# Geometry Processing on Voxel Objects

David Coeurjolly
CNRS, Université de Lyon

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## Digital geometry model in one slide

Topology and geometry processing of objects defined on regular lattices

• Digital Objects = subsets of $\mathbb{Z}^d$
• Geometrical predicates = integer only computations
• $\Rightarrow$ strong impact of arithmetical properties
• Digital Topology = adjacency relationship induced by the lattice
• Digital surfaces = cells of a Cartesian cubical complex

## Material Sciences Applications

Non-invasive snow micro-tomographic images
X-ray Micro-tomograph
3SR Lab and CEN/CNRM - GAME URA 1357/Météo-France - CNRS
Up to $2048^3$

## Voxel objects in computer graphics

• Widely used in geometric modeling and rendering to represent complex and interactive scenes
• Nice mathematical modeling framework

$64K^3$ voxel grid!
Villanueva, Alberto Jaspe, Fabio Marton, and Enrico Gobbetti. "SSVDAGs: symmetry-aware sparse voxel DAGs." In Proceedings of the 20th ACM SIGGRAPH Symposium on Interactive 3D Graphics and Games, pp. 7-14. ACM, 2016.

## Outline

• Digital surface reconstruction

• Curvature tensor estimation

• Variational geometry processing: Ambrosio-Tortorelli functional

Collaborators: Jacques-Olivier Lachaud (Chambéry), Tristan Roussillon (Lyon), Nicolas Bonneel (Lyon), Jérémy Levallois (Lyon), Thomas Caissard (Lyon), Marion Foare (Lyon)

C., Jacques-Olivier Lachaud, Jérémy Levallois. "Multigrid Convergent Principal Curvature Estimators in Digital Geometry". Computer Vision and Image Understanding, 129(1):27-41, June 2014.
Thomas Caissard, C., Jacques-Olivier Lachaud, Tristan Roussillon. "Laplace–Beltrami Operator on Digital Surfaces". Journal of Mathematical Imaging and Vision, 2018.
C., Marion Foare, Pierre Gueth, Jacques-Olivier Lachaud. "Piecewise smooth reconstruction of normal vector field on digital data". Computer Graphics Forum (Proceedings of Pacific Graphics), 35(7), September 2016.
Nicolas Bonneel, C., Pierre Gueth, Jacques-Olivier Lachaud. "Mumford-Shah Mesh Processing using the Ambrosio-Tortorelli Functional". Computer Graphics Forum (Proceedings of Pacific Graphics), 37(7), October 2018.

## Objectives

 Regularization with same combinatorics ... with voxel attributes ... in a multilabel context
[Marching-Cubes, Dual Contouring, Volumetric reconstruction, Mesh fairing...]

## Variational approach

$$\mathcal{E}(\hat{P}) := \alpha \sum_{i=1}^{n} \|\mathbf{p}_i - \hat{\mathbf{p}}_i\|^2 + \beta \sum_{f\in F} \sum_{{\hat{\mathbf{e}}_j} \in \partial f } ( \hat{\mathbf{e}}_j \cdot \mathbf{n}_{f} )^2 + \gamma \sum_{i=1}^{n} \|\hat{\mathbf{p}}_i - \hat{\mathbf{b}}_i\|^2$$

$\mathbf{p}_i$: vertices of the input digital surface, $\mathbf{n}_f$: normal vector per face

$\hat{\mathbf{p}}_i$: regularized vertices, $\hat{\mathbf{e}}_j$: edge between two regularized vertices, $\hat{\mathbf{b}}_i$: barycenter of the four adjacent regularized vertices of $\hat{\mathbf{p}}_i$ vertices

## Variational approach

$$\mathcal{E}(\hat{P}) := {\color{blue}\alpha \sum_{i=1}^{n} \|\mathbf{p}_i - \hat{\mathbf{p}}_i\|^2 } + \beta \sum_{f\in F} \sum_{{\hat{\mathbf{e}}_j} \in \partial f } ( \hat{\mathbf{e}}_j \cdot \mathbf{n}_{f} )^2 + \gamma \sum_{i=1}^{n} \|\hat{\mathbf{p}}_i - \hat{\mathbf{b}}_i\|^2$$

$\mathbf{p}_i$: vertices of the input digital surface, $\mathbf{n}_f$: normal vector per face

$\hat{\mathbf{p}}_i$: regularized vertices, $\hat{\mathbf{e}}_j$: edge between two regularized vertices, $\hat{\mathbf{b}}_i$: barycenter of the four adjacent regularized vertices of $\hat{\mathbf{p}}_i$ vertices

Data attachment

## Variational approach

$$\mathcal{E}(\hat{P}) := {\color{blue}\alpha \sum_{i=1}^{n} \|\mathbf{p}_i - \hat{\mathbf{p}}_i\|^2 } + {\color{red}\beta \sum_{f\in F} \sum_{{\hat{\mathbf{e}}_j} \in \partial f } ( \hat{\mathbf{e}}_j \cdot \mathbf{n}_{f} )^2} + \gamma \sum_{i=1}^{n} \|\hat{\mathbf{p}}_i - \hat{\mathbf{b}}_i\|^2$$

$\mathbf{p}_i$: vertices of the input digital surface, $\mathbf{n}_f$: normal vector per face

$\hat{\mathbf{p}}_i$: regularized vertices, $\hat{\mathbf{e}}_j$: edge between two regularized vertices, $\hat{\mathbf{b}}_i$: barycenter of the four adjacent regularized vertices of $\hat{\mathbf{p}}_i$ vertices

## Variational approach

$$\mathcal{E}(\hat{P}) := {\color{blue}\alpha \sum_{i=1}^{n} \|\mathbf{p}_i - \hat{\mathbf{p}}_i\|^2 } + {\color{red}\beta \sum_{f\in F} \sum_{{\hat{\mathbf{e}}_j} \in \partial f } ( \hat{\mathbf{e}}_j \cdot \mathbf{n}_{f} )^2} + {\color{orange}\gamma \sum_{i=1}^{n} \|\hat{\mathbf{p}}_i - \hat{\mathbf{b}}_i\|^2}$$

$\mathbf{p}_i$: vertices of the input digital surface, $\mathbf{n}_f$: normal vector per face

$\hat{\mathbf{p}}_i$: regularized vertices, $\hat{\mathbf{e}}_j$: edge between two regularized vertices, $\hat{\mathbf{b}}_i$: barycenter of the four adjacent regularized vertices of $\hat{\mathbf{p}}_i$ vertices

Data attachment + Quad alignement + Fairness term

## Minimizing the energy

$$\frac{\partial \mathcal{E}(\hat{P})}{\partial \hat{\mathbf{p}}_i} := \alpha \sum_{i=1}^{n} 2(\hat{\mathbf{p}}_i - \mathbf{p}_i) + \beta \sum_{f\in F} \sum_{\hat{\mathbf{e}}_j \in \partial F} 2( \hat{\mathbf{e}}_j \cdot \mathbf{n}_{f} )\mathbf{n}_f + \gamma \sum_{i=1}^{n} 2(\hat{\mathbf{b}}_i - \hat{\mathbf{p}}_i)\,$$
$\Rightarrow$ The gradient is linear in the vertices position
$\Rightarrow$ Optimal position given by solving a linear system
$\Rightarrow$ Efficient interactive regularization on GPU 😀

## Multilabel case

$\Rightarrow$ Same formulation and trivial regularization thanks to the grid structure

## Digital Geometry point of view: theoretical guarantees

If $M$ is smooth (positive reach) and $M_h:=M\cap ( h\cdot\mathbb{Z}^3)$ its digitization on a grid with gridstep $h$, then $$\frac{1}{n}\sum_{i=1}^{n} \|\mathbf{p}^*_i - \mathbf{p}_i\| \leq C\cdot h \quad\quad \frac{1}{n}\sum_{i=1}^{n}\,d(\mathbf{p}^*_i, \partial M) \leq C'\cdot h$$

If normal vectors $\{\mathbf{n}_ƒ\}$ are multigrid convergent, normal vectors to regularized quads are also multigrid convergent and we have an even better approximation of $\partial M$

## Digitization model

Given $\Shape \subset \R^d$, its digitization at gridstep $h$ is $$\DigF{\Shape}{h} = \left(\frac{1}{h} \cdot \Shape\right) \cap \Z^d$$

$\Shape \in \Shapes$
$\DigF{\Shape}{1}$
$\DigF{\Shape}{0.5}$
$\DigF{\Shape}{0.25}$

Example: $h^2|\DigF{\Shape}{h}|$ converges to the measure of $\Shape$ as $h\rightarrow 0$

[Gauss, Dirichlet, Huxley...]

## Multigrid convergence of a local estimator

Given a digitization process $\AnyDig$, a local discrete geometric estimator $\hat{E}$ of some geometric quantity $E$ is multigrid convergent for the family of shapes $\Shapes$ if and only if, for any $\Shape \in \Shapes$, there exists a grid step $h_\Shape>0$ such that the estimate $\hat{E}(\AnyDig_{\Shape}(h),\hat{\vx},h)$ is defined for all $\hat{\vx} \in \Bd{\Body{\AnyDig_{\Shape}(h)}{h}}$ with $0< h< h_\Shape$, and for any $\vx \in \dS$,

$$\forall \hat{\vx} \in \Bd{\Body{\AnyDig_{\Shape}(h)}{h}} \text{ with } \| \hat{\vx} -\vx\|_\infty \le h, \quad\quad \color{myred}{| \hat{E}(\AnyDig_{\Shape}(h),\hat{\vx},h) - E(\Shape,\vx) | \le \tau_{\Shape,\vx}(h)},$$
where $\tau_{\Shape,\vx}: \R^{+} \setminus\{0\} \rightarrow \R^+$ has null limit at $0$. The convergence is uniform for $\Shape$ when every $\tau_{\Shape,\vx}$ is bounded from above by a function $\tau_\Shape$ independent of $\vx \in \Bd{\Shape}$ with null limit at $0$.

## Digital/Continuous mapping

Let $\Shape$ be a compact domain of $\R^d$ such that $\partial X$ has positive reach greater that $R$. Let $\partial_h\Shape\EqDef\partial\Body{\Dig_h(\Shape))}{h}$. Then for any $0 < h < 2R/\sqrt{2}$,

• The Hausdorff between $\partial \Shape$ and $\partial_h\Shape$ is bounded by $\sqrt{d}h/2$
• For $d=2$, there exists an homeomorphism between $\partial X$ and $\partial_h X$
• For $d\geq 3$, no homeomorphism ☹, but
• Projection operator $\xi:\,\partial_h\Shape\rightarrow\partial \Shape$ is surjective
• Area of non-injective parts of $\xi$ the tends to zero

## Digitization as an Hausdorff sampling of the continuous object

Can we estimate the curvature tensor on digital surfaces with multigrid convergence properties ?

## Huge literature on differential quantity estimators

• Generic framework
• Varifold approaches

→ accuracy depends on the mesh/point cloud quality
→ incompatible constraints on convergence theorem w.r.t. digital surface

## Main contributions: Integral Invariant approach

If $\Shape$ is compact and $\partial \Shape$ has positive reach $\rho$ and $C^3$-continuity then

• Mean curvature and principal curvature estimations converge in $O(h^{\frac{1}{3}})$
• Normal vector field estimation converges in $O(h^{\frac{2}{3}})$
• Principal Curvature directions converge in $\frac{1}{|\PrincCurv{1}(\Shape,\vx)-\PrincCurv{2}(\Shape,\vx)|} O(h^{\frac{1}{3}})$

## Overall proof scheme

${\kappa}(\Shape,\vx) \EqDef \underbrace{\frac{3\pi}{2{{R}}} - \frac{3\cdot { A_R(\Shape,\vx)}}{{{R}}^3}}_{ \tilde{\kappa}^{R}(\Shape,\vx) } + O(R)$ [Pottmann et al. 2007]

$A_R(\Shape,\vx) \rightarrow \AreaC(\Ball{R/h}{\vx/h} \cap \DigF{\Shape}{h})$

+ $\quad\CurvH{R}(\DSh,\vx,h)$

$\CurvH{R}(\DSh,\hat{\vx},h) \rightarrow \Curv(\Shape,\vx)$

## Area estimator "à la" Gauss

${\color{myblue}\AreaC(\DSh)} \EqDef h^2 \Card (\DSh)$
$\quad \AreaC(Y) = \Area(\Shape) + {\color{myblue}O(h^\beta)}\quad\quad\text{with}\quad \beta \ge 1$
• $\beta = 1$ for $\Shapes^C$ [Gauss, Dirichlet]
• $\beta = \frac{15}{11}-\epsilon$ for $\Shapes^{3-SC}$ (non null curvature) [Huxley 1990]

## Area estimator "à la" Gauss

${\color{myblue}\AreaC(Y)} \EqDef h^2 \Card (Y) \quad\quad\text{with}\quad Y = \Ball{R/h}{\vx/h} \cap \DigF{\Shape}{h}$
$\quad \AreaC(Y) = A_R(\Shape,\vx) + {\color{myblue}O(h^\beta R^{2-\beta})} \quad\quad\text{with}\quad \beta \ge 1$
$\vx$
$\vx$
$\AreaC(Y) = 12 \times 1^2 = {\color{myblue}12}$
$\AreaC(Y) = 50 \times (0.5)^2 = {\color{myblue}12.5}$
$\color{myred} A_R(\Shape,\vx) \approx 12.9$

## Overall proof scheme - 1/3

$A_R(\Shape,\vx) \rightarrow \AreaC(\Ball{R/h}{\vx/h} \cap \DigF{\Shape}{h})$

+ $\quad\CurvH{R}(\DSh,\vx,h)$

$\CurvH{R}(\DSh,\hat{\vx},h) \rightarrow \Curv(\Shape,\vx)$

## Curvature estimation by digital integration

$\hat{\kappa}^{R}(\DigF{\Shape}{h},\vx,h) \EqDef \frac{3\pi}{2{{R}}} - \frac{3 {\color{myblue}\AreaC(Y)}}{{{R}}^3} {\color{mydarkred}+O(R)} \quad\text{with}\quad Y = \Ball{R/h}{\vx/h} \cap \DigF{\Shape}{h}$
$\vx$
$\vx$
$\hat{\kappa}^{3}(\DigF{\Shape}{1},\vx,1) = \frac{3\pi}{2\times 3} - \frac{3 \times {\color{myblue}12}}{3^3} \approx 0.237$
$\hat{\kappa}^{3}(\DigF{\Shape}{0.5},\vx,0.5) = \frac{3\pi}{2 \times 3} - \frac{3 \times {\color{myblue}12.5}}{3^3} \approx 0.182$
$\color{myred} {\kappa}(\Shape,\vx) = \frac{1}{7.5} \approx 0.133$

## Overall proof scheme - 2/3

$A_R(\Shape,\vx) \rightarrow \AreaC(\Ball{R/h}{\vx/h} \cap \DigF{\Shape}{h})$

+ $\quad\CurvH{R}(\DSh,\vx,h)$

$\CurvH{R}(\DSh,\hat{\vx},h) \rightarrow \Curv(\Shape,\vx)$

## Positioning Error

$\pi^\Shape_h(\hat{\vx})$
For any 2D shape with positive reach, given $0 < h \le \reach{\Shape}$, let $\mathbf{n}(\Shape, \vx, l)$ be the segment centred at $\vx$, aligned with the normal vector at $\vx$ of length $2l$. Then : \begin{align} \pi^\Shape_h : \partial\Body{\DSh}{h} &\rightarrow \dS \,,\\ \hat{\vx} &\mapsto \vx = \pi^\Shape_h(\hat{\vx}) \,, \end{align} where $\vx$ is the unique point such that $\hat{\vx} \in \mathbf{n}\left(\Shape,\vx, {\color{myred}\frac{\sqrt{2}}{2}h}\right)$.
$\vx$
$\hat{\vx}$
${\color{mygreen}|| \vx - \hat{\vx} ||_\infty = O(h^{\alpha'})} \quad\quad\text{with}\quad\alpha' \ge 1$

## Multigrid convergence of the digital curvature estimator

Let $\Shape$ be a convex shape in $\R^2$ with a $C^3$ bounded positive curvature boundary. $\begin{split} \forall \vx \in \partial\Shape, \forall \hat{\vx} \in \partial\Body{\DSh}{h}, \| \hat{x} -x\|_\infty \le h \Implies\\ | \CurvH{R}(\DSh,\hat{\vx},h) - \Curv(\Shape,\vx) | =\quad & {\color{mydarkred} O(R)}\\ +\quad & {\color{myblue} O\left(\frac{h^\beta}{R^{1+\beta}}\right)}\\ +\quad & {\color{mygreen} O\left(\frac{h^{\alpha'}}{R^2}\right) + O\left(h^{\alpha'}\right) + O\left(\frac{h^{2\alpha'}}{R^2}\right)} \end{split}$

→ Setting $R=kh^\alpha$, we select $\alpha$ to minimize all errors.

$\color{myred} \left| \CurvH{R}(\DSh,\hat{\vx},h) - \Curv(\Shape,\vx) \right| \le O\left(h^\frac{1}{3}\right) \quad\text{setting}\quad R=kh^\frac{1}{3}$

## Curvature tensor on digital surfaces

\begin{align} &\PrincCurvH{1}{R}(\DSh,\vx,h) \EqDef \frac{6}{\pi R^6}\left({\color{myblue}\hat{\lambda}_2} - 3{\color{myblue}\hat{\lambda}_1}\right) + \frac{8}{5R}\\ &\PrincCurvH{2}{R}(\DSh,\vx,h) \EqDef \frac{6}{\pi R^6}\left({\color{myblue}\hat{\lambda}_1} - 3{\color{myblue}\hat{\lambda}_2}\right) + \frac{8}{5R}\\ &\PrincDirH{1}{R}(\DSh,\vx,h) \EqDef {\color{myblue}\hat{\nu}_1}\\ &\PrincDirH{2}{R}(\DSh,\vx,h) \EqDef {\color{myblue}\hat{\nu}_2}\\ &\NormalDirH{R}(\DSh,\vx,h) \EqDef {\color{myblue}\hat{\nu}_3} \end{align}

$\{{\color{myblue}\hat\nu_i,\hat\lambda_i}\}$ are the eigenvalues/eigenvectors of the covariance matrix of $B_r(\vx)\cap \Shape$

$\color{myred} \left| \PrincCurvH{i}{R}(\DSh,\hat{\vx},h) - \PrincCurv{i}(\Shape,\vx) \right| \le O\left(h^\frac{1}{3}\right) \quad\text{setting}\quad R=kh^\frac{1}{3}$

\begin{align} &\exists h_\Shape \in \R^+,\,\, \forall h \in \R,\,\, 0 < h < h_\Shape,\,\, \,\,\nonumber\quad \quad\quad \quad\\ &\forall \vx \in \Bd{\Shape}, \quad \forall \hat{\vx} \in \Bd{\Body{\DSh}{h}} \text{ avec } \| \hat{\vx} -\vx\|_\infty \le h, \nonumber\quad \quad\quad \quad \\ &\quad \quad\quad \quad\| \PrincDirH{1}{R}( \DSh, \hat{\vx}, h ) - \PrincDir{1}(\Shape, \vx) \| \le \frac{1}{|\PrincCurv{1}(\Shape,\vx)-\PrincCurv{2}(\Shape,\vx)|} O(h^{\frac{1}{3}})\,, \\ &\quad \quad\quad \quad\| \PrincDirH{2}{R}( \DSh, \hat{\vx}, h ) - \PrincDir{2}(\Shape, \vx) \| \le \frac{1}{|\PrincCurv{1}(\Shape,\vx)-\PrincCurv{2}(\Shape,\vx)|} O(h^{\frac{1}{3}})\,, \\ &\quad \quad\quad \quad\| \NormalDirH{R}( \DSh, \hat{\vx}, h ) - \NormalDir(\Shape, \vx) \| \le O(h^{\frac{2}{3}})\,. \end{align}

## In summary

• Robust, efficient implementation (convolutions)
• Parametrized by a integration radius $R$ or a grid step $h$
• Proof relies on digital/continuous relationships and geometrical moment estimation

## Problem statement

$\Rightarrow$ Piecewise smooth reconstruction of the normal vector field

## Ambrosio-Tortorelli functional

$$\mathcal{AT}_\eps(u,v) = {\alpha} \underbrace{\dint_{{M}} |u - g|^{{2}} \dx}_{\text{attachment term}} + \underbrace{\dint_{{M}} |{v} \grad u|^{{2}} \dx}_{\text{smoothness term}} + \underbrace{{\lambda} \dint_{{M}} {\eps} |\grad {v}|^{{2}} + \frac{1}{4 {\eps}} |1 - {v}|^{{2}} \dx}_{\text{discontinuity length}}$$

• Two functions to optimize: $u$ (scalar map) and $v$ (feature scalar map, $\mathcal{M}\rightarrow [0,1]$)
• $v\approx 1$ on smooth parts, $v\approx 0$ near features
• the AT functional $\Gamma-$converges to Mumford-Shah's functional $\mathcal{AT}_\eps \gammaconv{\eps \to 0} \mathcal{MS}$
• (integration domain does not change, no Hausdorff measure)

## Ambrosio-Tortorelli functional

$$\mathcal{AT}_\eps(u,v) = {\alpha} \underbrace{\dint_{{M}} |u - g|^{{2}} \dx}_{\text{attachment term}} + \underbrace{\dint_{{M}} |{v} \grad u|^{{2}} \dx}_{\text{smoothness term}} + \underbrace{{\lambda} \dint_{{M}} {\eps} |\grad {v}|^{{2}} + \frac{1}{4 {\eps}} |1 - {v}|^{{2}} \dx}_{\text{discontinuity length}}$$

• $\eps$ -- thickness of the feature set ($[distance]$)
• ${\alpha}$ -- attachment coefficient to control the smoothing strength ($[area^{-1}]$)
• $\lambda$ -- proportional to the length of discontinuities ($[distance^{-1}]$)

## Discretization

"à la" Discrete Exterior Calculus [Hirani, Desbrun, Grady...]:

$\mathcal{M}$ is a cellular complex ($\mathcal{M}'$ its dual), $\sigma^k$ are $k-$cells of $\mathcal{M}$ (resp. $\sigma'^k$ of $\mathcal{M}'$)

• $k-$forms are vectors of $|\{\sigma^k\}|$ scalars
• Linear operators are matrices
• e.g.
• $d_k$ (exterior derivative) maps primal $k-$forms to primal $(k+1)-$forms
• wedge product $\alpha\wedge \beta$ maps $k-$forms and $l-$forms to $(k+l)-$forms
• Hodge-star $\star_k$ operator to maps primal $k-$forms to dual $k-$forms
• ...

## Discretization (bis)

$$\mathcal{AT}_\eps(u,v) = {\alpha} \underbrace{\dint_{{M}} |u - g|^{{2}} \dx}_{\text{attachment term}} + \underbrace{\dint_{{M}} |{v} \grad u|^{{2}} \dx}_{\text{smoothness term}} + \underbrace{{\lambda} \dint_{{M}} {\eps} |\grad {v}|^{{2}} + \frac{1}{4 {\eps}} |1 - {v}|^{{2}} \dx}_{\text{discontinuity length}}$$

$v$ is a primal $0$-form, $u$ is a triple of dual $0$-forms $(u_1,u_2,u_3)$ and we discretize $\mathcal{AT}_\epsilon$

## Discretization (ter)

$$\mathcal{AT}^d_\epsilon(u,v) \EqDef \alpha \sum_{i=1}^{3} \langle{u_i - g_i},{u_i - g_i}\rangle_\bar{0} + \sum_{i=1}^{3} \langle{v \wedge d_\bar{0} u_i},{v \wedge d_\bar{0} u_i}_\bar{1}\rangle_\bar{1} \\ \quad\quad\quad\quad\quad\quad\quad\quad\quad+ \lambda \epsilon \langle{ d_{0} v },{ d_{0} v }\rangle_1 + \frac{\lambda}{4\epsilon} \langle{1 - v},{1 - v}\rangle_0$$

• if $\gamma$ is a primal $0-$form and $\beta$ a dual $1-$form, then $\gamma\wedge\beta=diag(\beta)M\gamma$ with $M\EqDef\frac{1}{2}|d_0|$
• $A$ is the matrix form of $d_0$
• $B$ is the matrix form of $d_\bar{0}$
• $\vu_i,\,\vv,\,\vg$ are column vectors containing associated $k-$form scalars
• $S_i$ is a diagonal matrix encoding the Hodge star $\star_i$

$$\mathcal{AT}^d_\epsilon(\vu,\vv)= \alpha (u-g)^T S_{\bar{0}} (u-g) + u^T B^T \operatorname{Diag}(M v) S_{\bar{1}} \operatorname{Diag}(M v) B u \\ + \lambda\,\epsilon\, v^T A^T S_1 A v + \frac{\lambda}{4\epsilon} (1-v)^T S_0 (1-v)$$

## Optimization

$\min_{u,v} \mathcal{AT}_\eps \Leftrightarrow \left( \nabla_u \mathcal{AT}_\eps = 0 \right) \wedge \left(\nabla_v \mathcal{AT}_\eps = 0 \right)$

For a given $\epsilon$:

$$\nabla_u AT_\epsilon[u, v] = 0 \Leftrightarrow \left[ \alpha\,S_{\bar{0}} - B^T \operatorname{Diag}(M v) S_{\bar{1}} \operatorname{Diag}(M v) B \right] u = \alpha\,S_{0}\,g$$

$$\nabla_v AT_\epsilon[u, v] = 0 \Leftrightarrow \left[ \frac{\lambda}{4\epsilon} S_0 + \lambda\,\epsilon\,A^T S_1 A + M^T \operatorname{Diag}(B u) S_{\bar{1}} \operatorname{Diag}(B u) M \right] v = \frac{\lambda}{4\,\epsilon} S_0$$

$\Rightarrow$ only linear system solves on sparse matrices !

## In summary

• Anisotropic normal vector field regularization with feature selection
• Sharp features
• Parameters make sense :)
• Variational problem discretization using a combinatorial representation of the digital surface

Chicken/egg problem: measure of quads $\mu(s)$ used in the DEC operators

## Ambrosio-Tortorelli on meshes

Nicolas Bonneel, C., Pierre Gueth, Jacques-Olivier Lachaud. Mumford-Shah Mesh Processing using the Ambrosio-Tortorelli Functional. Computer Graphics Forum (Proceedings of Pacific Graphics), 37(7), October 2018.

## Geometrical deep learning data fusion

Combining Voxel and Normal Predictions for Multi-View 3D Sketching Johanna Delanoy, David Coeurjolly, Jacques-Olivier Lachaud, Adrien Bousseau. Computers and Graphics, 82:65–72, 2019

## Curvature estimation

Integral Invariant approach: integration ball centers are projected onto the regularized surface.

$\Rightarrow$ better estimation on low resolution voxel set

## Subpixel Euclidean distance transformation

$\Rightarrow$ subpixel accuracy, fast, n-D...

## Multigrid convergent pixel/voxel coverage estimation

$\Rightarrow$ Key ingredient in many quantity estimation applications

• Using the regularized proxy: still $O(h^2)$ worst-case error in the coverage estimation :/, much lower error in practice
• E.g. projecting on maximal straight lines (2D) : $O(h^{7/3})$ (Conj.)

## DGtal Packages

Kernel Arithmetic Geometry Shapes Topology $$\tiny \mathbb{Z}^d$$ $$\tiny u_0+\frac{1}{u_1+\frac{1}{\ldots+\frac{1}{u_k}}}$$ $$Ax=b$$ DEC Graph Mathematic Image IO

## Conclusion

• Nice geometrical model with many interactions (arithmetic's, theory of words, computational geometry, discrete mathematics...)
• Very specific discrete/continuous properties
• Related to various areas (image processing, material sciences, geometrical modeling, rendering...) data

• Multigrid convergent framework
• for discrete-continuous
• to validate estimators: length, surface area, coverage, volumetric integrals, curvature tensor, Laplace-Beltrami on surfaces...
• Discrete Exterior Calculus on digital surfaces with intrinsic metric information