Cube, Cube-Octahedron, Or Rhombododecahedron

CUBE,CUBE, CUBE-OCTAHEDRON,CUBE-OCTAHEDRON,

OROR

RHOMBODODECAHEDRONRHOMBODODECAHEDRON ??

J. SERRA Centre de Morphologie Mathématique, Ecole des Mines de Paris France

IWVF3, Capri, May 28-30, 1997 IWVF3, Capri, May 28-30, 1997

Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 1 TheThe ThreeThree CubicCubic SystemsSystems of of Grids Grids (I) (I)

PlacePlace a a point point at at the the center center of of a a Euclidean Euclidean unit unit cube. cube. You You cancan project project it: it: 1/1/ onon all all of of the the six six faces;faces; 2/2/ onon all all of of the the eighteight vertices;vertices; 3/3/ on on all all of of the the twelve twelve edges. edges.

InIn each each case, case, by by symmetry, symmetry, all all projections projections have have a a same same length. length. SoSo byby translation translation each each systemsystem generatesgenerates a a netnet ofof vertices vertices and and edgesedges through through the the whole whole space, space, that that we we shall shall call call a a grid. grid.

Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 2 TheThe ThreeThree CubicCubic SystemsSystems of of Grids Grids (II) (II)

InIn these these three three grids, grids, the the families families of of vertices vertices admit admit the the followingfollowing interpretations interpretations : : •• 1rst1rst casecase:: translations translations of of the the unit unit cube cube vertices, vertices, ⇒⇒ CUBICCUBIC GRIDGRID •• 2nd2nd casecase:: the the centres centres of of the the previous previous cubes cubes are are added added to to their their vertices,vertices, ⇒⇒ CENTEREDCENTERED CUBICCUBIC GRIDGRID (( cccc grid grid ) ) •• 3rd3rd casecase:: the the centres centres of of the the faces faces are are added added to to the the vertices vertices of of thethe cubic cubic grid. grid. ⇒⇒ FACE-CENTEREDFACE-CENTERED CUBICCUBIC GRIDGRID (( fcc fcc gridgrid ) )

Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 3 FirstFirst NeighboursNeighbours

8 neighbours 6 neighbours 12 neighbours for the cc grid for the cubic grid for the fcc grid

Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 4 CubicCubic GridGrid NeighboursNeighbours Patterns Patterns

•• The The sixtsixt firstfirst neighboursneighbours generategenerate an an octahedron octahedron of of 77 voxels voxels ; ; by by addingadding : : -- the the 12 12 secondsecond neighbours, neighbours, one one findsfinds a a cuboctahedron cuboctahedron of of 1919 voxelsvoxels (dist. (dist. ratio:ratio: 1.41)1.41) -- and and again again thethe 8 8 third third ones,ones, one one finallyfinally obtains obtains the the cube cube of of 27 27 voxelsvoxels (dist. (dist. ratio:ratio: 1.73).1.73).

Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 5 CCCC Grid Grid NeighboursNeighbours Patterns Patterns

•• The The 8 8 first first neighbours neighbours generategenerate a a cube cube of of 7 7 voxels; voxels; •• by by adding adding the the 6 6 second second neighbours,neighbours, one one findsfinds a a rhombododecahedronrhombododecahedron of of 1919 voxelsvoxels (distance (distance ratioratio 1.15);1.15); •• these these rhombododecahedrarhombododecahedra partitionpartition the the space space ; ; •• The The mediansmedians of of the the rhombsrhombs are are notnot edges.edges.

Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 6 FCCFCC Grid Grid NeighboursNeighbours Patterns Patterns

•• The The 12 12 first first neighboursneighbours generategenerate a a cube-octahedron cube-octahedron of of 1313 voxels. voxels. •• The The cube-octahedra cube-octahedra do do not not fillfill thethe spacespace (they (they leaveleave octahedricoctahedric holesholes betweenbetween them)them) •• However, However, they they generategenerate a a regularregular net net where where allall edgesedges havehave the the samesame lenght.lenght.

Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 7 CCCC andand FCCFCC Grids Grids : : Square Square SectionsSections

•• In In both both cc-grid cc-grid and and cfc-grid, cfc-grid, the the sections sections normalnormal to to –– the the cube cube diagonals diagonals are are composed composed of of hexagons, hexagons, –– the the cube cube edges edges are are composed composed of of squares.squares. •• The The ratios ratios between between square square andand interplane interplane spacingsspacings have have aa valuevalue 22 forfor cc-grid, cc-grid, and and 1.41 1.41 for for fcc fcc grid grid

11

1

Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 8 CCCC andand FCCFCC Grids Grids : : HexagonalHexagonal SectionsSections

•• For For the the sections sections normalnormal to to thethe diagonal diagonal ofof the the cube, cube, we we findfind againagain aa staggering staggering effect, effect, with with now now a a perodicity perodicity of of three. three.

•• The The ratios ratios between between hexagonal hexagonal and and interplane interplane spacings spacings are are 0.400.40 forfor cc-gridcc-grid and and 0.81 0.81 forfor fcc fcc gridgrid

Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 9 SteinerSteiner DecompositionsDecompositions

•• Consider Consider the the centre centre x x of of a a cube cube C C . . By By dilating dilating –– the the three three vectors vectors from from x x to to the the faces faces of of C, C, one one obtains obtains a a CubeCube;; –– the the four four vectors vectors fromfrom x x to to thethe verticesvertices of of C,C, one one generatesgenerates a a RhombododecahedronRhombododecahedron;; –– the the six six vectors vectors fromfrom x x to to thethe edgesedges of of C,C, one one obtainsobtains a a TetrekaidecahedronTetrekaidecahedron;; –– the the two two tetrehedra tetrehedra built built on on the the diagonals diagonals of of C, C, one one obtains obtains a a Cube-octahedron.Cube-octahedron.

Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 10 SteinerSteiner DecompositionsDecompositions (cont (contdd))

•• Cube Cube :: 1 1 1 (1,0) CC == (( ))⊕⊕ (( 1 )) ⊕⊕ (( ))

•• Cube-octahedron Cube-octahedron : : 1 0 0 1 DD == (( 0 1 ))⊕⊕ (( 1 0 ))

•• Rhombododecahedron Rhombododecahedron : : 1 . 0 . . -1 . 0 RR == (( . 0 ))⊕⊕ (( . 1 )) ⊕⊕ (( 0 . ))⊕⊕ (( -1 . ))

Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 11 SteinerSteiner DecompositionsDecompositions (cont (contdd))

By dilation of the six basic vectors of the f.c.c. grid , one obtains the tetrakaidecahedron ( and not the cube-octahedron, which is not Steiner).

0 1 0 -1 0 . 0 . . 0 0 . ( . .)⊕( . . )⊕( 1 .)⊕( -1 .)⊕( 0 . )⊕( . 0 )

Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 12 StaggeringStaggering thethe GridsGrids

•• For For both both polyhedrapolyhedra R R andand D,D, the the previousprevious decompositionsdecompositions generategenerate thethe staggeringstaggering byby a a 45°45° turn turn of of the the grid,grid,e.g.e.g. . 1 . 1 . 1 . 1 . D = 1 . 1 ∪ . 1 . ∪ 1 . 1 D = ()(). 1 . ∪ ()()1 . 1 ∪ ()(). 1 . •• By By so so doing, doing, we we leave leave inside inside holes, holes, or, or, equivalently,we equivalently,we drop drop half half of of thethe voxels.voxels. Alernatively, Alernatively, we we cancan buildbuild more more condensed condensed R R andand DD asas followsfollows 1 1 . . 1 . upper and lower planes 1 1 . odd central plane 1 1 1 upper and lower planes()(). . . odd central plane ()(). 1 . . . . . 1 . . 1 1 upper and lower planes even central plane 1 1 1 upper and lower planes()(). 1 1 even central plane ()(). 1 .

Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 13 DirectionsDirections

•• Unlike Unlike the the unit unit disc, disc, the the unit unit sphere sphere cannot cannot be be divided divided into into as as manymany sectorssectors as as we we want.want. •• The The 3-D 3-D sphere sphere can can be be partitionned partitionned only only into into 2, 2, 4, 4, 6, 6, 8, 8, 12, 12, and and 20 20 equalequalsolidsolid angles. angles. •• The The axes axes ofof the the 6, 6, 88 andand 1212 solid solid angles angles coincide coincide withwith thethe directionsdirections ofof the the firstfirst neighboursneighbours in in the the cubic,cubic, cccc andand fcc fcc gridsgrids respectively.respectively. Therefore, Therefore, they they cancan bebe accessedaccessed digitally digitally.. •• The The finest finest equidistributed equidistributed digital digital set set of of directions directions is is written written : : 0 1 0 -1 0 . 0 . . 0 0 . (( . . )) ;; (( . . )) ;; (( 1 . )) ;; ((-1 . )) ;; (( 0 . )) ;; (( . 0 ))

Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 14 Cube-octahedralCube-octahedral DilationDilation

a ) : Series of 74 cross sections of cube-octahedral dilation expanded Polystyrene ( from Shell) of size 5 of a )

Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 15 Cube-octahedralCube-octahedral closingclosing

a ) Initial image cube-octahedral closing of size 12 of a )

Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 16 CubicCubic DilationDilation

a ) cubic dilation of size 5 of a )

Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 17 SpaceSpace GraphsGraphs

•In•In RR33 , , aa space space graph graph is is –a–a setset X Xofof pointspoints;; –– a a collectioncollection E Eofofedgesedges,, i.e.i.e.ofof lines lines homotopic homotopic to to the the segment segment [0,1][0,1] .. Both Both ends ends ofof each each edgeedge belongbelong totoXX,, andand two two edgesedges maymay possiblypossibly meet meet at at their their extremities extremities only only ; ; –– a a collectioncollection F Fofof faces faces,,i.e.i.e.ofof surfaces surfaces homotopic homotopic to to the the closed closed unitunit disc. disc. The The contours contours of of the the faces faces are are exclusively exclusively edges, edges, and and twotwo faces faces may may meetmeet alongalong edgesedges onlyonly ; ; –a–a setset P Pofof blocks blocks,, formed formed by by the the connected connected components components of of the the spacespace that that remain remain when when all all points points edges edges and and faces faces have have been been removed.removed.

Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 18 EulerEuler -Poincaré -Poincaré NumberNumber

•• Space Space graphsgraphs are arethetheturningturning point point between between Euclidean Euclidean and and Digital Digital spaces.spaces. Defined Defined in in RR33 , , they they cancan bebe reinterpretedreinterpreted in in Z Z33 ,andthe ,andthe derivedderived notions notions areare meaningful meaningful in in both both spaces.spaces. •• It It isis in in particular particular thethe case case forfor the the Euler-PoincaréEuler-Poincaré NumberNumber νν (Y)(Y) ofof the the set set YY== XX∪∪EE∪∪FF formedformed byby thethe points, points, edges edges and and facesfaces ofof space space graph graph X X . . It It is is equal equal to to

νν ( (YY)) == N N (vertices)(vertices)++ N N (faces)(faces)--NN (edges)(edges)--NN (blocks)(blocks) •• From From a a digital digital point point of of view, view, the the problem problem consists consists then then to to associateassociate convenient convenient space space graphs graphs with with the the sets sets under under study. study.

Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 19 CommentsComments on on Graphs Graphs and and Connectivity Connectivity

AccordingAccording to to the the type type of of operation, operation, three three different different levels levels of of connectivityconnectivity maymay bebe involved:involved: ⇒⇒Erosions,Erosions, dilations, dilations, and and derived derived filters, filters, as as well well as as skeletons skeletons (by(by erosion/opening)erosion/opening) dodo not not needneed edges edges:: one one alwaysalways dilates dilates aa setset ofof points pointsbyby a a setset ofof points points .. ⇒⇒Geodesy,Geodesy, reconstructions,reconstructions, connected connected componentscomponents ((e.g.e.g.ultimateultimate erosion)erosion) require require grids grids,, i.e. i.e.verticesvertices + + edges edges . . ⇒⇒thinnings,thinnings, thickenings, thickenings, oror any any operatoroperator whichwhich preservespreserves homotopyhomotopy requiresrequires thethe datumdatum of of aaspacespace graphgraphonon the the grid.grid.

Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 20 SpaceSpace GraphGraph for for the the FCC FCC Grid Grid (I) (I)

WeWe vill vill associate associate with with the the F.C.C F.C.C grid grid the the space space graph graph whose: whose: ⇒⇒ verticesvertices and and edges edges are are those those of of the the grid,grid, ⇒⇒ facesfaces areare the the elementaryelementary triangles triangles (( butbutnotnotthethe squares squares !),!), ⇒⇒ blocks blocks areare the the smallestsmallest tetrahedratetrahedra and and octahedra, octahedra, i.e. i.e.

Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 21 SpaceSpace GraphGraph for for the the FCC FCC Grid Grid (II) (II)

•• In In such such a a graph, graph, the the basic basic cube-octahedron cube-octahedron is is not not a a convex convex polyhedron,polyhedron, but but admits admits six six conic conic sinks. sinks. Its Its convex convex enveloppe enveloppe for for the the graph,graph, is is its its stellation stellation,, i.e. i.e.itsits circumscribedcircumscribed octahedron.octahedron.

•• Given Given a a set set X X (the (the 1’s 1’s ) ) the the vertices, vertices, and and the the edges, edges, faces, faces, and and blocs blocs whosewhose all all vertices vertices are are 1’s 1’s generate generate the the space space graph graph of of X. X. •• The The graph graph associated associated with with the the 0’s 0’s is is obtained obtained by by applying applying the the samesame ruleruletoto set set X Xcc ..

Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 22 Euler-PoincaréEuler-Poincaré CountCount on on FCCFCC Grid Grid

•• Euler-Poincaré Euler-Poincaré count count illustrates illustrates the the role role of of the the Space Space Graph. Graph. The The eventsevents to to be be checked checked are are the the six six following following subsets subsets of of basic basic blocks blocks

•• The The occurrences occurrences ofof each each of of them them are are computed computed and and added, added, the the firstfirst twotwo ones ones positively, positively, and and the the last last four four ones ones negatively. negatively.

Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 23 ThinningsThinnings and and Homotopy Homotopy

•• Consider Consider a a neighborhood neighborhood based based thinning thinning . . It It is is homotopic homotopic when, when, byby changingchanging 1 1→→0,0, we we do do not not locally locally modify modify the the genus genus of of the the boundaries,boundaries, i.e. i.e.thethe change change mustmust not not –– open open aa hole, hole, neither neither createcreate a a new new particle;particle; –– generate generate a a donut donut of of grains,grains, oror ofof pores;pores; –– suppress suppress a a graingrain oror aa pore;pore; etc.. etc.. •• When When the the neighborhood neighborhood is is based based on on the the unit unit digital digital cube- cube- octahedron,octahedron, then then oneone findsfinds fivefive candidate candidate configurationsconfigurations only, only, up up toto rotations rotations and and complement. complement.

Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 24 ExamplesExamples of of homotopic homotopic thinningsthinnings

•• Here Here are are two two candidate candidate configurations configurations for for cube-octahedral cube-octahedral homotopichomotopic thinning thinning

a) b)

••thethe 1’s 1’s ; ; •• thethe 0’s 0’s ; ; and and in in between, between, a a no no man’s man’s land. land. ForFor configurationconfiguration b b)) three three 0’s 0’s havehave to to bebe addedadded in in the the upperupper plane, plane, andand three three 1’s 1’s in in the the lower lower plane, plane, to to get get the the stellations. stellations.

Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 25 ReferencesReferences

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Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 27