This note describes the implementation of the basic structuring elements in the Mamba-Image library. These structuring elements are composed of those which are defined on the elementary neighborhood of a point (hexagon on an hexagonal grid, square on a square one) but also of the octogonal (square grid) and dodecagonal (hexagonal grid) ones

Beucher, Serge

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Algorithmic description of erosions and dilations inMambaSerge BeucherTo cite this version:Serge Beucher. Algorithmic description of erosions and dilations in Mamba. 2011. <hal-00879628>HAL Id: hal-00879628https://hal.archives-ouvertes.fr/hal-00879628Submitted on 4 Nov 2013HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Algorithmic description of erosions and dilations in MambaSerge BEUCHERCMM/ARMINES/MinesParisTechhttp://www.mamba-image.orgThis document is part of the Mamba algorithmic documentationThis note describes the implementation of the basic structuring elements in the Mamba-Image library [1].  These structuring elements are composed of  those which are defined on theelementary neighborhood of a point (hexagon on an hexagonal grid, square on a square one) butalso of  the octogonal (square grid) and dodecagonal (hexagonal grid) ones.1. Basic structuring elements on the elementary neighborhoodA class named structuringElement allows to define structuring elements on the elementaryneighborhood  of  a  point.  A structuring  element  is  simply  defined  by  a  list  of  points  of  theneighborhood and by the grid  which is  used.  The points  are coded according to  the followingscheme (Figure 1):Figure 1: Directions coding for the hexagonal and square grids.For instance, an hexagonal structuring element is defined by:HEXAGON = structuringElement([0, 1, 2, 3, 4, 5, 6], mamba.HEXAGONAL)A square one by:SQUARE3x3 = structuringElement([0, 1, 2, 3, 4, 5, 6, 7, 8], mamba.SQUARE)The  following  definition  corresponds  to  a  centered  segment  of  size  2  in  the  60°  direction  onhexagonal grid:SEGMENT60 = structuringElement([0, 1, 4], mamba.HEXAGONAL)In  any case,  the origin of  the structuring element is  always the center  point  although it  is  notcompulsory that this center point belongs to the structuring element.1An erosion or a dilation by such a structuring element is obtained by performing the infimum (forthe erosion) or the supremum (for the dilation) of the different shifts of the initial image which areimposed by (derived by the position of) the points belonging to the structuring element.  Thesesuccessive  operations  can  be  obtained  with  the  infNeighbor or  supNeighbor transformationsavailable in the Mamba library.Note that, with some structuring elements, it would be possible to achieve erosions and dilations ina faster way by using successive operations with segments. A dilation by a 3x3 square for instancecan be obtained with four successive dilations by size 1 segments in the directions 1, 3, 5 and 7 ofthe square grid which is twice as fast as the implementation based on suprema of shiftings. Thesame approach could be used with the elementary hexagon (that is, performing three successivedilations in directions 1, 3 and  5 instead of six shifts). However, in this case, it is not a good idea,as it produces errors on the edge of the image. These edge effects are illustrated in Figure 2.Figure 2: Edge effects when computing an hexagonal dilation with three successive dilations bysegments: on the left, points generated by dilations in directions 1 and 3 fall outside the imagewindow and do not contribute to the next dilations.They come from the fact that a point generated by a dilation by a segment may fall outside of theimage window which prevents it to be used at the next step to generate another point inside thewindow. The same phenomenon may occur also with erosions. As this bias does not occur whenusing suprema or  infima of  shiftings,  this  implementation is  therefore used systematically withoperators (erosion, dilation) using the structuringElement class (even on a square grid, for the sakeof simplicity, although this bias does not appear).The structuring elements already defined in the Mamba library are given in Figure 3. Note thatthese  structuring  elements  can  be  easily  rotated,  transposed  through  specific  methods  of  thestructuringElement class.Note also that a specific implementation of large structuring elements exists in the library (moduleerodilLarge.py).  This  implementation  is  described  in  [2].  As  this  implementation  actually  usessuccessive operations with segments (with some computation tricks to cope with edge effects), it islikely that these operators be faster on the square grid even for small size structuring elements.2Figure 3: Some structuring elements present in the Mamba library.2. Octogonal structuring elements2.1. Octogons designOctogons  can  be  obtained  by  concatenating  dilations  with  squares  and  with  conjugatesquares (squares turned by pi/4), also called diamonds.Figure 4: size of the octogon compared to the respective sizes of the square and of the diamondused to get it.The size n of the octogon is equal to:n = n1 + n2where n1 is the size of the square and n2 the size of the diamond. These sizes determine the size of3the sides of the octogon, n1 determining the size s1 of the horizontal and vertical sides and n2  thesize s2 of the oblique sides (Figure 4). We have:s1=2n1s2=2 n2In order to obtain isotropic octogons, we put:s1=s2which leads to:n1=2−1n=0.41421 nn2=n−n1=2−2nFinally, as n1 and n2 must be integer numbers, we write:n1=E0.41421 n0.5where E(x) stands for the integer part of x.n2=n−n1The precision of the formula is sufficient for sizes of operations up to 100,000.The following figure (Figure 5) shows the increasing sequence of octogons generated by theseformulas.Figure 5: Successive octogons generated by the above algorithm.2.2. Generating a sequence of octogons of increasing sizesThe following table (Table 1) contains the respective sizes of the squares and diamonds forthe octogons of sizes 1 to 10. Note that these respective sizes do not increase according to a periodicscheme.  We show that,  to  obtain an  octogon of  size  i+1 from an  octogon of  size i,  either  anoperation with a square or with a conjugate square is performed. Many morphological transformations (distance functions, residual transforms) need the elaborationof  sequences  of  basic  operations  (erosions  or  dilations).  So,  it  is  necessary to  exhibit  the  ruleallowing to obtain a size i+1 transformation from the previous size i result. In the case of octogons,this rule is simple:• calculate n=E 0.41421i0.5• calculate n '=E0.41421 i10.5• if n = n', the size i + 1 octogon is obtained by an operation with a diamond. Otherwise, it is4obtained by an operation with a square.As an example, let us determine how to produce a dilation by an octogon of size 73 of a set X fromthe dilation of size 72. Let us calculate n and n' (see below). We see that n =  n'. Therefore the size73 octogonal dilation will be obtained by applying a dilation by a diamond to the previous dilatedset.n=E 0.41421×720.5=30n '=E0.41421×730.5=30n n1 n21 0 12 1 13 1 24 2 25 2 36 2 47 3 48 3 59 4 510 4 5Table 1: Squares and diamonds sizes  for the  generation of size n octogons. 3. Dodecagonal structuring elements3.1. Generating dodecagonsDodecagons  can  be  obtained  on  the  hexagonal  grid  by  concatenating  dilations  withhexagons and with conjugate hexagons. These latter ones are produced by two transposed tripodstructuring elements (Figure 6).Note that the size of the conjugate hexagon which corresponds to the number of iterations of bothtripod  transformations  is  half  the  size  of  the  circumscribed  hexagon  (Figure  6a).  As  for  theoctogons, the respective sizes n1 of the hexagon and n2 of the conjugate hexagon control the sizes s1and s2 of the 60° and 90° sides of the dodecagon. In order to generate isotropic dodecagons, wemust have s1 = s2. This condition leads to the following values for the size n1 of the hexagon and n2of the conjugate hexagon (Figure 6b) needed to generate a size n dodecagon.s1=n1s2=3n2s1=s2 leads to n1=3 n2The size n of the dodecagon (which is also the size of the embedding hexagon) is given by:n=n12n2Then:n1=32−3n≈0.46410 nn2=2−3n≈0.26795 nAs for octogons, the values n1 and n2 must be integer values. However, contrary to the octogonalcase, it is not sufficient to take the closest integer to obtain an appropriate value. Indeed, if n is, forinstance, equal to 5, n1 and n2 are respectively equal to 2.32 and 1.34 before truncation, which leadsto n1 = 2 and n2 = 1 if the nearest integer rule was enforced. But n would be equal to 4 and not to 5...5In order to cope with this problem, a more sophisticated rule must be applied.(a)(b)Figure 6: (a) generation of a conjugate hexagon of size 1 by two successive tripods. Note the sizeof the conjugate hexagon compared to its embedding hexagon. (b) Relations between the size of thedodecagon and the sizes of the hexagon and conjugate hexagon.Firstly, let us calculate n1. Its real value is bounded by two successive integer values, p and p+1.Now,  if  the  size  n  of  the  dodecagon is  an  even  number,  as  n=n12n2 ,  n1 must  be  even.Therefore, the approximation of n1 must be chosen among p and p+1 depending on the parity ofthese two numbers.Figure  7  shows  successive  dodecagons  produced  by  this  algorithm  and  Table  2  gives  thecorresponding hexagon and conjugate hexagon sizes for the first ten values.Figure 7: Successive dodecagons generated by the above procedure.6n n1 n21 1 02 0 13 1 14 2 15 3 16 2 27 3 28 4 29 5 210 4 3Table 2: Hexagons and conjugate hexagons sizes  for the  generation of size n dodecagons..3.2. Iterating dodecagonsAs for octogons, it is interesting to provide a rule for producing dodecagonal operations ofsize n + 1 from the result of the operation at size n. However, in the dodecagonal case, the rule isnot as simple as it was in the octogonal case because, as it can be shown in the previous table, toobtain the result of a dodecagonal operation of size n + 1, the results of size n but also of size n - 1operations are needed. For instance, if we consider the size 5 dodecagonal operation, we see that itcan be performed by concatenating a size 3 hexagonal operation and a size 1 operation with aconjugate hexagon. But the result of size 6 can only be produced by performing an operation withan elementary hexagon on the size 4 result. On the contrary,  to get  an operation of size 7,  theprevious result of size 6 is sufficient: we just have to perform an elementary hexagonal dilation toget it.So, to perform successive dodecagonal operations of increasing sizes, we must preserve the twoprevious results. Then, from the results of the size i and size i - 1 operations, getting the result of thesize i + 1 operation is achieved by the following procedure:• calculate  n1 for  the  size  i  operation,  with  the  parity  correction  described  above  (in  aniterative procedure, this value is supposed to have been calculated at the previous step).• calculate n'1 for the size i + 1 operation.• if n'1 = n1 + 1, perform an elementary hexagonal operation on the size i result to obtain thesize i + 1 dodecagonal operation.• else, perform an operation by an elementary conjugate hexagon on the size i - 1 result toobtain  the size i + 1 dodecagonal operation.This implementation has been used to realise the dodecagonal distance function of a set. It can befound in the mambaComposed.miscellaneous.py module.4. ConclusionErosions and dilations with basic structuring elements are realised in Mamba in such a waythat  no  edge  effects  are  likely  to  occur,  which  would  be  unacceptable  for  these  elementarytransformations. This exactness is obtained after a small reduction of the performances.Regarding octogonal and dodecagonal operators, their design has been made in order to favour theirsimplicity. However, it is of the outmost importance to keep in mind some of their characteristics7and,  in  particular,  the  fact  that  the  operations  are  not  associative.  It  is  generally not  true  thatδm+n = δm(δn) if δn is a dodecagonal or an octogonal dilation (the same problem holds for erosions).Therefore, it is wise to handle these transformations cautiously to avoid erroneous results.5. References[1] Beucher, Nicolas (2010): Mamba Image Library Python Reference - http://mamba-image.org.[2]  Beucher  Serge  (2010):  Fast  implementation  of  large  erosions  and  dilations  in  Mamba- http://cmm.ensmp.fr/~beucher/publi.html(Publication date: February 22, 2011)This document is  copyrighted under the  Creative Commons "Attribution Non-Commercial  NoDerivatives" license. For terms of use, see http://creativecommons.org/about/licenses/.8

Algorithmic description of erosions and dilations in Mamba

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Algorithmic description of erosions and dilations in Mamba

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