Instituto de Computação
INSTITUTO DE COMPUTAÇÃO
Darwin Danilo Saire Pilco
Multi-scale Morphological Image Simplification Based
on Extremum Relationships: Improvements and
Applications
Simplificações Morfológicas Multiescala Baseada em
Relações Extremas: Melhoras e Aplicações
CAMPINAS
2017
Darwin Danilo Saire Pilco
Multi-scale Morphological Image Simplification Based on
Extremum Relationships: Improvements and Applications
Simplificações Morfológicas Multiescala Baseada em Relações
Extremas: Melhoras e Aplicações
Dissertação apresentada ao Instituto de Computação da Universidade Estadual de Campinas como parte dos requisitos para a obtenção do título de Mestre em Ciência da Computação.
Thesis presented to the Institute of Computing of the University of Campinas in partial fulfillment of the requirements for the degree of Master in Computer Science.
Supervisor/ Orientador: Prof. Dr. Gerberth Adín Ramírez Rivera Co-supervisor/Coorientador: Prof. Dr. Neucimar Jerônimo Leite
Este exemplar corresponde à versão final da Dissertação defendida por Darwin Danilo Saire Pilco e orientada pelo Prof. Dr. Gerberth Adín Ramírez Rivera.
CAMPINAS
2017
Ficha catalográfica
Universidade Estadual de Campinas
Biblioteca do Instituto de Matemática, Estatística e Computação Científica Ana Regina Machado - CRB 8/5467
Saire Pilco, Darwin Danilo,
Sa28m SaiMulti-scale morphological image simplification based on extremum relationships : improvements and applications / Darwin Danilo Saire Pilco. – Campinas, SP : [s.n.], 2017.
SaiOrientador: Gerberth Adín Ramírez Rivera. SaiCoorientador: Neucimar Jerônimo Leite.
SaiDissertação (mestrado) – Universidade Estadual de Campinas, Instituto de Computação.
Sai1. Morfologia matemática. 2. Processamento de imagens. 3. Análise de imagem. 4. Teoria dos grafos. I. Ramírez Rivera, Gerberth Adín,1986-. II. Leite, Neucimar Jerônimo,1961-. III. Universidade Estadual de Campinas. Instituto de Computação. IV. Título.
Informações para Biblioteca Digital
Título em outro idioma: Simplificações morfológicas multiescala baseadas em relações
extremas : melhoras e aplicações
Palavras-chave em inglês:
Mathematical morphology Image processing
Image analysis Graph theory
Área de concentração: Ciência da Computação Titulação: Mestre em Ciência da Computação Banca examinadora:
Gerberth Adín Ramírez Rivera [Orientador] Hélio Pedrini
Roberto Hirata Junior
Data de defesa: 31-03-2017
Programa de Pós-Graduação: Ciência da Computação
Universidade Estadual de Campinas Instituto de Computação
INSTITUTO DE COMPUTAÇÃO
Darwin Danilo Saire Pilco
Multi-scale Morphological Image Simplification Based on
Extremum Relationships: Improvements and Applications
Simplificações Morfológicas Multiescala Baseada em Relações
Extremas: Melhoras e Aplicações
Banca Examinadora:
• Prof. Dr. Gerberth Adín Ramírez Rivera IC-UNICAMP
• Prof. Dr. Hélio Pedrini IC-UNICAMP
• Prof. Dr. Roberto Hirata Junior IME-USP
A ata da defesa com as respectivas assinaturas dos membros da banca encontra-se no processo de vida acadêmica do aluno.
Acknowledgements
• Firstly, I would like to thank my parents Florence and Maximiana for supporting me at all times, for the values they have implanted in me, and for giving me the opportunity to have an excellent education. Especially for being an extraordinary example of life to follow.
• I would also like to thank my brothers for supporting me spiritually and being an important part of my life. To all of them for filling my life with great moments that we have shared. In special to my sister Maria for giving me her emotional support in the development of this research.
• My sincere thanks go to my co-advisor Prof. Dr. Neucimar, deceased in the course of this thesis, for the continuous support of my Master study and related research, for his patience, motivation, and immense knowledge. His guidance helped me in all the time of research of this thesis.
• I would like to thank my advisor Prof. Dr. Adín for believing in me and whose guidance helped me in the writing of this thesis.
• Last but not the least, I thank my friends for trust and believe in me and to have made this stage of my life, a journey of experiences that I will never forget.
Resumo
A abordagem de simplificação de imagem visa reduzir a quantidade de informação presente nas imagens, isto através da filtragem de componentes de ruído e eliminação de detalhes não significativos, mantendo as informações necessárias para a obtenção do resultado dese-jado. Além disso, a simplificação de imagens tem mostrado ser útil em diversas aplicações de processamento de imagens como um passo adicional para tarefas mais complexas, como a segmentação e extração de características.
Nesta Dissertação de Mestrado, criamos um método de simplificação de imagens, que aproveita os múltiplos níveis de observação (multi-escala), que utiliza uma medida de persistência e ainda tem demanda computacional reduzida. Para isso, exploramos um método de simplificação baseado em grafos que garante o bom comportamento da supres-são dos nós extremos da imagem (máximas / mínimas), tendo em conta a informação da distância e contraste, bem como alguns aspectos interessantes da teoria do espaço-escala. Finalmente, apresentamos dois tipos de resultados: teóricos e experimentais. Os resul-tados teóricos são a criação e demonstração de novas propriedades no grafo que suporta todo o processo de simplificação. Com essas novas propriedades, nós definimos uma atua-lização local do grafo que implica um desvio interessante em toda a estrutura do algoritmo que, originalmente, é muito custos. Por outro lado, os resultados experimentais são dividi-dos em duas partes. A primeira parte são as comparações de tempo de execução, usando as bases de dados “Berkeley Segmentation Dataset and Benchmark 300” e “Berkeley Segmen-tation Dataset and Benchmark 500”, demonstrando que nosso método contribui para uma redução no custo computacional. E a segunda parte é a apresentação de aplicações onde mostramos como combinar este processo de simplificação com ferramentas morfológicas bem conhecidas para abordar problemas relacionados, principalmente, com segmentação de imagens multiescala e homogeneização.
Abstract
Image simplification approach aims to reduce the amount of information present in the images, this by filtering out noise components and eliminating of non-significant details while keeping the information necessary to the achievement of the desired outcome. In addition, image simplification has been proved useful in several image processing appli-cations as an additional step for more complex tasks, such as segmentation and feature extraction.
In this Master’s Thesis, we create a method of images simplification, that takes ad-vantage of the multiple levels of observation (multi-scale), that uses a persistence measure and still be of reduced computational demand. To achieve it, we explore a graph-based simplification method that guarantees a well-behaved suppression of the image extrema (maxima/minima) by taking into account both information of distance and contrast, as well as some interesting aspects of the scale-space theory.
Finally, we present two types of results: theoretical and experimental. The theoretical results are the creation and demonstration of new properties on the graph that supports the whole process of simplification. With this properties, we define a local update of the graph which implies an interesting bypass in the whole algorithm structure which, orig-inally, is very time-consuming. On the other hand, the experimental results are divided into two parts. The first part is the run-time comparisons, using the “Berkeley Segmenta-tion Dataset and Benchmark 300” and “Berkeley SegmentaSegmenta-tion Dataset and Benchmark 500” databases, demonstrating that our method contributes to a reduction in computa-tional cost. And the second part is the presentation of applications where we illustrate how to combine this simplification process with well-known morphological tools to address problems related mainly to multi-scale image segmentation and homogenization.
List of Figures
1.1 Application of successive simplifications using image segmentation . . . 15
1.2 Application of successive simplifications using image homogenization . . . . 15
2.1 Threshold operator . . . 19
2.2 Discrete distances . . . 21
2.3 Examples of structuring elements . . . 21
2.4 Example erosion operator . . . 23
2.5 Example dilation operator . . . 24
2.6 Examples of opening and closing . . . 25
2.7 Watershed transform . . . 26
2.8 Watershed example . . . 26
2.9 Image simplification using mosaic approach . . . 27
2.10 Image simplification using leveling . . . 27
2.11 Dynamic representation . . . 28
2.12 Dynamic simplification . . . 28
3.1 Multi-scale approach . . . 30
3.2 Representation multi-scale and morphological scale-space . . . 31
3.3 Two dimensional scale-space . . . 32
3.4 Difference between multi-scale and morphological scale-space approach . . 33
3.5 Example morphological scale-space Properties . . . 34
3.6 Non-flat pyramid-shaped structuring function . . . 35
3.7 Example of MMDE approach . . . 36
3.8 Examples of opening and closing operators scale-dependent . . . 37
3.9 Structuring Function SMMT . . . 38
3.10 Examples of SMMT . . . 38
4.1 The original approach and our approach . . . 42
4.2 Image extrema . . . 43
4.3 The extrema relationship . . . 44
4.4 Example of the extrema relationship . . . 45
4.5 Representation of SF pyramid shape . . . 46
4.6 Example of the transformation for image simplification . . . 47
4.7 Example of simplification on a 1D signal . . . 48
4.8 Shortest paths between regional extrema . . . 49
4.9 Simultaneous simplification . . . 51
4.10 Relationship update . . . 53
5.1 Applications process . . . 57
5.4 Successive simplifications using morphological reconstruction . . . 59
5.5 Comparison segmentation on original and simplified image . . . 60
5.6 Examples of image simplification using morphological recostruction . . . . 60
5.7 Image homogenization using regions growing . . . 61
5.8 Monotonic reduction using watershed segmentation . . . 62
5.9 Representation of SF pyramid shape . . . 63
5.10 Gradient improvement . . . 63
5.11 Gradient improvement on fish female cells . . . 63
5.12 Samples of BSDS300 . . . 64
5.13 Samples of BSDS300 . . . 65
5.14 Time execution of our approach against the original approach . . . 66
5.15 Time execution of our approach against the original approach . . . 67
List of Tables
Contents
Acknowledgements 5
Resumo 6
Abstract 7
1 Introduction 13
1.1 Image Simplification Methods . . . 14
1.2 Principal Contributions . . . 16
1.3 Organization . . . 16
2 Mathematical Morphology 18 2.1 Morphological Image Transformation . . . 18
2.1.1 Image Transformation Properties . . . 19
2.1.2 Discrete Distance Functions . . . 20
2.2 Element and Structuring Function . . . 21
2.3 Basic Morphological Operators . . . 21
2.3.1 Erosion . . . 22
2.3.2 Dilation . . . 23
2.4 Another Morphological Operators . . . 24
2.5 Watershed transform . . . 25
2.6 Techniques for image simplification . . . 26
2.6.1 Mosaic approach . . . 26
2.6.2 Leveling approach . . . 27
2.6.3 Dynamic measure approach . . . 28
2.7 Summary . . . 28
3 Morphological Scale-Space 30 3.1 Linear Scale-Space . . . 31
3.2 Scale-Space on Two Dimensions . . . 32
3.3 Morphological Scale-Space . . . 33
3.3.1 Morphological Scale-Space Properties . . . 33
3.3.2 Scale-Dependent Structuring Function . . . 34
3.3.3 Morphological Scale-Space Operators . . . 35
3.4 Techniques for Image Simplification . . . 36
3.4.1 Multiscale Morphological Dilation Erosion - MMDE . . . 37
3.4.2 Method Based on Opening and Closing . . . 37
3.4.3 Method Based on Scale-Dependent Non-Flat Structuring Function . 38 3.5 Summary . . . 39
4 The Proposed Method 40
4.1 General Description of the Method . . . 41
4.1.1 Image Extrema . . . 43
4.1.2 Relationship between Extrema . . . 43
4.1.3 Set of Tuples S and Order Relation R . . . 45
4.1.4 Morphological Simplification . . . 46
4.2 New Theoretical Properties . . . 48
4.2.1 Shortest Paths Between Regional Extrema . . . 49
4.2.2 Simultaneous Simplification . . . 51
4.2.3 Local Update of The Graph . . . 52
4.3 Summary . . . 53 5 Results 54 5.1 Theoretical Results . . . 55 5.2 Experimental Results . . . 56 5.2.1 Applications . . . 57 5.2.2 Comparison . . . 64
5.3 Other Segmentation Examples . . . 66
5.4 Summary . . . 67
6 Conclusions and Future Work 69
Chapter 1
Introduction
Nowadays, many applications of computer, cell phone or another device make use of the images as an essential resource. This makes images a common part of people’s daily lives, as well as being used in different fields and areas. Important examples are medicine, video production, photography, remote sensing, and security monitoring.
The vast majority of these applications to achieve their goal (e.g., face detection, fin-gerprint recognition, cell nucleus detection) share common steps such as image acquisition and image processing. For this reason, they also present common problems for the treat-ment of images. One of these problems present in these applications is the noise, which hinders the correct visualization of the image and it is generally associated with the acqui-sition process. Another important problem is to achieve improvement of certain features of the image, such as contour enhancement, which are useful for correctly delimiting the regions within the image and it is associated with image processing. These two problems, noise, and feature enhancement, present different importance for distinct areas, becoming relevant in, e.g., medical applications and video surveillance. Finally, an important point to consider is the problem of high computational cost employed in complex tasks, since these tasks are often used in a large dimensional space that requires a higher processing and memory resource of the computer.
Due to these problems, the correct treatment of the images is necessary, emphasizing a pre-processing step in order to clean the image of unwanted components (e.g., the noise) and highlight the most important features of the image (e.g., the edges), in addition to taking into account the response time of the applications.
Image processing and analysis area covers a large number of applications, ranging from lower-level tasks (e.g., extremum point detection) to more specialized tasks, such as segmentation and classification, that require the suppression of unnecessary details present in the input images.
Some techniques addressed these problems as part of the algorithm, but in many cases, it is common to apply a pre-processing step in the original image prior to any further processing. Within the pre-processing area, we focus on addressing these problems with the “image simplification approach”, whereby the elimination of non-significant details is obtained in the image while keeping the information necessary to the achievement of the desired outcome, besides the reduction of response time.
CHAPTER 1. INTRODUCTION 14
1.1
Image Simplification Methods
The Image simplification method is commonly referred to as a pre-processing step and, in particular, Mathematical Morphology [32],[34],[12],[28] introduces interesting low-level simplification filters exhibiting well-known properties [33]. The typical filtering by opening and closing [34] and their combination as alternate sequential filters are commonly used to eliminate undesirable components of an image while preserving its main content.
Morphologically, the leveling approach [26, 27] defines a reconstruction-based simpli-fication without changes in the final contours w.r.t., the original image. Another example of morphological simplification is the dynamic measure [2] which selects components of an image according to the notion of extinction values (e.g., area or volume) [35]. This pro-cedure is closely related to the measure of persistence of a signal and is used to eliminate image components regarded as non-significant. The main limitation that shares the works mentioned above is the presence of a simple level of observation, limiting its capacity to extract information.
To address this problem, the multi-scale approach is created which has multiple levels of observation. Within this approach, the scale-space theory [38] defines a multi-level processing (from finer to coarser scales) related to different representations of the original signal. In such a case, the simplification should be well-behaved in the sense that, given a certain feature of interest (e.g., the zero-crossing of a function), one seeks to track its representation along the different scales. This multi-level transformation should satisfy some properties of monotonicity, continuity, fidelity and euclidean invariance [38]. The monotonicity concept guarantees the non-inclusion of new interest features at different scales; the continuity states that a continuous path should be defined by the remaining features along these scales; the fidelity ensures that the signal tends to its original form as the scale tends to zero, and, finally, the euclidean invariance asserts that translation and rotation transformations result in translated and rotated signals.
The method discussed here using this approach is based on scale-dependent non-flat structuring function [7], where a toggle-like transformation simplifies an image in a self-dual way. The drawback of using multi-scale methods is the lack of persistence measures in simplification.
Recently, a non-self-dual simplification method is introduced that taken into account the relationship between image extrema [29]. More specifically, it considers the distance and contrast between the various regional maxima (minima) and define a total order relation closely linked to the degree of simplification one wants to impose. As we will see later, this multi-level process establishes a non-decreasing and well-behaved activity from which the least significant extrema merge successively with the most significant ones, in a given neighborhood. This image simplification method presents interesting advantages (e.g., the measure of the persistence of the regional maxima or minima ) but its high computational cost makes this method an unfeasible option.
In this Master Thesis, we explore a graph-based simplification method that guaran-tees a well-behaved suppression of the image extrema (regional maxima or minima) by taking into account both information of distance and contrast. To achieve this simpli-fication, we use some interesting properties of an approach that handles multiple levels
(a) (b) (c) (d)
Figure 1.1: Successive image simplification in different levels of observation from (a) to (d), combined with image segmentation.
(a) (b) (c) (d) (e)
Figure 1.2: Successive image homogenization in different levels of observation from (a) to (e), using image simplification.
of observation, called morphological scale-space. By highlighting some new properties of the method, we define a local update of the graph which implies an interesting bypass in the whole algorithm structure reducing the time-consuming of the original algorithm. Finally, we present some morphological applications where is illustrated how to combine this simplification process with well-known morphological tools to approach problems re-lated mainly to multi-scale (multiple levels of observation) image segmentation, which is the process of partitioning a image into multiple segments. This combination is illustrated in Fig. 1.1, where at each level from Fig. 1.1(a) to Fig. 1.1(d) is increased the amount of less significant components that are eliminated. This application results in Fig. 1.1(d), which is composed only of significant components.
Another application developed in this research is related with image homogenization, shown in Fig.1.2, where from the Fig. 1.2(a)to Fig. 1.2(e)the similar components of the image are merged dependent of the observation.
CHAPTER 1. INTRODUCTION 16
1.2
Principal Contributions
The main contributions of this work are the following:
• The creation of a new image simplification method, based on the algorithm defined by Leite and Polo [29]. Our method removes noise and preserver the edges of the images.
• The introduction of new properties concerned with the graph created using the extrema relationship, resulting in the local update of a graph.
• The reduction of the steps necessary to perform the images simplification is per-formed by the creation of a bypass between the steps of the original approach. • The reduction of the computational cost from the original approach, keeping its
advantages.
• Several morphological applications are presented and discussed. In these applica-tions we showed how to combine the simplification step with well-known morpho-logical tools in applications related to image segmentation and homogenization.
1.3
Organization
The text of this Master’s Thesis is organized into six chapters, as follows. Chapter 2 : Mathematical Morphology
In this chapter, we make a brief introduction to the basic concepts of mathematical morphology, which are necessary for proper understanding of the remaining text. Initially, some basic definitions of image transformations are presented, as well as some concepts of structuring functions and elements. Also are presented the morphological operators used to define image transformations, in addition to a basic segmentation algorithm, but quite studied in the literature. Finally, the techniques of image simplification are presented which use morphological operators.
Chapter 3 : Morphological Scale-Space
The main problems related to the use of multi-scale methods are due to the difficulty of linking significant and relevant information of a signal in 1D or a 2D image, through the different levels (scales). In order to avoid these problems, Witkin [38] proposes a new approach, called scale-space, that formalizes a set of properties allowing the consistent manipulation of different structures of the image. With this representation, a feature of interest in the image describes a continuous path and it is possible to relate the information obtained in the different levels of observation, as well as to determine its exact location in the original signal.
In this chapter, we present the most relevant concepts of the space-scale approach, either linear or 2D, as well as its main properties, which make possible the manipulation of the image, in a consistent way. Finally, the techniques of images simplification are
shown which make use of the morphological scale-space approach. Chapter 4 : The Proposed Method
One of the most recent approaches to simplifying images in mathematical morphology is given by Leite and Polo [29], with the creation of a simplification method that takes into account the relationship between the extrema of the image (regional minima and maxima), based on separation and contrast.
In this chapter, we present a new image simplification method, based on the method of Leite and Polo [29]. Although our method presents similar stages, we managed to address the main drawbacks of the original approach. These problems are: 1) the high computational cost demanded at the time of executing the method and 2) the almost null exploration in the experimental area using this simplification method.
We solve the first problem by a local update in the graph that supports whole the simplification process. This is achieved by the creation of new properties, which have a direct influence on the graph.
Initially, we discussed the method proposed by Leite and Polo [29] showing drawbacks in some stages and presenting our solutions. Subsequently, we present the new properties along with their demonstrations.
Chapter 5 : Results
We in this chapter, address the problem of almost no exploration in the experimental area using our method of simplification. Here we illustrate some examples where we combine the process of image simplification of our method, with well-known morphological tools to address problems mainly related to the segmentation and homogenization of images multi-scale images.
We show how to combine the image simplification method with watershed transforma-tion, with morphological reconstructransforma-tion, with the regions growth algorithm and finally, we present a gradient improvement by combining the gradients of the different scales generated.
In addition, we present two types of results, the theoretical results, where we present the properties created in this research and the experimental results, where we compare the computational cost used by the Leite and Polo algorithm [29] and our algorithm. These experimental results are obtained using the databases BSDS300 [25] and BSDS500 [1] databases.
Chapter 6 : Conclusions and Future Work
Finally, we present the conclusions obtained in the development of this research work, as well as the perspectives of future works.
Chapter 2
Mathematical Morphology
In the introduction chapter, we said that our method makes use of scale-space theory. However, to understand this approach, we need to know the theory about mathematical morphology MM, as well as its basic concepts and some of its main operations. This is because the morphological scale-space theory is based on MM. Besides, MM will be useful when we want to explain our different applications proposed, where we combine our method with some morphological tools.
The approach of MM is widely used as a pre-processing step and aims to highlight the features required to solve different problems. MM consists of a non-linear process to images processing [12, 32, 34, 28] and it is defined by a set of transformations, which supports the tasks of analysis, segmentation, and recognition, based on the information of the geometry of the image and the shape of its objects.
The result of such transformation depends essentially on the comparison of the content of an image of interest with a smaller one, and of known shape, called as structuring element SE, or structuring function SF. Generally, this another image contains geometric or topological characteristics related to the information to be extracted or removed from the image of interest.
In this chapter we describe some concepts related to MM theory, as follows. Initially, in Section 2.1 we discuss the operators used to define image transformations and its main algebraic and topological properties. Subsequently, in Section2.2 we introduce the concept and the different shapes of the SF and SE, while Section2.3, we describe two basic morphological operations, named erosion and dilation, as well as, some of their properties. Another derivative morphological operations is defined in Section 2.4. We will show the transformed of Watershed in Section2.5and finally, techniques of simplification of images using MM are presented in Section2.6.
2.1
Morphological Image Transformation
Initially, we need the concept of image transformations, since it encompasses all the operations performed on the images. The Morphological image transformations are “image to image transformations,” i.e., the transformed image has the same definition domain as the input image and it is still a mapping of this definition domain onto the set of
(a) Input image f (b) T[0,150](f ) (c) T[156,235](f )
Figure 2.1: The threshold T[r](f ) on (a) the input images f using the ranges r equal to
(b) [0, 150] and (c) [156, 235].
negative integers [34]. We use the generic notation Ψ to refer to an image transformation. In order to get a better understanding of this concept, we present a trivial example of image to image transformation is the identity transformation, denoted by,
I(f ) = f, (2.1)
where f represent a signal on 1D or 2D dimensions, and I is the idempotent function. Another widely used, image to image transformation is the threshold operator which sets all pixels of the input image lying in a given range of gray tone values to the value 1 and the remaining ones to the value 0. Formally,
h
T[ti,tj](f )
i
(x) = 1, if ti ≤ f (x) ≤ tj,
0, otherwise, (2.2)
where T is the threshold operator and ti, tj are the ranges of values minima and
maxima respectively for a result positive.
Fig. 2.1 shows the threshold operator on the input image, Fig. 2.1(a). Where, the threshold for Fig.2.1(b)and Fig.2.1(c)are in the ranges of [0, 150] and [156, 235], respec-tively.
One of the bases used in MM is set theory, which uses the basic set operators, the union,S, and the intersection, T. When grey scale images are considered, such operations are represented by the operators of point-wise the maximum ∨, and the minimum ∧, respectively [16, 31]. In two images (also represented as signals) f1 and f2 the point-wise
maximum and minimum are defined as follows for each point x,
(f1∨ f2)(x) = max{f1(x), f2(x)}, (2.3)
(f1∧ f2)(x) = min{f1(x), f2(x)}. (2.4)
2.1.1
Image Transformation Properties
These transformations have important properties [28], allowing them to define aspects of their behavior and help in choosing an appropriate transformation for a given problem. Some of these properties are presented below.
CHAPTER 2. MATHEMATICAL MORPHOLOGY 20 to any image f is equivalent to applying it only once,
Ψ(n) is idempotent ⇔ Ψ(n−1)Ψ = Ψ.
• Extensivity: A transformation Ψ is extensive if, for all images f , the transformed image is greater than or equal to the original image,
Ψ(f ) ≥ f.
• Anti-extensivity: A transformation Ψ is anti-extensive if, for all images f , the transformed image is less than or equal to the original image,
Ψ(f ) ≤ f.
• Increasingness: A transformation Ψ is increasing if it preserves the ordering rela-tion between images,
Ψ is increasing ⇔ ∀f, g, f ≤ g ⇒ Ψ(f ) ≤ Ψ(g).
• Duality: Two transformations Ψ and Φ are dual with respect to complementation if applying Ψ to an image is equivalent to applying Φ to the complement of the image and taking the complement of the result,
Ψ(f ) = Φ?(f ) ⇒ Ψ(f ) =hΦ(fc)ic.
Where c is the complement and ? is the negative operator. If the duality between two transformations is established, some properties of a transformation are directly inherited from its dual transformation if any. For instance, the following rules apply,
Φ idempotent = Ψ idempotent, (2.5)
Φ extensive = Ψ anti-extensive, (2.6)
Φ anti-extensive = Ψ extensive, (2.7)
Φ increasing = Ψ increasing. (2.8)
2.1.2
Discrete Distance Functions
Another important part of our work is the concept of discrete distance, which is widely used in image analysis and especially in MM. The distance provides the image definition domain with a metric or measure of separation of its points.
nonnega-4 3 2 3 nonnega-4 3 2 1 2 3 2 1 0 1 2 3 2 1 2 3 4 3 2 3 4 (a) 4-connected 2 2 2 2 2 2 1 1 1 2 2 1 0 1 2 2 1 1 1 2 2 2 2 2 2 (b) 8-connected
Figure 2.2: Discrete distances.
tive real number with any two points p and q and satisfying the three following conditions,
d(p, q) ≥ 0 and d(p, q) = 0 ⇔ p = q; (2.9)
d(p, q) = d(p, q) (symmetry); (2.10)
d(p, q) ≤ d(p, r) + d(r, q) (triangle inequality). (2.11) There exist many discrete distances satisfying the axioms of a metric. The Fig. 2.2
shows 4-connected distance in Fig.2.2(a), and 8-connected distance in Fig. 2.2(b), calcu-lated from the central pixel of a discrete image.
2.2
Element and Structuring Function
In the introduction of this chapter, we say that transformations in MM are given by comparing the content of an input image with a smaller one. This small image is called structuring element SE, associated with the binary case or structuring function SF, for grey scale case. The SF and SE are used to extract relevant information from an image associated with the neighborhood at each pixel.
In general, these SE and SF define, amongst other things, the size and shape of the region to be considered by the morphological transformation. We can see different shapes of SE in Fig.2.3, where the black box represent the central element.
2.3
Basic Morphological Operators
The fundamental operations for MM are erosion and dilation, constituting the basis for the definition of more complex transformations. We need to know these operators since we manipulate the input image with a variation of these operators.
(a) Diagonal (b) Line segment (c) Diamond (d) Square
CHAPTER 2. MATHEMATICAL MORPHOLOGY 22
2.3.1
Erosion
Definition 2.2. (Erosion operator [34]). The erosion of a set X by a structuring element B, is denoted by εB(X) and is defined as the locus of points x such that B is included in
X when its origin is placed at x,
εB(X) = {x : Bx ⊆ X}. (2.12)
When we use a SE for binary case then, the Eq. 2.12 can be rewritten in terms of an intersection of set translations (math of Minkowski), and the translations being determined by the SE,
(X B) = \
b∈B
X−b. (2.13)
Eq. 2.13 can be directly extended to binary and grey scale images. The erosion of an image f , by a structuring element B, is denoted by εB(f ) and is defined as the minimum
of the translations of f by the vectors −b of B, εB(f ) =
^
b∈B
f−b. (2.14)
In the Fig. 2.4 we can see an example of the erosion operator X B, on a binary image X, using the SE denoted by B.
Properties of erosion operator
The main morphological properties of the erosion operator are given by Haralick, Heijmans and Jackway [10, 11, 15].
• Non-commutative: The erosion operator ε is non-commutative if, applying the erosion on the binary image X, using the SE represented by b, is not equivalent to apply this operation on the binary image B, using the SE described by x,
εb(X) 6= εx(B).
• Non-Associative: The erosion operator ε is non-associative when we need to keep the order of operations, for no affect the outcome of the transformation,
εC(εB(A)) 6= εB(εC(A)).
• Anti-Extensivity: The erosion operator ε is anti-extensive if, for all binary image X, the image result of the erosion operation using the SE represented by b, is less than or equal to the original image,
εb(X) ⊆ X,
(a) X (b) B (c) (X B)
Figure 2.4: Example on a binary image X, of erosion operator X B, using the SE denoted by B.
2.3.2
Dilation
Definition 2.3. The dilation of a set X by a structuring element B, is denoted by δB(X)
and is defined as the locus of points x such that B hits X when its origin coincides with x,
δB(X) = {x : Bx∩ X 6= ∅}. (2.15)
For binary images, we can use the dual-way of the Eq. 2.15, which result is given by the following equation,
(X ⊕ B) = [
b∈B
X−b. (2.16)
Note that the dilation uses ˇBx, defined as the symmetrical transpose of B, where
ˇ
Bx = {−b : b ∈ B}
We can extend the Eq. 2.16 to grey scale images, where the dilation of an image f , by a structuring element B, is denoted by δB(f ) and is defined as the maximum of the
translation of f by the vectors −b of B, δB(f ) =
_
b∈B
f−b. (2.17)
In Fig. 2.5 we can see an example of the dilation operator X ⊕ B, on a binary image X, using the SE denoted by B.
Properties of dilation operator
The main morphological properties of the dilation operator are defined by Haralick, Hei-jmans and Jackway [10, 11,15]:
• Commutative: The dilation operator δ is commutative if, applying the dilation on the binary image X, using the SE represented by b, is equivalent to apply this operation on the binary image B, using the SE described by x,
CHAPTER 2. MATHEMATICAL MORPHOLOGY 24
(a) X (b) B (c) (X ⊕ B)
Figure 2.5: Example on a binary image X, of dilation operator X ⊕ B, using the SE denoted by B.
• Associative: The dilation operator δ is associative when we do not need to keep the order of operations, for no affect the outcome of the transformation,
δδC(B)(A) = δC(δB(A)).
• Extensivity: The dilation operator δ is extensive if, for all binary image X, the image result of the dilation operation using the SE represented by b, is greater than or equal to the original image,
δb(X) ⊇ X,
if the SE contain the center.
2.4
Another Morphological Operators
The transformations presented in the previous Section2.3have some limitations, however, if they are properly combined, interesting properties can be obtained, which are used by some authors in the relation works. These combinations can be the opening and closing. Definition 2.4. (Opening [34]) The opening γ of an image f , by a SE B, is denoted by γB(f ) and is defined as the erosion of f by B followed by the dilation with the reflected
SE ˇB,
γB(f ) = δBˇεB(f ). (2.18)
Definition 2.5. (Closing [34]) The closing of an image f , by a SE B, is denoted by φB(f )
and is defined as the dilation of f with a B followed by the erosion with the reflected SE ˇ
B,
φB(f ) = εBˇδB. (2.19)
In general, these operations are used to recover, in the second transformation, certain structures of the image, which were removed by the first transformation.
For the case of binary images, the aperture is used to regularize the contours and eliminate small “islands” and “capes,” on the other hand, the lock suppresses small “lakes.”
(a) Binary image. (b) Opening. (c) Closing.
Figure 2.6: Examples of (b) opening and (c) closing on (a) the binary image.
These cases can be observed in the Fig.2.6. Where the original binary image, seen in Fig.2.6(a), undergoes the transformation of opening shown in Fig. 2.6(b), and closing in Fig.2.6(c), using a square SE of size 3 × 3.
In the case of grey scale, the opening eliminates the clear structures and the closing work on the dark structures.
It is important to note that these transformations are dual, increasing, and idempotent. On the other hand, the opening is extensive and the closing is anti-extensive.
2.5
Watershed transform
Another important point that we need to know is the image segmentation (the division of the image in regions and join the similar regions), by reason of we used this algorithm in our proposed applications and they are seen in later chapters.
In the area of image processing, and more specifically in MM, any grey scale image can be viewed as a topographic surface. Where, high intensity denotes peaks and hills while low intensity denotes valleys.
Similar to other techniques of segmentation, the aim of the Watershed technique [3] is to divide the analyzed gray level image into regions.
Generally, the image is divided into background and foreground with the objects or regions that are to be extracted. First, we extract the markers from the significant structures of the image, and then use the watershed transform to obtain the contours of such structures as accurately as possible. The objective of this technique is to determine the contours that define these objects.
We can notice that a grey scale image is a topographic surface, 2D, and can be seen as a signal, 1D. With that in mind, we can see the watershed process in Fig. 2.7, where a signal is presented by the function f and each minimum of this signal corresponds to a perforation (markers), in Fig.2.7(a), that it is placed in the water vertically at a constant velocity. Starting at the lowest minimum altitude, the water progressively fill the basins of the images, shown from Fig. 2.7(b) to Fig. 2.7(h). The set of water divisors built in places where two distinct water flows meet, representing the basin partition of the original image, is called watershed divisors, present in Fig.2.7(h).
CHAPTER 2. MATHEMATICAL MORPHOLOGY 26 f Markers (a) f (b) f divisor (c) f divisor (d) f divisor divisor (e) f
divisor divisor divisor
(f)
f
divisor divisor divisor divisor
(g)
f
divisor divisor divisor divisor
(h)
Figure 2.7: Illustration to watershed transform, where (a) is the signal 1D and (b)-(h) show the process of flooding and the construction of water divisors.
(a) Original. (b) Gradient. (c) Markers. (d) Segmentation
Figure 2.8: Example of watershed segmentation on the nucleus of Arabidopsis thaliana. Image taken from Legland,Arganda and Philippe [20].
2.6
Techniques for image simplification
In this section, we describe of the image simplification method using MM as the principal process.
The image simplification aims not only to filter out noisy components but also simplify the image through the elimination of non-significant details while keeping the information necessary to the achievement of the desired outcome. In this sense, the morphological transformations are ideal, since its primary function is extracted relevant information from an image. In other words, this can be interpreted as removing unwanted details from the image. Following are some MM techniques that were used for image simplification.
2.6.1
Mosaic approach
A task that is often related to image segmentation is concerned with the “hierarchization” of an image, that is, the production of a series of images with decreasing level of detail: between two successive images of the series, details of least importance are suppressed while the important features are preserved. For example, Fig. 2.9(b) has been obtained by computing the catchment basins of the gradient of Fig. 2.9(a) and assigning to each
(a) Original. (b) Mosaic. (c) Mosaic order 2.
Figure 2.9: Successive image simplification using mosaic approach. Image taken from Heijmans and Vincent [13].
basin the mean gray level of the corresponding pixels in the original image. This image is often referred to as a mosaic image.
The over-segmentation of Fig. 2.9(b) can be clearly noted. To get rid of this problem while producing a series of images with decreasing level of detail, the method considers the adjacency graph of the catchment basins. Then determining the watersheds of this graph allows one to merge catchment basins into catchment basins of second order. The resulting image, called a mosaic image of order 2 (see Fig.2.9(c)), has less detail than the previous one but the main features have been preserved.
The procedure can be iterated to produce mosaic images of order 3, 4, etc. Unlike the classical Gaussian pyramids the present method has the advantage that it avoids blurring effects and preserves the most significant contours at best.
2.6.2
Leveling approach
The leveling approach proposed by Meyer et al. [26, 27] defines a reconstruction-based simplification without changes in the final contours w.r.t., the original image. These works propose morphological filters that simplify an image without blurring or alterations of its contours. The filters use morphological operators of dilation, erosion and superior or inferior reconstruction. The contours of the simplified image are accurate as in the initial image, and the complex tasks can be performed in the simplified image without having to resort to the original image.
The basic steps to the develop algorithm are the following: First we used the transfor-mations of closing and dilation, then employed a morphological reconstruction (minimum).
(a) Original. (b) Mosaic. (c) Mosaic order 2. (d) Mosaic order 3.
Figure 2.10: Successively image simplification, using 3 increasing levelings. Image taken from Meyer and Fernand [26].
CHAPTER 2. MATHEMATICAL MORPHOLOGY 28
(a) 1-D sinal. (b) Dynamic of sinal. (c) Image.
(d) Dynamic of image.
Figure 2.11: Examples of minima dynamic (b), (d). On (a) a 1D sinal and (c) an image respectively. Images taken from Meyer et al. [26, 34].
To this result, we applied the erosion transformation and finally, a morphological recon-struction (maximum). These steps are applied for different SE, given as result, different leveling, shown in Fig.2.10.
2.6.3
Dynamic measure approach
The Dynamic was defined by Bertrand [2] and expanded by Soille [34]. The lowest image minima have a dynamic equal to the difference between the highest and lowest image grey scale values.
We show the dynamic of the minima in Fig. 2.11(b), obtained from a 1D signal, Fig. 2.11(a). Furthermore, we see that the dynamic of a minimum can be interpreted as a measure of its depth. Similarly, for a 2D signal 2.11(c), also represented as a grey scale image, the dynamic of the image is shown in the Fig.2.11(d).
The dynamics are also used as a measure extinction by Bertrand [2] and as seeds for morphological reconstruction. Fig. 2.12 shows a result of these applications where the dynamics are used as marker function to watershed segmentation.
2.7
Summary
This chapter has provided some examples of MM, as well as a brief review of concepts needed to understand the rest of the text. A greater emphasis was placed on the erosion and dilation operations of MM, and the watershed algorithm, which are an important
(a) (b)
Figure 2.12: An original image f (a), and (b) the highest components associated to the minima of f which have a dynamics greater than 30 [2].
part to understand the method developed in this work.
However, one must also consider the analysis of different levels of representation, which has been widely used to deal with the multi-scale nature of images. Through such ap-proaches, it is possible to extract the features of interest that become explicit at each level. Note that the related works explained in this chapter lack these multiple levels of observation. The next chapter discusses in detail this approach with multiple levels of information.
Chapter 3
Morphological Scale-Space
In the introduction, we say that work with the morphological scale-space approach, this is due to the presence of interesting properties that make possible our simplification method. But to understand this approach, we first need to understand scale-space theory, its basic concepts, and its main properties.
Remember, the images simplification aims to eliminate the non-significant components while maintaining the necessary details to achieve the desired outcome. In this sense, the basic morphological transformations on which the previous techniques are performed do not provide the level of detail necessary in their operations, since such details are only visible at a level of observation. For this reason, the scale-space approach was created, combining the multi-scale approach and the transformations of MM, to obtain different levels of observation.
In image processing, is crucial that the features extraction from an image, and certain information only makes sense under some observation conditions (e.g., the distance of observation). This interpretation is related to the concept to scale.
For a discrete function (e.g., a digital image), there are two different concepts to scale; the external scale used to indicate the size of the image and the internal scale to represent the sampling density. However, choosing the appropriate observation scale is not a trivial task. For that reason, the multi-scale approach was developed.
The multi-scale approach, shown in Fig3.1, produces a set of signals whose structures are successively suppressed. In this way, it is possible to analyze the different levels of representation and to use those that only exhibit the interest features.
Fig.3.2(a) shows one of the major problems of the multi-scale method. The difficulty
Coarse levels of scale
Original signal
Figure 3.1: A multi-scale representation of a signal is an ordered set of derived signals intended to represent the original signal at different level of scale. Image taken from Lindeberg [23]
(a) Multi-Scale (b) Scale-Space
Figure 3.2: The representation of the approach (a) multi-scale and (b) morphological scale-space.
of relating significant information of the signal through the different levels. Accordingly, the scale-space approach [38] was created, showing in Fig.3.2(b)that with the continuity property, guarantees a continuous path, of the remaining features, along these scales. Thus, it is possible to relate information obtained at different levels of observation, as well as to determine its precise location in the original signal.
In this chapter, we describe some concepts and properties related to the scale-space theory [37,35,4], which later on help us to understand our method of simplification. This chapter is distributed as follows.
In Section 3.1 we introduce the linear scale-space concept, with some properties. A two-dimensional scale-space is shown in Section 3.2. Subsequently, in Section 3.3, we explain the morphological scale-space, as well as, its main properties. Finally, image simplification techniques using this morphological scale-space approach are presented in Section3.4.
3.1
Linear Scale-Space
The scale-space definition was developed for one-dimensional continuous signals, which generate a set of signals obtained by convolving the original signal with a Gaussian zero-mean kernel whose standard deviation represents the concept of scale. This concept is generally known as gaussian scale-space [38, 14] and is defined as follows.
Definition 3.1. (Gaussian scale-space [38, 14]).
Let f (x) : Rn→ R be the original signal and let p(x, σ) : Rn× R → R be the kernel of
smoothing. The image scale-space L : Rn× R → R in the scale σ is given by
h
L(x, σ)i(f ) = f (x) ∗ p(x, σ), (3.1) where the symbol ∗ denotes the convolution operation and p(x, σ) is the gaussian ker-nel [16], p(x, σ) = (2πσ2)−1/2exp − 1 2σ2x 2 . (3.2)
CHAPTER 3. MORPHOLOGICAL SCALE-SPACE 32 Let us recall that, the convolution process [8] is defined as,
w(x, y) ∗ h(x, y) = a X s=−a b X t=−b w(s, t)h(x − s, y − t), (3.3)
where w(x, y) is a kernel of size m×n and a, b are the values used to move within the kernel, besides, h(x, y) is the image. This equation evaluated for all values of the displacement variables x and y so that all elements of w visit every pixel in h.
The smoothing process should satisfy a set of properties. The conditions for the one-dimensional linear case, called scale-space theory axioms, are summarized below [22, 24]:
• Invariance to translation, rotation and scale. • Non-highlighting of local extremes.
• Non-creation of new extremes.
• Linearity and separability properties.
3.2
Scale-Space on Two Dimensions
Lifshitz, Lawrence and Pizer [21] observed that when we want to extend the linear scale-space properties for two or more dimensions, any convolution kernel used to create a representation scale-space introduces new features in the scale changes. For this reason, a set of harder properties were considered, which constrain the composition of scale-space. Even so, it is not possible to completely prevent the increasing of new local extremes [23]. One of these principal properties in a two-dimensional scale-space is created to preserve the monotonicity of the components. This property is called causality defined for Koen-derink [18]. According to which new level surfaces (curves) should not be created with increasing scale. In the discrete case, the local extremes must satisfy the non-enhancement property, where the value of a maximum must not increase or change (and a minimum should not decrease) for increasing scales.
Even with this property, scale-space on two dimensions continue to present some prob-lems. We can see in the Fig.3.3, one the principal problems of the gaussian kernel for the
(a) (b) (c) (d)
Figure 3.3: Example of gaussian two dimensional scale-space on (a) original image, using a scale equal to (b) two, (c) four and (d) eight.
(a) (b) (c)
Figure 3.4: Difference between multi-scale and morphological scale-space approach, where (a) is the image pyramid, (b) is the image pyramid rescaled to the same size and (c) is the morphological scale-space. Images taken from Lindeberg [23].
two dimensional scale-space is the smoothing presented in the image after the convolution process.
3.3
Morphological Scale-Space
The morphological scale-space approach is created in response to the problems mentioned in the previous sections, by extending linear scale-space to two dimensions. This morpho-logical approach address the following problems:
1. The addition of new extremes in the different scales.
2. The smoothing of the contours by the use of the gaussian kernel. 3. The translation of the extremes in the different scales.
In Fig. 3.4, we can observe a comparison the multi-scale approach in Fig. 3.4(a)
(rescaled to the same size in Fig.3.4(b)) and the morphological scale-space in Fig.3.4(c). In contrast to multi-scale approaches, such as pyramids, the scale-space representation preserves the same spatial resolution (same number of samples) at all scales. This feature allows immediate access to the data of interest at any scale without the need for additional processing.
3.3.1
Morphological Scale-Space Properties
The morphological scale-space approach address the problems previously mentioned, using some of its main properties [38], which we describe below.
CHAPTER 3. MORPHOLOGICAL SCALE-SPACE 34
(a) Monotonicity (b) Continuity (c) Fidelity and
Euclidean Invariance
Figure 3.5: Morphological scale-space properties.
• Monotonicity: The monotonicity concept guarantees the non-inclusion of new interest features at different scales.
• Continuity: The continuity states that a continuous path should be defined by the remaining features along these scales.
• Fidelity: The fidelity ensures that the signal tends to its original form as the scale tends to zero.
• Euclidean Invariance: The euclidean invariance asserts that translation and ro-tation transformations result in translated and rotated signals.
Note, these properties ensure that when suppressing an extreme, this does not reappear on higher scales and that the remaining extremes not suffer translation effect or change of value. We can see this properties in Fig. 3.5.
3.3.2
Scale-Dependent Structuring Function
In order to perform the morphological scale-scape operations, we need to know a variation of the flat structuring functions presented in the section 2.2, which due to the multi-scale nature of the multi-scale-space approach must be structuring functions dependent on the observation scale.
In the case of grey images, most applications consider only flat structuring functions. However, non-flat structuring functions gσ, can be used to define scale-dependent
mor-phological transformations, which allow multi-scale representation of the analyzed signal. The gσ : Gσ ⊂ R2 → R is defined as,
gσ(x) = |σ|g(|σ| −1
x) : x ∈ Gσ, ∀σ 6= 0, (3.4)
where Gσ = {x : kxk < R} is te support regional of the function gσ. To guarantee
the behavior of a monotonically decreasing function along any radial direction from the origin (anti-convex), it must satisfy posterior conditions [16],
|σ| → 0 ⇒ gσ(x) →
0, if x = 0,
−5 0 5 −5 0 5 −5 −4 −3 −2 −1 0
Figure 3.6: Non-flat pyramid-shaped structuring function gσ.
0 < |σ1| < |σ2| ⇒ gσ1(x) ≤ gσ2(x) : ∀x ∈ Gσ1. (3.6)
|σ| → ∞ ⇒ gσ(x) → 0. (3.7)
Moreover, to prevent the level shifting, we need to take into account, Sup
t∈Gσ
{gσ(t)} = 0. (3.8)
To avoid horizontal translation effects, one must consider
gσ(0) = 0. (3.9)
A non-flat structuring function that we need to take into account, for the development of our work is the non-flat pyramid-shaped structuring function defined by Leite and Polo [29] and shown in Fig.3.6, which equation is defined by, as
gσ(x, y) = −σ max
h
|x|, |y| i. (3.10)
3.3.3
Morphological Scale-Space Operators
At the beginning of this Master Thesis, we say that our method carries out a suppres-sion of extrema, that is, the elimination of maxima and minima regional in the im-age. To understand how we perform this process, we need to know the definition of the non-flat scale-dependent structuring function gσ, introduced in Subsection 3.3.2, the
scale-space operators presented in this subsection, and their main properties referred in Subsection3.3.1.
The creation of the set of morphological scale-space transformations is achieved by em-ploying of morphological operators of erosion ε, and dilation σ, presented in Section 2.3, using a structuring function gσ. We now describe the adaptation of the basic
CHAPTER 3. MORPHOLOGICAL SCALE-SPACE 36 Definition 3.2 (Erosion scale-dependent [16]). The erosion of the function f (x) with the structuring function gσ(x), noted [εgσ(f )](x), is defined by,
[εgσ(f )](x) = Inf
t∈G∩D−x
{f (x + t) − gσ(t)}. (3.11)
Definition 3.3 (Dilation scale-dependent [16]). The dilation of the function f (x) with the structuring function gσ(x), noted [δgσ(f )](x) is defined by,
[δgσ(f )](x) = Sup
t∈G∩ ˇDx
{f (x − t) + gσ(t)}. (3.12)
where f : D ⊂ R2 → R is the image function, D
xis the translate of D, Dx = {x+t : t ∈ D},
and ˇDx is the reflection of D. Finally, gσ : Gσ ⊂ R2 → R is the scaled structuring
function. An example of such a function is given by Van Den Boomgaard et al. [36] where gσ(x) = −2σ1 x2, and σ > 0 is the scale parameter.
The result of these operations depends on the origin of the structuring function gσ.
To avoid level shifting and horizontal translation effects, respectively, we have to consider Supx∈Gσ{gσ(x)} = 0 and gσ(0) = 0, defined by Jackway et al. [16] and introduced in
Subsection3.3.2.
Also was proved the conditions of scale-space to ensure the non-changes in the original gray-scale and position of the remaining extrema, as well as the non-introduction of new extrema in the simplified signal, are also obtained from this type of structuring function whose shape is concave downward and monotonic decreasing along any radial direction from its origin [16].
3.4
Techniques for Image Simplification
In this section, we describe of the image simplification method using the morphological scale-space as the principal process.
In the context of scale-space theory, combinations of erosion and dilation operations using non-flat structuring functions have proved to be efficient for the elimination of un-wanted components. The following discusses some approaches proposed in the literature.
(a) (b) (c)
Figure 3.7: The transformation of the image (a) with the MMDE method using the scales (b) −0.5 and (c) 0.5 in the structuring functions.
(a) (b) (c) (d)
Figure 3.8: The morphological transformation scale-dependent of (a) the orignal image to achieved (b) opening, (c) closing and (d) a combination.
3.4.1
Multiscale Morphological Dilation Erosion - MMDE
Jackway et al. proposed a new scale-space method called Multi-scale Morphological Dila-tion and Erosion [15,16], where the morphological scale-space is defined for both positive and negative scales. For positive scales is considered the morphological operation of dila-tion and for negative scales, the erosion operadila-tion. To achieve this separadila-tion, Jackway et al. make use of a morphological operator type toggle, defined by,
(f ⊗ gσ)(x) = δgσ(f )(x), if σ > 0, f (x), if σ = 0, εgσ(f )(x), if σ < 0, (3.13)
where f is the original image,δgσ(f )(x) denotes a dilation operator and εgσ(f )(x)
the erosion operator, the pixel in the position x using the scale-dependent structuring function gσ.
Fig.3.7illustrates a possible scale-space representation generated by the MMDE. Note that, while the “negative” scales enhances the dark structures of the image. The positive scales have the opposite effect.
3.4.2
Method Based on Opening and Closing
To generate a scale-space from the morphological operations of opening and closing [5], we need to use non-flat scale-dependent structuring functions, which have to possess the necessary and sufficient conditions to guarantee the monotonicity of the components, to achieves these properties; the structuring functions have to be of the anti-convex shape [17].
The fact that opening and closing operations constitute non-auto-dual morphological filters can make them sensitive to noise. Since the opening eliminates positive components and closing has the same effect on negative components. To address this problem, opening and closing can operations be sequentially alternated.
Fig. 3.8 shows a possible transformation of the morphological operations of opening in Fig.3.8(b) and closing in Fig.3.8(c) using a non-flat structural function dependent on the scale and combinations of both operators in Fig3.8(d).
CHAPTER 3. MORPHOLOGICAL SCALE-SPACE 38
Figure 3.9: Structuring function of SMMT. Image taken from Dorini and Leite [7].
(a) (b) (c) (d)
Figure 3.10: Examples of SMMT on (a) the original image, using scales σ, of (b) −0.2, (c) −1.5 and (d) 3.5.
3.4.3
Method Based on Scale-Dependent Non-Flat Structuring
Function
Dorini and Leite’s works [7,6] introduce a new image processing operator based on scaled versions of a signal defined by the morphological scale-space transformations for image segmentation. This operator, defined on the scope of a toggle transformation, considers local pixel information (not only scale knowledge) to determine if each pixel should be processed by erosion or dilation operator scale-dependent.
In their work is used as structuring function g(x, y) = −max{x2, y2}, which
scale-dependent representation is given by, gσ(x, y) = −
1
|σ|max{x
2
, y2}, (3.14)
where σ is the scale of the structuring function and which shape is shown in Fig.3.9. Using toggle operators provides two main advantages [34], (1) the primitives and (2) the decision rule. The approach uses as primitives an extensive and an anti-extensive transformation, namely, the scale-dependent dilation and erosion. The decision rule in-volves, at a point x, the value f (x) and the primitive results.
Formally the toggle operator defined by Dorini and Leite is given by,
(f gσ)n(x) = ψn 1(x), if ψ1n(x) − f (x) < f (x) − ψ2n(x), f (x), if ψn 1(x) − f (x) = f (x) − ψ2n(x), ψ2n(x), otherwise, (3.15)
where ψ1n(x) = (f ⊕ gσ)n, that is the dilation of f with the scale-dependent structuring
function gσ, n times. In the same way, ψn2(x) = (f gσ)n.
different scales, using the morphological operators with the scale-dependent structuring function previous defined in Eq.3.14.
The previously image simplification techniques, using the morphological space-scale approach, have the main drawback of not having any condition or measure of persistence that guarantees that the remaining ends are the most significant, making it difficult to select a scale.
3.5
Summary
In this chapter, we presented a brief review of space-scale theory, ranging from the nec-essary properties to some of the existing approaches in the literature. A greater empha-sis was placed on the erosion and dilation operations multi-scale using a non-flat scale-dependent structuring function, which is an important part to understand the method developed in this work.
Finally, we presented techniques of simplification of images using the space-scale mor-phological approach, which lack a measure of persistence that guarantees the remaining extrema are the most significant.
Polo and Leite addressed this problem, with the creation of an image simplification method that first eliminates the least significant extremes, guaranteeing that the most significant ends are the last ones to be eliminated. The next chapter discusses in detail this technique on which our simplification method is based.
Chapter 4
The Proposed Method
The image processing and analysis area includes a large number of applications ranging from the lowest level tasks (e.g., extremum point detection) to the most specialized ones, such as segmentation and classification, requiring the deletion of unnecessary details of the input images. In such a case, it is common to apply a pre-processing step to the original image before any further consideration. Often, this step aims not only to filter out noisy components but also to simplify the image through the elimination of non-significant details, while keeping the information necessary to the achievement of the desired outcome.
As we have seen in the previous chapters, the mathematical morphology, MM, and the scale-space approach provide the necessary characteristics for image simplification. The area of MM seeks to highlight the most significant components, which in a dual way can be interpreted as the suppression of least significant components. For this reason, MM has become the focus of images simplification for excellence. The constraints presented by the basic transformations of MM are addressed by the morphological scale-space ap-proach, with the presentations of multiple observation levels, called multi-scale, where each level is known as a scale. For this reason, in the previous chapters, we saw the scale-dependent structuring functions in Subsection 3.3.2, as well as the morphological scale-space operators in Subsection3.3.3. It should be noted that the main contributions of the morphological scale-space approach are its properties of monotonicity, continuity, fidelity and euclidean invariance, which make possible the consistent manipulation of the extrema used in our method of images simplification.
Note that, one of the constraints in the previous simplification methods is the lack of a measure of the persistence of extrema in the input image, thus performing with each input parameter the elimination of many extrema.
Polo and Leite in “From Extrema Relationships to Image Simplification Using Non-flat Structuring Functions” [29] address this problem creating a simplification method which takes into account the measure of the persistence of extrema. This method performs well-behaved elimination in the sense that the remaining extrema are the most significant.
This simplification method was only defined in theory, showing its advantages and properties, but in the experimental part, it had not been explored yet. Furthermore, the method proposed by Polo and Leite [29], which we call the original approach, has the following drawbacks: (1) the presence of some intermediate steps with a high
tional cost required to perform the creation of the graph and (2) the little or no amount of applications of the method in the experimental field, which can serve as a guide about the possible uses of the method.
In this Master’s Thesis, we accept the challenge of addressing the drawbacks of the original approach mentioned above, creating an algorithm that, although sharing similar stages, is faster in the execution and still preserves all the advantages of the original approach. This is achieved through the creation of a series of properties developed in this research work. In addition, we also propose some applications, which combine our simplification method with some well-behaved morphological tools, giving a better idea of how this simplification method can be used.
In this chapter, we describe and solve the first drawback of the original approach. In Section 4.1 we discuss the main steps of the original approach [29] on morphological scale-space using extrema relationships and in the Section 4.2, we explore and created new properties for the extrema relationships, for achieving a local update which reduced the computational cost of the original approach.
4.1
General Description of the Method
The main concepts behind the simplification process explored in this work consider the notion of scale-dependent erosion and dilation, as stated in the works [16,36] and further explored, for Dorini and Leite [7] and Polo and Leite [29]. These operations were defined in Section3.3.
Jackway et al. [16] proved that the morphological scale-space conditions ensure (1) the non-changes in the value of original gray-scale, (2) the non-translation of the position of the remaining extrema. As well as (3) the non-introduction of new extrema in the simplified signal. These conditions are the basis for the morphological image simplification approach (Section 3.3) and are obtained by using a type of scale-dependent structuring function, whose shape is concave downward and monotonic decreasing along any radial direction from its origin.
In Fig. 4.1, we show the sequence of steps that the algorithm of image simplification executes to merge the extrema of the input image. The original approach is presented by the white blocks and our proposal by the dark blocks. Then we introduce and discuss each of the steps.
Following, we give a simple explanation of how our method of simplification is carried out, and then in the later sections, go deeper into each of the steps.
Initially, we obtain an input image and extract its extrema (regional maxima or min-ima), because these are the ones we want to simplify. Subsequently, we need to get a relationship between these extracted extrema; this is achieved by creating a graph, which connects each of the extrema, with their closest extrema. Once we have constructed the structure of the graph, we are going to get the separation between each pair of connected extrema in the graph, as well as the contrast of the extrema. With these two features (separation and contrast), we were able to generate a scale σ, using the difference of contrasts divided by the separation, for the extrema connected in the graph. When we