Digital Geometry: Estimators§

author:	David Coeurjolly

Introduction§

Objectives

Define estimators to compute digital version of quantities defined in Continuous space
Demonstrate the stability of the proposed estimators
Prove their convergence
Check if estimated quantities are consistent within the digital space: regions with positive curvature match with a digital definition of convexity

e.g.

Length, surface area, …
Tangents / normal vectors
Curvature, …
Area / Volume, geometrical moments
…

Algorithmic point of view

Evaluate/compare estimators also in terms of complexity/efficiency

Mathematical Context§

Multigrid analysis Gauss digitization scheme parametrized by a grid-step

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

Definition§

Idea

Single scalar quantity attached to a digital object $Dig(X,h)$
E.g.:
- Area in 2D (resp. 3D)
- Geometrical moments:
  
  $m_{pq}(\mathcal{X}) = \int_\mathcal{X} x^py^q dx dy$

Multigrid convergence definition

Def.

A discrete geometric estimator $\tilde{E}$ of some geometric quantity $E$ is multigrid convergent for a family of shapes $\mathbb{X}$ and a digitization process $Dig$ iff for all shape $X \in \mathbb{X}$ , there exists a grid step $h_X>0$ such that the estimate $\tilde{E}(Dig(X,h),h)$ is defined for all $0<h < h_X$ and: $| \tilde{E}(Dig(X,h),h) - E(X) | \le \tau_X(h)$

where $\tau_X : \mathbb{R}^+ \rightarrow \mathbb{R}^{+}$ with null limit at $0$ . This function is the speed of convergence of the estimator.

Area/Volume§

From previous lectures…

If $\tilde{E}(Dig(X,h)) = h^2|Dig(X,h)|$ , this estimator converges for convex shapes $\mathbb{X}$ with speed $O(h)$ [Gauss, Dirichlet]
If $\tilde{E}(Dig(X,h)) = h^2|Dig(X,h)|$ , this estimator converges for $C^3$ convex shapes $\mathbb{X}$ with speed $O(h^{\frac{15}{11}+\epsilon})$ [Huxley]

Quantity	1	0.1	0.01	0.001	…
h	1	0.1	0.01	0.001	…
$h^{\frac{15}{11}}$	1	0.04328	0.00187	0.00008	…

Step-based Length Estimation§

Idea

We specify:

A set of elementary displacement or pattern (e.g. $\rightarrow$ and $\nearrow$ )
A weight per displacement vector
Length Estimation sum of weighed occurrences of each pattern

$\Rightarrow$ statistical analysis to optimize the weights to minimize errors for random distribution of segments of length $n$

BLUE (Best Linear Unbiaised Estimator): $\tilde{E}(Dig(X,h))= h( 0.948n_{\rightarrow} + 1.343n_{\nearrow})$

Step-based Perimeter Estimation (bis)§

Generalizarion

We decompose the $\partial Dig(X,h)$ into pattern of length $m$

$\partial Dig(Dig(X,h)) = w_1w_2\ldots w_n\lambda$
For each pattern $w_i$ , we consider a weight $p(\cdot)$
The estimator is thus

$\tilde{E}(Dig(X,h))= h \sum_{i=1}^n p(w_i)$

Main Result

Thm.

$\forall\,n$ and $\forall \, p(\cdot)$ the set of slopes $\alpha$ such that the estimator is convergent is countable $\Rightarrow$ most of the time, the estimator does not converge

[Tajine,Daurat]

Solution locally adapt the parameter m ? set m as a function of h ? ( $\rightarrow DSS$ )

DSS Based Perimeter Estimation§

Basic Idea

Compute the decompostion of the contour $\partial Dig(X,h)$ into maximal DSS $\{D_i\}_N$ (with thus $\bigcup_N D_i = \partial Dig(X,h)$ )
Estimate the length by:

$\tilde{E}_{DSS}(Dig(X,h))= h\cdot \sum_{i=1}^n d(first_{D_i}, first_{D_{(i+1)\, mod\, N}} )$

Main result

Thm.

$\tilde{E}_{DSS}(Dig(X,h))$ is multigrid convergent for convex shapes with speed $4.5h$

Experimental evaluation§

We need

A family of analytical shapes on which $E$ is defined
A multigrid digitization scheme

Useful tool: Maximal Segment Covering§

Principle

Let $S_{i,j}$ defines a piece of the contour from index $i$ to $j$
Given a property $P:\{ S_{i,j}\,|\,i,j\in \partial Dig(X,h) \} \rightarrow \{0,1\}$ (e.g. being a DSS, a DCA,…)
P is incremental if
- $\forall i \in \partial Dig(X,h),\, P(S_{i,i})$
- $\forall i,j \in \partial Dig(X,h),\, P(S_{i,j})\leq P(S_{next(i),j})$
- $\forall i,j \in \partial Dig(X,h),\, P(S_{i,j})\leq P(S_{i,prev(j)})$
$S$ is maximal if $\forall S',\, S\subset S'\, \neg P(S')$

Maximal covering = set of all maximal segment of $\partial Dig(X,h)$

Algorithmic point of view

If the property is dynamic (insertion/dilation at each extremity) with $O(p)$ per operations $\Rightarrow\, O(p\cdot n)$
DSS is dynamic $O(1)$
DSS with thickness parameter is dynamic $O(\log n)$

Illustration§



X	$Dig(X,1)$	$Dig(X,\frac{1}{2})$	$Dig(X,\frac{1}{4})$

Useful to

Define local differential estimators
Detect local concavity/convexity

From normal vector local estimator to length/surface area estimator§

Idea digital version of

$l(\gamma) =\int_\gamma n(s)\cdot ds$

Hence:

$\tilde{E}_{norm}(Dig(X,h)) = h\cdot \sum_{s\in \partial Dig(X,h)} \tilde{n}(s)\cdot\vec{n}_{elem}(s)$

Main result: If $\tilde{n}$ uniformly converges in $O(h^\alpha)$ , $\tilde{E}_{norm}(Dig(X,h))$ converges in $O(h^\alpha)$

Definition of multigrid convergence§

Multigrid convergence for local geometric quantities

Def.

The estimator $\tilde{Q}$ is multigrid-convergent for the family $\mathbb{X}$ if and only if, for any $X \in \mathbb{X}$ , there exists a grid step $h_X>0$ such that the estimate $\tilde{Q}(Dig(X,h),y,h)$ is defined for all $y \in \partial{Dig(X,h)}$ with $0<h < h_X$ , and for any $x \in \partial{X}$ ,

$\forall y \in \partial{Dig(X,h)}$

with

$\| y - x \|_\infty \le h, \quad \left | \tilde{Q}(Dig(X,h),y,h) - Q(X,x) \right | \le \tau_{X,x}(h),$

where $\tau_{X,x}: \mathbb{R}^{+*} \rightarrow \mathbb{R}^+$ has null limit at 0. This function defines the speed of convergence of $\tilde{Q}$ toward $Q$ at point x of $\partial{X}$ . The convergence is uniform for $X$ when every $\tau_{X,x}$ is bounded from above by a function $\tau_X$ independent of $x \in \partial{X}$ with null limit at 0

$\Rightarrow$ we need a mapping $\partial X\rightarrow\partial Dig(X,h)$

$\Rightarrow$ Uniform convergence is a strong constraint

Normal/Tangent estimation§

Generic fitting approach

Fix a neighborhood $m$ around a point $p\in \partial Dig(X,h)$
Fit the $2m+1$ digital points $\{p_i(x_i,y_i)\}$ by a function $f(x_i,\vec{\beta})$ with parameter vector $\vec{\beta}$
Least-square fitting : Minimize quadratic error:

$S = \sum_i^{2m+1} (y_i - f(x_i,\vec\beta)^2$

E.g., $f(x_i,\vec\beta) = \beta_0 + \beta_1 x$ for linear fitting.

Example: tangent vector estimator

$\tilde{Q}_{tan-fitting}(Dig(X,h), p, m) = \beta_1$ where $\beta_1$ is the result of a least-square linear fitting

$\Rightarrow$ No convergence results for fixed $m$

DSS based Normal vector§

Trivial idea Use a kind of symmetric maximal DSS to estimate the tangent

Algorithmic

Since DSS recognition is O(1)-dynamic, all tangents can be obtained in O(n)

Maximal Segment Analysis§

More flexible approach Maximal segment from mxaimal covering

Given a maximal DSS with slopde $\alpha_h\in[0,1]$ containing point $P\in\partial Dig(X,h)$
We consider a local parametrization of $\partial X = \gamma(s) = (x,f(x))$
Suppose P=(0,0), we have, $\forall x\in[0,l]$

$\alpha_hx- 2h \leq f(x) \leq \alpha_hx + 2h\, \Leftrightarrow\, \alpha_h=\frac{f(x) \pm O(h)}{x}$

Maximal Segment Analysis (bis)§

If the curve is locally linear, $f(l) = f'(0)l$ , and

$\alpha_h=f'(0) + \frac{O(h)}{l}$

If the curve has curvature greater than $k_{min}>0$ , Taylor decomposition gives us:

$\alpha_h = f'(0) + O(l) + \frac{O(h)}{l}$

Convergence Result

Prop.

$lim_{h\rightarrow 0} \alpha_h = f'(0)\,\Leftrightarrow\, \Omega(h^a)\leq l \leq O(h^b)$

with $0 < b \leq a < 1$

$\Rightarrow$ Length of maximal DSS is crucial !

Curvature Estimation§

Fitting an order-2 polyonmial

No convergence for fixed $m$ value

Chord length approach

Idea: estimate the curvature from digital tangent length $\tilde{k} = \frac{1}{\tilde{r}}$ with $\tilde{r} = \frac{l^2}{8h} + \frac{h}{2}$
The neighborhood is locally adapted but convergence if $l = \Theta(h^{-1/2})$ (see below)
Noise sensitive

Circumscribing circle from two half-tangent

Idea: locally, estimate left/right half-tangents (resp. OP and PQ) and estimate $\tilde{r}=h\left(\frac{|OP| |PQ| |QO|}{4\mathcal{A}_{OPQ}}\right)$
The neighborhood is locally adapted but convergence if $l = \Theta(h^{-1/2})$ (see below)
Noise sensitive

Nice but $l$ is not in $\Theta(h^{-1/2})$

Maximal Segment Again§

We need to consider the following quantities

$n_e$ the number of edges of the convex hull of $Dig(X,h)$
$n_{MS}$ the number of maximal segments in the covering of $\partial Dig(X,h)$
$l_e$ length of convex hull edge $e$ ( $l_1$ metric for (1)-contours)
$l_{MS}$ length of a maximal segment ( $l_1$ metric for (1)-contours)

Then, we want to compute:

sup of $l_{MS}$
inf of $l_{MS}$
mean $l_{MS}$ value

Everything as functions of h and considering specific shape family $\mathbb{X}$

Step 1: Digital Convex Hull§

Convex hull of X smallest convex set containing point set X. As consequence of the Def, the convex hull is polygonal convex set with vertices in X

Main result in Lattice polytope in 2D [Barany, Zunic, Balog, Acketa,…]

Thm.

$c_1(X) h^{-\frac{2}{3}} \leq n_e \leq c_2(X)h^{-\frac{2}{3}}$

$\Rightarrow n_e = \Theta\left(h^{-\frac{2}{3}}\right)$

(Similar results in n-D exist)

Step 2: Number of maximal segments§

Main result [Lachaud, de Vieilleville, Feschet]

If $\partial X$ is convex and $C^3$

$\frac{n_e}{\Theta(\log(h^{-1}))} \leq n_{MS} \leq 3 n_e$

Hence: $\Theta\left (\frac{h^{-\frac{2}{3}}}{\log(h^{-1})}\right ) \leq n_{MS} \leq \Theta(h^{-\frac{2}{3}})$


$n_e=16\,, n_{MS}=24$	$n_e=24\,, n_{MS}=4$

Step 3: Length of convex hull edges/MS§

(Length = $l_1$ distance for (1)-contours)

Results on the sum of lengths

$\sum_e t_e = |\partial Dig(X,h)| = \Theta(h^{-1})$

From [Lachaud, de Vieilleville, Feschet]

$|\partial Dig(X,h)| < \sum_e t_{MS} \leq 19|\partial Dig(X,h)|$

Hence

$\sum_e t_{MS} = \Theta(h^{-1})$

Summary§

If $\partial X$ is convex and $C^3$

Quantity	Smallest MS length	Average MS length	Largest MS length
$t_{MS}$	$\Omega(h^{-\frac{1}{3}})$	$\Theta(h^{-\frac{1}{3}}) \leq \cdot \leq \Theta(h^{-\frac{1}{3}}log(h^{-1})$	$O(h^{-\frac{1}{2}})$
$ht_{MS}$	$\Omega(h^{\frac{2}{3}})$	$\Theta(h^{\frac{2}{3}}) \leq \cdot \leq \Theta(h^{\frac{2}{3}}log(h^{-1}))$	$O(h^{\frac{1}{2}})$

(Hints for $\bar{t}_{MS}$ , the lower bound = lower bound on $\sum t_{MS}$ / upper bound $n_{MS}$ , results for smallest/largest MS require couple of more steps)

$\Rightarrow$ Any slope of MS containing P provides multigrid convergent estimation of tangent at P

Local Differential Estimator Summary§

Tangent Estimation in 2D

Curvature Estmiation in 2D

Tangent Estimation in Dimension 3§

Slice based approaches

Each surfel belong to two 2d-curves in planes parallel to axis grids
Compute the tangent in the two curves
Return cross-product $\rightarrow$ normal vectors

Tangent Estimation in Dimension 3§

Convolution based approaches

Convolution of elementary normal vectors in a given neighborhood

$\Rightarrow$ still have to fix a neighborhood parameter

Tangent Estimation in Dimension 3§

Digital Plane Recognition approaches

Around each point, recognize maximal digital plane
Kind of adaptive approach for the neighborhood parameter

… but …

Properties on maximal DPS are difficult to obtain

Curvature in dimension 3§

Mean and Gaussian curvature

Mean curvature $H =\frac{1}{2}( k_1 + k_2)$
Gaussian curvature $K = k_1 k_2$

Integral based Approaches§

Fitting an implicit polynomial surface is still doable but we need information on the neighborhood

Integral Invariant approach neighborhood in $O(h^\frac{1}{3})$ seems to be required

Idea compute area of the intersection between a ball $B_r(x)$ and $X$ at $x\in\partial X$

$A_r(x) = \int_{B_r(x)} \chi(p)dp$

Then, from Taylor expansion and for $r\rightarrow 0$

$A_r(x) = \frac{\pi}{2} r^2 - \frac{\kappa(X,x)}{3}r^3 + O(r^4)$

$V_r(x) = \frac{2\pi}{3} r^3 - \frac{\pi H(X,x)}{4}r^4 + O(r^5)$

Hence,

$\tilde{H}_r(X,x)\stackrel{def}{=} \frac{8}{3r} - \frac{4V_r(x)}{\pi r^4}$

$\tilde{H}_r(X,x)\rightarrow H(X,x)$ by definition when $r\rightarrow 0$

Toward Digital Version of Integral Invariants§

Volume by $|Dig(X\cap B_r(x),h)|$ $\Rightarrow$ first error term induced by [Gauss,Huxley] (O(h))
plus error term induced by the Taylor expansion
plus error term induced by difference between $x$ and $\hat{x}$ (back-projection used here)

Main Result

Thm.

For a family of shape with onvex $C^3$ -boundary and bounded curvature, $\exists h_0 \in \mathbb{R}^+$ , for any $h \le h_0$ , setting $r=k h^{\frac{1}{3}}$ , we have

$\forall x \in \partial X, \forall \hat{x} \in \partial Dig(X,h), \| \hat{x} -x\|_\infty \le h \Rightarrow$

$| \tilde{Q}(Dig(X,h),\hat{x}) - k(X,x) | \le K h^{\frac{1}{3}}.$

(similar bound in 3D)

What about Gaussian curvature§

Idea

Instead of computing the volume of $B_r(x)\cap X$ , we compute its covariance matrix

$M_r(x)= \int_{B_r(x)} pp^T\chi(p)dp - V_r(x)\mu_r(x)\mu_r^T(x)\,,$
Eigenvalues of $M_r(x)$ are such that:

$M_1 = \frac{\pi}{48}(3\kappa_1(x) + \kappa_2(x))r^6 + O(r^7)$

$M_2 = \frac{2\pi}{15}r^5 - \frac{\pi}{48}(\kappa_1(x) + 3\kappa_2(x))r^6 + O(r^7)$

$M_3 = \frac{19\pi}{480}r^5 - \frac{9\pi}{512}(\kappa_1(x) + \kappa_2(x))r^6 + O(r^7)$

Result

Similar convergence results exist with speed $O(h^\frac{1}{3})$

Experimental analysis confirms the $h^\frac{1}{3}$ neighborhood size

Parameter-free Curvature Estimator§

First, let’s have a look to the theorem statement

Thm.

For a family of shape with convex $C^3$ -boundary and bounded curvature, $\exists h_0 \in \mathbb{R}^+$ , for any $h \le h_0$ , setting $r=k h^{\frac{1}{3}}$ , we have …………

To have the convergence, we need the radius to be in $O(h^{\frac{1}{3}})$

We know that

$\Theta(h^{\frac{1}{3}}) \leq h\bar{t}^2_{MS} \leq \Theta(h^{\frac{1}{3}}log^2(h^{-1}))$

$\Rightarrow$ Let’s use (square of) average MS length to define r

$\Rightarrow$ Parameter-free convergence in $\Theta(h^{\frac{1}{3}}log(h^{-1}))$ !

$\Rightarrow$ Automatic selection of the scale parameter

Feature Selection§

Idea Use scale-space behavior of II estimators to classify surfels into flat,smooth,edge regions

Other quantities/properties§

Digital Convextiy§

Simple definition

$S \subset\mathbb{Z}^d$ is digitally convex iff there exists convex shape $X\subset\mathbb{R}^d$ such that $S = X\cap\mathbb{Z}^d$

Convexity and DSS§

Let $S\subset \mathbb{Z}^2$ be a (1)-curve

Properties on convex hull

Each edge of $Conv(S)$ is a period of some DSS
Each vertex of $Conv(S)$ is a leaning point for some period of some DSS

Main result [Debled-Rennesson, Doerksen-Reiter]

Thm.

S is digitally convex $\Leftrightarrow$ slopes of maximal segment in the covering are monotonic

Digital Geometry: Estimators§

Principles§

Introduction§

Mathematical Context§

Global Quantities§

Definition§

Area/Volume§

Step-based Length Estimation§

Step-based Perimeter Estimation (bis)§

DSS Based Perimeter Estimation§

Experimental evaluation§

Useful tool: Maximal Segment Covering§

Illustration§

From normal vector local estimator to length/surface area estimator§

Local Quantities§

Definition of multigrid convergence§

Normal/Tangent estimation§

DSS based Normal vector§

Maximal Segment Analysis§

Maximal Segment Analysis (bis)§

Curvature Estimation§

Maximal Segment Again§

Step 1: Digital Convex Hull§

Step 2: Number of maximal segments§

Step 3: Length of convex hull edges/MS§

Summary§

Local Differential Estimator Summary§

Tangent Estimation in Dimension 3§

Tangent Estimation in Dimension 3§

Tangent Estimation in Dimension 3§

Curvature in dimension 3§

Integral based Approaches§

Toward Digital Version of Integral Invariants§

What about Gaussian curvature§

Illustrations§

Parameter-free Curvature Estimator§

Illustrations§

Feature Selection§

Other quantities/properties§

Digital Convextiy§

Convexity and DSS§