Chapter I : Basic Notions

Transcription

1 Chapter I : Basic Notions - Image Processing - Lattices - Residues - Measurements J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology I. 1

2 Image Processing (I) => Image processing may be divided into three classes of questions, namely Codification, Extraction of characteristics and Segmentation. 1) Codification : Codification concerns the representation of the images. It comprises: Acquisition : Transformation of analog images into numerical ones. Compression : Modification of image representation. Analog Image Numerical Image n 1 Acquisition Compression Numerical Image Numerical Image n 2 Synthesis : Generation of an image from a more symbolic representation. Numbers Synthesis Numerical Image J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology I. 2

3 Image Processing (II) 2) Extraction of Characteristics : The aim here is to improve image quality or to exhibit some of its features. This includes in particular measurements, noise reduction, and filtering. Image1 Image Tranformations Image2 Numbers 3) Segmentation : Segmentation consists in partitioning the images into zones which are homogeneous according to a given criterion. Pixels Segmentation Regions J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology I. 3

4 Four Approaches in Image Processing Linear Non linear Geometric Space Linear: Convolution, Fourier,Wavelets Tomography Kriging, Splines Morphological: Morph. Filtering Hierarchies (e.g. Granulometry) Random Sets Watersheds Abstract Spaces Statistical: Multivariate Analysis Neural Nets Stereology Syntactical: semantic approaches Grammars Indexation J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology I. 4

5 Definitions of Mathematical Morphology Mathematics Lattice theory for objects or operators in continuous or discrete spaces; Topological and stochastic models. Physics Signal analysis techniques based on set theory aiming at the study of relations between physical and structural properties. Signal Processing Nonlinear signal processing technique based on minimum and maximum operations. Engineering Algorithms and software / hardware tools for developing signal processing applications. J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology I. 5

6 Basic Structures Linear signal processing : The basic structure in linear signal processing is the vector space i.e. a set of vectors V and a set of scalars K such that 1) - K is a field ; - V is a commutative group 2) - There exists a multiplicative law between scalars and vectors. Mathematical morphology : The basic structure is a complete lattice i.e. a set L such that: 1) L is provided with a partial ordering, i.e. a relation with A A A B, B A A = B A B, B C A C 2)For each family of elements {Xi} P, there exists in L : a greatest lower bound {Xi}, called infimum ( or inf.) a smallest upper bound {Xi}, called supremum ( or sup.) J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology I. 6

7 Basic Operations Linear Signal Processing Since the structure is that of a vector space, whose fundamental laws are addition and scalar product, then The basic operations are those which preserve these laws, i.e. which commute under them: Ψ( λ i f i ) = λ i Ψ (f i ) The resulting operator is called convolution. Mathematical Morphology Since the the Lattice structure lies on on the the ordering relation, on on the the sup and the the inf, the the basic operations are are those which preserve these fundamental laws, namely Ordering Preservation :: {X Y Ψ Ψ(X) Ψ Ψ(Y)} increasingness Commutation under Supremum.: Ψ ( X i ) i ) = Ψ (X (X i ) i ) Dilation Commutation under Infimum: Ψ ( X i ) i ) = Ψ (X (X i ) i ) Erosion J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology I. 7

8 Examples of Lattices Lattice of subsets P(E) of a set E: The partial ordering is defined by the inclusion law: X Sup: Inf: Y X Extremes: E, Y Lattices of real or integer numbers: This total ordering is given by the succession of the values: Sup: (usual sense) Inf: Extremes: -, + - Order + Lattice of convex sets: The order is defined by the inclusion law: Sup : Convex hull of the union Inf : Intersection X Sup Y J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology I. 8

9 Lattices of Functions If If E is is an an arbitrary set, and if if T designates R, R, Z one of of their closed subsets, then the the functions f f :: E T generate in in turn a new lattice, denoted by by T E,, for for the product ordering f f g iff iff f(x) g(x) for for all all x E,, where sup and inf infderive directly from those of of T, T, i.e. i.e. ( ( ff i )(x) i = ff i (x) i (x) ( ( ff i )(x) i = ff i (x) i (x).. By By convention, the the same symbol 0 stands for for the the minimum in in T and in in T E.. In In T E,, the the pulse functions :: k x,t (y) x,t (y) = t t when x = y ;; k x,t (y) x,t (y) = 0 when x y are are sup-generators, i.e. i.e. any f f :: E T is is a supremum of of pulses. The approach extends directly to to the the products of of T type lattices, i.e. i.e. to to multivariate functions (( e.g. color images, motion). J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology I. 9

10 Lattices of Partitions Definition :: A partition of of space E is is a mapping D: D: E P(E) such that (i) (i) x E,, x D(x) (ii) (ii) (x, (x, y) y) E, E, either D(x) = D(y) or or D(x) D(y) = The partitions of of E form a lattice D for for the the ordering according to to which D D' D' when each class of of D is is included in in a class of of D'. D'. The largest element of of D is is E itself, and the the smallest one is is the the pulverizing of of E into all all its its points. The sup of the two types of cells is the pentagon where their boundaries coincide. The inf, simpler, is obtained by intersecting the cells. J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology I. 10

11 Atoms, Co-primes and Complement A subset L L of of lattice L is is called a sub-lattice if if it it closed under and and contains the the two extremes 0 and m of of L. L. A lattice L is is complemented when for for every a L,, there exists one b L at at least such that a b = m ;; a b = 0.. A non zero element a of of a lattice L is is an an atom if if x a x = 0 or or x = a.. An An element x L is is said to to be be a co-prime when x a b x a or or x b.. Moreover, element x L is is strongly co-prime when for for any (finite or or not) family {b {b i i,, i I} x { b i, i, i I } x b for some b { b i } i.. J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology I. 11

12 Sup-generators ; Distributivity A lattice L is is sup-generated when it it has a subset X, X, called a sup-generator, such that every a L is is the the supremum of of the the elements of of X that it it majorates a = { x X,, x a } When the the sup-generators are are co-prime (resp. atomic), then lattice L is issaid to to be beco-prime (resp. atomic).. Lattice L is is distributive if, if, for for all all a,, y,, z L a (( y z )) = (( a y )) (( a z )) a (( y z )) = (( a y )) (( a z )).. or, or, equivalently When theses conditions extend to to infinity, lattice L is is infinite distributive a (( y i, i, i i I )) = { (( a y i ) i ),, i i I } a (( y i, i, i i I )) = { (( a y i ) i ),, i i I } (( NB :: the the two conditions are no no longer equivalent!)!) J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology I. 12

13 Characterisation of P(E) Lattices Theorem (( G.Matheron): The four following statements are are equivalent L is is complemented and generated by by the the class Q of of its its co-primes ;; L is is atomic (( of of class Q a a )) and generated by by the the class Q ss of of its its strong co-primes; L is is isomorphic to to a P(E) type lattice ;; L is is isomorphic to to lattice P(Q).. When they are are satisfied, L is is infinite distributive and we we have Q = Q a = a Q s. s. Other lattices The function lattice T E is is infinite distributive but not complemented. _ The pulses are are sup-generating co-primes, and even strong co-primes when T is is discrete (T (T finite, T = Z ), ), but they are are not atoms.. The lattice D of of the the partitions is is sup-generated, but neither distributive nor complemented. J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology I. 13

14 Notion of duality The two laws of Sup. and Inf. play a symmetrical role. Each involution (c) that permutes them generates a duality. More precisely, Definition: Two operators ψ and ψ are dual with respect to the involution (c) when: ψ ( X c ) = [ ψ (X)] c Examples of involution : Lattice of subsets of a set: The involution is the complement. It translates to the classical notion of foreground and background: Lattice of real functions bounded by [0,M]: The involution is the reflection with respect to M/2. M 0 Sets E Compl. Involution M 0 E J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology I. 14

15 Self duality Linear processing : The convolution operation is self dual, that is dual of itself: f * (-g) = - (f * g) This means that positive or negative (bright and dark) components are processed in a symmetrical way. Mathematical morphology : The fundamental duality between Sup. and Inf. translates to all morphological tools. In general, morphological operations go by pair and correspond to each other by duality: as examples erosion and dilation, opening and closing. However,operators may also be - self-dual, i.e. ψ (X c ) = [ ψ (X)] c ( e.g. morph. centre) -or invariant under duality, i.e. ψ (X c ) = ψ (X) (e.g. boundary set in R n ) J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology I. 15

16 Input Output Comparison Extensivity anti-extensivity : A transformation is extensive if its output is always greater than its input. By duality, it is anti-extensivity when the output is always smaller than the input. Extensivity : X Ψ (X) anti-extensivity : X Ψ (X) X Ε X ϕ(x) ϕ(f) f Set (extensivity) Function (extensivity) Idempotence : A transformation is idempotent if its output is invariant with respect to the transformation itself: Idempotence: Ψ [ Ψ (X) ] = Ψ (X) J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology I. 16

17 Lattices of Operators With every lattice L is is associated the the class L' L' of of the the operations α: α: L L.. Now, L' L' turns out to to be be a lattice where :: α β (in (in L' L' )) α(a) β(a) for for all all A L ( α i ) i )(A) = α i (A) i ( α i ) i )(A) = α i (A) i (( in in L' L' )) (( in in L )) (( in in L' L' )) (( in in L )) for for example, The mappings which are are :: -- increasing,, --or or extensive,, --or or anti-extensive,, over L are are each a sub-lattice of of L' L' ;; More generally, we we shall meet lattices for for :: --openings --filters --activity etc... J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology I. 17

18 Notion of Residues in Morphology The theory of morphological filters has highlighted the increasing and idempotence properties, as well as the ordering rules between transformations. There is a family of transformations which studies the difference between two (or many) basic transformations. Their common basis relies on the notion of difference also called residue. Primitive: ψ Primitive: θ Difference Residue J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology I. 18

19 Classification of residues The residues that are used in practice can be classified in three groups: 1) Residues of two primitives 2) Residues of two family of primitives 3) Residues relying on "hit or miss" transformations Residues Two primitives: θ, ψ Two families of primitives: {θ }, {ψ } i i Residue with "hit or miss" transformations Examples: Gradient Top hat Examples: Ultimate erosion Skeleton Conditional bisector Examples: Thinning Thickenning SKIZ J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology I. 19

20 Magnification and Operators It is often necessary to modify the working scale, i.e. to perform a space similitude X λx. Here two situations can be distinguished: 1) Transformations which commute under magnification : ψ (X) = (1 /λ) ψ (λ X) Examples: boundary, skeleton, center of gravity. 2) Family of transformations that are compatible with magnification : ψ 1 ( X) = (1 / λ) ψ λ (λ X) Examples: Granulometries, FAS, associated with similar structuring elements B(λ) = λb. We then have ψ B (X) = (1/λ) ψ λβ (λ X). N.B. - Family {ψ B } is globally invariant under magnification; - Mutatis mutandis, the above notions are also valid for functions ; - They model Multi-resolution Decomposition ( i.e. pyramids ). J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology I. 20

21 Curves and Measurements Every Image processing ends either by by a new image (e.g. filtering) or or by by a measurement, i.e. i.e. by by numbers. Measurements The simplest and the the most often used measurement is is the the labelling of of presence // absence. The second one is is Lebesgue measure, or or its its digital versions. There exist a few other ones, topological or or metrical. Families depending on on a positive parameter In In this case, the the two usual representations consist in in either associating a measurement with each transformation of of the the family and obtaining a curve depending on on parameter λ. λ. Examples: histogram, size distributions; or or considering the the family of of transformations as as the the sections of of a numerical function. Example :: distance function. J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology I. 21

22 References On Roots :: Mathematical morphology has two main roots: lattice theory and random geometry. It It was created by by G. G. Matheron and J.Serra in in 1964 and known by by their three basic publications :: {MAT75} mainly deals with sets (topological framework, random sets, boolean models, convexity, granulometry, representation of of increasing mapping by by dilations), {SER82} concentrates on on translation invariant mapping (extension to to functions, discrete morphology, thinning/thickening, combination of of operators) and {SER88} enlarges the the approach the the lattice framework (dilation, theory of of morphological filtering, connectivity, skeletons, boolean functions).this lattice approach was pursued by by H.Heijmans and Ch. Ronse {HEI90}{RON91}. In In addition, there exists three excellent treatises on on the the subject {COS89} {SCH93} {HEI 94}. Instructive overviews of of mathematical morphology can be be found in in {SER87},{HAR87},{GIA88},{DOU92b} and {SER97}. J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology I. 22