Chapter VI. Set Connection and Numerical Functions

Transcription

1 Chapter VI Set Connection and Numerical Functions Concepts : -> Extension to functions -> Connected operators -> Numerical Geodesy -> Leveling and self-duality Applications : -> Extrema analysis -> Contour preservation -> Strong filters -> Segmentation J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology VI. 1

2 Passage to Numerical Functions Three passages from binary to to grey tone images must be be viewed. Geodesy It It is is the the simplest one. Dilation and erosion being increasing, it it suffices to to define numerical operations from binary ones, applied level by by level. Applications They are are not the the same as as the the binary case. Priority is is now given to to the the processing of of the the extrema and to to contours preservation. Connections This task is is more difficult. We can either :: -- generalise the the concept of of a connection to to lattices, and find connections which are areadapted to to numerical functions, -- or or use functions to to induce set set connections on on their supports. This simpler (but less powerful) approach will be be adopted here. J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology VI. 2

3 Lattice of Numerical Functions (reminder) In In order _ to to _ ovoid the the distinction between continuous and digital cases, the the axes!,!, ", ", or or any of of their close subsets are are all all denoted by by the the generic symbol T. T. Set T is is a totally ordered lattice of of extrema 0 and m. m. When E is is an an arbitrary set, the the functions f f :: E T form in in turn a totally distributive lattice, denoted by by T E,, for for the theproduct ordering: f f g iff iff f(x) g(x) for all all x E,, In In this lattice, and,, called numerical, are are defined by: ( f i )(x) i = ff i (x) i (x) ( ( ff i )(x) i = ff i (x) i (x).. Moreover, in in T E the theimpulse functions :: k x,t (y) x,t (y) = t t if if x = y ;; k x,t (y) x,t (y) = 0 if if x y,, are are sup-generator, i.e. i.e. each f f may be be written f f = k x,t x,t for for convenient x's x's and t's. t's. J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology VI. 3

4 Lattice of the Partitions (reminder) Reminder :: 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 VI. 4

5 Set Connections induced by Functions Goal :: Let C be be a connection on on P(E) and f f :: E T. T. We look for for a regional criterion σ on on f f such that :: (i) (i) x E,, f(x) fulfils σ ;; (ii) (ii) A, A, B C,, with A B,, if if f f fulfils σ on on A and B, B, then f f fulfils σ on on A B.. Result :: Hence criterion σ generates a subclass C σ of of C which is is a second connection on on P(E). In In particular, C σ partitions set set E in in maximal classes satisfying criterion σ. σ. Example :: The zones where f f is is constant. The connected components of P(R 1 ) according C σ are either - the red segments; - or the points, elsewhere J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology VI. 5

6 A Smooth induced Connection x z y B λ λ{ { x y Smooth connection :: E =! n n,, provided with the the arcwise connection, and function f f :: E T is is fixed. The class C P(! n n )) composed of of i) i) the the singletons plus the the empty set set ;; ii) ii) all all connected open sets Y P(! n n )) such that f f is is k-lipschitz along all all paths included in in Y, Y, forms a second connection on on P(! n n ), ), called smooth connection. Implementation :: H(x) stands the the unit disc of of " 22 at at point x. x. The partition which is is associated with C has for for non point classes the the connected components of of set set X = { x E ;; sup{ f(x) --f(y),, y H(x)} k } J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology VI. 6 T f E

7 An Example of Smooth Connection Comment : the two phases of the micrograph cannot be distinguished by means of thresholds. The smooth connectivity classifies them according to their roughnesses a) Initial image: rock electron micrograph b) smooth connection of slope 7 c) smooth connection of slope 6 (- in dark, the point connected components - in white, each particle is the base of a cylinder) J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology VI. 7

8 Jump induced Connection Jump connection :: E =! n n,, provided with the the arcwise connection, and function f f :: E T is is fixed. The class C P(! n n )) composed of of i) i) the the singletons plus the the empty set set ;; ii) ii) all all connected sets around each minimum, and where the the value of of f f is is less than k above the the minimum ;; forms a second connection on on P(! n n ), ), called jump connection from minima. Similarly, one can start from the the maxima, or or take the the intersection of of both connections k { T m 0 Y A connected component in the jump connection of range k from the minima. f E J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology VI. 8

9 An Example of symmetrical Jump Connection a) Initial image: polished section of alumine grains b) Jump connection of size 12 : - in dark, the point connected components ; - in white, the other ones c) Skiz of the set of the dark points of image b) J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology VI. 9

10 Connected Operators Definition :: A function operator ψ :: T E T E is is said to to be be connected (for criterion σ) σ) when the the partition of of E by by ψ(f) is is larger than that of of E by by f f.. a) b) c) d) Three mosaic images, due to C. Vachier, obtained by merging the watershed of the gradient of a): b) by dynamics ; c) by areas ; d) by volumes J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology VI. 10

11 Flat and Increasing Connected Operators From now on, we we focus exclusively i i )) on on the the criterion σ of of flat zones ;; ii ii )) and on on those operators ψ :: T E T E that are are flat and increasing. Basic Properties :: Every binary increasing connected (resp. and increasing) operator induces on on T E,, via via the the cross sections, a unique increasing connected (resp. and increasing) operator (( H. H. Heijmans )) ;; In In particular, the the geodesic implementations extend to to the the numerical case ;; The properties of of the the set set case, to to be be strong filters, to to constitute semigroups, etc.. are are transmitted to to the the connected operators induced on on T E.. Note that a mapping may be be anti-extensive on on T E,, but extensive on on the the lattice D of of the the partitions (e.g.reconstruction openings). J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology VI. 11

12 Numerical Geodesic Dilations (I) Let f f and g be be two numerical functions from! dd into T, T, with g f. f. The binary geodesic dilation of of size λ of of each cross section of of g inside that of of f f at at the the same level induces on on g a dilation δ f,λ f,λ (g) (g)(s.beucher).. Equivalently, (L.Vincent) the the subgraph of of δ f,λ f,λ (g) (g) is is the the set set of of those points of of the the sub-graph of of f f which are are linked to to that of of g by by --a non descending path --of of length λ. λ. numerical geodesic dilation of g with respect to f f g δ f,λ (g) J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology VI. 12

13 Numerical geodesic Dilations (II) The digital version starts from the the unit geodesic dilation: δ f (g) f (g) = inf inf (g (g B, B, f) f) which is is iterated n times to to give that of of size n δ f,n f,n (g) (g) = δ (n) f (n) f (g) (g) = δ f f (δ (δ f f......(δ (δ f f (g))). The Euclidean and digital erosions derive from the the corresponding dilations by by the the following duality ε f (g) f (g) = m --δ f (m f (m--g) g),, which is is different from the the binary duality. numerical geodesic erosion of f with respect to g : f g ε f,λ (g) J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology VI. 13

14 Numerical Reconstruction The reconstruction opening of of f f from g is is the the supremum of of the the geodesic dilations of of g inside f, f, this sup being considered as as a function of of f: f: γ rec rec (f (f;; g) g) = { δ f,λ (g) f,λ (g),, λ>0 } The dual closing for for the the negative is is ϕ rec rec (f (f;; g) g) = m --γ rec rec (m- f f ;; m- m-g) g) The three major applications are are :: --swamping, or or reconstruction of of a function by by imposing markers for for the the maxima; --reconstruction from an an erosion --contrast opening,which extracts and filters the the maxima. Numerical Reconstruction of g inside f : f g γ rec (f ; g) J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology VI. 14

15 Reconstruction Opening by Erosion Goal : contour preservation Whereas the adjunction opening modifies contours, this transform is aimed to efficiently and precisely reconstruct the contours of the objects which have not been totally removed by the filtering process. Algorithm - the mask is the original signal, - the marker is an eroded of the mask. Structuring element Original Erosion γ rec ( f ; ε B (f) ) = { δ f (n) (ε B (f)), n > 0 } :B Reconstruction Dilation opening by a disc Opening by reconstruction J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology VI. 15

16 Application to Retina Examination Comment: The aim is to extract and to localise aneurisms. Reconstruction operators ensure that one can remove exclusively the small and isolated peaks ( case study due to F. Zana and J.C.Klein ). a) Initial image b) closing by c) difference a) minus b) dilatation-reconstruction followed by a threshold followed by opening by érosion- reconstruction J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology VI. 16

17 Reconstruction of a Function from Markers Goal To remove the useless maxima (or minima) of a function. Algorithm The "marker" is a bi-valued (0,m) function identifying the peaks of interest. Rec. Markers: m f The reconstruction process result is the largest function f and admitting maxima at the marked points only. It is called the swamping of f by opening (S.Beucher, F.Meyer), Swamping of f by markers m ( by opening ) J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology VI. 17

18 An Example of Swamping : Contrast Opening Goal Both morphological and reconstruction openings reduce the functions according to size criteria which work on their cross sections. In opening by dynamics, the criterion holds on grey tones contrast (M.Grimaud). Algorithm - Shift down the initial function f by constant c; - Rebuilt f from function f - c, i.e. γ rec (f ; f-c) = { { δ (n) f (f-c), n > 0 } The associated top-hat extracts all peaks of dynamics c Original Résultat C J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology VI. 18

19 Application to Maxima Detection The maxima of of a numerical function on on a space E are are the the connected components of of E where f f is is constant and surrounded by by lower values. Therefore they are are given by by the the residues of of contrast opening of of shift c = 1 More generally, the the residuals associated with a shift c extract the the maxima surrounded by by a descending zone deeper than c. c. They are are called Extended Maxima (S. (S. Beucher). J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology VI. 19

20 Strong Filters by Reconstruction Here are a few nice properties of of the filters by by reconstruction Proposition(J.Serra) :: Let γ rec rec be be a reconstruction opening on on T E that does not create pores and ϕ rec rec be be the the dual of of such an an opening (( not necessarily γ rec rec ). ). Then :: ν = ϕ rec rec γ rec rec and µ = γ rec rec ϕ rec rec are are strong filters. [[ Any pore of of X, X, firstly blocked by by ϕ rec rec,, and then recovered by by γ rec rec,, is is a pore of of X γ rec rec ϕ rec rec (X), hence µ (Ι (Ι µ) µ) = µ.] µ.] In In particular, Ι γ rec rec ϕ rec rec is is an an opening (appreciated for for its its top-hat when the the notion is is extended to to numerical functions, see IV IV -9). Proposition (J.Crespo, J.Serra) :: Let {γ {γ rec rec i {ϕ rec i } and {ϕ rec i } i denote a granulometry and a (not necessarily dual) anti-granulometry, then -- the the corresponding alternating sequential filters N ii and M ii are are strong ;; and --both operators Ψ n = {ϕ i γ i i, i, 1 i n} and Θ n = {γ i ϕ i i, i, 1 i n} are are strong filters. J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology VI. 20

21 Semi-groups of Filters by Reconstruction Proposition (Ph. Salembier, J.Serra): Let γ rec rec be be a reconstruction opening on on E and ϕ be be a closing that does not create particles. Then :: ϕ γ rec rec γ rec rec ϕ (( γ rec rec ϕ γ rec rec = ϕ γ rec rec ϕ γ rec rec ϕ=γ rec rec ϕ) ϕ) [ϕγ rec rec is is invariant under γ rec rec since ϕ, ϕ, can only enlarge the the particles of of γ rec rec (X)] Proposition (Ph. Salembier, J.Serra): Let{γ rec rec i } i be be a granulometry and {ϕ {ϕ i } i be be an ananti-granulometry of of the the above types. Then: a) a) for for all all i, i, both products ν i i rec i i rec i = ϕ i γ rec i and µ i = γ rec i ϕ i i satisfy i the the relations j i j i ν i ν i j = j ν j and j µ i µ i j = j µ jj [we always have j i j i µ j µ j i i µ j j ;; in in addition, here γ rec rec j j =γ rec i rec j j =γ rec i j rec j j γ rec i i rec j ϕ j =γ rec i γ rec j ϕ j =γ rec i ϕ j γ rec j ϕ j γ rec i ϕ i γ rec jj ϕ j j ]] b) b) Therefore, the the associated A.S.F. N i et i etm i form i a semi group N j N j i =N i i N i j = j N ;; M sup(i,j) sup(i,j) j M j i = i M i M i j = j M sup(i,j) sup(i,j) J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology VI. 21

22 An Example of a Pyramid of Connected A.S.F.'s Flat zones connectivity, (i.e. ϕ = 0 ). Each contour is preserved or suppressed, but never deformed : the initial partition increases under the successive filterings, which are strong and form a semi-group. ASF of size 8 ASF of size 4 Initial Image ASF of size 1 ( hexagonal structuring elements) J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology VI. 22

23 Adjacency Marker based opening allows to to design a self-dual operator, called leveling, and due to to F.Meyer. We will introduce first the the notion of of adjacency :: Adjacency: Let C be be a connection on on P(E). Sets X,Y P(E) are are said to to be be adjacent when X Y is is connected, whereas X and Y are are disjoint. Note that for for the the digital connection by by a 2x2 square opening, the the point marker M of of the the figure is is adjacent to to no no grain of of set set X, X, but to to X itself. M X Adjacency Prevention: Connection C is is adjacent preventing when, for for any element M P(E) and any family {B {B i i ;; i I} in in C, C, to to say that M is is adjacent to to none of of the the B ii is is equivalent to to saying that M is is not adjacent to to B i. i. J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology VI. 23

24 Levelling Given marker M, M, consider A P(E).. Let ~ γ M (A) be be the the union of of the the grains of of A that hit hit M or or that are are adjacent to to it it ;; ~ ϕ M (A) be be the the union of of A and of of its its pores that are are included in in M and non adjacent to to M cc Definition (F. (F. Meyer) :: leveling λ is is defined as as the theactivity supremum λ = γ M ( ϕ M i.e. i.e. λ(a) A = γ M A, A, and λ(a) A c c = ϕ M A c c.. Leveling λ acts as as opening γ M inside A,, and as as closing ϕ M inside A c c.. Self-duality: The mapping (A,M) λ(a,m) from P(E) P(E) P(E) is is selfdual. If If M itself depends on on A, A, i.e. i.e. if if M = µ(a), then λ, λ, as as a function of of A only, is is self-dual iff iff µ is is already self-dual. The extension to to functions (via their cross-sections) will be be denoted by by (f, (f, g) g) Λ (f, (f, g) g) J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology VI. 24

25 Properties of Levelling Here are are a few nice properties of of levelling :: Proposition (F.Meyer): The levelling (A,M) λ(a,m) is is an an increasing mapping from P(E) P(E) P(E); it it admits the the equivalent expression: λ = γ M (( ) ϕ M ) ) Proposition (G.Matheron): The two mappings A λ Μ (A),, given M, M, and M λ Α (M),, given A, A, are are idempotent (hence are are connected filters on on P(E) ). ). Proposition (J. (J. Serra): Levelling A λ Μ (A) is is a strong filter, and is is equal to to the the commutative product of of its its two primitives λ = γ M ϕ M = ϕ M γ M iff iffconnection C is is adjacency preventing. Then, λ satisfies the the stability relation :: γ x x (( II λ )) = γ xx γ xx λ,, i.e. i.e. preserves the the sense of of variation at at the the grains/pores junctions.. J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology VI. 25

26 An Example Initial image : «Joueur de fifre», by E. MANET Markers : hexagonal alternated filters, (non self-dual) Initial image, pp flat zones : Marker ϕ 1 γ 2 flat zones : Marker γ 1 ϕ 2 flat zones : J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology VI. 26

27 Duality for Functions If If 0 and m stand for for the the two extreme bounds of of the the gray axis T, T, then the the set set complement operation is is replaced by by its its function analogue f m --f f and we we have for for levelling Λ m --Λ (( m --f f,, m -g -g)) = Λ (( f, f, g )) (1) (1) which means that f, f, g Λ(f, g) g) is is always a self dual mapping. In In addition, if if g derives from f f by by a self-dual operation, i.e. i.e. g = g(f) with m -g( m --f f )) = g (( f f )) (2) (2) (convolution, median element), then levelling f f Λ(f, g(f)) is is self-dual. Observe that rel. (2) (2) is is distinct from that of of invariance under complement g( g( m --f f )) = g (( f f )) which is is satisfied by by the the module of of the the gradient, or or by by the the extended extrema, for for example, and which does not imply self-duality for for f f Λ(f). J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology VI. 27

28 An Example of Duality Marker: extrema with a dynamics h ( invariance under complement). Initial image flat zones : h = 80 flat zones : J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology VI. 28

29 Levelling as function of the marker We now fix fix set set A and study the the mapping M λ Α (M) as as marker M varies. Set A generates on on P(E) the the A-activity ordering, A by by the the relations M 1, 1 A M 22 i.e. i.e. if if M 11 meets A or or is isadjacent to toa, then M 22 meets A or or is is adjacent to toa and if if M 22 meets A cc or oris is adjacent to toa c c,, then M 11 meets A cc or or is is adjacent to to A c c.. M 1 M2 A Proposition (J. (J. Serra): If If M 11, A M 22,, then we we have λ M1 M1 λ M2 M2 (A) = λ M2 M2 λ M1 M1 (A) = λ M2 M2 (A) In In practice, this granulometric pyramid allows to to grade markers activities.. J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology VI. 29

30 An Example of Pyramid Marker: Initial image, where the h-extrema are given value zero (self-dual marker) Initial image flat zones : Levelling for h = 50 flat zones : Levelling for h = 80 flat zones : J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology VI. 30

31 An Example of Noise Reduction Marker: Gaussian convolution of size 5 of the noisy image a) Initial image, plus noise points b) Gaussian convolution of a) c) Levelling of a) by b) flat zones : J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology VI. 31

32 References On Numerical Geodesy :: The essentials of of numerical geodesy were created by by S.Beucher and F.Meyer during the the eighties {BEU90},{MEY90}. Efficient developments are are found in in L.Vincent {VIN93}, P.Soille {SOI91}, C.Vachier {VAC95}. Ch. Ronse and J. J. Serra studied the the links between symmetry, geodesy and connection {RON99}. On Connected Operators :: In In {MEY90} and in in {SAL92}, reconstruction is is used as as a tool to to modify the the homotopy of of a function, for for multi-resolution purposes. The contrast opening is is defined in in {GRI92}. A systematic investigation of of semi-groups and pyramids, by by Ph.Salembier and J.Serra, is is given in in {SER93a} and used for for sequences compression and filtering in in {MGT96}, {SAL96}, {PAR94}, {CAS97}, and {DEC97}. Nice properties of of and were found by by J.Crespo and Al Al {CRE95}. The notion of of a levelling is is due to to F.Meyer {MEY98}, see also G.Matheron {MAT97}, and J.Serra {SER98b}. The larger class of of the the grains operators has been introduced and studied by by H. H. Heijmans {HEI97}. J. Serra Ecole des Mines de Paris ( 2000 ) Course on Math. Morphology VI. 32