Digital Geometry: Contour Analysis§

author:	David Coeurjolly

Contour representation§

Contour Encoding§

Objectives

Compression : compression ratio between the contour and its code
Reversibility : from the encoding, we could reconstruct the contour (loss-less compression)
Robustness : invariant with respect to a given class of transformations
Information capacity : can geometrical or topological information be deduced from the code

Freeman Chain Code§

Idea

Starting point + differential code according to adjacency displacement and an orientation

Example§

$P_0=(2,1),\, C=\{2,1,0,1,0,7,7,6,5,4,3,5,4,3\}$

Properties§

Reversible and unique encoding

Code driven geometrical transformations

Translation: just move $P_0$
Rotations by multiple of $\frac{\pi}{2}$ $\quad c'_i = c_i+2\, mod(8)$ (for (0)-adjacency)

Basic length estimator

$L += 1$ for even codes
$L +=\sqrt{2}$ for odd codes

Properties (Ctd.)§

Given a (1)-adjacency, let $n_i$ be the number of ith-code

Path is closed iff: $n_i = n_{i+2\,mod(4)}$

(but may be self-intersecting)

Representation as a word on an alphabet

$W=\{a,b,\bar{a},\bar{b}\}$

Closed path: words such that number of a equals number of $\bar{a}$ …
Let suppose that $\bar{(u\cdot v)} = \bar{v}\cdot \bar{u}$

A Polyomino $P$ tiles the plane if and only if its contours has the following structure (up to cyclic rotations) $X\cdot Y\cdot Z\cdot \bar{X}\cdot \bar{Y}\cdot \bar{Z}$ [Beauquier-Nivat]

Fundamendal Objets: DSS§

Digitization based definition§

Given a digitization model:

Def.

A digital straight line (resp. segment) is the result of a digitization and an Euclidean line (resp. segment)

First Historical Construction§

Digital Segment tracing by Bresenham’s algorithm

GIQ based definition: For each abscissa we consider the closest grid point

Incremental construction

Process the abscissa one by one
Propagate errors

Properties

The digitization is a (0)-arc (proof it)
If slope is in [0,1], only two codes occur

Implementation details§

Segment between $(x_1,y_1)$ and $(x_2,y_2)$ in $\mathbb{Z}^2$ with slope in [0,1]

Naive approach

y = y_s
for(i = x_1; i <= x_2; ++i)
   display(i,y);
   y_real = (y_2-y_1)/(x_2-x_1) * (i+1) + y_1;
   dy = y_real - y;
   if ( dy > 0.5)
     y++;

Mix between integers (i,x_s,x_e,y_prev) and real (y,dy) numbers
6 operations on real numbers + comparison

Integer only implementation§

e = x_2 - x_1
dx = 2*e
dy = 2*(y_2 - y_1)
while (x_1 < x_2)
  display(x_1,y_1);
  x_1 ++;
  e  -= dy;
  if (e <= 0)
     y_1 ++;
     e += dx:

Rationale

Error e can be propagated when moving to the next point (by an increment given by the slope)
Instead of computing the slope, we keep its numerator/denominator
To handle 0.5 numbers, we multiply everything by 2
$\Rightarrow$ inside the loop, 4 integer based operations and one comparison, no possible round-off errors

First arithmetical results§

Let us consider GIQ digitization scheme (similar to Bresenham’s) and a straight line $y=ax +b$ with $0\leq a \leq 1$ . Let $S$ be the digitization of the straight line and C its Freeman code

Thm

$\exists p,q\in\mathbb{Z}, a=\frac{p}{q} \Leftrightarrow\text{ C is periodic with minimal period } \frac{q}{gcd(p,q)}$

<demo>

Example§

Periodic structure $\rightarrow$ canonical pattern

Fast drawing algorithms (run-based, pattern-based, …)
Fast recognition algorithms
…

Euclid’s Axioms§

Any two points define a unique straight line

Two non-parallel straight lines intersects at a single point

	Single point
	Empty set
	Non-connected set
	Connected set

Euclid’s Axioms (bis)§

$\Rightarrow$ we need to redefine parallelism, intersections, …

Switching to another digitization scheme, we may avoid some situations.

E.g. with analytical models, empty set case will never occur

DSS Characterizations§

Chordal Property [Rosenfeld]

Def.

$\forall p \forall q\text{ and }\forall m\in[pq]\text{, there exists M(i,j) such that }\max(|i-x|,|j-y|)<1$

Evenness Property [Veelaer]

Def.

$\forall a,b,c,d\in[pq] \text{ such that }\vec{ba}_x=\vec{dc}_x,\, |\vec{ba}_y-\vec{dc}_y|=1$

Diophantine Equations§

Definition

Def.

Diophantine equation = Equation with integer parameters and solutions in $\mathbb{Z}$

Example

$ax + by = c$

(with $a,b,c\in\mathbb{Z}$ )

What is the general form of solutions of such linear diophantine equation ?

Solving Linear Diophantine Equation§

$ax + by = c$ , and let $S$ denote the solutions

Existence

$S\neq\emptyset \Leftrightarrow gcd(a,b) | c$

Homogeneous case

$ax+by=0 \Rightarrow S=\{(-bk,ak), k\in\mathbb{Z}\}$

General case

$a' = \frac{a}{gcd(a,b)},\, b'=\frac{b}{gcd(a,b)},\, c'=\frac{c}{gcd(a,b)}$

$a'x + b'y = c'$

Since $gcd(a',b')=1$ , $\exists u,v\in\mathbb{Z}$ such that $a'u +b'v = 1$ [Bezout]

$\Rightarrow S=\{(uc' - b'k, vc' + a'k), k\in\mathbb{Z}\}$

Your turn: 12x+15y=51

Analytical DSS§

Idea Design a analytical definition instead of combinatorial ones

Def.

$S\subset\mathbb{Z}^2$ is a analytical DSS with parameters $(a,b,\mu,\omega)\in\mathbb{Z}$ with $gcd(a,b)=1$ , if for all $(x,y)\in S$ , we have

$0 \leq ax -by + \mu < \omega$

$\omega$ acts as a thickness (arithmetical thickness, >1)
$\frac{a}{b}$ is the DSS slope
$\mu$ is the arithmetical intercept
S is the union of solutions of $\omega-1$ diophantine equations:

$S = \bigcup_{k=1..\omega-1} \{(x,y)\in\mathbb{Z}^2\,|\, ax -by = k - \mu\}$

Connectivity§

Thm.

$D(a,b,\mu,\omega)$

If $\omega < \max(|a|,|b|)$ , D is not a (k)-path
If $\omega = \max(|a|,|b|)$ , D is a (0)-arc
If $\omega = |a|+|b|$ , D is a (1)-arc
If $\max(|a|,|b|) < \omega < |a|+|b|$ , D is (*)-connected
If $\omega > |a|+|b|$ , D is thick


D(3,7,0,5)	D(3,7,0,7)	D(3,7,0,8)	D(3,7,0,10)	D(3,7,0,16)

Links between DSS and classical digitization scheme§

Given an Euclidean straight line $d:\, ax - by + \mu=0$ with $\{a,b,\mu\}\in\mathbb{Z}$ Without loss of generality, we suppose $b=|b|=\max(|a|,|b|)$

$D(a,b,\mu,b)$ corresponds to a kind of OBQ of d

$\{(x,y)\,|\, y= \left \lfloor \frac{-ax-\mu}{b} \right \rfloor \}$
$D(a,b,\mu+b-1,b)$ corresponds to a kind of BBQ of d

$\{(x,y)\,|\, y=\left \lceil \frac{-ax-\mu}{b} \right\rceil \}$
$D(a,b,\mu+[\frac{b}{2}],b)$ corresponds to the GIQ quantization of d

$\{(x,y)\,|\, y=\left [ \frac{-ax-\mu}{b} \right ] \}$
$D(a,b,\mu+[\frac{a+b}{2}],a+b+1)$ corresponds to the supercover digitization of d (analytical digitization with squares)

Periodicity§

Thm.

Given $D(a,b,\mu,\omega)$ , D is invariant by translation with vector $k.(b,a)^T$ ( $k\in\mathbb{Z}$ )

<proof>

Corollary

Coro.

Any sequence of $b$ pixels defines a DSS pattern, the pattern is minimal iff $gcd(a,b)=1$

Similar results than the one obtained by straight segment digitization but more generic

Toward stronger arithmetical results on DSS§

Farey Series§

Def.

The Farey Series $\mathcal{F}_m$ of order m is the ordered sequence of irreducible fractions with denominator less or equal to m

e.g.:: $\mathcal{F}_5=\left \{\frac{0}{1},\frac{1}{5},\frac{1}{4},\frac{1}{3},\frac{2}{5},\frac{1}{2},\frac{3}{5},\frac{2}{3},\frac{3}{4},\frac{4}{5},\frac{1}{1}\right \}$

Properties

if $\frac{h}{k}$ et $\frac{h'}{k'}$ are two successive fractions in $\mathcal{F}_m$ (with $\frac{h}{k}<\frac{h'}{k'}$ ), then $kh'-hk'=1$
if $\frac{h}{k}$ , $\frac{h''}{k''}$ and $\frac{h'}{k'}$ are three successive fractions in $\mathcal{F}_m$ (with $\frac{h}{k}<\frac{h''}{k''}<\frac{h'}{k'}$ ), then $\frac{h''}{k''}=\frac{h+h'}{k+k'}$ (median fraction)
$\mathcal{F}_{m+1}$ is given from $\mathcal{F}_m$ inserting median of successive fractions with denominator $\leq m+1$

$\mathcal{F}_6=\left \{\frac{0}{1},{\bf \frac{1}{6}},\frac{1}{5},\frac{1}{4},\frac{1}{3},\frac{2}{5},\frac{1}{2},\frac{3}{5},\frac{2}{3},\frac{3}{4},\frac{4}{5},{\bf \frac{5}{6}},\frac{1}{1}\right \}$

Stern-Brocot Tree§

Binary search tree on $\mathcal{F}_m$ fractions

Composition of DSS pattern§

Given

$D_1$ with slope $\frac{u_1}{v_1}$ and Freeman code of one of its pattern $C_1$
$D_2$ with slope $\frac{u_2}{v_2}$ and Freeman code of one of its pattern $C_2$
$\frac{u_1}{v_1}< \frac{u_2}{v_2}$
and $u_2v_1 - u_1v_2 = 1$ (unimodularity)

Then

$\Rightarrow$ $C_1C_2$ is the Freeman code of one period of the DSS with slope $\frac{u_1+u_2}{v_1+v_2}$

All patterns: applications to DSS pattern intersection§

Object Recognition§

Recognition Algorithm§

Idea

Given a digital set $S$ , and a geometrical primitive (DSS, Digital circle, digital plane, …), decide if $S$ is a subset of such primitive

Detection Yes/No answer
Recognition a valid parametrization of the primitive, the complete set of primitives containing $S$ -> preimage

Useful to

Perform contour polygonalization/decomposition into primitives
Define multigrid convergent differential estimators

Geometrical Recognition (1)§

Idea

Revert the digitization process and solve a dual problem

DSS: Linear dual space

Geometrical Recognition (2)§

Let us consider the following OBQ digitization scheme of $l: ax - by + \mu=0$

Each pixel $(x_0,y_0)$ contributes to two linear constraints:

$0 \leq \alpha x_0 - y_0 + \beta < 1$

Geometrical Recognition (3)§

Algorithm principle

Given a set S of pixels

Start with a domain in the dual space $\bar{S}$ (i.e. we consider DSS in the first octant)
For each pixel, compute the intersection between $\bar{S}$ and the two associated constraints

$\Rightarrow$ if $\bar{S}$ is empty, S is not a DSS (for OBQ scheme)

$\bar{S}$ is called the preimage of S

Example§

$a=(2,2)$ , $D_a: 2\alpha - 2+\beta=0$ and $D'_a: 2\alpha - 2+\beta=1$

Example§

$b=(4,3)$ , $D_b: 4\alpha - 3+\beta=0$ and $D'_b: 4\alpha - 3+\beta=1$

Example§

$c=(8,4)$ , $D_c: 8\alpha - 4+\beta=0$ and $D'_c: 8\alpha - 4+\beta=1$

Example§

$d=(9,5)$ , $D_d: 9\alpha - 5+\beta=0$ and $D'_d: 9\alpha - 5+\beta=1$

Example§

$p=\left(\frac{1}{3},2\right)$ in the dual space leads to $l: \frac{1}{3}x -y +2 = 0$

Computational cost§

Preimage = Classical linear programming problem

We only have linear constraints
$\bar{S}$ is convex
on-line recognition algorithm in $O(n\log(n))$ to maintain $\bar{S}$ ( $O(\log(n))$ per constraint)
Detection in $O(n)$

If $S$ is a (0)-path

Thm. $\bar{S}$ has at most 4 vertices in the dual space

$\Rightarrow$ On-line DSS recognition in $O(n)$ (O(1) per constraint)

Rationale of the preimage structure§

Farey Fan

The Farey Fan of order $n$ is a decomposition of the space into cells where each cell corresponds to a DSS preimage of length $n+1$


Order 2	Order 3	Order 6

DSS and Farey Fan Cell§

Arithmetical Recognition§

Idea

Use the analytical representation of DSS:

$D(a,b,\mu)$ a DSS in the first octant
the reminder of a pixel $(x,y)$ is the quantity

$r = ax- by$
the k-th net is defined by the solutions of

$ax -by =k$
the DSS is the union of nets with $k\in\{\mu,\ldots,\mu + b - 1\}$

Definitions (again)

Defs.

$(x,y)$ is exterior to D if its reminder is lesser than $\mu-1$ or greater than $\mu+b$
$(x,y)$ is weakly exterior to D if its reminder is equal to $\mu-1$ or $\mu+b$
The net with order $\mu$ is the upper leaning net
The net with order $\mu+b-1$ is the lower leaning net

Main Recognition Theorem§

Let U (resp. U’) be the upper leaning point with minimal abscissa (maximal abscissa) and L (resp. L’) be the lower leaning point with minimal abscissa (maximal abscissa)

Given a DSS digital set S w and a point M (with reminder r), we have to decide if $S\cup\{M\}$ is still a DSS

Thm.

if $\mu \leq r < \mu+b$ then $S\cup\{M\}$ is a DSS with parameter $D(a,b,\mu)$
if M is exterior, $S\cup\{M\}$ cannot be a DSS
if M is weakly exterior with $r=\mu+1$ , $S\cup\{M\}$ is a DSS where the slope is given by $\vec{UM}$
if M is weakly exterior with $r=\mu+b$ , $S\cup\{M\}$ is a DSS where the slope is given by $\vec{LM}$

At each step, we maintain/update U,U’,L,L’ and DSS parameters a,b, $\mu$

Analysis§

Computational cost

$O(n)$ algorithm with $O(1)$ per pixel
possible to design dynamic bidirectionnal algorithm (we can add/remove points on both sides) with same complexity

Key point of the proof: unimodular vectors

Given two vectors $\vec{u},\vec{v}\in\mathbb{Z}^2$ , $\vec{u}$ and $\vec{v}$ are unimodular iff $det(u,v) \pm 1$

E.g. $\vec{UU'}$ and $\vec{UM}$ are unimodular if $M$ is weakly superior

(if vectors define fractions, fractions are neighbors in a given Farey series)

Unimodular means that there is no integer point in the parallelogram ( $\vec{u},\vec{v}$ )

Digital Plane§

Similar approaches

Definition as the result of a digitization process
Analytical definition: $P(a,b,c,\mu,\omega): \mu \leq ax + by + cz \leq \mu + \omega$ with similar topological results according to $\omega$ , with similar arithmetical structure (leaning points, unimodular vectors, …)


P(6,13,27,0,15)	P(6,13,27,0,27)	P(6,13,27,0,46)

General Approach for Recognition Algorithm Design§

Digitization -> Constraints

Given the primitive (plan, circle, …), digitization usually implies inequality system we have to solve

Usually:

Coefficients in the inequalities are usually integers
Easier if we can linearize the system (even in higher dimensions, e.g. 3 for circles)

Regularity/arithmetic in analytical representation

Leads to more efficient algorithms
Difficult for high order primitives or in higher dimension

Design the recognition as a separation problem with tools from Computational Geometry

E.g. disc recognition as a separability problem with Euclidean arc between interior/exterior point sets.

In practice: a kind of mix between all these approaches…

Contour Polygonalization§

Contour Decomposition into maximal DSS§

Idea

Starting from a contour point (and given a direction), decompose the contour into maximal DSS adding pixels one by one

$\Rightarrow$ O(n) algorithm

$\Rightarrow$ Changing the starting point, decompositions differs by one (N, N+1)

In 3D…§

Similar approach but

No trivial order
Heuristic to scatter seeds for the incremental recognition
Parallel or sequential process

Usually

Geodesic one to generate as anisotropic as possible DPS patches
Using curvature information in parallel mode

Back in 2D: Polygonalization problem§

Principle

Convert a digital contour into a polygon such that its digitization is the input digital set

First approach: use DSS decomposition

Indeed, DSS segments are “reversible” … but not the vertices..

$\Rightarrow$ We need to constraint the preimages to ensure that successive DSSs have euclidean representative segment with “reversible” intersection vertex

Example of a solution in 2D§

Reversible reconstruction in dimension 3§

More difficult because of topological issues.

Idea: start from a topologically valid reversible triangulation (Marching Cubes), merge triangles using digital plane decomposition information and constraints on preimage

(Marching Cubes)§

Reversible polyhedron
Topologically correct (combinatorial 2-manifold)
… but many facets

Digital Geometry: Contour Analysis§

Contour representation§

Contour Encoding§

Freeman Chain Code§

Example§

Properties§

Properties (Ctd.)§

Fundamendal Objets: DSS§

Digitization based definition§

First Historical Construction§

Implementation details§

Integer only implementation§

First arithmetical results§

Example§

Euclid’s Axioms§

Euclid’s Axioms (bis)§

DSS Characterizations§

Diophantine Equations§

Solving Linear Diophantine Equation§

Analytical DSS§

Illustration§

Illustration§

Illustration§

Illustration§

Illustration§

Connectivity§

Links between DSS and classical digitization scheme§

Periodicity§

Toward stronger arithmetical results on DSS§

Farey Series§

Stern-Brocot Tree§

Composition of DSS pattern§

Illustration§

All patterns: applications to DSS pattern intersection§

All patterns: applications to DSS pattern intersection§

All patterns: applications to DSS pattern intersection§

All patterns: applications to DSS pattern intersection§

All patterns: applications to DSS pattern intersection§

All patterns: applications to DSS pattern intersection§

All patterns: applications to DSS pattern intersection§

All patterns: applications to DSS pattern intersection§

All patterns: applications to DSS pattern intersection§

All patterns: applications to DSS pattern intersection§

Object Recognition§

Recognition Algorithm§

Geometrical Recognition (1)§

Geometrical Recognition (2)§

Geometrical Recognition (3)§

Example§

Example§

Example§

Example§

Example§

Computational cost§

Algorithm§

Rationale of the preimage structure§

DSS and Farey Fan Cell§

Arithmetical Recognition§

Illustration§

Main Recognition Theorem§

Theorem cases§

Algorithm§

Analysis§

Example§

Digital Plane§

General Approach for Recognition Algorithm Design§

Contour Polygonalization§

Contour Decomposition into maximal DSS§

In 3D…§

Back in 2D: Polygonalization problem§

Example of a solution in 2D§

Example of a solution in 2D§

Example of a solution in 2D§

Example of a solution in 2D§

Example of a solution in 2D§

Reversible reconstruction in dimension 3§

(Marching Cubes)§