Digital Geometry: Digital Model and Elementary Digital Topology§

author:	David Coeurjolly

Principles§

Idea

Take benefit from the regular structure of the lattice to enhance geometrical analysis of shapes

Integer based computations
Combinatorial algorithms
high performance algorithms
…

What do we need ?§

Grid/Lattice to define our cells
Digital Topology to add, at least, a combinatorial structure on cells
Digitization schemes to fill the gap between Euclidean/Digital worlds
Fundamental objects (straight lines, circles, planes, …)
Axioms
Specific Algorithms

Lattice and Tilings§

Requirements

Simplicity of the structure (storage, …)
Pixels access (coordinate system)
Neighbors pixel access
Consistency w.r.t. acquisition devices or processing pipeline
Capacity (Information theory)

More formally

Packing Density: fraction of the volume filled by a given collection of solids. Given a large number of equal Euclidean balls centered in the grid points, this quantity say how efficiently they are packed
Kissing Number: the number of neighboring balls that each ball touch
Covering: the least dense way to cover the space with equal overlapping balls

Example in dimension 2§

Definitions§

Lattice

Given a basis $(v_1,\ldots,v_n)$ of $\mathbb{R}^n$ ,

$\Lambda= \left \{ \sum_{i=1}^n a_iv_i \,|\, a_i \in \mathbb{Z} \right \}$

(finitely-generated free abelian group, symmetry group, …)

Five fundamental lattices in the Euclidean plane

Paving§

Definition (dual lattice structure): The cell of a lattice point $\lambda\in\Lambda$ is its Voronoi cell

Following fundamental lattice classification: pavings by squares, hexagons, triangles, rhombi and parallelograms.
By definition, the paving induced by a lattice is periodic
Triangular/Hexagonal lattice/paving are dual

Speaking of density packing/kissing number and covering, hexagonal lattice is optimal

Dimension 3§

Regular cubic grid

$V=\left ( \begin{array}{ccc} 1 &0 &0\\ 0 & 1 &0\\ 0 & 0 & 1 \end{array} \right )$

Face-centered cubic grid

$V_{fcc}=\left ( \begin{array}{ccc} 1 &1 &0\\ 1 & 0 &1\\ 0 & 1 & 1 \end{array} \right )$

Body-centered cubic grid

$V_{bcc}=\left ( \begin{array}{ccc} 1 &1 & -1\\ 1 & -1 &1\\ -1 & 1 & 1 \end{array} \right )$

Dimension 3 (bis)§

BCC has optimal covering
FCC has highest packing density and largest kissing number
FCC and BCC are dual

$V_{fcc}^{-1} = V_{bcc}^T$

Hexagonal grid

$V_{hexa}=\left ( \begin{array}{ccc} \sqrt{3}/2 & 0 \\ 1/2 &1 \end{array} \right )$

Semi-regular pavings§

Definition

Several convex tiles paving the plane
No T-junctions
Same configuration of tiles around each vertex

Lattice Encoding§

Square lattice

Lattice points: $\mathbb{Z}^2$ , direct access to direct neighbors $(i\pm 1, j)$ , $(i,j\pm 1)$

Triangular grid

Lattice points: special encoding of lattice points to $\mathbb{Z}^2$ , three direct neighbors (two local configurations when mapped to $\mathbb{Z}^2$ )

Hexagonal grid

Lattice points: special encoding of lattice points to $\mathbb{Z}^2$ , six direct neighbors (two local configurations when mapped to $\mathbb{Z}^2$ )

Lattice Encoding (ctd.)§

Cubic grid

Trivial

FCC/BCC grid

$(x,y,z) \text{ is generated by } V_{fcc} \Leftrightarrow (x,y,z)\in \mathbb{Z}^3, x+y+z=0\, (mod\, 2)$

$(x,y,z) \text{ is generated by } V_{bcc} \Leftrightarrow (x,y,z)\in \mathbb{Z}^3, x=y=z\, (mod\, 2)$

Elongated grids

E.g. square/cubic grid with different resolutions
$\Rightarrow$ Remapping $\mathbb{Z}^3\rightarrow \mathbb{R}^3$ on-the-fly

Fundamental Topology Elements§

Adjacency relationships in the square/cubic grid§

Combinatorial approach

In 2D:

4-neighbors: pixels sharing an edge $(i\pm 1, j)$ and $(i,j\pm 1)$
8-neighbors: pixels sharing an edge or a vertex $(i\pm 1, j\pm 1)$

In 3D:

6-neighbors: voxels sharing a face
18-neighbors: voxels sharing a face or an edge
26-neighbors: voxels sharing a face, an edge or a vertex

Topological approach

Two pixels/voxels are (k)-adjacent is the intersection of their (closed) cell is of dimension $k$

Mixing all dimensions:

4-neighbor, 18-neighbor $\equiv$ (1)-adjacent
8-neighbor, 26-neighbor $\equiv$ (0)-adjacent
6-neighbor $\equiv$ (2)-adjacent

More definitions§

(k)-path

A sequence of digital points $:\{p_i\}_{i=0\ldots n}$ is a (k)-path if for each point, $p_i$ is (k)-adjacent to $p_{i-1}$ (except for $i=0$ )

(k)-arc

A (k)-arc is a (k)-path such that each $p_i$ has exactly two (k)-adjacent neighbors (except for extremities)

(k)-curve

A (k)-curve is a (k)-arc such that $p_0 = p_n$

(k)-object

A set S of digital point is a (k)-object iff for any pair of points, there exists a (k)-path in S

Illustrations (ctd.)§

Can you spot (k)-arcs/(k)-objects/(k)-curves for $k=\{0,1\}$ ?

Toward a definition of contour§

Objective: define a notion of object contour/boundary matching with Jordan theory

$C$ is a Jordan curve (or simple closed curve) in $\mathbb{R}^2$ if $C$ is the image of a continuous and injective map from the circle to $\mathbb{R}^2$

Jordan theorem states that:

$\mathbb{R}^2\setminus C$ has two connected components, one is bounded (aka interior) and the other one is unbounded (exterior)
Each continuous path $\pi$ from an interior point to an exterior one crosses $C$ (with an odd number of intersections)
Implies an orientation of $C$

Idea mimic a digital version of Jordan framework replacing $C$ by a (k)-curve ?

Digital paradox§

Given the following (0)- and (1)-curves, do they define Jordan-like curve ?

$\Rightarrow$ It depends…. we need a pair of adjacency relationships !

Example: Filling holes§

Adjacency pairs§

$(\kappa,\lambda)$ Jordan pair such that k is the adjacency for the object and l the adjacency for the complementary

In dimension 2

(0,1) and (1,0)

In dimension 3

(2,1), (2,0) (1,2) and (0,2)

Border definition§

Border: Given a $(\kappa,\lambda)$ Jordan pair, the border of $X$ is the set of $(\kappa)$ -adjacent digital points which are $(\lambda)$ -adjacent to points in $X^C$

The result is a $(\kappa)$ -object but we need more constraints to have a $(\kappa)$ -curve
Hard to generalize to 3D
We do not have $\partial X= X\setminus X^C$ (or kind of, both are considered as open sets)

_images/bubble-object-color-borders-48.png

Cellular grid space and topology§

Idea embed the digital space $\mathbb{Z}^n$ into a cellular space (cartesian cubic space) to represent oriented inter-pixel elements

In 2D

0-cells are called pointels (element of dimension 0), by convention, Digital points in $\mathbb{Z}^n$ are embedded into 0-cells
1-cells are called linels (element of dimension 1)
2-cells are called pixels
each cell can be signed

In nD

0-, … , n-cells
by convention: n-cells are called spels and (n-1)-cells are called surfels

(two 0-cells, two 2-cells and four 1-cells)

Digital Surface§

Principle defines digital surface as a set of (n-1)-cells (surfels)

Let us consider a $\beta$ relationship on surfels (anti-reflexive, local transitive closure, locally defined)
Given a $(\kappa,\lambda)$ Jordan adjacency pair, $(\kappa,\lambda,\beta)$ triplet is a Jordan triplet

$\beta$ is anti-reflexive

$\beta$ is a relationship on surfels

$\beta$ can extract all surfels (informally)

We can demonstrate that such Jordan triplets leads to well-defined digital Jordan surface

Illustration in 2D (here, k=1)

_images/digital-surface-IntAdjacency.png

Approach is valid for various digital structures

$\beta$ in 3D§

Two valid $\beta$ relationships on (2,1)- or (2,0)- pairs on closed objects

_images/digital-surface-SurfaceTracking2.png

_images/digital-surface-SurfaceTracking.png

$\beta$ relationship + graph traversal (depth first, breadth first,…) $\Rightarrow$ digital surface tracker

Efficiency of the tracker is guided by the `beta`:math: complexity

Generic Breadth-first tracker§

For (2)-object

$\beta$ is not oriented, each surfel is processed 4 times
$\beta^*$ with oriented arcs (2 arcs per surfel), each surfel is processed 2 times
$\beta^{*+}$ with oriented arcs (1 or 2 arcs per surfels), each surfel is processed 4/3 times (on average)
The graph is 4-regular
$\Rightarrow$ Hamiltonian path exists if $X$ is homeomorphic to a ball

Digital Surface Extraction§

Overall algorithm (for single connected surface)

Scan the image until we find a first surfel
Apply the tracker to traverse all surfels

Complex Objects

Several connected components, holes, …

$\Rightarrow$ Scan the complete volume, mark all surfels as potential starting surfels and apply the tracker on each starting surfel (removing traversed surfels)

Digitization schemes§

Main ideas§

Formalize the embedding $\mathbb{R}^n\rightarrow\mathbb{Z}^n$

To be able to generate digital objects by digitization of Euclidean ones
Important if we want to propagate properties (roughly, the digitization of a Jordan curve should be a $(\kappa,\lambda)$ - Jordan for some grid steps)
When we define a differential estimator (e.g. curvature) on digital contour, allows us to design multigrid convergent estimators
…

Gauss model§

Let $\mathcal{X}\subset \mathbb{R}^n$ and $X$ its digitization

$X = \mathcal{X}\cap\mathbb{Z}^n$

Easy to implement
Mostly used on objects with 2-measure $\neq$ 0 but even in this case, $X$ may be empty
By definition $X \subseteq \mathcal{X}$
We need more constraints on $\mathcal{X}$ to ensure topological properties on $X$ or $\partial X$

Gauss models (bis)§

This model was first used to approximate

$\int_\mathcal{X} ds$

$|X|$

Grid intersection models§

Idea Defined for oriented contours $\partial\mathcal{X}$

For each intersection $\partial\mathcal{X}$ with a grid edge, we select the {closer,inner,outer} grid point

(resp. GIQ -Grid Intersect Quantization-, OBQ -Object Boundary Quantization-, BBQ -Background Boundary Quantization-)

We construct (0)-objects which could also be (0)-arcs or (0)-curves
Less generic than Gauss model
bubbles can appear on ties (when $\partial\mathcal{X}$ crosses the edge at its mid-point) $\Rightarrow$ global Oracle to remove ties
Distance properties between $X$ and $\partial\mathcal{X}$

Analytic models§

Generic definition

Let $d_*$ be a metric, $B$ its unit ball and $M=\check{B}$

$\mathbb{A}(\mathcal{X})&=\{p\in\mathbb{Z}^n~|~ B(p)\cap \mathcal{X} \neq \emptyset\} \\&=\{p\in\mathbb{Z}^n~|~ d_*(p,\mathcal{X})\leq \frac{1}{2} \} \\&=(\mathcal{X}\oplus M)\cap\mathbb{Z}^n$

Still bubbles may exist

Properties§

Following the definition (F,G $\in \mathbb{R}^n$ ):

prop.

$\mathbb{A}(F\cup G)= \mathbb{A}(F) \cup \mathbb{A}(G)$

$\mathbb{A}(F)= \bigcup_{p\in F} \mathbb{A}(p)$

$\mathbb{A}(F\cap G)\subseteq \mathbb{A}(F) \cap \mathbb{A}(G)$

$\text{if } F\subseteq G\text{ then } \mathbb{A}(F) \subseteq \mathbb{A}(G)$

$\Rightarrow$ Allows modeling of digital objects but CSG approach (Constructive Solid Geometry)

Interesting theoretical tool: multigrid digitization§

Idea Digitization parametrized by a grid step

E.g. for Gauss digitization

$X_h = \left (\frac{1}{h}\cdot \mathcal{X}\right )\cap \mathbb{Z}^n$

Mathematical results can be obtained with constraints on $\mathcal{X}$ , for example

thm.

If $\partial \mathcal{X}$ is $C^2$ with bounded curvature, there exists a grid step $h_0$ such that for $0<h<h_0$ , $X_h$ is topologically equivalent to $\mathcal{X}$

thm.

If $\partial \mathcal{X}$ is $C^2$ with bounded curvature, the retro-projection from $\hat{x}\in\partial X_h$ onto $\partial \mathcal{X}$ at $x$ along its normal direction is continuous, mono-valuated and surjective (for $0<h<h_0$ ) and $\| \hat{x} -x\|_\infty \le h$ )

Another example§

Question 1 Given a digital object, How to estimate its areas ?

Answer Well, let’s count the number of grid points (unit squares) (estimator denoted E)

Question 2 Is this estimator multigrid convergent ? What is the convergence speed ?

Answer

Let’s consider the estimator $E_h$ at grid-step h defined on the digitization of the Euclidean shape $\mathcal{X}$ from a given class of shapes
If $\mathcal{X}$ is a finite convex shape, there exists a grid step $h_0$ such that for $0<h<h_0$ we have:

$| E_h( X_h ) - \mu(\mathcal{X}) | \leq O(h)$

[Gauss, Dirichlet]

If $\mathcal{X}$ is $C^3$ (or finitely piece-wise $C^3$ with positive curvature almost everywhere…) then

$| E_h( X_h ) - \mu(\mathcal{X}) | \leq O(h^{\frac{15}{11}+\epsilon})$

[Huxley,…]

Would there be better approaches ?

Digital Geometry: Digital Model and Elementary Digital Topology§

Digital Model§

Principles§

Approach§

What do we need ?§

Lattice and Tilings§

Example in dimension 2§

Definitions§

Paving§

Dimension 3§

Dimension 3 (bis)§

Semi-regular pavings§

Lattice Encoding§

Lattice Encoding (ctd.)§

Fundamental Topology Elements§

Adjacency relationships in the square/cubic grid§

More definitions§

Illustrations§

Illustrations (ctd.)§

Toward a definition of contour§

Digital paradox§

Example: Filling holes§

Adjacency pairs§

Border definition§

Cellular grid space and topology§

Digital Surface§

$\beta$ in 3D§

Generic Breadth-first tracker§

Digital Surface Extraction§

Digitization schemes§

Main ideas§

Gauss model§

Gauss models (bis)§

Grid intersection models§

Analytic models§

Properties§

Interesting theoretical tool: multigrid digitization§

Another example§