For the purposes of object or defect identification required in industrial vision applications, the operations of mathematical morphology are more useful than the convolution operations employed in signal processing because the morphological operators relate directly to shape. The tutorial provided in this paper reviews both binary morphology and gray scale morphology, covering the operations of dilation, erosion, opening, and closing and their relations. Examples are given for each morphological concept and explanations are given for many of their interrelationships.

Image Analysis Using Mathematical Morphology

This paper examines the set-theoretic interpretation of morphological filters in the framework of mathematical morphology and introduces the representation of classical linear filters in terms of morphological correlations, which involve supremum/infimum operations and additions. Binary signals are classified as sets, and multilevel signals as functions. Two set-theoretic representations of signals are reviewed. Filters are classified as set-processing (SP) or function-processing (FP). Conditions are provided for certain FP filters that pass binary signals to commute with signal thresholding because then they can be analyzed and implemented as SP filters. The basic morphological operations of set erosion, dilation, opening, and closing are related to Minkowski set operations and are used to construct FP morphological filters. Emphasis is then given to analytically and geometrically quantifying the similarities and differences between morphological filtering of signals by sets and functions; the latter case allows the definition of morphological convolutions and correlations. Toward this goal, various properties of FP morphological filters are also examined. Linear shift-invariant filters (due to their translation-invariance) are uniquely characterized by their kernel, which is a special collection of input signals. Increasing linear filters are represented as the supremum of erosions by their kernel functions. If the filters are also discrete and have a finite-extent impulse response, they can be represented as the supremum of erosions only by their minimal (with respect to a signal ordering) kernel functions. Stable linear filters can be represented as the sum of (at most) two weighted suprema of erosions. These results demonstrate the power of mathematical morphology as a unifying approach to both linear and nonlinear signal-shaping strategies.

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Morphological filters--Part I: Their set-theoretic analysis and relations to linear shift-invariant filters

From the Publisher:
Handbook of Computer Vision Algorithms in Image Algebra provides engineers, scientists, and students with an introduction to image algebra and presents detailed descriptions of over 80 fundamental computer vision techniques. The book also introduces the portable iac++ library, which supports image algebra programming in the C++ language.

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Handbook of Computer Vision Algorithms in Image Algebra

A family of nonlinear filters based on order statistics is presented. A mathematical tool derived through robust estimation theory, order statistics has allowed engineers to develop nonlinear filters with excellent robustness properties. These filters are well suited to digital image processing because they preserve the edges and the fine details of an image much better than conventional linear filters. The probabilistic and deterministic properties of the best known and most widely used filter in this family, the median filter, are discussed. In addition, the authors consider filters that, while not based on order statistics, are related to them through robust estimation theory. A table that ranks nonlinear filters under a variety of performance criteria is included. Most of the topics treated are very active research areas, and the applications are varied, including HDTV, multichannel signal processing of geophysical and ECG/EEG data, and a variety of telecommunications applications. >

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Order statistics in digital image processing

Accurate assessment of estrogen receptor (ER), progesterone receptor (PR), and Ki-67 is essential in the histopathologic diagnostics of breast cancer. Commercially available image analysis systems are usually bundled with dedicated analysis hardware and, to our knowledge, no easily installable, free software for immunostained slide scoring has been described. In this study, we describe a free, Internet-based web application for quantitative image analysis of ER, PR, and Ki-67 immunohistochemistry in breast cancer tissue sections. The application, named ImmunoRatio, calculates the percentage of positively stained nuclear area (labeling index) by using a color deconvolution algorithm for separating the staining components (diaminobenzidine and hematoxylin) and adaptive thresholding for nuclear area segmentation. ImmunoRatio was calibrated using cell counts defined visually as the gold standard (training set, n = 50). Validation was done using a separate set of 50 ER, PR, and Ki-67 stained slides (test set, n = 50). In addition, Ki-67 labeling indexes determined by ImmunoRatio were studied for their prognostic value in a retrospective cohort of 123 breast cancer patients. The labeling indexes by calibrated ImmunoRatio analyses correlated well with those defined visually in the test set (correlation coefficient r = 0.98). Using the median Ki-67 labeling index (20%) as a cutoff, a hazard ratio of 2.2 was obtained in the survival analysis (n = 123, P = 0.01). ImmunoRatio was shown to adapt to various staining protocols, microscope setups, digital camera models, and image acquisition settings. The application can be used directly with web browsers running on modern operating systems (e.g., Microsoft Windows, Linux distributions, and Mac OS). No software downloads or installations are required. ImmunoRatio is open source software, and the web application is publicly accessible on our website. We anticipate that free web applications, such as ImmunoRatio, will make the quantitative image analysis of ER, PR, and Ki-67 easy and straightforward in the diagnostic assessment of breast cancer specimens.

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ImmunoRatio: a publicly available web application for quantitative image analysis of estrogen receptor (ER), progesterone receptor (PR), and Ki-67

medical fields has led to the automated processing ofpictorial data. Here, a device called the cytocomputer searches for genetic mutations. A computer revolution has occurred not only in technical fields but also in medicine, where vast amounts of information must be processed quickly and accurately. Nowhere is the need for image processing techniques more apparent than in clinical diagnosis or mass screening applications where data take the form of digital images. New high-resolution scanning techniques such as computed tomography, nuclear magnetic resonance, po-sitron emission tomography, and digital radiography produce images containing immense amounts of relevant information for medical analysis. But as these scanning techniques become more vital to clinical diagnosis, the work for specialists who must visually examine the resultant images increases. In many cases, quantitative data in the form of measurements and counts are needed to supplement nonimage patient data, and the manual extraction of these data is a time-consuming and costly step in an otherwise automated process. Furthermore, subtle variants of shade and shape can be the earliest clues to a diagnosis, placing the additional burden of complete thoroughness on the examining specialist. For the last five years, the University of Michigan and the Environmental Research Institute of Michigan have conducted a unique series of studies that involve the processing of biomedical imagery on a highly parallel computer specifically designed for image processing. System designers have incorporated the requirements of extracting a verifiable answer from an image in a reasonable time into an integrated approach to hardware and software design. The system includes a parallel pipelined image processor, called a cytocomputer, and a high-level language specifically created for image processing, C-3PL, the cytocomputer parallel picture processing language. These studies have involved a great many people from both the medical and engineering communities and have highlighted the interdisciplinary aspects of biomedical image processing. The methods have been tested in anatomy, developmental biology, nuclear medicine, car-diology, and transplant rejection. The general consensus is that quantification by automated image analysis not only increases diagnostic accuracy but also provides significant data not obtainable from qualitative analysis alone. One study in particular, on which descriptions in this article are based, involves a joint effort by the University of Michigan's human genetics and electrical and computer engineering departments and is supported by a grant from the National Cancer Institute. Basically, automated image analysis is being applied via sophisticated biochemical and computer techniques to derive …

Sternberg

Papers

Biomedical Image Processing