759 research outputs found
Logarithmic mathematical morphology: a new framework adaptive to illumination changes
A new set of mathematical morphology (MM) operators adaptive to illumination
changes caused by variation of exposure time or light intensity is defined
thanks to the Logarithmic Image Processing (LIP) model. This model based on the
physics of acquisition is consistent with human vision. The fundamental
operators, the logarithmic-dilation and the logarithmic-erosion, are defined
with the LIP-addition of a structuring function. The combination of these two
adjunct operators gives morphological filters, namely the logarithmic-opening
and closing, useful for pattern recognition. The mathematical relation existing
between ``classical'' dilation and erosion and their logarithmic-versions is
established facilitating their implementation. Results on simulated and real
images show that logarithmic-MM is more efficient on low-contrasted information
than ``classical'' MM
Interval-valued and intuitionistic fuzzy mathematical morphologies as special cases of L-fuzzy mathematical morphology
Mathematical morphology (MM) offers a wide range of tools for image processing and computer vision. MM was originally conceived for the processing of binary images and later extended to gray-scale morphology. Extensions of classical binary morphology to gray-scale morphology include approaches based on fuzzy set theory that give rise to fuzzy mathematical morphology (FMM). From a mathematical point of view, FMM relies on the fact that the class of all fuzzy sets over a certain universe forms a complete lattice. Recall that complete lattices provide for the most general framework in which MM can be conducted.
The concept of L-fuzzy set generalizes not only the concept of fuzzy set but also the concepts of interval-valued fuzzy set and Atanassov’s intuitionistic fuzzy set. In addition, the class of L-fuzzy sets forms a complete lattice whenever the underlying set L constitutes a complete lattice. Based on these observations, we develop a general approach towards L-fuzzy mathematical morphology in this paper. Our focus is in particular on the construction of connectives for interval-valued and intuitionistic fuzzy mathematical morphologies that arise as special, isomorphic cases of L-fuzzy MM. As an application of these ideas, we generate a combination of some well-known medical image reconstruction techniques in terms of interval-valued fuzzy image processing
Digital Morphometry : A Taxonomy Of Morphological Filters And Feature Parameters With Application To Alzheimer\u27s Disease Research
In this thesis the expression digital morphometry collectively describes all those procedures used to obtain quantitative measurements of objects within a two-dimensional digital image. Quantitative measurement is a two-step process: the application of geometrical transformations to extract the features of interest, and then the actual measurement of these features. With regard to the first step the morphological filters of mathematical morphology provide a wealth of suitable geometric transfomations. Traditional radiometric and spatial enhancement techniques provide an additional source of transformations. The second step is more classical (e.g. Underwood, 1970; Bookstein, 1978; and Weibull, 1980); yet here again mathematical morphology is applicable - morphologically derived feature parameters. This thesis focuses on mathematical morphology for digital morphometry. In particular it proffers a taxonomy of morphological filters and investigates the morphologically derived feature parameters (Minkowski functionals) for digital images sampled on a square grid. As originally conceived by Georges Matheron, mathematical morphology concerns the analysis of binary images by means of probing with structuring elements [typically convex geometric shapes] (Dougherty, 1993, preface). Since its inception the theory has been extended to grey-level images and most recently to complete lattices. It is within the very general framework of the complete lattice that the taxonomy of morphological filters is presented. Examples are provided to help illustrate the behaviour of each type of filter. This thesis also introduces DIMPAL (Mehnert, 1994) - a PC-based image processing and analysis language suitable for researching and developing algorithms for a wide range of image processing applications. Though DIMPAL was used to produce the majority of the images in this thesis it was principally written to provide an environment in which to investigate the application of mathematical morphology to Alzheimer\u27s disease research. Alzheimer\u27s disease is a form of progressive dementia associated with the degeneration of the brain. It is the commonest type of dementia and probably accounts for half the dementia of old age (Forsythe, 1990, p. 21 ). Post mortem examination of the brain reveals the presence of characteristic neuropathologic lesions; namely neuritic plaques and neurofibrillary tangles. They occur predominantly in the cerebral cortex and hippocampus. Quantitative studies of the distribution of plaques and tangles in normally aged and Alzheimer brains are hampered by the enormous amount of time and effort required to count and measure these lesions. Here in a morphological algorithm is proposed for the automatic segmentation and measurement of neuritic plaques from light micrographs of post mortem brain tissue
Inf-structuring functions and self-dual marked flattenings in bi-Heyting algebra
International audienceThis paper introduces a generalization of self-dual marked flattenings defined in the lattice of mappings. This definition provides a way to associate a self-dual operator to every mapping that decomposes an element into sub-elements (i.e. gives a cover). Contrary to classical flattenings whose definition relies on the complemented structure of the powerset lattices, our approach uses the pseudo relative complement and supplement of the bi-Heyting algebra and a new notion of \textit{inf-structuring functions} that provides a very general way to structure the space. We show that using an inf-structuring function based on connections allows to recover the original definition of marked flattenings and we provide, as an example, a simple inf-structuring function whose derived self-dual operator better preserves contrasts and does not introduce new pixel values
The Lattice Overparametrization Paradigm for the Machine Learning of Lattice Operators
The machine learning of lattice operators has three possible bottlenecks.
From a statistical standpoint, it is necessary to design a constrained class of
operators based on prior information with low bias, and low complexity relative
to the sample size. From a computational perspective, there should be an
efficient algorithm to minimize an empirical error over the class. From an
understanding point of view, the properties of the learned operator need to be
derived, so its behavior can be theoretically understood. The statistical
bottleneck can be overcome due to the rich literature about the representation
of lattice operators, but there is no general learning algorithm for them. In
this paper, we discuss a learning paradigm in which, by overparametrizing a
class via elements in a lattice, an algorithm for minimizing functions in a
lattice is applied to learn. We present the stochastic lattice gradient descent
algorithm as a general algorithm to learn on constrained classes of operators
as long as a lattice overparametrization of it is fixed, and we discuss
previous works which are proves of concept. Moreover, if there are algorithms
to compute the basis of an operator from its overparametrization, then its
properties can be deduced and the understanding bottleneck is also overcome.
This learning paradigm has three properties that modern methods based on neural
networks lack: control, transparency and interpretability. Nowadays, there is
an increasing demand for methods with these characteristics, and we believe
that mathematical morphology is in a unique position to supply them. The
lattice overparametrization paradigm could be a missing piece for it to achieve
its full potential within modern machine learning
Inf-semilattice approach to self-dual morphology
Today, the theoretical framework of mathematical morphology is phrased in terms of complete lattices and operators defined on them. That means in particular that the choice of the underlying partial ordering is of eminent importance, as it determines the class of morphological operators that one ends up with. The duality principle for partially ordered sets, which says that the opposite of a partial ordering is also a partial ordering, gives rise to the fact that all morphological operators occur in pairs, e.g., dilation and erosion, opening and closing, etc. This phenomenon often prohibits the construction of tools that treat foreground and background of signals in exactly the same way. In this paper we discuss an alternative framework for morphological image processing that gives rise to image operators which are intrinsically self-dual. As one might expect, this alternative framework is entirely based upon the definition of a new self-dual partial ordering
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