# Scale-Space Feature extraction on Digital Surfaces

### Jérémy Levallois1,2, David Coeurjolly1, and Jacques-Olivier Lachaud2

1-Université de Lyon, CNRS/LIRIS, UMR5205, F-69621, France
2-Université de Savoie, CNRS/LAMA, UMR5127, F-73776, France

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## Feature extraction on 3D geometrical objects

Huge litterature... but various definitions.

We will focus on detecting singularities...
... that must be preserved regardless of scale or noise.

## Digital data

• isothetic surface
(set of pixels/voxels)
• arithmetical noise

Why digital data ?

→ Direct result of scanners (e.g. X-ray tomography)
→ multigrid convergence of estimated quantity

## Objectives

Feature extraction :

• Working on digital data
• Scale-space approach:
- robustness to noise
- scale independent features
• With geometrical/theoretical interpretation of features

## (Selected) State of the Art

Name Technique Selection Classification Scale-Space Usability for digital data

Moment Analysis[1]

(Clarenz et al.)

Threshold

VCM[2][3]

(Mérigot et al.)

$\frac{\lambda'_2}{\lambda'_1 + \lambda'_2 + \lambda'_3}$ Threshold

Sphere Fitting[4]

Threshold

Surface Variation[5]

(Pauly et al.)

$\frac{\lambda_2}{\lambda_1 + \lambda_2 + \lambda_3}$ Threshold

Tensor Voting[6]

(Park et al.)

$\frac{\lambda_3+\lambda_2}{\lambda_1}$ Threshold

Integral Invariant

Scale-Space Curvature $\frac{d k}{d r}$ Fitting

Spectral approaches[7]

(Song et al.)

Laplacian eigenvalues Threshold

## Previously... Digital Curvature Estimators

Continuous : $\tilde{\kappa}_{{{R}}}(x) := \frac{3\pi}{2{{R}}} - \frac{3 {\color{myred}A_R(x)}}{{{R}}^3}$

Digital : $\PCurvHat{{{R_d}}}(\hat{x}) := \frac{1}{h} ( \frac{3\pi}{2{{R_d}}} - \frac{3 {\color{myred}\sharp_n}}{{{R_d}}^3})$

## Previously... Uniform multigrid convergence of Digital Curvature Estimators

($R=kh^{1/3}$)
2d $O(h^{1/3})$
3d mean $O(h^{1/3})$
3d principal $O(h^{1/3})$

## ⇨ Radius as Scale-Space parameter

$R=4$ $R=7$ $R=10$ $R=12$

$R=14$ $R=17$ $R=20$ $R=25$

## Scale-Space Curvature based Feature Estimator

Curvature Plot (in logscale)
$\begin{eqnarray} G_{X,x}(R) & := & {\color{myblue}\frac{3 \pi}{2 R} - \frac{3 A(R,x)}{R^3}} + {\color{mygrey}O(R)} \\ & := & {\color{myblue}\kappa_0} + {\color{mygrey}O(R)} \end{eqnarray}$

Intercept : $\kappa_0$

Slope : $0$

$\begin{eqnarray} G_{X,x}(R) & = & {\color{myred}\frac{3}{2} \frac{1}{R} ( \pi - \alpha_0 )}\\ & & + {\color{mygreen}\frac{\kappa_{-} + \kappa_{+}}{6}} + {\color{mygrey}O(R)} \end{eqnarray}$

Intercept : depending to $\alpha_0$

Slope : $-1$

## Influence of digitization

For a constant radius $R$, there exists an infinity of curves which lead to the same digitization of $B_R(x)\cap \Shape$:

$\tilde{\kappa}(R) = 0$
${\color{myred}\kappa_{max} = \frac{2h}{R^2 + h^2}}$
For fixed $h$, estimated values $\hat{\kappa}_R \le \kappa_{max}$ are considered as flat zone, or outliers (in the sense that doesn't capture the feature with current radius).

## Influence of digitization

Slope : $-2$

## Scale-Space Analysis

For each point, for a range of radii:
1. Plot the scale-space curvature (abs: log of kernel radius, ord: log of mean curvature).
2. Remove all data is below $\kappa_{max}$ (we can't determine if it is a flat region or too small radius to detect a feature).
3. Compute the distance to models[1].
4. We classify the point by choosing the minimal distance. Here we have a smooth surface...

[1]- with Least Square Linear Fitting

Smooth
Edge
Flat

## Comparisons

Moment Analysis VCM Sphere
Fitting
Surface
Variation
Tensor
Voting
Our
$R_1$ $R_2$ $R_1$ $R_2$

## Comparisons

Moment Analysis Our
$R_1$ $R_2$

## Conclusion

• Robust / Stable feature estimation
• Classification into linear/smooth/singularities parts
• ... with some theoretical tools and geometrical interpretation

## Perspectives

• Post-treatment
• Theoretical proofs behind transitions of models