Fast computation of morphological operations with arbitrary structuring elements

Mathematical Morphology (MM) is a general method for image processing based on set theory. The two basic morphological operators are dilation and erosion. From these, several non linear filters have been developed, usually with polynomial complexity and this because the two basic operators depend strongly on the definition of the structural element. Most efforts to improve the algorithm's speed for each operator are based on structural element decomposition and/or efficient codification. In this second part, the concepts developed in part I (see Diaz de Leon and Sossa Azuela, “Mathematical morphology based on linear combined metric spaces on Z^1 (part I): Fast distance transforms,” Journal of Mathematical Imaging and Vision, Vol. 12, No. 2, pp. 137–154, 2000) are used to prove that it is possible to reduce the complexity of the morphological operators to zero complexity (constant time algorithms) for any regular discrete metric space and to eliminate the use of the structural element. In particular, this is done for an infinite family of metric spaces further defined. The use of the distance transformation is proposed for it comprises the information concerning all the discs included in a region to obtain fast morphological operators: erosions, dilations, openings and closings (of zero complexity) for an infinite (countable) family of regular metric spaces. New constant time, in contrast with the polynomial time algorithms, for the computation of these basics operators for any structural element are next derived by using this background. Practical examples showing the efficiency of the proposed algorithms, final comments and present research are also given here.

Mathematical morphological operations are commonly used as a tool in image processing for extracting image components that are useful in the representation and description of region shape. In this paper, six basic morphological operations are investigated to remove noise and enhance the appearance of binary images. Dilation, erosion, opening, closing, fill and majority operations are tested on twenty-five images and subjectively evaluated based on perceived quality of the enhanced images. Results of the experiments showed that noise can be effectively removed from binary images using combinations of erode-dilate operations. Also, the binary images are significantly enhanced using combinations of majority-close operations.

ABSTACT Image segmentation has been an important and challenging issue in the field of computer vision over decades. It plays a critical role for most image analysis tasks, such as object detection and recognition. The aim of this paper is to obtain segmented image of Printed Circuit Board (PCB)’s track using mathematical morphological operation. Morphological image analysis is based on the idea that images represent a collection of spatial patterns that can be analyzed based on some predefined patterns called structuring elements. Most basic morphological operation, dilation and erosion have been employed in this research. Detail elaboration on this method will be discussed and experimental results are presented.

Mathematical morphology is widely used in image processing. Since all morphological operations are based on evaluating values contained in a local neighborhood, calculations on large neighborhood are particularly computationally intensive. Fast implementations for morphological operations do exist but are mostly only adapted to structuring elements of a limited size and/or a specific shape. Therefore, difficulties appear when an arbitrary or complex shape like a circle of a large size is needed. Since dilation and erosion can be used for the representation of all morphological operations, an efficient implementation of those operators is fundamental. Thus, this paper presents a new efficient algorithm for these basic operators of mathematical morphology. It enables the use of any shape of flat structuring element for grayscale images up to 3 dimensions. Also, we have defined this method as an iterative algorithm so it can be used on any parallel architecture and in particular, for GPU (Graphic Processor Unit) computing.

This paper presents an 2-D systolic array architecture for an efficient implementation of gray-scale morphological dilation using arbitrary structuring element and used it to design 2-stage pipelined architecture for morphological opening. The proposed systolic array architecture processes pixels on stream that eliminates the need for buffering image data prior to processing. Additionally, unlike existing systolic array architecture, it does not necessarily adds delay element for processing pixels of an image thereby increasing the processing frame rate. The 2-stage pipelined architecture, on the other hand, provides better performance compared to naive hardware implementation to compute morphological opening. The architecture has been synthesized using Xilinx Design Suite 14.2 ISE and prototyped on Virtex 6 FPGA Board (XC6VLX240T-3FF1156) and verified using Xilinx ISIM Simulator. The architecture provides real time performance when tested for high resolution images using moderate size non-rectangular SEs and results outperforms existing systolic array implementation.

Summary Mathematical morphological filtering (MMF) is a powerful tool for image processing based on the shape of the structure element (SE). It was introduced into seismic data processing to suppress noise and enhance signal quality. We explain the basic mathematic morphology concepts with set theory and define the basic and advanced morphological operations in seismic data processing. Unlike conventional seismic filtering techniques, MMF is a nonlinear operator so that it can more effectively isolate and attenuate seismic noise based on their shape differences from the signal. We apply different types of morphological filter on field (and various stages of processed) data to demonstrate their effectiveness for suppression of both coherent and incoherent noise and results in an improvement of signal to noise ratio.

The two fundamental operations in morphological image processing are dilation and erosion. These processes are defined via structuring elements. It is of practical interest to consider a variety of structuring element shapes. The realisation of dilation/erosion for convex structuring elements by use of partial differential equations (PDEs) allows for digital scalability and subpixel accuracy. However, numerical schemes suffer from blur by dissipative artifacts. In our paper we present a family of so-called flux-corrected transport (FCT) schemes that addresses this problem for arbitrary convex structuring elements. The main characteristics of the FCT-schemes are: (i) They keep edges very sharp during the morphological evolution process, and (ii) they feature a high degree of rotational invariance. We validate the FCT-scheme theoretically by proving consistency and stability. Numerical experiments with diamonds and ellipses as structuring elements show that FCT-schemes are superior to standard schemes in the field of PDE-based morphology.

Purpose – The purpose of this paper is to present a new geometric model based on the mathematical morphology paradigm, specialized to provide determinism to the classic morphological operations. The determinism is needed to model dynamic processes that require an order of application, as is the case for designing and manufacturing objects in CAD/CAM environments. Design/methodology/approach – The basic trajectory-based operation is the basis of the proposed morphological specialization. This operation allows the definition of morphological operators that obtain sequentially ordered sets of points from the boundary of the target objects, inexistent determinism in the classical morphological paradigm. From this basic operation, the complete set of morphological operators is redefined, incorporating the concept of boundary and determinism: trajectory-based erosion and dilation, and other morphological filtering operations. Findings – This new morphological framework allows the definition of complex three-dimensional objects, providing arithmetical support to generating machining trajectories, one of the most complex problems currently occurring in CAD/CAM. Originality/value – The model proposes the integration of the processes of design and manufacture, so that it avoids the problems of accuracy and integrity that present other classic geometric models that divide these processes in two phases. Furthermore, the morphological operative is based on points sets, so the geometric data structures and the operations are intrinsically simple and efficient. Another important value that no excessive computational resources are needed, because only the points in the boundary are processed.

In this paper, a novel image denoising method with a morphological opening is proposed. In many image recovery, the regularization has been introduced to prevent overfitting between a recovered image and a noisy observation. In the regularization of an image, the smoothness of the image is assumed. The cost function, which is minimized during the regularization, is defined as a sum of the approximation error term and the smoothness penalty of the image. In morphological image processing, an image is approximated as a union of structuring elements (SEs) by the morphological image opening. In the approximation with the opening, the SE represents the local intensity surface of the image. We hence impose the smoothness penalty on the SE, instead of the recovered image itself, and propose an image opening method with the regularization penalty for the SE. In denoising, the approximation error is minimized under the constraint of the penalty for the SE. In experiments, we show that the proposed filtering method has ability to remove impulsive noises from degraded texture images.

The mathematical morphology was applied to high-speed automaton fault diagnosis in this paper, which a new method was proposed to diagnosis high-speed automaton fault based on mathematical morphology and the combination of fuzzy clustering. Take a high-speed automaton fault data in laboratory tests for example; various types were analyzed for the fault signal. The results demonstrated that the combination of mathematical morphology and fuzzy clustering applied on a high-speed automaton can effectively diagnose faults, providing a new way for high-speed automata fault diagnosis.

Basic mathematical operations are fundamental for numeracy learning in elementary education, yet many fourth-grade students still struggle with addition, subtraction, and multiplication, especially in complex tasks. Previous methods often focus on final results and rarely apply systematic formative assessment at the beginning of instruction to support both numeracy skills and learning effort. This study examined Mini Quizzes as an early formative assessment strategy for Grade IV students at SD Muhammadiyah 4 Pucang Surabaya. Using a quantitative one-group pretest–posttest design with 62 students, results showed a 36.34% improvement in basic operation skills, with the largest gain in multiplication (+73.85%). Learning effort, measured through summative score growth, increased by 17.47%, and effect sizes ranged from moderate to high. The study demonstrates the novelty of integrating Mini Quizzes routinely at the start of lessons to reinforce conceptual understanding and sustain learning effort. Mini Quizzes are therefore recommended as a practical tool to strengthen numeracy achievement and continuous mathematics learning outcomes in elementary schools. Highlights: Students demonstrated substantial growth in arithmetic skills after systematic early-session assessments. The strongest improvement occurred in higher-complexity multiplication tasks. Summative score progression reflected consistent increases in student learning effort. Keywords: Mini Quiz, Basic Mathematical Operations, Formative Assessment, Numeracy Elementary School

A novel clustering algorithm based on mathematical morphology for wind power generation prediction

Mathematical morphology is a relatively new approach that is being applied to image processing and analysis. Mathematical morphology is based on shape. Hence morphological tools are well suited to problems of shape analysis. Shape is conceptually quite abstract, and it is rather difficult to quantify. In this paper a simple quantitative shape descriptor based on a modified image-skeleton spectrum is defined, and experiments to validate its use are described. We attempted to incorporate human ideas of shape into the new shape descriptor. Throughout this paper it is assumed that the reader has a good knowledge of the basic morphological operations and of the skeletonization procedure.

APPLICATION OF MORPHOLOGICAL OPERATORS AS ENVELOPE EXTRACTORS FOR IMPULSIVE-TYPE PERIODIC SIGNALS

A nonlocal operator model for morphological image processing is proposed in this paper. Compared with the previous nonlocal morphological operators, the derivations of the proposed model are easier to inherit useful properties and physical interpretations from the traditional morphology. Two important properties of the model have been clarified in theory. And for instances, based on the model, two basic morphological operators (i.e., erosion and dilatation) are developed with convincing experimental results. In addition, the generality and usability of the model are also discussed.