Differential geometry of sampled smooth surfaces: principal frames, umbilics and ridges. Frédéric Cazals and Marc Pouget INRIA SOPHIA-ANTIPOLIS, GEOMETRICA PROJECT
I. Estimating Differential Quantities Using Polynomial Fitting of Osculating Jets Symp. on Geometry Processing 2003. II. Classification of umbilics and ridges Smooth surfaces, umbilics, lines of curvatures, foliations, ridges and the medial axis: a concise overview. Research report INRIA RR-5138. III. Algorithm for umbilics and ridges detec- tion Ridges and umbilics of a sampled smooth surface: a complete picture gearing toward topological coherence. Submitted.
Problem addressed Input: • S is a smooth surface embedded in R 3 . • A triangulation or a point cloud sampled over S is provided. Output: • Estimate local differential properties of S : normal, curvatures, higher order... • Convergence results.
Contributions • Method: polynomial fitting of arbitrary degree. • Convergence analysis for interpolation and least square approxima- tion. • Error analysis of two stages methods. • Experimental study.
Height functions and jets y y’ x x’ Height function = jet + h.o.t: f ( x , y ) = J B , n ( x , y )+ O ( || ( x , y ) || n + 1 ) J B , n ( x , y ) = B 00 + B 10 x + B 01 y + B 20 x 2 + B 11 xy + B 02 y 2 + ··· + B 0 n y n . J B , n , the Taylor expansion of f called the osculating n -jet, involves N n = 1 + 2 + ··· +( n + 1 ) = ( n + 1 )( n + 2 ) / 2 coefficients.
Differential Quantities Unit normal vector � n S = ( − B 10 , − B 01 , 1 ) t / 1 + B 2 10 + B 2 01 . Second order info using the Weingarten map . . . { B 10 , B 01 , B 20 , B 11 , B 02 } Higher order info Retrieve the Monge form of the surface. z = 1 2 ( k 1 x 2 + k 2 y 2 )+ 1 6 ( b 0 x 3 + 3 b 1 x 2 y + 3 b 2 xy 2 + b 3 y 3 ) + 1 24 ( c 0 x 4 + 4 c 1 x 3 y + 6 c 2 x 2 y 2 + 4 c 3 xy 3 + c 4 y 4 )+ ...
Fitting methods Let p i ( x i , y i , f ( x i , y i )) be N sample points around the origin. Interpolation: find a n -jet J A , n : f ( x i , y i ) = J B , n ( x i , y i )+ O ( || ( x i , y i ) || n + 1 ) = J A , n ( x i , y i ) , i = 1 ... N . Least-Square Approximation: find a n -jet J A , n minimizing: N ∑ ( J A , n ( x i , y i ) − f ( x i , y i )) 2 . i = 1 Well poisedness of the problem and nearness to non-poised problems is measured by condition numbers.
Convergence issues Surface: O ( h ) p z = B 00 + B 10 x + B 01 y + ... S p i Polynomial fi tting: z z = A 00 + A 10 x + A 01 y + ... y x The convergence is sought for a sequence of sample points function of the parameter h Wish: A i j = B i j + O ( r ( h ))
Approximation results Hypothesis: N points p i ( x i , y i , z i ) , with x i = O ( h ) , y i = O ( h ) Theorem [Interpolation or Approximation of degree n ] Accuracy on coefficients of degree k of the Taylor expansion of f is O ( h n − k + 1 ) : A k − j , j = B k − j , j + O ( h n − k + 1 ) ∀ k = 0 ,..., n ∀ j = 0 ,..., k . Corollary • unit normal coeffs estimated with accuracy O ( h n ) , • principal curvatures and directions estimated with accuracy O ( h n − 1 )
Remarks • A two step method (estimating normal first and curvatures later) is not relevant. • One has to fit a polynomial with all its coefficients, neglecting first order coefficients leads to a loose of accuracy. • Intuition: higher the degree ⇐ ⇒ more degrees of freedom = ⇒ better fit. • Error bounds instead of asymptotic convergence can be derived. Cf. P .G. Ciarlet and P .-A. Raviart.
✁ � Algorithm 1. Collecting N samples • Mesh case: ith rings • PC case: local mesh, Power Diag. in the tangent plane, keep the neighbors and discard weights. T TX X N Natural neighbors and tangent-neighbors Figure from J.D.Boissonnat and J.Flototto
Algorithm 2. Chose of the coordinate system • Implicit plane fitting (PCA) • the z -direction is the world coordinate axis with least angle w.r.t. the normal of the fitted plane. 3. Fitting problem, degenerate cases —almost singular matrices • Interpolation: choose samples differently • Approximation: decrease degree, increase # pts 4. Retrieve Differential quantities • Classical calculus.
Classical surfaces (a)Elliptic paraboloid (16k points) (b)Surface of revolution (8k points)
Robustness on noisy model
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I. Estimating Differential Quantities Using Polynomial Fitting of Osculating Jets II. Classification of umbilics and ridges 1. Introduction 2. Monge patch 3. Umbilics 4. Contact and ridges 5. Medial Axis III. Algorithm for umbilics and ridges detec- tion
1. Introduction: umbilics and lines of curvature
1. Introduction: ridges and crest lines
2. Monge patch The surface is locally given by a height function at the origin in the principal frame: z = 1 2 ( k 1 x 2 + k 2 y 2 )+ 1 6 ( b 0 x 3 + 3 b 1 x 2 y + 3 b 2 xy 2 + b 3 y 3 ) + 1 24 ( c 0 x 4 + 4 c 1 x 3 y + 6 c 2 x 2 y 2 + 4 c 3 xy 3 + c 4 y 4 )+ ... In this Monge coordinate system, the Taylor expansion of the blue (max) curvature along the blue curvature line going through the origin has ex- pansion: P 2 ( k 1 − k 2 ) x 2 + ... 1 = 3 b 2 1 +( k 1 − k 2 )( c 0 − 3 k 3 1 k 1 ( x ) = k 1 + b 0 x + 1 ) . P Rk: switching the orientation of the principal directions reverts the sign of odd degree coefficients.
3. Umbilics: generic classifi cation A curvature line is an integral curve of the principal direction field. Umbilics are singularities of these fields, they are classified by: • the index, which is the number of turns of the principal field along a circuit around the umbilic. It is ± 1 / 2 . • the limiting principal directions, which are tangent directions to lines of curvature ending at the umbilic. There are 1 or 3 such directions. Rk: the principal field is not orientable close to an umbilic
3. Umbilics: Lemon, Monstar and Star • Lemon: index +1/2, 1 limiting principal direction. • Monstar: index +1/2, 3 limiting principal directions. • Star: index -1/2, 3 limiting principal directions. Figures from Porteous, Geometric differentiation
4. Contact: defi nition C The surface S is parameterized by a Monge patch z ( x , y ) . c C is the sphere of center c ( 0 , 0 , r ) passing through the origin. S z(x,y) (x,y) Definition. 1 The contact function at the origin between the surface and the sphere C is: g ( x , y ) = distance ( z ( x , y ) , c ) 2 − ( radius of C ) 2 = x 2 ( 1 − rk 1 )+ y 2 ( 1 − rk 2 ) − r 3 C M ( x , y )+ ... Rk: g ( x , y ) = 0 ⇐ ⇒ ( x , y , z ( x , y )) ∈ C ∩ S
✆ ☎ ✞ ✟ ✆ ✆ ✂ ✂ � ✁ ✂ ✄ ✆ ✄ ✄ ✝ ✆ ✂ � ✁ ✂ ✄ ✠ ✆ ✝ ✆ ✝ ✁ ✝ ✄ ✡ ✟ ☛ ✝ ✄ ✡ ✂ � ✂ ✠ ✄ ☎ ✆ ✄ ✝ ✞ ✟ � ✁ ✂ ✄ ✄ 4. Contact: singularities Definition. 2 Let f ( x , y ) be a smooth bivariate function. Function f has an A k or D k singularity if, up to a diffeomorphism, it can be written as: A k : f = ± x 2 ± y k + 1 , k ≥ 0 , � (1) D k : f = ± yx 2 ± y k − 1 , k ≥ 4 . The singularity is further denoted A ± k or D ± k if the product of the coefficients of the monomials is ± 1 . Zero level sets of the A k : f = x 2 ± y k + 1 singularities
5. Contact: A 2 The sphere is a sphere of curvature of radius 1 / k 1 or 1 / k 2 Contact with the blue sphere Contact with the red sphere
4. Contact: A 3 with the blue sphere The blue sphere of curvature has radius 1 / k 1 and b 0 = 0 . Recall the expansion of k 1 along the blue line: P 2 ( k 1 − k 2 ) x 2 + ... 1 1 = 3 b 2 1 +( k 1 − k 2 )( c 0 − 3 k 3 k 1 ( x ) = k 1 + b 0 x + 1 ) . P The contact point is called a ridge point: • elliptic if P 1 < 0 then this is a A + 3 singularity and the blue curvature is maximal along its line; • hyperbolic if P 1 > 0 then this is a A − 3 singularity and the blue curvature is minimal along its line.
4. Contact: A 3 with the blue sphere Blue elliptic ridge point Blue hyperbolic ridge point on the blue curve on the green curve A + A − 3 contact 3 contact Maximum of k 1 Minimum of k 1
� ☎ ✆ ✝ ✆ ✞ ✆ 4. Contact: A 4 blue The blue sphere of curvature has radius 1 / k 1 , b 0 = 0 and P 1 = 0 . The blue curvature has an infection along the blue line. This is a ridge turning point and the ridge changes from elliptic to hyper- bolic. ✁✄✂ ✁✄✂ ✁✄✂
4. Contact: at umbilics Umbilics are distinguished by the number of ridges passing through: • Elliptic or 3-ridge umbilic. k 1 is minimal and k 2 is maximal. • Hyperbolic or 1-ridge umbilic. There are curves passing through the umbilic on which k 1 and k 2 are constant. Note that at a ridge point, the line of curvature is tangent to the level set of the principal curvature. Hyperbolic umbilic Symmetric elliptic umbilic unsymmetric elliptic umbilic Ridges change color at umbilic and are hyperbolic
4. Contact: Elliptic umbilics Symmetric elliptic umbilic Unsymmetric elliptic umbilic Figures from P . Giblin, 2 and 3 dimensional patterns of the face. Elliptic umbilics are stars. Dashed lines are level sets of the blue curvature, Thin lines are blue curvature lines, Thick lines are blue ridges.
4. Contact: Hyperbolic umbilics Star Lemon monstar Figures from P . Giblin, 2 and 3 dimensional patterns of the face.
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