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 detection 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 R3 . • 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 approximation. • 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) = JB,n(x, y) + O(||(x, y)||n+1 ) JB,n(x, y) = B00 + B10 x + B01y + B20 x2 + B11 xy + B02 y2 + · · · + B0n yn.
JB,n , the Taylor expansion of f called the osculating n-jet, involves Nn = 1 + 2 + · · · + (n + 1) = (n + 1)(n + 2)/2 coefficients.
Differential Quantities Unit normal vector q
nS = (−B10, −B01, 1)t / 1 + B210 + B201. Second order info using the Weingarten map . . . {B10, B01, B20, B11, B02} Higher order info Retrieve the Monge form of the surface. 1 1 z = (k1 x2 + k2y2 ) + (b0x3 + 3b1x2 y + 3b2xy2 + b3y3 ) 2 6 1 + (c0 x4 + 4c1x3 y + 6c2x2 y2 + 4c3xy3 + c4y4 ) + . . . 24
Fitting methods Let pi (xi , yi , f (xi , yi )) be N sample points around the origin. Interpolation: find a n-jet JA,n : f (xi , yi ) = JB,n(xi , yi ) + O(||(xi, yi )||n+1) = JA,n(xi , yi ), i = 1 . . . N. Least-Square Approximation: find a n-jet JA,n minimizing: N
2 (J (x , y ) − f (x , y )) ∑ A,n i i i i .
i=1
Well poisedness of the problem and nearness to non-poised problems is measured by condition numbers.
PSfrag replacements
Convergence issues Surface: O(h)
z = B00 + B10x + B01y + . . .
p S pi
Polynomial fitting:
z
z = A00 + A10x + A01y + . . .
y x
The convergence is sought for a sequence of sample points function of the parameter h Wish: Ai j = Bi j + O(r(h))
Approximation results Hypothesis: N points pi(xi , yi , zi ), with xi = O(h), yi = O(h) Theorem [Interpolation or Approximation of degree n] Accuracy on coefficients of degree k of the Taylor expansion of f is O(hn−k+1): Ak− j, j = Bk− j, j + O(hn−k+1)
∀k = 0, . . . , n ∀ j = 0, . . . , k.
Corollary • unit normal coeffs estimated with accuracy O(hn), • 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 N
X
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
.
I. Estimating Differential Quantities Using Polynomial Fitting of Osculating Jets II. Classification of umbilics and ridges 1. 2. 3. 4. 5.
Introduction Monge patch Umbilics Contact and ridges Medial Axis
III. Algorithm for umbilics and ridges detection
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: 1 1 z = (k1 x2 + k2y2 ) + (b0x3 + 3b1x2 y + 3b2xy2 + b3y3 ) 2 6 1 + (c0 x4 + 4c1x3 y + 6c2x2 y2 + 4c3xy3 + c4y4 ) + . . . 24 In this Monge coordinate system, the Taylor expansion of the blue (max) curvature along the blue curvature line going through the origin has expansion: P1 k1 (x) = k1 + b0x + x2 + . . . 2(k1 − k2)
P1 = 3b21 + (k1 − k2)(c0 − 3k13).
Rk: switching the orientation of the principal directions reverts the sign of odd degree coefficients.
3. Umbilics: generic classification 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: definition C
The surface S is parameterized by a Monge patch z(x, y). C is the sphere of center c(0, 0, r) passing through the origin.
c
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 r = x2 (1 − rk1) + y2(1 − rk2) − CM (x, y) + . . . 3 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 Ak or Dk singularity if, up to a diffeomorphism, it can be written as: ( Ak : f = ±x2 ± yk+1 , k ≥ 0, (1) 2 k−1 Dk : f = ±yx ± y , k ≥ 4.
± if the product of the coefficients The singularity is further denoted A± or D k k of the monomials is ±1.
Zero level sets of the Ak : f = x2 ± yk+1 singularities
5. Contact: A2 The sphere is a sphere of curvature of radius 1/k1 or 1/k2
Contact with the blue sphere
Contact with the red sphere
4. Contact: A3 with the blue sphere The blue sphere of curvature has radius 1/k1 and b0 = 0. Recall the expansion of k1 along the blue line: P1 k1 (x) = k1 + b0x + x2 + . . . 2(k1 − k2)
P1 = 3b21 + (k1 − k2)(c0 − 3k13).
The contact point is called a ridge point: singularity and the blue curvature is • elliptic if P1 < 0 then this is a A+ 3 maximal along its line; singularity and the blue curvature • hyperbolic if P1 > 0 then this is a A− 3 is minimal along its line.
4. Contact: A3 with the blue sphere
Blue elliptic ridge point on the blue curve contact A+ 3 Maximum of k1
Blue hyperbolic ridge point on the green curve A− contact 3 Minimum of k1
4. Contact: A4 blue
The blue sphere of curvature has radius 1/k1, b0 = 0 and P1 = 0. The blue curvature has an infection along the blue line. This is a ridge turning point and the ridge changes from elliptic to hyperbolic.
4. Contact: at umbilics Umbilics are distinguished by the number of ridges passing through: • Elliptic or 3-ridge umbilic. k1 is minimal and k2 is maximal. • Hyperbolic or 1-ridge umbilic. There are curves passing through the umbilic on which k1 and k2 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
Figures from P. Giblin, 2 and 3 dimensional patterns of the face.
monstar
5. Medial Axis: Definition Let S be a closed surface embedded in R3 The medial axis MA(S) consists of the points of the open set R3 \S having two or more nearest points on S. The contact sphere must not intersect + singularities. the surface then its contact points are A+ or A 1 3
The MA is composed of several sheets, its boundary corresponds to some elliptic ridge point of the surface.
5. Medial Axis: Stratified structure
5. Medial Axis: missing elliptic ridge points The center of the sphere of curvature of a ridge point can fail to be on the boundary of the MA in two cases: • the surface is inside the sphere which cannot be maximal (e.g. min. of k2 > 0), • the surface intersects the sphere away from the ridge point.
I. Estimating Differential Quantities Using Polynomial Fitting of Osculating Jets II. Classification of umbilics and ridges III. Algorithm for umbilics and ridges detection 1. 2. 3. 4. 5. 6.
Preliminaries Umbilics Orientation and ridges Rough algorithm for ridges Correct ridges at umbilics Crest lines and filtered ridge points
1. Preliminaries: Input of algorithm S is a triangulated surface. Each vertex of S has a Monge patch, that is: • the normal vector, • the principal directions with an orientation, • the coefficients of the Monge patch (up to 4th order). Such a surface can be generated by: • a smooth surface and analytical calculus, • a triangulation and our jet fitting algorithm
Output of algorithm • Detect ridge color and type. • Find the right connection of ridges at umbilics and other crossings.
1. Preliminaries: Patches around vertices
Patch = topological disk neighborhood around a vertex It is a union of triangles: • initialized with the 1-ring, • then add the nearest incident triangle without changing the topology of the patch (priority queue).
2. Umbilics 1. Define a patch around each vertex, 2. collect vertices minimizing (k1 − k2) on their patch, 3. compute the index on the contour of the patch.
Moreover, 3rd order Monge coefficients gives: • the number and the directions of the limiting principal directions, • the number of ridges crossing at the umbilic, hence the type elliptic or hyperbolic of the umbilic.
3. Ridge point and Orientation of an edge b>0 Sign of b0 on an oriented blue line crossing a blue ridge P1 k1(x) = k1 +b0x + x2 +. . . 2(k1 − k2)
Ridge max of k1 b=0 b0 R2
b0>0 V1
R1 b0>0
V2
Blue elliptic ridge (Max of Kmax)
4. Rough algorithm for ridges
Models of 40k points.
5. Correct ridges at umbilics • Define patches around umbilics so that 1 or 3 ridge points are detected on edges of the contour. • Connect these ridge points to the umbilic. Observation. 2 If the topology of the ridges is generic and all edges not in umbilic patches have a correct orientation, then the topology of ridges is recovered.
6. Crest lines and sharp ridge points A crest line is a ridge line of maximum of max(|k1|, |k2|). Crest lines are a subset of elliptic ridges.
One may see these lines as the visually most salient curves on a surface.
6. Crest lines and sharp ridge points If the principal curvatures are constant on a surface (for example a plane or a cylinder) then all points are ridge points.
Sharp ridge point: one can avoid ridge points with a small variation of curvature requiring P/(k1 − k2) to be greater than a threshold. P1 k1 (x) = k1 + b0x + x2 + . . . 2(k1 − k2)
6. Crest lines and sharp ridge points: Mechanical model
All ridges
Unfiltered crest lines
Filtered crest lines
6. Crest lines and sharp ridge points: Mechanical model
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