Deep Morphological Neural Networks
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Morphological Image Processing Introduction
Morphological Image Processing Introduction • In many areas of knowledge Morphology deals with form and structure (biology, linguistics, social studies, etc) • Mathematical Morphology deals with set theory • Sets in Mathematical Morphology represents objects in an Image 2 Mathematic Morphology • Used to extract image components that are useful in the representation and description of region shape, such as – boundaries extraction – skeletons – convex hull (italian: inviluppo convesso) – morphological filtering – thinning – pruning 3 Mathematic Morphology mathematical framework used for: • pre-processing – noise filtering, shape simplification, ... • enhancing object structure – skeletonization, convex hull... • segmentation – watershed,… • quantitative description – area, perimeter, ... 4 Z2 and Z3 • set in mathematic morphology represent objects in an image – binary image (0 = white, 1 = black) : the element of the set is the coordinates (x,y) of pixel belong to the object a Z2 • gray-scaled image : the element of the set is the coordinates (x,y) of pixel belong to the object and the gray levels a Z3 Y axis Y axis X axis Z axis X axis 5 Basic Set Operators Set operators Denotations A Subset B A ⊆ B Union of A and B C= A ∪ B Intersection of A and B C = A ∩ B Disjoint A ∩ B = ∅ c Complement of A A ={ w | w ∉ A} Difference of A and B A-B = {w | w ∈A, w ∉ B } Reflection of A Â = { w | w = -a for a ∈ A} Translation of set A by point z(z1,z2) (A)z = { c | c = a + z, for a ∈ A} 6 Basic Set Theory 7 Reflection and Translation Bˆ = {w ∈ E 2 : w -
Lecture 5: Binary Morphology
Lecture 5: Binary Morphology c Bryan S. Morse, Brigham Young University, 1998–2000 Last modified on January 15, 2000 at 3:00 PM Contents 5.1 What is Mathematical Morphology? ................................. 1 5.2 Image Regions as Sets ......................................... 1 5.3 Basic Binary Operations ........................................ 2 5.3.1 Dilation ............................................. 2 5.3.2 Erosion ............................................. 2 5.3.3 Duality of Dilation and Erosion ................................. 3 5.4 Some Examples of Using Dilation and Erosion . .......................... 3 5.5 Proving Properties of Mathematical Morphology .......................... 3 5.6 Hit-and-Miss .............................................. 4 5.7 Opening and Closing .......................................... 4 5.7.1 Opening ............................................. 4 5.7.2 Closing ............................................. 5 5.7.3 Properties of Opening and Closing ............................... 5 5.7.4 Applications of Opening and Closing .............................. 5 Reading SH&B, 11.1–11.3 Castleman, 18.7.1–18.7.4 5.1 What is Mathematical Morphology? “Morphology” can literally be taken to mean “doing things to shapes”. “Mathematical morphology” then, by exten- sion, means using mathematical principals to do things to shapes. 5.2 Image Regions as Sets The basis of mathematical morphology is the description of image regions as sets. For a binary image, we can consider the “on” (1) pixels to all comprise a set of values from the “universe” of pixels in the image. Throughout our discussion of mathematical morphology (or just “morphology”), when we refer to an image A, we mean the set of “on” (1) pixels in that image. The “off” (0) pixels are thus the set compliment of the set of on pixels. By Ac, we mean the compliment of A,or the off (0) pixels. -
Hit-Or-Miss Transform
Hit-or-miss transform • Used to extract pixels with specific neighbourhood configurations from an image • Grey scale extension exist • Uses two structure elements B1 and B2 to find a given foreground and background configuration, respectively C HMT B X={x∣B1x⊆X ,B2x⊆X } • Example: 1 Morphological Image Processing Lecture 22 (page 1) 9.4 The hit-or-miss transformation Illustration... Morphological Image Processing Lecture 22 (page 2) Objective is to find a disjoint region (set) in an image • If B denotes the set composed of X and its background, the• match/hit (or set of matches/hits) of B in A,is A B =(A X) [Ac (W X)] ¯∗ ª ∩ ª − Generalized notation: B =(B1,B2) • Set formed from elements of B associated with B1: • an object Set formed from elements of B associated with B2: • the corresponding background [Preceeding discussion: B1 = X and B2 =(W X)] − More general definition: • c A B =(A B1) [A B2] ¯∗ ª ∩ ª A B contains all the origin points at which, simulta- • ¯∗ c neously, B1 found a hit in A and B2 found a hit in A Hit-or-miss transform C HMT B X={x∣B1x⊆X ,B2x⊆X } • Can be written in terms of an intersection of two erosions: HMT X= X∩ X c B B1 B2 2 Hit-or-miss transform • Simple example usages - locate: – Isolated foreground pixels • no neighbouring foreground pixels – Foreground endpoints • one or zero neighbouring foreground pixels – Multiple foreground points • pixels having more than two neighbouring foreground pixels – Foreground contour points • pixels having at least one neighbouring background pixel 3 Hit-or-miss transform example • Locating 4-connected endpoints SEs for 4-connected endpoints Resulting Hit-or-miss transform 4 Hit-or-miss opening • Objective: keep all points that fit the SE. -
Digital Image Processing
Digital Image Processing Lecture # 11 Morphological Operations 1 Image Morphology 2 Introduction Morphology A branch of biology which deals with the form and structure of animals and plants Mathematical Morphology A tool for extracting image components that are useful in the representation and description of region shapes The language of mathematical morphology is Set Theory 3 Morphology: Quick Example Image after segmentation Image after segmentation and morphological processing 4 Introduction Morphological image processing describes a range of image processing techniques that deal with the shape (or morphology) of objects in an image Sets in mathematical morphology represents objects in an image. E.g. Set of all white pixels in a binary image. 5 Introduction foreground: background I(p )c I(p)0 This represents a digital image. Each square is one pixel. 6 Set Theory The set space of binary image is Z2 Each element of the set is a 2D vector whose coordinates are the (x,y) of a black (or white, depending on the convention) pixel in the image The set space of gray level image is Z3 Each element of the set is a 3D vector: (x,y) and intensity level. NOTE: Set Theory and Logical operations are covered in: Section 9.1, Chapter # 9, 2nd Edition DIP by Gonzalez Section 2.6.4, Chapter # 2, 3rd Edition DIP by Gonzalez 7 Set Theory 2 Let A be a set in Z . if a = (a1,a2) is an element of A, then we write aA If a is not an element of A, we write aA Set representation A{( a1 , a 2 ),( a 3 , a 4 )} Empty or Null set A 8 Set Theory Subset: if every element of A is also an element of another set B, the A is said to be a subset of B AB Union: The set of all elements belonging either to A, B or both CAB Intersection: The set of all elements belonging to both A and B DAB 9 Set Theory Two sets A and B are said to be disjoint or mutually exclusive if they have no common element AB Complement: The set of elements not contained in A Ac { w | w A } Difference of two sets A and B, denoted by A – B, is defined as c A B { w | w A , w B } A B i.e. -
Grayscale Mathematical Morphology Václav Hlaváč
Grayscale mathematical morphology Václav Hlaváč Czech Technical University in Prague Czech Institute of Informatics, Robotics and Cybernetics 160 00 Prague 6, Jugoslávských partyzánů 1580/3, Czech Republic http://people.ciirc.cvut.cz/hlavac, [email protected] also Center for Machine Perception, http://cmp.felk.cvut.cz Courtesy: Petr Matula, Petr Kodl, Jean Serra, Miroslav Svoboda Outline of the talk: Set-function equivalence. Top-hat transform. Umbra and top of a set. Geodesic method. Ultimate erosion. Gray scale dilation, erosion. Morphological reconstruction. A quick informal explanation 2/42 Grayscale mathematical morphology is the generalization of binary morphology for images with more gray levels than two or with voxels. 3 The point set A ∈ E . The first two coordinates span in the function (point set) domain and the third coordinate corresponds to the function value. The concepts supremum ∨ (also the least upper bound), resp. infimum ∧ (also the greatest lower bound) play a key role here. Actually, the related operators max, resp. min, are used in computations with finite sets. Erosion (resp. dilation) of the image (with the flat structuring) element assigns to each pixel the minimal (resp. maximal) value in the chosen neighborhood of the current pixel of the input image. The structuring element (function) is a function of two variables. It influences how pixels in the neighborhood of the current pixel are taken into account. The value of the (non-flat) structuring element is added (while dilating), resp. subtracted (while eroding) when the maximum, resp. minimum is calculated in the neighborhood. Grayscale mathematical morphology explained via binary morphology 3/42 It is possible to introduce grayscale mathematical morphology using the already explained binary (black and white only) mathematical morphology. -
Abstract Mathematical Morphology Based on Structuring Element
Abstract Mathematical morphology based on structuring element: Application to morpho-logic Marc Aiguier1 and Isabelle Bloch2 and Ram´on Pino-P´erez3 1. MICS, CentraleSupelec, Universit´eParis Saclay, France [email protected] 2. LTCI, T´el´ecom Paris, Institut Polytechnique de Paris, France [email protected] 3. Departemento de Matematicas, Facultad de Ciencias, Universidad de Los Andes, M´erida, Venezuela [email protected] Abstract A general definition of mathematical morphology has been defined within the algebraic framework of complete lattice theory. In this frame- work, dealing with deterministic and increasing operators, a dilation (re- spectively an erosion) is an operation which is distributive over supremum (respectively infimum). From this simple definition of dilation and ero- sion, we cannot say much about the properties of them. However, when they form an adjunction, many important properties can be derived such as monotonicity, idempotence, and extensivity or anti-extensivity of their composition, preservation of infimum and supremum, etc. Mathemati- cal morphology has been first developed in the setting of sets, and then extended to other algebraic structures such as graphs, hypergraphs or sim- plicial complexes. For all these algebraic structures, erosion and dilation are usually based on structuring elements. The goal is then to match these structuring elements on given objects either to dilate or erode them. One of the advantages of defining erosion and dilation based on structuring ele- arXiv:2005.01715v1 [math.CT] 4 May 2020 ments is that these operations are adjoint. Based on this observation, this paper proposes to define, at the abstract level of category theory, erosion and dilation based on structuring elements. -
Computer Vision – A
CS4495 Computer Vision – A. Bobick Morphology CS 4495 Computer Vision Binary images and Morphology Aaron Bobick School of Interactive Computing CS4495 Computer Vision – A. Bobick Morphology Administrivia • PS7 – read about it on Piazza • PS5 – grades are out. • Final – Dec 9 • Study guide will be out by Thursday hopefully sooner. CS4495 Computer Vision – A. Bobick Morphology Binary Image Analysis Operations that produce or process binary images, typically 0’s and 1’s • 0 represents background • 1 represents foreground 00010010001000 00011110001000 00010010001000 Slides: Linda Shapiro CS4495 Computer Vision – A. Bobick Morphology Binary Image Analysis Used in a number of practical applications • Part inspection • Manufacturing • Document processing CS4495 Computer Vision – A. Bobick Morphology What kinds of operations? • Separate objects from background and from one another • Aggregate pixels for each object • Compute features for each object CS4495 Computer Vision – A. Bobick Morphology Example: Red blood cell image • Many blood cells are separate objects • Many touch – bad! • Salt and pepper noise from thresholding • How useable is this data? CS4495 Computer Vision – A. Bobick Morphology Results of analysis • 63 separate objects detected • Single cells have area about 50 • Noise spots • Gobs of cells CS4495 Computer Vision – A. Bobick Morphology Useful Operations • Thresholding a gray-scale image • Determining good thresholds • Connected components analysis • Binary mathematical morphology • All sorts of feature extractors, statistics (area, centroid, circularity, …) CS4495 Computer Vision – A. Bobick Morphology Thresholding • Background is black • Healthy cherry is bright • Bruise is medium dark • Histogram shows two cherry regions (black background has been removed) Pixel counts Pixel 0 Grayscale values 255 CS4495 Computer Vision – A. Bobick Morphology Histogram-Directed Thresholding • How can we use a histogram to separate an image into 2 (or several) different regions? Is there a single clear threshold? 2? 3? CS4495 Computer Vision – A. -
Morphological Operations on Matrix-Valued Images
Morphological Operations on Matrix-Valued Images Bernhard Burgeth, Martin Welk, Christian Feddern, and Joachim Weickert Mathematical Image Analysis Group Faculty of Mathematics and Computer Science, Bldg. 27 Saarland University, 66041 Saarbruc¨ ken, Germany burgeth,welk,feddern,weickert @mia.uni-saarland.de f g http://www.mia.uni-saarland.de Abstract. The output of modern imaging techniques such as diffusion tensor MRI or the physical measurement of anisotropic behaviour in materials such as the stress-tensor consists of tensor-valued data. Hence adequate image processing methods for shape analysis, skeletonisation, denoising and segmentation are in demand. The goal of this paper is to extend the morphological operations of dilation, erosion, opening and closing to the matrix-valued setting. We show that naive approaches such as componentwise application of scalar morphological operations are unsatisfactory, since they violate elementary requirements such as invariance under rotation. This lead us to study an analytic and a geo- metric alternative which are rotation invariant. Both methods introduce novel non-component-wise definitions of a supremum and an infimum of a finite set of matrices. The resulting morphological operations incorpo- rate information from all matrix channels simultaneously and preserve positive definiteness of the matrix field. Their properties and their per- formance are illustrated by experiments on diffusion tensor MRI data. Keywords: mathematical morphology, dilation, erosion, matrix-valued imaging, DT-MRI 1 Introduction Modern data and image processing encompasses more and more the analysis and processing of matrix-valued data. For instance, diffusion tensor magnetic reso- nance imaging (DT-MRI), a novel medical image acquisition technique, measures the diffusion properties of water molecules in tissue. -
Chapter 7 Mathematical Morphology with Noncommutative Symmetry Groups Jos B.T.M
Chapter 7 Mathematical Morphology with Noncommutative Symmetry Groups Jos B.T.M. Roerdink Centre for Mathematics and Computer Science, Amsterdam, The Netherlands Mathematical morphology as originally developed by Matheron and Serra is a theory of set mappings, modeling binary image transformations, that are invar iant under the group of Euclidean translations. Because this framework turns out to be too restricted for many practical applications, various generalizations have been proposed. First, the translation group may be replaced by an arbitrary com mutative group. Second, one may consider more general object spaces, such as the set of all convex subsets of the plane or the set of gray-level functions on the plane, requiring a formulation in terms of complete lattices. So far, symmetry properties have been incorporated by assuming that the allowed image transfor mations are invariant under a certain commutative group of automorphisms on the lattice. In this chapter we embark on another generalization of mathematical morphology by dropping the assumption that the invariance group is commuta tive. To this end we consider an arbitrary homogeneous space (the plane with the Euclidean translation group is one example, the sphere with the rotation group another), that is, a set 2e on which a transitive but not necessarily commutative transformation group f is defined. As our object space we then take the Boolean algebra 0P(2e) of all subsets of this homogeneous space. First we consider the case in which the transformation group is simply transitive or, equivalently, the basic set 2e is itself a group, so that we may study the Boolean algebra '2?(f). -
A Fast Algorithm for Morphological Erosion and Dilation
A FAST ALGORITHM FOR MORPHOLOGICAL EROSION AND DILATION C. Jeremy Pye J. A. Bangham School of Information Systems, University of East Anglia, Norwich, NR4 7TJ, UK. Tel: +44 0 1603 456161; Fax: +44 0 1603 453345 Email:[email protected],[email protected] ABSTRACT maximum and minimum values as the histogram adapts it k l This paper describes a new algorithm for performing erosion is possible to reduce the computation required by a l and dilation which is suitable for flat line-segment structur- structuring function from kl down to computations per el- ing functions, and which has a computational complexity ement. A drawback of this type of method is that the data that is independent of the structuring function size. Unlike must be represented by a fixed number of bits in order to a other proposed algorithms, the computation time required by build a histogram. Astola [4] presents a similar method based this method is directly proportional to the number of extrema upon a double heap structure which elevates this constraint. within the signal being processed. This makes it particularly Recently several algorithms designed to perform the fast cal- suitable for signals and images that have large and slowly culation of the running maximum or minimum on a one varying segments. dimensional signal have been presented. Pitas [5, 6] de- velops an algorithm based upon the ”divide and conquer” 1 INTRODUCTION principle designed to exploit the redundancy in the running calculations, resulting in a reduced number of comparisons Since its evolution in the late 1970's, mathematical morphol- k per element log . -
International Journal of Advance Engineering and Research Development Volume 1, Issue 12, December -2014
Scientific Journal of Impact Factor (SJIF): 3.134 ISSN (Online): 2348-4470 ISSN (Print) : 2348-6406 International Journal of Advance Engineering and Research Development Volume 1, Issue 12, December -2014 A Survey Report on Hit-or-Miss Transformation Wala Riddhi Ashokbhai1 Department of Computer Engineering, Darshan Institute of Engineering and Technology, Rajkot, Gujarat, India [email protected] Abstract- Hit or miss is a field of morphological digital image processing which is basically used to detect shape of regular as well as irregular objects. Initially, it was defined for binary images, as a morphological approach to the problem of template matching, but can be extended for gray scale images. For gray-scale it has been debatable, leading to multiple definitions that have been only recently unified by means of a similar theoretical foundation. Various structuring elements are used which are disjoint by nature in order to determine the output. Many authors have proposed newer versions of the original binary HMT so that it can be applied to gray-scale data. Hit or miss transformation operation takes a binary image as input and a kernel (also known as structuring element), and the result is another binary image as output. Main motive of this survey is to use Hit-or-miss transformation in order to detect various features in a given image. Keywords-Hit-and-Miss transform, Hit-or-miss transformation, Morphological image technology, digital image processing, HMT I. INTRODUCTION The hit-or-miss transform (HMT) is a powerful morphological technique that was among the initial morphological transforms developed by Serra (1982) and Matheron (1975). -
Application of Mathematical Morphology to the Analysis of X-Ray NDE Images Mathew Scaria Chackalackal Iowa State University
Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1991 Application of mathematical morphology to the analysis of X-ray NDE images Mathew Scaria Chackalackal Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Electrical and Electronics Commons Recommended Citation Chackalackal, Mathew Scaria, "Application of mathematical morphology to the analysis of X-ray NDE images" (1991). Retrospective Theses and Dissertations. 112. https://lib.dr.iastate.edu/rtd/112 This Thesis is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Application of Mathematical Morphology to the analysis of X-ray NDE Images by Mathew Scaria Chackalackal A Thesis Suhn1itted to the Graduate Faculty in Partial Fulfilln1ent of the Requiren1ents for t.he Degree of lVIASTER OF SCIENCE Depart1nent: Electrical Engineering and Computer Engineering Major: Electrical Engineering Approved: In Charge of Major Work For the lVlajor Department For the Graduate College Iowa State University Ames, Iowa 1991 11 TABLE OF CONTENTS ACKNOWLEDGEMENTS ..... X CHAPTER 1. INTRODUCTION 1 CHAPTER 2. BINARY MORPHOLOGY . .5 ~Iinkowski Algebra .5 Translation . .; Minkowski Addition 6 Minkowski Subtraction 8 Illustration of Minkowski Subtraction . 8 Binary Morphology . 9 Dilation ...... 10 Properties of Dilation . 12 Erosion .......... 14 Properties of Erosion . 17 Openings and Closings . 21 Properties of Opening and Closing 25 CHAPTER 3. GRAY-SCALE MORPHOLOGY 26 Three Dimensional Binary Image 27 Top Surface and Umbra .