Regularity and Reconstruction Andrew du Plessis (describing joint work with Sabrina Tang Christensen) Matematisk Institut, Aarhus Universitet VVG60 Liverpool, 30th March, 2016
Digital Image Let L be a cubical lattice in R 3 , of side-length d . Let X ⊂ R 3 ; then X ∩ L is the digital image of X with repect to L , often written D L ( X ) . A much-studied problem in computer vision is to determine conditions under which the geometry, or at least the topology, of X can be recovered from a digital image.
Regularity A closed subset X of R 3 is r -regular if both X and R 3 − X are the unions of the closed r -balls contained in them. There are two equivalent conditions: 1 X is r -regular if and only if ∂ X is a C 1 -surface such that, for any x ∈ ∂ X , one of the two closed r -balls touching ∂ X at x is contained in X , whilst the other is contained in R 3 − X . 2 X is r -regular if and only if, for all points y ∈ R 3 with distance < r from ∂ X , the nearest point to y in ∂ X is unique. √ We will need to assume that X is r -regular for some r > 1 2 d 3, otherwise features of X may not be captured by the lattice L ; for example, a ball of radius less than √ 1 3 can have empty 2 d digital image with repect to L :
Voxel reconstruction The Voxel reconstruction (or Voronoi reconstruction) of D L ( X ) is the union of all closed cubes of side d with centres at the points of D L ( X ) and edges parallel to the lattice axes. (The cubes involved here are, for each x ∈ D L ( X ) the set of points in R 3 at least as close to x as to any other lattice point.)
Connected components Proposition 1 (Stelldinger, Köthe (2005)) √ Let X ⊂ R 3 be r -regular, with r > 1 2 d 3. Then there is a 1-1-correspondence between connected components of X and connected components of V L ( X ) , and between connected components of R 3 − X and R 3 − V L ( X ) . Key ingredients in the proof: 1 The distance between connected components of X is ≥ 2 r , 2 The distance between connected components of V L ( X ) is ≥ d , √ 3 ∂ V L ( X ) ⊂ { x ∈ R 3 | d ( x , ∂ X ) ≤ 1 2 d 3 } , and similar statements for complements. Addendum Constituent cubes of V L ( X ) in the same component of V L ( X ) can be joined by a sequence of face-adjacent cubes.
Configurations Consider a cube of side d with vertices points of the lattice L . What are the possible configurations of black vertices (those contained in X ) and white vertices (those contained in R 3 − X )? There are 2 8 = 256 possibilities; however, allowing rotation, reflection and complementarity (switching black and white) reduces this to 14. Seven of these correspond to V L ( X ) having a manifold point at the centre of the cube, seven to singularities there: dis- of xels e e (1) (2) (3) (4) (5) - (6) (7) (8) (9) (10) (11) (12) (13) (14)
Configurations and regularity 1 Configuration (8) can occur regardless of regularity; consider a saddle point at the centre of a lattice face, with parametrization ( x , y ) �→ ( x , y , cxy ) with repect to coordinates centred at that point and axes parallel to the lattice axes. In fact, this is the only singularity possible if X is r -regular with r > 1 . 5581214782 ... , but with less regularity, more singularities can appear.
Configurations and regularity 2 √ It turns out that with regularity 1 3 + ǫ , configurations (9) and 2 d (10) can also occur, (9) (10) but configurations (11)-(14) cannot. (11) (12) (13) (14)
Configurations and regularity 3 Configurations (8) and (9) can only occur, however, when paired with their own or the other’s complement: (a) (b) (c) Proving these claims requires some fairly subtle geometry; discussion is deferred for now.
Refining the reconstruction We refine the voxel reconstruction, constructing W L ( X ) by adding wedges to (or subtracting wedges from) configurations (8) and (9), and cutting corners in configuration (10).
The refined reconstruction √ Thus, when X is 1 2 d 3 + ǫ -regular, W L ( X ) is a closed submanifold in R 3 , whose components are in 1-1 correspondence with those of X ; similarly with components of their complements. It follows that the components of its boundary are in 1-1 correspondence with those of X , essentially by repeated application of the Jordan-Brouwer separation theorem. Also, if C is a component of ∂ X , then the corresponding component D of ∂ W L ( X ) is contained in the interior of N , the √ 3 + ǫ ′ -disc bundle of C , for any ǫ ′ ∈ ( 0 , ǫ ) . normal closed 1 2 d Main Theorem √ Suppose that X is compact and 1 3 + ǫ -regular. Then W L ( X ) is 2 d ambient isotopic to X .
Line fields 1 We begin by constructing a vector field transverse to D by interpolating lines. Intersect ∂ W L ( X ) with each lattice cube, and choose in a canonical way lines along the resulting curve transverse to the faces traversed: and interpolate to an appropriate canonically chosen line at the centre of the lattice cube.
Line fields 2
Lines and vector fields This produces a continuous field of lines on D such that the lines defined on D intersected with a configuration cube have intersection of length at least d with that cube, and (apart from configuration (10)) meet D just once in the cube. Doubling up, these lines have intersection with the 8-voxel cube surrounding the configuration cube of length at least 2 d , and (again apart from configuration (10)) meet D just once in this larger cube. This yields a continuous non-zero vector field ψ 1 on a small neighbourhood of D pointing along lines from white to black. There is also a continuous non-zero vector field ψ 2 on N pointing along normals from white to black.
Vector fields Using a partition of unity we construct a continuous vector field ψ agreeing with ψ 2 near D and with ψ 1 near the boundary of N . Indeed, ψ has no zeroes. For if ψ had a zero, then a normal line ℓ underlying ψ 2 would coincide with one of the lines underlying ψ 1 , but with opposite vector direction. The line ℓ meets N in a segment of length � ( 3 ) + 2 ǫ ′ ; but this segment contains the intersection of ℓ with d an 8-voxel cube surrounding an intersection of ℓ with D . This intersection has length 2 d . Contradiction! D can be approximated by a smooth surface ˜ D arbitrarily close to it, so that ψ 1 , so ψ , is transverse to ˜ D . ˜ D is isotopic to D . By a corollary to the Poincaré-Hopf theorem, C and ˜ D have the same Euler characteristic, so they are homeomorphic.
Isotopy Indeed, C and ˜ D are smoothly ambient isotopic in N , with the isotopy the identity near the boundary of N . If C and D are 2-spheres, this is essentially the 3-dimensional annulus theorem. For other orientable surfaces it follows from a result of Chazal and Cohen-Steiner from 2005. We note that N ≡ C × [ − 1 , 1 ] , with ˜ D embedded in C × ( − 1 , 1 ) and separating C × {− 1 } and C × { 1 } . Chazal and Cohen-Steiner’s argument identifies C × {− 1 } and C × { 1 } , giving C × S 1 . They argue that ˜ D is incompressible, and thus, by a result of Jaco and Shalen on Seifert manifolds, is isotopic to either a horizontal or a vertical surface. An intersection number argument rules out the second possibility, and a covering space argument shows the horizontal surface is isotopic to C × { 0 } . Extending such isotopies by the identity away from normal-bundle neighbourhoods of all boundary components of X gives an ambient isotopy between X and W L ( X ) .
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