of morphological filters with gray scale structuring elements

Mathematical morphology with gray scale structuring elements has attracted much attention, since combinations of the operations in this class can realize almost all noise-removing filters. However, the optimization method for the combination is still uncertain. In this paper, an optimization method for a mathematical morphological filter with gray scale structuring elements is proposed. This method is based on the concept of a neural network with morphological operations and on learning using simulated annealing. The method is also applied to gray scale bipolar morphological filters for image differentiation.

Fast recursive algorithms for morphological operators based on the basis matrix representation

A real-time implementation method for the most general morphological system, the so-called grayscale function processing (FP) system is presented. The proposed method is an extension of our previous works (1993, 1995) using the matrix representation of the FP system with a basis matrix (BM) and a block basis matrix (BBM) composed of grayscale structuring elements (GSE). In order to further improve the computational efficiency of the basis matrix representation, we propose recursive algorithms based on the observation of the BM and BBM. The efficiency of the proposed algorithms is gained by avoiding redundant steps in computing overlapping local maximum or minimum operations. It is shown that, with the proposed scheme, both opening and closing can be determined in real time by 2N-2 additions and 2N-2 comparisons, and OC and CO by 4N-4 additions and 4N-4 comparisons, when the size of the GSE is equal to N. It is also shown that the proposed recursive opening and closing require only 3N-3 memory elements.

Gray-scale structuring element decomposition

Efficient implementation of morphological operations requires the decomposition of structuring elements into the dilation of smaller structuring elements. Zhuang and Haralick (1986) presented a search algorithm to find optimal decompositions of structuring elements in binary morphology. We use the concepts of Top of a set and Umbra of a surface to extend this algorithm to find an optimal decomposition of any arbitrary gray-scale structuring element.

Decomposition of additively separable structuring elements with applications

The techniques of decomposing gray-scale structuring elements with emphasis on the additively separable structuring elements are presented. Based on the concept of additively separable, we proposed the square Euclidean distance as the measure in the distance transformation. Since the structuring element corresponding to square Euclidean distance is additively separable, the developed techniques can be applied to decompose the structuring element into successive dilation of small structuring elements with window size 3 × 3.

Statistical analysis of morphological openings

Statistical analysis of morphological openings is carried out to study noise-suppression and edge-preserving properties of binary and gray-scale structuring elements. Based on the fact that basis functions are a general representation of any morphological mapping that is translation-invariant and increasing, it is shown that a statistical analysis using this representation is feasible and more general than the threshold decomposition approach.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Efficient neural network implementation of morphological operations

This paper introduces a neural network implementation of gray scale operators. In this structure, synaptic weights are represented by a gray scale structuring element and trained by a learning algorithm based on an optimal criterion called the overall equality index. The proposed algorithm leads to a computationally simple implementation, with numerical examples to illustrate its performance.

Optimal mean-square N-observation digital morphological filters : II. Optimal gray-scale filters

The second part of this two-part study interprets digital gray-scale morphological operations as numerical functionals on integer-valued vectors. The result is a gray-scale morphological statistical estimation theory based on N observation random variables and a consequent theory of mean-square optimization. Whereas in the binary setting the search for optimal structuring elements is clearly restricted by the requirement of binary N-vector structuring elements, gray-scale structuring elements are N-vectors of integers (not confined to the range of the images). Thus, it is necessary to find a set of structuring elements from which both singleand multiple-erosion filters can be optimized, the latter being characterized within the framework of the gray-scale Matheron representation. Consequently, a significant part of the paper is concerned with the derivation of the fundamental set, this set being a minimal family of structuring elements from which it is always possible to select the erosions comprising an optimal filter. As in the case of binary optimization, we treat constrained optimality; however, here we apply it to construction of optimal linear filters. Finally, we demonstrate that erosion optimization is equivalent to dilation optimization.