Convergence of binomial-based derivative estimation for C 2 noisy discretized curves

Journées RAIM 09, Lyon, 26-28 octobre 2009 Convergence of binomial-based derivative estimation for C 2 noisy discretized curves H-A. Esbelin, R. Malgouyres, C. Cartade and S. Fourey Univ. Clermont 1, LAIC, France {esbelin,remy.malgouyres,colin}@laic.u-clermont1.fr GREYC, ENSICAEN Sebastien.Fourey@greyc.ensicaen.fr

1 Introduction Discrete Estimators from lines and planes 2 1D-2D Binomial Masks 3 Equations différentielles 4 Convolutions on surfaces First Order Partial Derivatives 5 Experiments and Results Normal Estimation Curvature Estimation

Normal and Tangent Estimation DSS (Digital Straight Segments) based methods. [Debled-Rennesson, Klette, Kovalesky, Reveillès, Andrès, Lachaud.]

Same idea applied to surfaces... Approach based on digital plane recognition. [Sivignon, Dupont, Chassery]

Chen et al. (1985), Papier and Françon (1998) [Chen et al., 1985] Estimate the normal vector on each surfel by averaging the normals of their 4 e-neighbors. [Papier and Françon, 1998] Same averaging process but taking into account an arbitrarily large neighborhood (umbrella) with decreasing weights.

What does an umbrella look like?

The Binomial Derivative Estimator Input : Discrete function Γ width hγ(i) = f (ih) + ɛ h (i) f is the sampled C 2 function, h the pixel s size and ɛ h the noise. ( ( 2m 1 u)(n) = 1 i=m ( ) ) 2m 1 2 2m 1 (u(n + i) u(n 1 + i)) m 1 + i 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 i= m+1 Smoothed finite differences Binomial smoothing kernel

The Error Model Rounded case ] : ɛ h (i) 1 2h which is equivalent to Γ(i) = [ f (ih) h Floor case : 0 ɛ h (i) h which is equivalent to Γ(i) = f (ih) h Uniform Noise case : 0 ɛ h (i) Kh α with 0 < α 1 and K a positive constant. Note that the round case and the floor case are particular cases of uniform noise with α = 1.

Prior Results (DGCI 08) Theorem Suppose that f : R R is a C 3 function and f (3) is bounded, α ]0, 1], K R + and h R +. Suppose Γ : Z Z is such that hγ(i) f (hi) Kh α (uniform noise case). Then for m = h 2(α 3)/3, we have ( 2m 1 Γ)(n) f (nh) O(h 2α/3 ) Theorem Under the assumptions of Theorem 1, for some constant K and m sufficiently large, 2m 1 Γ(n) f (nh) h2 m 4 f (3) + K h α 1 m

More recent results (DGCI 09) Works for C 2 functions ; Uniform convergence for noisy parametrized curves. Theorem Suppose that f is a C 2 function and f (2) is bounded. Suppose Γ : Z Z is such that hγ(i) f (hi) Kh α (uniform noise case). Then if m = h (α 2)/1,01 we have ( 2m 1 Γ)(n) f (nh) O(h (0,51(α 0,01)/1,01 ). Theorem Under the assumptions of Theorem 3, for some constant K and m sufficiently large, 2m 1 Γ(n) f (nh) hm 0,51 f (2) + K h α 1 m

Convergence Result For Parametrized Discretizations Definition The binomial discrete tangent at M i is the real line going through M i directed by the vector ( m+1 (x i ), m+1 (y i )), when this vector is nonzero. Theorem Let g be a C 2 parametrization of a simple closed curve C. Suppose that for all i we have g(ih) hσ h (i) Kh α 1. Then for some constant K and m sufficiently large, 2m 1 Σ h (n) g (nh) hm 0,51 g (2) + K h α 1 m

Pixel-length Parametrization Definition A parametrization of a real curve γ is pixel-length if for all u et u such that γ x and γ y are monotonic between u and u, we have γ(u) γ(u ) 1 = u u. The idea is that for a curve with a pixel-length parametrization, the speed on the curve is the same as the speed of a discretization of the curve with each edgel taking the same time y x

Tangent Estimation for a general C 2 curve Theorem Let C be a simple closed curve C and M 0 C. Suppose that g is a regular C 2 parametrization of C and wlog M 0 = g(0). Let Σ h : Z Z 2 be a 4-connected discrete parametrized curve, lying in a tube of C of width Kh α. Suppose that Σ 2 h (0) M 0 Kh α (up to a translation on the parameters of Σ, this is alway possible). Then 2m 1 Σ h (0) T 0 K hm 0,51 + K h α 1 m

Discrétisation d une EDO On discrétise l équation différentielle par x [a, b], y (x) = f (x, y(x)), y(x 0 ) = y 0 (D m u)(i) = hf (x i, u i ) m N, D m masque de dérivation

Exemples m = 2 u n+3 + u n+2 u n 1 u n 4h = u n+1 ou u n+2 m = 3 u n+4 + 2u n+3 2u n+1 u n 8h = u n+2

Calculs exacts suite récurrente linéaire d équation caractéristique (x 1)(x + 1) 2m 1 = 2 2m 1 hx m. une racine x 1 proche de 1 et 2m 1 racines x 2, x 3,..., x 2m proches de 1

Calculs exacts Équation y = y Développement limité de la solution numérique : Comme la n ième itérée est de la forme u n = a 1 x n 1 + a 2x n 2 + + a 2mx n 2m x n 1 = (1 + x n + O( h 2 )) n = exp(x) + O(h) On choisit les conditions initiales pour que a 1 = 1 et a i = 0 pour i 1.

Calculs exacts

Iterated classical smoothing masks Classical 2D averaging filter ( 16)

1D sampling, Parametrized Curves, Digital Surfaces

Adjacency Between Surfels e-adjacency v-adjacency e-neighborhood v-neighborhood

How To Define Convolutions on Surfaces? 155 000 surfels

The Non-Planar Convolution Operator Let X Z 3 et Σ = δ(x). Furthermore, let E a vector space over R and two functions f and F : f : Σ E F : Σ Σ R Definition We define the operator Ψ, acting on f and F as follows : Ψ F (f ) : Σ E s f (s ) F(s, s ) (1) s Σ

Properties and Non Properties Non abelian : Ψ F (Ψ G (f )) Ψ G (Ψ F (f )) in general. Associative : Ψ ΨH (G)(f ) = Ψ H (Ψ G (f ))

The Iterated Convolution Operator X Z 3, Σ = δ(x), E is a vector space over R. f : Σ E, F : Σ Σ R. Definition Let n N, we define Ψ (n) F { Ψ (0) F Ψ (n) F (f ) = f (f ) = Ψ F(Ψ (n 1) F (f )) if n > 0. (2)

The Our Local Kernel

Classical 2D averaging filter ( 16) The local averaging mask H(s, s)

The Our Local Kernel

Over a plane with normal n = (1, 2, 3) ( 50 000 surfels)

With Lauren-Papier method (breadth first search)

Over a plane with normal n = (1, 2, 3) ( 50 000 surfels)

Over a hyperbolic paraboloid Response Breadth-first traversal

We define two derivative masks D u (s, s) and D v (s, s) : D u (s, s) D v (s, s) Orientations and slice curves.

Partial derivative operators Using Ψ (n), H, D u and D v we define u and v : Le n Z, where (n) u (n) v = Ψ Du (Ψ (n) H (σ)) (3) = Ψ Dv (Ψ (n) H (σ)) (4) σ : Σ R 3 s (x,y,z), the center of the surfel s

Normal vectors estimation Let s Σ. We define Γ (n) (s), the estimated normal vector of Σ at the center of s (after n on-surface convolutions). Γ (n) (s) = (n) u (n) u (s) (n) v (s) (s) (n) v (s) (5)

Second order derivative operations, curvatures Using Ψ (n) H, u, and v (eventually twice ), we estimate the first and second order partial derivatives : S u, S v, 2 S u 2, 2 S v 2, and 2 S u v Then, using the coefficients of the first and second fundamental forms, we compute the Gaussian or mean curvatures.

On digitized spheres Average error (in degree) and standard deviation on spheres with increasing radii.

On digitized tori Average error (in degree) an standard deviation on tori with increasing large and small radii

Gaussian Curvature on a Torus (35000 surfels)

Gaussian Curvature on a Torus Estimated and exact Gaussian curvatures on a torus. (Large radius is 80, small radius is 40.)

Gaussian Curvature on Bunny

Mean Curvature on Bunny

Questions