Chapter IV : Morphological Filtering

Transcription

1 Chapter IV : Morphological Filtering Serial composition -> Products θψ, θψθ -> Alternating sequential filters Parallel composition -> Morphological centers -> Contrast enhancement J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology IV. 1

2 Definition of a Morphological Filter (I) In signal processing, the word "filters" is somewhat fuzzy and its meaning is context dependent. It may have the connotation of a convolution, or may denote any system which transforms one image into another. In mathematical morphology, the word has the following precise meaning: Definition (G. Matheron, J. Serra): A morphological filter is an increasing and idempotent transformation of a complete lattice into itself. Increasingness This requirement is the most fundamental. It ensures the preservation of the order relation, after filtering. This property implies some information loss during the filtering process. J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology IV. 2

3 Idempotence Definition of a Morphological Filter (II) By definition, an idempotent operator transforms any original signal, or image, into an invariant one for the said operator (Ch. I). This property appears often, but in an implicit way, in descriptions. An optical filter will be qualified of «red», an amplifier of «band-pass», for example. In Morphological filtering, the property becomes explicit, and compulsory. Idempotence may be reached after a single pass, or as as limit under iteration. It may concern a unique operation, or a whole sequence. Finally, note that when linear filters are idempotent, then they do not admit inverses : they loose information, which make them a bit more «morphological». J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology IV. 3

4 How to obtain Morphological Filters? Up to now, only two types of morphological filters have been seen : opening and closing. To create new other ones, two modes of combination may be used: Serially: Composition products of basic filters. It results in the alternating (sequential) filters. Parallel: Combination of basic filters with sup.and inf. It results in the morphological centre. Two primitives Filters: ψθ, ψθψ Composition Basic filters: opening, closing Two families of primitives Alternating sequential filters Combination with sup. and inf. Center, Contrast, etc J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology IV. 4

5 Alternating Filters (I) Let ζ and ψ be two ordered filters (with ζ ψ). By combining them, one can generate four filters which satisfy the following properties: Theorem (G.Matheron): 1) ζψ, ψζ, ζψζ, ψζψ are morphological filters (i.e. increasing and idempotent mappings); 2) ψ ψζψ ψζψ ζψ ψζ ζψζ ζ 3) ζψζ is the smaller filter larger than ζψ ψζ ψζψ is the larger filter smaller than ζψ ψζ 4) We have the following equivalences ζψζ =ψζ ψζψ = ζψ ψζ ζψ N.B.: 1-The composition of more than three operators does not give another filter. Since, as a filter, ζψ is idempotent, hence ζψζψ = ζψ. 2- the result extends to all ordered pairs of under-filters ζ( i.e.ζ ζζ), and over-filter ψ (i.e. ψψ ψ) (H.Heijmans ) J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology IV. 5

6 Proof of of the the theorem :: Alternating Filters (II) 1) 1) ζ ζζζ ζψζ ψψζ ψζ ψζ ψζζ ψζψ ψ 2) 2) idempotence of of ζψ,ψθζ,ψζψ,ζψζ: ψζ ψζ ψζζζ ψζψζ ψψψζ ψζ ψζ ;; ψζψ (ψζψζ)ψ ψζψψζψ ψψψψζψ ψζψ ;; 3) 3) smallest majorant: if if filter ϕ ψζ ζψ, then ϕ = ϕϕ ϕϕ ζψ ζψ ψζ ψζ = ζψζ ;; 4) 4) ψζ ψζ = ζψζ ψζ ψζ ζψψ ζψ ζψ ;; ψζ ψζ ζψ ζψ ζψζ ψψζ ψζ ψζ ψζζ ζψζ.. J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology IV. 6

7 Ordering relations between alternating filters _ R ζ ζψζ I ψζψ ψ Space J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology IV. 7

8 Comparison between γϕ and ϕγ The more often, the two basic primitives used in the composition are an opening γ and a closing ϕ. Then the previous theorem states that, in general, there is no order relation between γϕ and ϕγ: closing ϕ γϕ ϕγϕ = ϕ opening γ ϕγ no order Structuring element: γϕγ = γ There is a "classical" case which illustrates point 4) of the previous theorem, namely that of the filters by reconstruction (Ch VI-13). Then the products γϕ and ϕγ are ordered, with γϕ ϕγ J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology IV. 8

9 "Top hat" Extension Goal The basic "Top hat" highlights peaks but does not remove dense fluctuations, such as noise. If we wish to compensate both smooth signal variations and also noise, then we can start from operator: γϕ I and take its residue ρ(f) = f - { γϕ(f) f } (for a given function f). In particular, when γϕ is a strong filter(e.g.1-d opening and closing, or reconstruction ones) then γϕ I turns out to be an algebraic opening. Level Opening of closing: γϕ Original Difference Sample J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology IV. 9

10 Inf-filter, Sup-filter and Strong filter Goal : The notions of sup-, inf- and strong filters are introduced to describe the properties of composition products such as θψ, and their robustness i.e. the sensitivity of the result with respect to variations of the input signal. Definition : - filter if ψ = ψ(ι ψ) Filter ψ is - filter if ψ = ψ(ι ψ) strong if ψ = ψ(ι ψ) = ψ(ι ψ) Theorem (G.Matheron) : A mapping is a -filter (resp. -filter ) if and only if it can be decomposed into a product γϕ of an opening with a closing ( resp. ϕγ ) If ϕ is a closing and γ an opening, then ϕ and γ are strong filters, γϕ and ϕγϕ are -filters, ϕγ and γϕγ are -filters. J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology IV. 10

11 Geometrical Interpretation The -filter property ensures that any signal g between f and f ψ(f) gives the same filtered version as f itself, i.e. ψ(g) = ψ(f). The -filters yield the dual result. Finally, in case of strong filters, this robustness is extended to all signals which are comprised between f ψ(f) and f ψ(f). ψ -filter f g (f ψ(f)) ψ(g)=ψ(f) ψ -filter (f ψ(f)) g f ψ(g)=ψ(f) ψ strong filter f ψ(f) g f ψ(f) ψ(g)=ψ(f) f g ψ(f)=ψ(g) f g ψ(f)=ψ(g) f g ψ(f)=ψ(g) f g f g f g J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology IV. 11

12 Alternating Sequential Filters (I) The number of distinct filters that can be created by composing two filters is rather restricted. If we want to go further, the composition must involve not two filters but two families of filters. This idea leads to alternating sequential filters. Theorem (J.Serra) : Consider two families {ζ i } of under-filters and {ψ i } of over-filters such that : {ζ i } increases with i, } {ψ i } decreases with i,... ψ n... ψ 2 ψ 1 ζ 1 ζ 2... ζ n... ψ 1 ζ 1 then the two products of composition N i = ζ i ψ i... ζ i ψ 2 ζ 1 ψ 1 M i = ψ i ζ i... ψ 2 ζ ψ 2 1 ζ 1 are idempotent, and called Alternating Sequential Filters J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology IV. 12

13 Alternating Sequential Filters (II) Proof of of the the theorem 1) 1) Put µ i = i ψ i ζ i i. i... For.. j j i i,, one has µ j µ j i µ i j j µ i µ i j,since j ψ i ζ i i ζ i j j ψ i ζ i i i ζ i ζ i i i ζ i i ζ j j ψ j ζ j j ψ j i ζ i i i ψ j ζ j j ζ j j j ψ j ζ j jj (same proof, mutatis mutandis, for for the the second inequality) 2) 2) M i M i i = i (µ (µ i µ i i-1...µ i-1 1 )(µ 1 )(µ i µ i i-1...µ i-1 1 ) 1 ) µ i (µ i (µ i µ i i-1...µ i-1 1 ) 1 ) (µ (µ i µ i i-1...µ i-1 1 )= 1 )= M ii but also :: M i M i i = i (µ (µ i µ i i-1...µ i-1 1 ) 1 )(µ i µ i i-1...µ i-1 1 ) 1 ) µ i (µ i (µ i µ i i-1...µ i-1 1 )= 1 )= M i. i. Absorption Law j j i i M j M j i = i (µ (µ j...µ j i+1 ) i+1 )(µ i µ i i-1...µ i-1 1 ) 1 )(µ i µ i i-1...µ i-1 1 ) 1 ) =M jj ((...but M i M i j j M j! j!)) J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology IV. 13

14 Properties of alternating sequential filters Duality Theoretically, alternating sequential filters are not self-dual, that is: M i ( m - f ) m - M i ( f ) In practice, the difference is often negligible. Compatibility with magnifications If the families of primitives are compatible with scale modifications, that is: ψ k (h k (.)) = h k (ψ 1 (.)), ζ k (h k (.)) = h k (ζ 1 (.)) where h k (.) represents a spatial or temporal similarity, then the resulting alternating sequential filters are also compatible. Intuitively, this property indicates that the filter of order k works at a scale k as a filter of order 1 at a scale 1. J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology IV. 14

15 Usefulness of alternating sequential filters (I) Comment: In general, the two basic families transformations are granulometries. The F.A.S. are very useful when dealing with noisy signals.the following example shows that a simple ϕγ is not appropriate to cancel noise, whatever the size of the structuring element is. Original signal ϕ γ 2 2 Signal + noise Noise Sum ϕ γ 5 5 J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology IV. 15

16 Usefulness of alternating sequential filters (II) An alternating sequential filter allows noise cancellation and smoothly produces a good approximation of the signal without noise. ϕ γ 2 2 ϕ γ ϕ γ ϕ γ Signal + noise ϕ γ ϕ γ ϕ γ ϕ γ ϕ γ ϕ γ J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology IV. 16

17 Pyramids The absorption law (1), below, of of the the ASF s, suggests we we focus on on families {ψ {ψ λ } λ of of operators that depend on on a positive parameter, λ say. For generating nice properties, the the {ψ {ψ λ } λ must satisfy some consistency as as λ varies. In In particular, they constitute a pyramid of of operators when each ψ λ (f) λ (f) may be be obtained from any transform ψ µ (f), for for 0 < µ λ,, i.e. i.e. λ µ > 0 there exists ν > 0 such that ψ λ =ψ λ ν ψ ν µ The two basic types of of pyramids are are the the following :: λ µ > 0 ψ λ λ ψ µ =ψ λ (1) λ (1) (the stronger imposes its its law) λ µ > 0 ψ λ λ ψ µ =ψ λ λ + + µ (2) (2) (the effects are are additive) λ + µ Note that the the condition of of being granulometric semi-group (( see III-15) is is more demanding than that of of forming a pyramid, since then we we have λ,, µ > 0 ψ λ λ ψ µ =ψ sup(λ, sup(λ, µ ) ).. λ, µ ). J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology IV. 17

18 An Example of a Pyramid d) e) a) b) c) ψ λ := λ := Hexagonal alternating sequential filter of of size λ begining by by a closing ;; (a) (a):= := Initial image f f ;; (b) (b):= := ψ 2 (f) 2 (f) ;; (c) (c):= := ψ 4 (f) 4 (f);; (d) (d):= := ψ 7 (f) 7 (f) = ψ 7 [ψ 7 [ψ 4 (f)] 4 (e) (e):= := ψ 2 [ψ 2 [ψ 4 (f)] 4 ψ 4 (f) 4 (f) (( the the non identical pixels are are in in white )) J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology IV. 18

19 The semi-group ψ λ ψ µ = ψ µ ψ λ = ψ sup(λ,µ λ,µ) λ,µ) The absorption law λ µ > 0 ψ λ λ ψ µ =ψ (1) λ (1) implies λ that :: -the ψ λ are λ are idempotent (not necessarily increasing), of of invariant class B λλ such that λ µ > 0 ψ λ (B λ (B µ ) ) = B λ (2) λ (2) --however, rel. (2) (2) does not imply relation (1). For generating a semi-group, the the additional law λ µ > 0 ψ µ ψ λ =ψ λ (3) λ (3) λ is is equivalent to to B λ λ B µ (4) (4) Rel. (1) (1) has a markovian meaning, since it it suffice to to know level µ for for being able to to determine al al the the upper ones (( i.e. i.e. those ψ λ for λ for which λ µ ));; When combined with rel.(3), it it becomes similar to to waveband :: ψ λ smoothes λ more than ψ µ since it it amends the the same fine details without introducing new ones. J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology IV. 19

20 Combination with sup. and inf. Methodology With the objective of creating new filters without compositions, let us consider now a family of primitives and a filter whose output is always one of the primitives values. In a first step, the selection rule relies on sup. and inf. but it can be extended to more complex rules such as those leading to "toggle mapping". ψ 1 Criterion Primitives ψ ψ ψ 2 n-1 n Combination with sup. and inf. Example: Morphological center Complex rules "toggle mapping" Example: Contrast enhancement J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology IV. 20

21 The Morphological Centre The aim of this transform is to create self-dual filters. It is a kind of gravity centre for lattice structures. Definition (J.Serra) : The morphological centre with respect to the family {ψ i,i J} is defined as: β= [I ( {ψ i }) ] ( {ψ i } In other words, if at point x - all (ψ i f)(x) are greater than the original, the centre is the smallest of them, -all(ψ i f)(x) are smaller than the original, the centre is the largest one. Otherwise the centre is the original signal. An example of morphological centre f ψ (f) 1 ψ (f) 2 J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology IV. 21

22 Properties of the Morphological Centre Self-duality If the family of primitives is self-dual as a whole (each primitive has its dual in the family), then the centre is self-dual. Strength If the primitives are strong filters, the centre is also a strong filter (e.g. reconstruction filters) Idempotence In general, the centre is not idempotent. But with any opening γ and closing ϕ, the idempotent iteration α of the centre between γϕγ and ϕγϕ is a strong filter (and self-dual if the opening and closing are dual for each other) α = β n = [ (I γϕγ γϕγ) ϕγϕ ] n Moreover, at every point x the convergence of the sequence (βf) n (x) is monotonous. Note the difference with the median operator which is also self-dual but not idempotent, and may oscillate indefinitely. J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology IV. 22

23 Contrast Operator This transformation is a sort of anticentre. Its goal is not to create a signal which is close to the original one but to produce a highly contrasted signal. Definition Given two anti-extensive and extensive transformations and an original signal f, the results is the closest transformation value of f. Property(F.Meyer, J.Serra) When the primitives are opening and closing, the resulting contrast is idempotent (but not increasing). An example of contrast enhancement ψ 1 f ψ 2 J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology IV. 23

24 Set Activity Lattice Definition (J.Serra) :: let let L be be the the family of of the the mappings from P(E) into itself. Consider two elements α et etβ of of L.. If If we we have α β Ι Ι α Ι Ι β and Ι Ι α Ι Ι β, β, i.e. i.e. if if β modifies more points than α does; then β is is said to to be be more active than α and one writes α&β.. Activity turns out to to be be an an ordering relation on on class L.. It It induces the the so so called Activity Lattice, where the the sup ' and the theinf ( of of family {ψ {ψ i,i I} i are are given by by 'ψ i = i [* [* ( ψ i )] i ( ψ i ) i ) and (ψ i = i [I [I ( ψ i )] i ( ψ i ) i ) When family {ψ {ψ i }is i }is self-dual, i.e. i.e. when for for any i I we we have ψ i i = * ψ j * j for for some j I, then 'ψ ii and (ψ ii become self-dual. J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology IV. 24

25 Function Activity Lattice Although the the activity ordering yields a complete lattice in in the the set set case, it it is is an aninf semi-lattice only in in case of of numerical functions. Typically, the the centre β of of family (ψ (ψ i ) i ) is is nothing but the the activity inf infof of the the (ψ (ψ i ) i ).. Supremum :: The activity sup ω of of mappings (ψ, ζ) ζ) exists if if and only if if { x: x: ψ x < x ΙΙ x } x { x: x: ζ x > x ΙΙ x } x =.. Then it it has a value ω x = x ψ x if x if ψ x < x ΙΙ x ; x ; ω x = x ζ x if x if ζ x > x ΙΙ x ; x ; ω x = x ΙΙ x if x if ψ x = x ζ x = x ΙΙ xx Examples :: ~ Contrast enhancements by by toggle mappings; ~ function levelling (Ch. VI). J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology IV. 25

26 References On theory of of Morphological Filtering :: The theory of of morphological filtering is is due to to G.Matheron {MAT88a}. Two simpler, and illustrated, presentations can be be found in in {MEY89c} {SER92c} and a systematic one in in {RON91}. The links with linear filters are are studied in in {MAR87b}. On Alternating Filters :: Alternating sequential filters were proposed by by S.R.Sternberg {STE83} and the the corresponding theory made by by J.Serra {SER88}. The top hat hat extension comes from {SAL91}. On Activity Lattice :: Centre and activity lattice appear for for the the first time in in {SER88,ch.8}; the the two major derivations of of this approach are are toggle mappings {SER89} and contrast operators {MEY89b}. J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology IV. 26