Optimum non linear binary image restoration through linear grey-scale operations

Non-linear image processing operators give excellent results in a number of image processing tasks such as restoration and object recognition. However they are frequently excluded from use in solutions because the system designer does not wish to introduce additional hardware or algorithms and because their design can appear to be ad hoc. In practice the median filter is often used though it is rarely optimal. This paper explains how various non-linear image processing operators may be implemented on a basic linear image processing system using only convolution and thresholding operations. The paper is aimed at image processing system developers wishing to include some non-linear processing operators without introducing additional system capabilities such as extra hardware components or software toolboxes. It may also be of benefit to the interested reader wishing to learn more about non-linear operators and alternative methods of design and implementation. The non-linear tools include various components of mathematical morphology, median and weighted median operators and various order statistic filters. As well as describing novel algorithms for implementation within a linear system the paper also explains how the optimum filter parameters may be estimated for a given image processing task. This novel approach is based on the weight monotonic property and is a direct rather than iterated method

Morphological filter mean-absolute-error representation theorems and their application to optimal morphological filter design

The present thesis derives error representations and develops design methodologies for optimal mean-absolute-error (MAE) morphological-based filters. Four related morphological-based filter-types are treated. Three are translation-invariant, monotonically increasing operators, and our analysis is based on the Matheron (1975) representation. In this class we analyze conventional binary, conventional gray-scale, and computational morphological filters. The fourth filter class examined is that of binary translation invariant operators. Our analysis is based on the Banon and Barrera (1991) representation and hit-or-miss operator of Serra (1982). A starting point will be the optimal morphological filter paradigm of Dougherty (1992a,b) whose analysis de scribes the optimal filter by a system of nonlinear inequalities with no known method of solution, and thus reduces filter design to minimal search strategies. Although the search analysis is definitive, practical filter design remained elu sive because the search space can be prohibitively large if it not mitigated in some way. The present thesis extends from Dougherty\u27s starting point in several ways. Central to the thesis is the MAE analysis for the various filter settings, where in each case, a theorem is derived that expresses overall filter MAE as a sum of MAE values of individual structuring-element filters and MAE of combinations of unions (maxima) of those elements. Recursive forms of the theorems can be employed in a computer algorithm to rapidly evaluate combinations of structuring elements and search for an optimal filter basis. Although the MAE theorems provide a rapid means for examining the filter design space, the combinatoric nature of this space is, in general, too large for a exhaustive search. Another key contribution of this thesis concerns mitigation of the computational burden via design constraints. The resulting constrained filter will be suboptimal, but, if the constraints are imposed in a suitable man ner, there is little loss of filter performance in return for design tractability. Three constraint approaches developed here are (1) limiting the number of terms in the filter expansion, (2) constraining the observation window, and (3) employing structuring element libraries from which to search for an optimal basis. Another contribution of this thesis concerns the application of optimal morphological filters to image restoration. Statistical and deterministic image and degradation models for binary and low-level gray images were developed here that relate to actual problems in the optical character recognition and electronic printing fields. In the filter design process, these models are employed to generate realizations, from which we extract single-erosion and single-hit-or-miss MAE statistics. These realization-based statistics are utilized in the search for the optimal combination of structuring elements

Morphological Operations Applied to Digital Art Restoration

This paper provides an overview of the processes involved in detecting and removing cracks from digitized works of art. Specific attention is given to the crack detection phase as completed through the use of morphological operations. Mathematical morphology is an area of set theory applicable to image processing, and therefore lends itself effectively to the digital art restoration process

A graph-based mathematical morphology reader

This survey paper aims at providing a "literary" anthology of mathematical morphology on graphs. It describes in the English language many ideas stemming from a large number of different papers, hence providing a unified view of an active and diverse field of research

Genetic programming applied to morphological image processing

This thesis presents three approaches to the automatic design of algorithms for the processing of binary images based on the Genetic Programming (GP) paradigm. In the first approach the algorithms are designed using the basic Mathematical Morphology (MM) operators, i.e. erosion and dilation, with a variety of Structuring Elements (SEs). GP is used to design algorithms to convert a binary image into another containing just a particular characteristic of interest. In the study we have tested two similarity fitness functions, training sets with different numbers of elements and different sizes of the training images over three different objectives. The results of the first approach showed some success in the evolution of MM algorithms but also identifed problems with the amount of computational resources the method required. The second approach uses Sub-Machine-Code GP (SMCGP) and bitwise operators as an attempt to speed-up the evolution of the algorithms and to make them both feasible and effective. The SMCGP approach was successful in the speeding up of the computation but it was not successful in improving the quality of the obtained algorithms. The third approach presents the combination of logical and morphological operators in an attempt to improve the quality of the automatically designed algorithms. The results obtained provide empirical evidence showing that the evolution of high quality MM algorithms using GP is possible and that this technique has a broad potential that should be explored further. This thesis includes an analysis of the potential of GP and other Machine Learning techniques for solving the general problem of Signal Understanding by means of exploring Mathematical Morphology

Modeling of evolving textures using granulometries

This chapter describes a statistical approach to classification of dynamic texture images, called parallel evolution functions (PEFs). Traditional classification methods predict texture class membership using comparisons with a finite set of predefined texture classes and identify the closest class. However, where texture images arise from a dynamic texture evolving over time, estimation of a time state in a continuous evolutionary process is required instead. The PEF approach does this using regression modeling techniques to predict time state. It is a flexible approach which may be based on any suitable image features. Many textures are well suited to a morphological analysis and the PEF approach uses image texture features derived from a granulometric analysis of the image. The method is illustrated using both simulated images of Boolean processes and real images of corrosion. The PEF approach has particular advantages for training sets containing limited numbers of observations, which is the case in many real world industrial inspection scenarios and for which other methods can fail or perform badly. [41] G.W. Horgan, Mathematical morphology for analysing soil structure from images, European Journal of Soil Science, vol. 49, pp. 161–173, 1998. [42] G.W. Horgan, C.A. Reid and C.A. Glasbey, Biological image processing and enhancement, Image Processing and Analysis, A Practical Approach, R. Baldock and J. Graham, eds., Oxford University Press, Oxford, UK, pp. 37–67, 2000. [43] B.B. Hubbard, The World According to Wavelets: The Story of a Mathematical Technique in the Making, A.K. Peters Ltd., Wellesley, MA, 1995. [44] H. Iversen and T. Lonnestad. An evaluation of stochastic models for analysis and synthesis of gray-scale texture, Pattern Recognition Letters, vol. 15, pp. 575–585, 1994. [45] A.K. Jain and F. Farrokhnia, Unsupervised texture segmentation using Gabor filters, Pattern Recognition, vol. 24(12), pp. 1167–1186, 1991. [46] T. Jossang and F. Feder, The fractal characterization of rough surfaces, Physica Scripta, vol. T44, pp. 9–14, 1992. [47] A.K. Katsaggelos and T. Chun-Jen, Iterative image restoration, Handbook of Image and Video Processing, A. Bovik, ed., Academic Press, London, pp. 208–209, 2000. [48] M. K¨oppen, C.H. Nowack and G. R¨osel, Pareto-morphology for color image processing, Proceedings of SCIA99, 11th Scandinavian Conference on Image Analysis 1, Kangerlussuaq, Greenland, pp. 195–202, 1999. [49] S. Krishnamachari and R. Chellappa, Multiresolution Gauss-Markov random field models for texture segmentation, IEEE Transactions on Image Processing, vol. 6(2), pp. 251–267, 1997. [50] T. Kurita and N. Otsu, Texture classification by higher order local autocorrelation features, Proceedings of ACCV93, Asian Conference on Computer Vision, Osaka, pp. 175–178, 1993. [51] S.T. Kyvelidis, L. Lykouropoulos and N. Kouloumbi, Digital system for detecting, classifying, and fast retrieving corrosion generated defects, Journal of Coatings Technology, vol. 73(915), pp. 67–73, 2001. [52] Y. Liu, T. Zhao and J. Zhang, Learning multispectral texture features for cervical cancer detection, Proceedings of 2002 IEEE International Symposium on Biomedical Imaging: Macro to Nano, pp. 169–172, 2002. [53] G. McGunnigle and M.J. Chantler, Modeling deposition of surface texture, Electronics Letters, vol. 37(12), pp. 749–750, 2001. [54] J. McKenzie, S. Marshall, A.J. Gray and E.R. Dougherty, Morphological texture analysis using the texture evolution function, International Journal of Pattern Recognition and Artificial Intelligence, vol. 17(2), pp. 167–185, 2003. [55] J. McKenzie, Classification of dynamically evolving textures using evolution functions, Ph.D. Thesis, University of Strathclyde, UK, 2004. [56] S.G. Mallat, Multiresolution approximations and wavelet orthonormal bases of L2(R), Transactions of the American Mathematical Society, vol. 315, pp. 69–87, 1989. [57] S.G. Mallat, A theory for multiresolution signal decomposition: the wavelet representation, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 11, pp. 674–693, 1989. [58] B.S. Manjunath and W.Y. Ma, Texture features for browsing and retrieval of image data, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 18, pp. 837–842, 1996. [59] B.S. Manjunath, G.M. Haley and W.Y. Ma, Multiband techniques for texture classification and segmentation, Handbook of Image and Video Processing, A. Bovik, ed., Academic Press, London, pp. 367–381, 2000. [60] G. Matheron, Random Sets and Integral Geometry, Wiley Series in Probability and Mathematical Statistics, John Wiley and Sons, New York, 1975

Convergence of adaptive morphological filters in the context of Markov chains

A typical parameterized r-opening *r is a filter defined as a union of openings by a collection of compact, convex structuring elements, each of which is governed by a parameter vector r. It reduces to a single parameter r-opening filter by a set of structuring elements when r is a scalar sizing parameter. The parameter vector is adjusted by a set of adaptation rules according to whether the re construction Ar derived from r correctly or incorrectly passes the signal and noise grains sampled from the image. Applied to the signal-union-noise model, the optimization problem is to find the vector of r that minimizes the Mean-Absolute-Error between the filtered and ideal image processes. The adaptive r-opening filter fits into the framework of Markov processes, the adaptive parameter being the state of the process. For a single parameter r-opening filter, we proved that there exists a stationary distribution governing the parameter in the steady state and convergence is characterized in terms of the steady-state distribution. Key filter properties such as parameter mean, parameter variance, and expected error in the steady state are characterized via the stationary distribution. Steady-state behavior is compared to the optimal solution for the uniform model, for which it is possible to derive a closed-form solution for the optimal filter. We also developed the Markov adaptation system for multiparameter opening filters and provided numerical solutions to some special cases. For multiparameter r-opening filters, various adaptive models derived from various assumptions on the form of the filter have been studied. Although the state-probability increment equations can be derived from the appropriate Chapman-Kolmogorov equations, the closed-form representation of steady-state distributions is mathematically problematic due to the support geometry of the boundary states and their transitions. Therefore, numerical methods are employed to approximate for steady state probability distributions. The technique developed for conventional opening filters is also applied to bandpass opening filters. In present thesis study, the concept of signal and noise pass sets plays a central role throughout the adaptive filter analysis. The pass set reduces to the granulometric measure (or {&r}-measure) of the signal and noise grain. Optimization and adaptation are characterized in terms of the distribution of the granulometric measures for single parameter filters, or in terms of the multivariate distribution of the signal and noise pass sets. By introducing these concepts, this thesis study also provides some optimal opening filter error equations. It has been shown in the case of the uniform distribution of single sizing parameter that there is a strong agreement between the adaptive filter and optimal filter based on analytic error minimization. This agreement has been also demonstrated in various r-opening filters. Furthermore, the probabilistic interpretation has a close connection to traditional linear adaptive filter theory. The method has been applied to the classical grain separation (clutter removal) problem. *See content for correct numerical representation

NDE data fusion using morphological approaches

The objective of most data fusion algorithms is to combine information made available by various sensors synergistically in order to enhance the overall level of information. Since information obtained from data sources such as sensors is often incomplete or imprecise in nature, the application of data fusion techniques has evoked interest in a number of fields ranging from robotics to nondestructive evaluation (NDE). In NDE applications, such techniques can be used to integrate and fuse data obtained using multiple inspection modalities to produce a more comprehensive picture of the condition of the test specimen. As an example, ultrasonic and eddy current imaging techniques are used very widely to inspect a variety of materials. Each technique offers inspection capabilities and limitations that are dictated by the underlying material/energy interaction process. The information generated using the two methods can be construed either as complementary or redundant in nature. Ideally it should be possible to utilize the redundant information to improve the signal-to-noise ratio. Likewise, it should be possible to fuse the complementary information from the two tests to increase the overall level of information made available to the analyst. Unfortunately the task of segmenting data as noise, redundant and complementary components of information can be frustrating. Consequently, most of the approaches proposed to date in NDE have relied on alternate methods;This dissertation proposes a new algorithm for fusing ultrasonic and eddy current images employing morphological imaging processing approaches. The fusion is accomplished in two stages. The first stage basically employs morphological approaches to reduce unwanted artifacts such as speckle noise in the ultrasonic image. The second stage extracts information about the locations and boundaries of defects on the basis of information contained in the morphological granulometric size distribution of the ultrasonic image. Data fusion is accomplished by combining information relating to the locations and boundaries of the defect obtained from the ultrasonic data with the defect depth information derived from the eddy current image. The validity of the approach is demonstrated using several experimentally derived ultrasonic and eddy current images

Image segmentation using genetic algorithm and morphological operations

Image segmentation is a fundamental component of picture processing and image analysis. Segmentation of an image entails the division or separation of the image into regions of similar attributes. The most basic attribute for segmentation is the image intensity (luminance for a monochromatic image). Several classical methods for image segmentation exist and it is well known that these methods are more or less heuristic and specific to a particular application. Genetic Algorithms (GA) are stochastic search methods, the functioning of which is inspired by laws of genetics, natural selection and evolution of organisms. Their main attractive characteristic is the ability to deal with hard combinatorial search problems efficiently, where parallel exploration of the search space, eliminates to a large extent the possibility of getting stuck in the local extrema. The basis of the theory is that individuals tend to pass on their traits to their offspring and the fittest of the individuals tend to have more offsprings. In effect, the tendency is to drive the population towards favorable traits. Over long periods of time, entirely new species are produced which are better adapted to a particular ecological condition. This thesis proposes a simple and robust method for image segmentation that is based on the application of Genetic Algorithm and Mathematical Morphology. The image is divided into nonoverlapping subimages and the genetic algorithm is applied to each subimage, starting with initial random populations. Each individual of the population is evaluated using an appropriate fitness function. The best-fit individuals are selected and mated to produce offsprings to form the next generation. Morphological operations are used to produce the next generation along with the crossover and mutation operators. The algorithm converges to yield the final segmented subimage. These segmented subimages then are combined to form the final result. The feasibility of applying genetic algorithm and morphological operations to an image segmentation problem is evaluated and results are presented and discussed