## 2017 |

Charisopoulos, Vasileios; Maragos, Petros Morphological perceptrons: Geometry and training algorithms Conference Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 10225 LNCS , 2017, ISSN: 16113349. Abstract | BibTeX | Tags: Machine learning, Mathematical morphology, Neural networks, Optimization, Tropical geometry | Links: @conference{346, title = {Morphological perceptrons: Geometry and training algorithms}, author = { Vasileios Charisopoulos and Petros Maragos}, booktitle = {Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)}, volume = {10225 LNCS}, pages = {3--15}, } Neural networks have traditionally relied on mostly linear models, such as the multiply-accumulate architecture of a linear perceptron that remains the dominant paradigm of neuronal computation. However, from a biological standpoint, neuron activity may as well involve inherently nonlinear and competitive operations. Mathematical morphology and minimax algebra provide the necessary background in the study of neural networks made up from these kinds of nonlinear units. This paper deals with such a model, called the morphological perceptron. We study some of its geometrical properties and introduce a training algorithm for binary classification. We point out the relationship between morphological classifiers and the recent field of tropical geometry, which enables us to obtain a precise bound on the number of linear regions of the maxout unit, a popular choice for deep neural networks introduced recently. Finally, we present some relevant numerical results. |

## 2013 |

Karianakis, Nikolaos; Maragos, Petros An integrated system for digital restoration of prehistoric theran wall paintings Conference 2013 18th International Conference on Digital Signal Processing, DSP 2013, 2013, ISBN: 9781467358057. BibTeX | Tags: Digital restoration, Image stitching, Inpainting, Mathematical morphology, Prehistoric wall paintings, Segmentation, Variational methods | Links: @conference{174, title = {An integrated system for digital restoration of prehistoric theran wall paintings}, author = { Nikolaos Karianakis and Petros Maragos}, booktitle = {2013 18th International Conference on Digital Signal Processing, DSP 2013}, } |

Maragos, Petros Chapter Two - Representations for Morphological Image Operators and Analogies with Linear Operators Book Chapter Hawkes, Peter W (Ed.): Advances in Imaging and Electron Physics, 177 , pp. 45 - 187, Elsevier, 2013, ISSN: 1076-5670. Abstract | BibTeX | Tags: image operators, lattices, Mathematical morphology, minimax algebra, nonlinear basis, representation, supremal convolution | Links: @inbook{MARAGOS201345, title = {Chapter Two - Representations for Morphological Image Operators and Analogies with Linear Operators}, author = {Petros Maragos}, editor = {Peter W Hawkes}, booktitle = {Advances in Imaging and Electron Physics}, volume = {177}, pages = {45 - 187}, publisher = {Elsevier}, } This chapter deals with representation theoretical issues of nonlinear image operators, mainly based on the methodology of mathematical morphology, and more generally operators on lattices. After a brief overview of developments in morphological image operators both chronologically and thematically, the chapter provides a survey of some main concepts and results in the theory of lattices and morphological operators, especially of the monotone type. It also provides comparisons with linear operator theory. Then, it introduces a nonlinear signal space called complete weighted lattice, which generalizes both mathematical morphology and minimax algebra. Afterwards, it focuses on the representation of translation-invariant and/or increasing operators either on Euclidean spaces (or their discretized versions) or on complete weighted lattices by using a nonlinear basis. The results are operator representations as a supremum or infimum of nonlinear convolutions that are either of the max-plus type or their generalizations in weighted lattices. These representations have several potential applications in computation, imaging and vision, and nonlinear functional analysis. |

## 2008 |

Pnevmatikakis, Eftychios A; Maragos, Petros An inpainting system for automatic image structure-texture restoration with text removal Conference Proceedings - International Conference on Image Processing, ICIP, 2008, ISSN: 15224880. Abstract | BibTeX | Tags: Inpainting, Mathematical morphology, Text detection, Texture synthesis | Links: @conference{201, title = {An inpainting system for automatic image structure-texture restoration with text removal}, author = { Eftychios A. Pnevmatikakis and Petros Maragos}, booktitle = {Proceedings - International Conference on Image Processing, ICIP}, pages = {2616--2619}, } In this paper we deal with the inpainting problem and with the problem of finding text in images. We first review many of the methods used for structure and texture inpaintings. The novel contribution of the paper is the combination of the inpainting techniques with the techniques of finding text in images and a simple morphological algorithm that links them. This combination results in an automatic system for text removal and image restoration that requires no user interface at all. Examples on real images show very good performance of the proposed system and the importance of the new linking algorithm. |

## 2002 |

Tzafestas, Costas S; Maragos, Petros Shape connectivity: Multiscale analysis and application to generalized granulometries Journal Article Journal of Mathematical Imaging and Vision, 17 (2), pp. 109–129, 2002, ISSN: 09249907. Abstract | BibTeX | Tags: Connected operators, Connectivity tree, Generalized granulometries, Hierarchical image representations, Mathematical morphology, Multiscale connectivity measures, Partitions, Reconstruction, shape analysis, Soilsection image analysis | Links: @article{117, title = {Shape connectivity: Multiscale analysis and application to generalized granulometries}, author = {Costas S Tzafestas and Petros Maragos}, journal = {Journal of Mathematical Imaging and Vision}, volume = {17}, number = {2}, pages = {109--129}, } This paper develops a multiscale connectivity theory for shapes based on the axiomatic definition of new generalized connectivity measures, which are obtained using morphology-based nonlinear scale-space operators. The concept of connectivity-tree for hierarchical image representation is introduced and used to define generalized connected morphological operators. This theoretical framework is then applied to establish a class of generalized granulometries, implemented at a particular problem concerning soilsection image analysis and evaluation of morphological properties such as size distributions. Comparative results demonstrate the power and versatility of the proposed methodology with respect to the application of typical connected operators (such as reconstruction openings). This multiscale connectivity analysis framework aims at a more reliable evaluation of shape/size information within complex images, with particular applications to generalized granulometries, connected operators, and segmentation. |

## 2000 |

Meyer, Fernand; Maragos, Petros Nonlinear scale-space representation with morphological levelings Journal Article Journal of Visual Communication and Image Representation, 11 (2), pp. 245–265, 2000, ISSN: 10473203. Abstract | BibTeX | Tags: Differential equations, Levelings, Mathematical morphology, Multiscale representation, Scale-space | Links: @article{113, title = {Nonlinear scale-space representation with morphological levelings}, author = {Fernand Meyer and Petros Maragos}, journal = {Journal of Visual Communication and Image Representation}, volume = {11}, number = {2}, pages = {245--265}, } In this paper we present a nonlinear scale-space representation based on a general class of morphological strong filters, the levelings, which include the openings and closings by reconstruction. These filters are very useful for image simplification and segmentation. From one scale to the next, details vanish, but the contours of the remaining objects are preserved sharp and perfectly localized. Both the lattice algebraic and the scale-space properties of levelings are analyzed and illustrated. We also develop a nonlinear partial differential equation that models the generation of levelings as the limit of a controlled growth starting from an initial seed signal. Finally, we outline the use of levelings in improving the Gaussian scale-space by using the latter as an initial seed to generate multiscale levelings that have a superior preservation of image edges. textcopyright 2000 Academic Press. |

## 1989 |

Maragos, Petros A Representation Theory for Morphological Image and Signal Processing Journal Article IEEE Trans. on Pattern Analysis and Machine Intelligence, 11 (6), pp. 586–599, 1989, ISSN: 01628828. Abstract | BibTeX | Tags: Imagelsignal processing, Mathematical morphology, nonlinearllinear filtering, shape analysis, systems representation | Links: @article{86c, title = {A Representation Theory for Morphological Image and Signal Processing}, author = {Petros Maragos}, journal = {IEEE Trans. on Pattern Analysis and Machine Intelligence}, volume = {11}, number = {6}, pages = {586--599}, } A unifying theory for many concepts and operations encountered in or related to morphological image and signal analysis is presented. The unification requires a set-theoretic methodology, where signals are modeled as sets, systems (signal transformations) are viewed as set mappings, and translational-invariant systems are uniquely characterized by special collections of input signals. This approach leads to a general representation theory, in which any translation-invariant, increasing, upper semicontinuous system can be presented exactly as a minimal nonlinear superposition of morphological erosions or dilations. The theory is used to analyze some special cases of image/signal analysis systems, such as morphological filters, median and order-statistic filters, linear filters, and shape recognition transforms. Although the developed theory is algebraic, its prototype operations are well suited for shape analysis; hence, the results also apply to systems that extract information about the geometrical structure of signals |

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