The notion of fuzziness as defined in this paper relates to situations in which the source of imprecision is not a random variable or a stochastic process, but rather a class or classes which do not possess sharply defined boundaries, e.g., the “class of bald men,” or the “class of numbers which are much greater than 10,” or the “class of adaptive systems,” etc. A basic concept which makes it possible to treat fuzziness in a quantitative manner is that of a fuzzy set, that is, a class in which there may be grades of membership intermediate between full membership and non-membership. Thus, a fuzzy set is characterized by a membership function which assigns to each object its grade of membership (a number lying between 0 and 1) in the fuzzy set. After a review of some of the relevant properties of fuzzy sets, the notions of a fuzzy system and a fuzzy class of systems are introduced and briefly analyzed. The paper closes with a section dealing with optimization under fuzzy constraints in which an approach to...

Fuzzy sets and systems

Preface 1. The approximation problem and existence of best approximations 2. The uniqueness of best approximations 3. Approximation operators and some approximating functions 4. Polynomial interpolation 5. Divided differences 6. The uniform convergence of polynomial approximations 7. The theory of minimax approximation 8. The exchange algorithm 9. The convergence of the exchange algorithm 10. Rational approximation by the exchange algorithm 11. Least squares approximation 12. Properties of orthogonal polynomials 13. Approximation of periodic functions 14. The theory of best L1 approximation 15. An example of L1 approximation and the discrete case 16. The order of convergence of polynomial approximations 17. The uniform boundedness theorem 18. Interpolation by piecewise polynomials 19. B-splines 20. Convergence properties of spline approximations 21. Knot positions and the calculation of spline approximations 22. The Peano kernel theorem 23. Natural and perfect splines 24. Optimal interpolation Appendices Index.

Approximation theory and methods, by M. J. D. Powell. Pp 339. £25 (hardcover), £8·50 (paperback). 1981. ISBN 0-521-22472-1/29514-9 (Cambridge University Press)

Optic disc (OD) detection is an important step in developing systems for automated diagnosis of various serious ophthalmic pathologies. This paper presents a new template-based methodology for segmenting the OD from digital retinal images. This methodology uses morphological and edge detection techniques followed by the Circular Hough Transform to obtain a circular OD boundary approximation. It requires a pixel located within the OD as initial information. For this purpose, a location methodology based on a voting-type algorithm is also proposed. The algorithms were evaluated on the 1200 images of the publicly available MESSIDOR database. The location procedure succeeded in 99% of cases, taking an average computational time of 1.67 s. with a standard deviation of 0.14 s. On the other hand, the segmentation algorithm rendered an average common area overlapping between automated segmentations and true OD regions of 86%. The average computational time was 5.69 s with a standard deviation of 0.54 s. Moreover, a discussion on advantages and disadvantages of the models more generally used for OD segmentation is also presented in this paper.

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Detecting the Optic Disc Boundary in Digital Fundus Images Using Morphological, Edge Detection, and Feature Extraction Techniques

CNN templates are generated using a genetic algorithm to segment mammograms.An adaptive threshold is computed in region growing process by using ANN and intensity features.In tumor classification, CNN produces better results than region growing.MLP produces the highest classification accuracy among other classifiers.Results on DDSM images are more appropriate than those of MIAS. Breast cancer is regarded as one of the most frequent mortality causes among women. As early detection of breast cancer increases the survival chance, creation of a system to diagnose suspicious masses in mammograms is important. In this paper, two automated methods are presented to diagnose mass types of benign and malignant in mammograms. In the first proposed method, segmentation is done using an automated region growing whose threshold is obtained by a trained artificial neural network (ANN). In the second proposed method, segmentation is performed by a cellular neural network (CNN) whose parameters are determined by a genetic algorithm (GA). Intensity, textural, and shape features are extracted from segmented tumors. GA is used to select appropriate features from the set of extracted features. In the next stage, ANNs are used to classify the mammograms as benign or malignant. To evaluate the performance of the proposed methods different classifiers (such as random forest, naive Bayes, SVM, and KNN) are used. Results of the proposed techniques performed on MIAS and DDSM databases are promising. The obtained sensitivity, specificity, and accuracy rates are 96.87%, 95.94%, and 96.47%, respectively.

Benign and malignant breast tumors classification based on region growing and CNN segmentation

The problem of blind noise level estimation arises in many image processing applications, such as denoising, compression, and segmentation. In this paper, we propose a new noise level estimation method on the basis of principal component analysis of image blocks. We show that the noise variance can be estimated as the smallest eigenvalue of the image block covariance matrix. Compared with 13 existing methods, the proposed approach shows a good compromise between speed and accuracy. It is at least 15 times faster than methods with similar accuracy, and it is at least two times more accurate than other methods. Our method does not assume the existence of homogeneous areas in the input image and, hence, can successfully process images containing only textures.

Image Noise Level Estimation by Principal Component Analysis

Mammography is the most effective method for the early detection of breast diseases. However, the typical diagnostic signs such as microcalcifications and masses are difficult to detect because mammograms are low-contrast and noisy images. In this paper, a novel algorithm for image denoising and enhancement based on dyadic wavelet processing is proposed. The denoising phase is based on a local iterative noise variance estimation. Moreover, in the case of microcalcifications, we propose an adaptive tuning of enhancement degree at different wavelet scales, whereas in the case of mass detection, we developed a new segmentation method combining dyadic wavelet information with mathematical morphology. The innovative approach consists of using the same algorithmic core for processing images to detect both microcalcifications and masses. The proposed algorithm has been tested on a large number of clinical images, comparing the results with those obtained by several other algorithms proposed in the literature through both analytical indexes and the opinions of radiologists. Through preliminary tests, the method seems to meaningfully improve the diagnosis in the early breast cancer detection with respect to other approaches.

Mammographic Images Enhancement and Denoising for Breast Cancer Detection Using Dyadic Wavelet Processing

In this paper, we perform the assessment of a CAD for the tumoral masses classification in mammograms by the uncertainty propagation through the system. Carrying on the work of the authors concerning the metrological characterization of the developed CAD, we validate the features extraction, features selection, and classification steps in this paper. In particular, suitable metrics such as the Receiving Operating Curve (ROC) and the Area Under ROC (AUC) are widely used in order to provide a quantitative evaluation of the performance. Finally, we implement a Monte Carlo simulation in order to provide the confidence interval for some coverage probabilities for all involved parameters. The procedure is tested on mammographic images containing both malignant and benign breast masses.

Metrological Characterization of a CADx System for the Classification of Breast Masses in Mammograms

Breast cancer is the most common cause of death among women and the most effective method for its diagnosis is mammography. However, this kind of analysis is very difficult to interpret and for this reason radiologists miss 20-30% of tumors. We propose a module for the segmentation of masses that can be implemented in a complete CADx (Computer Aided Diagnosis) system. In particular, we implement a new version of the region growing algorithm specific for this kind of images and for the constraints on computation time of this application.

Breast Mass Segmentation in Mammographic Images by an Effective Region Growing Algorithm

Contour-independent detection and classification of mammographic lesions

We hypothesize that quantification of structural similarity or dissimilarity between paired mammographic regions can be effective in detecting asymmetric signs of breast cancer. Bilateral masking procedures are applied for this purpose by using automatically detected anatomical landmarks. Changes in structural information of the extracted regions are investigated using spherical semivariogram descriptors and correlation-based structural similarity indices in the spatial and complex wavelet domains. The spatial distribution of grayscale values as well as of the magnitude and phase responses of multidirectional Gabor filters are used to represent the structure of mammographic density and of the directional components of breast tissue patterns, respectively. A total of 188 mammograms from the DDSM and mini-MIAS databases, consisting of 47 asymmetric cases and 47 normal cases, were analyzed. For the combined dataset of mammograms, areas under the receiver operating characteristic curves of 0.83, 0.77, and 0.87 were obtained, respectively, with linear discriminant analysis, the Bayesian classifier, and an artificial neural network with radial basis functions, using the features selected by stepwise logistic regression and leave-one-patient-out cross-validation. Two-view analysis provided accuracy up to 0.94, with sensitivity and specificity of 1 and 0.88, respectively.

Marcello Salmeri

Papers

Mammographic Images Enhancement and Denoising for Breast Cancer Detection Using Dyadic Wavelet Processing

Metrological Characterization of a CADx System for the Classification of Breast Masses in Mammograms

Breast Mass Segmentation in Mammographic Images by an Effective Region Growing Algorithm

Contour-independent detection and classification of mammographic lesions

Analysis of Structural Similarity in Mammograms for Detection of Bilateral Asymmetry