MINISTÉRIO DA CIÊNCIA E TECNOLOGIA INSTITUTO NACIONAL DE PESQUISAS ESPACIAIS
INPE–7682–PUD/43
Formal introduction to digital image processing
Gerald Jean Francis Banon
INPE São José dos Campos July 2000
Preface
The objects like digital image, scanner, display, look–up–table, filter that we deal with in digital image processing can be defined in a precise way by using various algebraic concepts such as set, Cartesian product, binary relation, mapping, composition, operation, operator and so on. The useful operations on digital images can be defined in terms of mathematical properties like commutativity, associativity or distributivity, leading to well known algebraic structures like monoid, vector space or lattice. Furthermore, the useful transformations that we need to process the images can be defined in terms of mappings which preserve these algebraic structures. In this sense, they are called morphisms. The main objective of this book is to give all the basic details about the algebraic approach of digital image processing and to cover in a unified way the linear and morphological aspects. With all the early definitions at hand, apparently difficult issues become accessible. Our feeling is that such a formal approach can help to build a unified theory of image processing which can benefit the specification task of image processing systems. The ultimate goal would be a precise characterization of any research contribution in this area. This book is the result of many years of works and lectures in signal processing and more specifically in digital image processing and mathematical morphology. Within this process, the years we have spent at the Brazilian Institute for Space Research (INPE) have been decisive. This second edition contains many small improvements which are the result of our teaching experience at INPE during the years of 1998, 1999 and 2000. The major of them are the intoduction of poset graphs which give a better foundation to the Hasse diagram in Chaper 1 and the introduction of two types of window operators in Chapter 4. São José dos Campos, July 2000. Gerald Jean Francis Banon
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Content
Figure list 1 Digital image
1.1 Image definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Image characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Some mathematical definitions and properties . . . . . . . . . . . . . . . . 1 5 13 v
2
Image moments
2.1 2.2 2.3 2.4 Image as a vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Image as a random variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Principal component analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some mathematical definitions and properties . . . . . . . . . . . . . . . . 31 34 41 48
3
Pointwise enhancement
3.1 3.2 3.3 3.4 3.5 Look–up–table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spatially invariant pointwise operator . . . . . . . . . . . . . . . . . . . . . . . Spatially invariant pointwise morphism . . . . . . . . . . . . . . . . . . . . . Enhancement technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some mathematical definitions and properties . . . . . . . . . . . . . . . . 61 71 78 87 89
4
Filtering
4.1 4.2 4.3 4.4 4.5 Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spatially invariant window operator . . . . . . . . . . . . . . . . . . . . . . . . Spatially invariant window morphism . . . . . . . . . . . . . . . . . . . . . . . Linear and morphological window operators . . . . . . . . . . . . . . . . . Some mathematical definitions and properties . . . . . . . . . . . . . . . . 111 129 137 163 168 173 175
Bibliography Index
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