DETECTION OF RIDGES AND RAVINES BASED ON ... - CiteSeerX

International Journal of Shape Modeling, Vol. 1, No. 1 (1994) 13{21 cfWorld Scienti c Publishing Company

DETECTION OF RIDGES AND RAVINES BASED ON CAUSTIC SINGULARITIES ELENA V. ANOSHKINA Laboratory of Computer Science and Engineering, The University of Aizu Fukushima, 965-80, Japan ALEXANDER G. BELYAEV Center for Mathematical Sciences, The University of Aizu Fukushima, 965-80, Japan TOSIYASU L. KUNII Laboratory of Computer Science and Engineering, The University of Aizu Fukushima, 965-80, Japan Received 29 April 1994 Revised 6 June Communicated by Y. Shinagawa A new concept of ridges, ravines and related structures (skeletons) associated with a surface in three-dimensional space is introduced. The concept is based on the investigation of relationships between the singularities of the distance function from the surface and singularities of caustic of the normals to the surface. It involves both local and global geometric surface properties and leads to the eective procedure for ridges and ravines recognition. Keywords : Surfaces, skeletons, ridges, ravines, caustics, singularities.

1. Introduction

In this paper we study correlations between local and global surface properties. Due to the discovered relationships between the singularities of the distance function from a surface and singularities of caustic of the normals to the surface we gave an eective procedure for ridge and ravine recognition. Recent advances in pattern recognition, computer vision and impedance tomography have inspired fresh interest in such robust features of a smooth surface as its parabolic line, umbilics and others that have previously been known only to specialists. However, handling these surface features calls for nontrivial geometric ideas. Such visual and natural features of a surface as its ridge-ravine lines have only relatively recently come to be understood, following the development of the catastrophe theory and the associated branch of mathematics known as singularity theory by R. Thom, V. I. Arnold and others. 

Supported in part by International Science Foundation (Soros) grant. 1

The rst serious mention of the ridge-ravine lines on a surface as a view point invariant was done by A. Gullstrand [1] ninety years ago. These surface features turned out to be closely connected with such optical notations as caustics and caustic singularities (the intensity of light is greater near a caustic and still greater near its singularities). Gullstrand worked on the accommodation of the eye lens and created the necessary fourth order dierential geometry to explain the relevant optics. He was awarded the Nobel Prize for Physiology and Medicine in 1911 for this work. Since the description of the ridge-ravine lines is based on advanced dierential geometry and they were entirely disregarded in textbooks on dierential geometry, some of them were rediscovered a few times by researchers on structural geology [2], face recognition [3], and the interpretation of magnetic resonance scans of the surface of the brain [4, 5]. A short historic review and a detailed mathematical treatment concerning caustics, ridge-ravine lines, and related surface features may be found in very recent and excellent book of I. R. Porteous [6]. We also should note here that among applied mathematicians working in Japan industry the ridge-ravine lines are known as \principal curvature extremum curves" [7]. Another trend in shape analysis is concern with the famous medial axis transformation (MAT) introduced by H. Blum [8]. MAT is one of the earliest and probably the most widely studied systematic technique for global shape description and pattern recognition (see, as example, [9] on application of MAT to pattern recognition and [10] on 3D medial axis geometry for the nonsingular points of MAT skeletons). Our idea [16, 17] to de ne ridge and ravine lines in a purely geometric manner, so that no particular coordinate system is involved in the de nition, and by a manner useful for animation arose after the paper [18] where the extraction of geometric features is based on singularity theory and nets of primitives were used for garment wrinkles modeling and animation. Following the novel achievements of singularity theory [11, 12, 13] and visual geometry [14, 15] we developed the MAT approach and found the relationships between medial axis geometry (global surface features), caustic singularities, and ridge and ravine lines (local surface features). Roughly speaking, end points of the medial axis skeleton for a given surface are situated on caustic singularities and correspond to ridge-ravine lines on the surface. These investigations led us to new results in one of the most classical areas of mathematics: dierential geometry of curves on a surface. We don't present here numerical results and algorithms, they are expected to be published in our future works. The purpose of this paper is to provide mathematical tools for further study of the discovered relationships between global and local surface properties. Among the shape description motivations of the problem we mention the applications to pattern recognition [19, 9] and animation [18, 20]. A further motivation is given by recent applications of methods based on homotopy theory, Morse theory 2

[21], and especially the marked molecules and atoms method (originally developed for Hamiltonian mechanics [14, 15]) for coding complex shapes of natural objects [15, 22, 23].

2. Ridges, Ravines and Distance Function Singularities

Let us consider a piecewise smooth, oriented hypersurface M in the Euclidean space

Rd . Let us introduce the distance function from a point x of the Euclidean space

to M by

dist (x; M ) = yinfM kx ? yk: 2

The distance function to a piecewise smooth hypersurface is continuous, but generally not necessarily smooth, even if the initial hypersurface is smooth. Let us consider the set of all singular points of the distance function to a hypersurface. It turns out that this set of the singular points is a curvilinear polyhedron (hence admitting a CW -complex structure), whose dimension in general position is equal to dim M . We call this polyhedron an R-skeleton for M .

De nition 1. The R-skeleton of a given hypersurface M is the set of all singular points of the mapping dist (
; M ) : Rd ! R: The following proposition seems trivial, nevertheless, we formulate it as a theorem taking into account its signi cance.

Theorem 1.For each point x 2 Rd let us nd its closest neighbors on the hypersurface M . (There always exist at least one). The R-skeleton is the closure of all such points each of them has more than one closest neighbor.

Let us consider a closed hypersurface M bounding a gure F . The medial axis transformation (MAT) skeleton is de ned as the locus of centers of all maximum disks of F , that is, those disks contained in F but contained in no other disk in F . Thus, the R-skeleton concept is a generalization of the MAT skeleton concept [8]. We also note here that R-skeleton has a resemblance to the concept of spine (see [14], pp. 165{169). Following [16] (see also [9], pp. 224{230), we give a few simple examples. The R-skeleton of an ellipse is the linear segment between two foci. The R-skeleton of a parabola is the ray with the origin at the focal point and situated on the parabola symmetry axis. The R-skeleton of a polygon consists only of linear segments and parabolic arcs. The R-skeleton of an ellipsoid is the ellipse domain

x2 + y2 + z2 = 1; a > b > c; a2 b2 c2



 2 2 x y z = 0; a2 ? c2 + b2 ? c2 = 1 :

3

The R-skeleton of a paraboloid z = Ax2 + By2 ; A < B; is the parabola domain  2  2 y = 0; z  21B + BAB ? Ax : Let M be provided with an orientation n(p), where p is a point on M , and n(p) is the unit normal vector at the point p. In order to distinguish ridges and ravines, we also introduce the notions of a ridge-skeleton and a ravine-skeleton [16].

De nition 2. The ridge-skeleton of a given hypersurface M is the connected component of the R-skeleton to be pointed by the normal n(p). The ravine-skeleton

of a given hypersurface M is the connected component of the R-skeleton to be pointed by the normal ?n(p).

Of course in some cases, there exists only one connected component of the Rskeleton which is called either a ridge-skeleton or a ravine-skeleton in accordance with an orientation. If M is a closed manifold, then the hypersurface M separates Rd into two parts, and we can say that the ravine-skeleton is the ridge-skeleton for the complementary part. We call a point p of a polyhedron P singular if P is not a manifold at p; a singular point is internal if there is a set in P that contains p and is homeomorphic to an open (d ? 1)-dimensional Euclidean ball; otherwise we call the singular point boundary.

De nition 3. Pseudoridges and pseudoravines are the connected components of the exterior singular points of the ridge-skeleton and the ravine-skeleton respectively.

De nition 4. A ridge (ravine ) of a piecewise smooth hypersurface M in the Euclidean space Rd is the set of points on M that realize in mum of the distance

function at points in a pseudoridge (pseudoravine).

3. Caustics for Ridge and Ravine Recognition

Unfortunately De nition 4 is very complicated from the computational point of view. We need to nd another way to recognize the ridges and ravines on a given surface. It turns out that the introduced concepts of a ridge and ravine are closely connected with the concept of a caustic generated by the normals to a given hypersurface M .

De nition 5. The caustic is the surface generated by the centers of the principal curvatures of M .

One can see that there are therefore two caustics for a surface in a three-dimen4

sional space. Now let us remind a few basic notations from the Morse theory [21]. For a function on a manifold, a point is called critical i all partial derivatives are null. At such a point the function is thus approximated by a quadratic form based on the second order partial derivatives, the matrix of which is called the Hessian matrix or Hessian. A critical point is called a Morse critical point if the Hessian is nondegenerate at this point. Let us consider a family of distance functions (de ned on M ) from a point (considered as the parameter of the family) to a given surface M . The proof of the following theorem can be found in [11].

Theorem 2.The caustic is formed by the parameters (points) for which the distance function has non-Morse critical points.

Let us now apply caustics to ridge and ravine recognition. Let us consider a point in the R-skeleton. The distance function from this point to a point of M generally has two equal minima. Let us move this point to a pseudoridge (pseudoravine). When the point achieves a pseudoridge (pseudoravine), the two equal minima merge. It follows that a point in a pseudoridge (pseudoravine) corresponds to the distance function having non-Morse critical points, namely, a degenerate minimum. Therefore, all pseudoridges (pseudoravines) lie in caustics. Moreover, since the distance function has a degenerate minimum when the point (parameter) lies either on a pseudoridge or a pseudoravine, then the point belongs to caustic singularities. This gives us a key for explicit calculation of the ridges and ravines. Fig. 1 demonstrates the arguments given above. For practical reasons sometimes below we will use subscripts for partial derivatives notations. Without loss of generality we perform calculations for the three-dimensional case. Let M be a hupersurface in R3 de ned by the radius vector r = r(u; v) and let n = n(u0 ; v0 ) be the unit normal vector to M at the point with coordinates (u0 ; v0 ). We always can choose local coordinates u, v so that (ru ; rv ; n) is an orthonormal base at the point (u0 ; v0 ), and the tangent vectors ru , rv coincide with the principal directions corresponding to the principal curvature radii R1 , R2 respectively. Therefore, ru = k1 n and rv = k2 n where k1 = 1=R1 and k2 = 1=R2 are the principal curvatures of M at the point (u0 ; v0 ), and nu = ?k1 ru , nv = ?k2 rv by the Frenet formulas. Let us nd the singular points of the caustics. For a given smooth surface M , the equations of the caustics are

r = r(u; v) + Ri (u; v)
n(u; v); i = 1; 2: The equations for the singular points of the caustics have the form 

   @R @R i i ru + @u
nu  rv + @v
nv = 0;

5

i = 1; 2;

two equal minima degenerate critical point R−skeleton

caustic

pseudoridge (pseudoravine)

degenerate minimum

ridge (ravine)

Fig. 1. Zoo of distance functions.

where a  b is the crossproduct of vectors a and b. Using the Frenet formulas, we get

ru  rv = n; ru  n = ?rv ; ru  nv = ?ru  (?k2 rv ) = ?k2n; n  rv = ?ru ; n  nv = n  (?k2rv ) = k2ru ; nu  rv = (?k1ru )  rv = k1 n; nu  n = (?k1ru )  n = k1rv ; nu  nv = (?k1ru )  (?k2rv ) = k1 k2n: Thus, the equations for the singular points of the caustics has the form 

    2  R R R @R R @R R i i i i i i i n 1 ? R ? R + R R + ru @u R ? 1 + rv @v R ? 1 = 0; i = 1; 2: 2 1 1 2 2 1

It is evident that the rst term equals to zero. Since ru and rv are linearly independent, we obtain three equations, each of which determines points that correspond to the singular points in the caustics:

R1 = R2 ;

@R1 = 0; @u

@R2 = 0: @v

(1)

Thus we come to

Theorem 3.Pseudoridges and pseudoravines lie on caustic singularities. Ridges and ravines consist of the points on M where the principal curvatures are equal 6

(umbilic points) or the principal curvature has an extremum along the corresponding principal direction.

Let us note that not all such points lie in ridges or ravines. Actually, for an ellipse in a plane shown in Fig. 2

x2 + y2 = 1; a > b; a2 b2

the caustic (the evolute, or, in other words, the locus of the centers of curvature) is the astroid (ax)2=3 + (by)2=3 = (a2 ? b2 )2=3 with four semicubical cusps ?

?







?

?





a 1 ? b2 =a2 ; 0 and 0; b 1 ? a2 =b2 : ? ? 
But the ridge-skeleton is fy = 0; jxj < a(1 ? b2 =a2 )g, the points a 1 ? b2 a2 ); 0 are the? pseudoridges corresponding to the ridges (a; 0), and two other extreme ?
2
2 points 0; b 1 ? a =b do not correspond to any ridge or ravine.

ridge−skeleton

caustic

ridge

ridge

pseudoridges

Fig. 2. Example with an ellipse.

Nevertheless, situation in a plane is clear. Let us consider a simple closed curve M in a plane. Let M be oriented anticlockwise and be given by a vector function r = r(s) where s is the arclength parametrization of M . The curvature of M at a point s is the number 2 k(s) = d dsr(2s)
n(s); 7

where a
b is the scalar product of vectors a and b.

Theorem 4.The function k(s) has a local positive maximum at a point i the point

belongs to a ridge, and k(s) has a local negative minimum at a point i the point belongs to a ravine.

General situation in a plane is illustrated in Fig. 3.

ridge ravine−skeleton

. r (s)

pseudoravine ravine

n (s) ridge

ridge−skeleton

pseudoridges

ridge

Fig. 3. General situation in a plane.

The situation in a three-dimensional case is more complicated. We calculated the critical points of the caustics and the points of the surface that correspond to the singularities of the caustics. These points of the surface are umbilical points (where the principal curvatures are equal) and the points where the derivatives of the principal curvatures along the respective principal directions equal to zero (see (1)). But not all such points correspond to the ridges and ravines (see the example with an ellipse). Let kmax and kmin be the principal curvatures (kmax  kmin), vmax and vmin be the corresponding principal directions. Consider also Gaussian Kg = kmax
kmin and mean Km = 12 (k1 + k2 ) curvatures. Let us note that if Kg is positive in some region on M , then the caustics generated by the normals of this region lie at the same side of M . If Kg is negative, then the caustics lie at dierent sides of M . If 8

Kg = 0, then the normals of the region form only one caustic (the other one \moves to in nity"). By analogy with two-dimensional case, we give

De nition 6. A ridge is the set of points where kmax has a local positive maximum along vmax . A ravine is the set of points where kmin has a local negative minimum along vmin . Taking into account theorems 3, 4 and the arguments as above we imply the equivalence between this de nition of ridge and ravine curves and the previous one. Nevertheless, it still remains an open question. A ridge and ravine extraction procedure may be given as follows. We go through the points of M , extracting principal curvatures and corresponding principal directions. 1. Calculate the rst and the second fundamental forms at the point: I = E (u; v)du2 + 2F (u; v)du dv + G(u; v)dv2 ; II = L(u; v)du2 + 2M (u; v)du dv + N (u; v)dv2 ; E = ru
ru ; F = ru
rv ; G = rv
rv ; L = ruu
n; M = ruv
n; N = rvv
n: 2. Find the Gaussian, mean and principal curvatures ? M 2 ; K = 1 EN + LG ? 2MF ; Kg = LN m 2 EG ? F 2 EG ? F 2 p p kmax = Km + Km2 ? Kg ; kmin = Km ? Km2 ? Kg : 3. Find the principal directions vmax and vmin as the eigenvectors corresponding to the eigenvalues kmax and kmin respectively from the spectral problem 

Ek ? L Fk ? M Fk ? M Gk ? N



v=0

and return to 1. Thus, we get the data as a quadruple

(vmax (p); vmin (p); kmax (p); kmin(p)) where a point p ranges over M . Tracing each line of curvature (integral curves of the pricipial directions vmax and vmin) we nd points on a ridge and a ravine in accordance with De nition 6.

4. Directions for Further Research 4.1. Correlation between local and global surface properties

In [24, 25] edges (shape variational points of an image intensity) are detected as points where the modulus of the gradient vector is maximum in the direction of the 9

gradient vector. In [26], ridges and ravines of an image intensity are de ned as the points where the directional derivative of the image intensity function equals zero along the direction in which the second directional derivative of the image intensity function has either the greatest or smallest magnitude. We would like to note here surprising resemblances between these de nitions and our one. Possibly there exist skeletons (re ecting global image intensity function properties) correspond to these image intensity features (re ecting local properties). Such skeletons may be used to derive information about disposition of these features and may be applied for image coding.

4.2. Multiscale approach

The above introduced R-skeleton is very sensitive to surface shape noise changes. One way of reducing the eects of noise is oered by an application of multiscale transforms like wavelet transforms. It allows to reduce the eects of noise and to extract essential ridges and ravines which are de ned by setting relevant thresholds (see [24] and [25] where such approach was successfully applied for edge extraction and image coding).

Acknowledgements

We are grateful to Sergei Duzhin, Oleg Okunev and Inna Scherback for stimulating and helpful discussions.

References 1. A. Gullstrand, \Zur Kenntnis der Kreispunkte", Acta Mathematica 29 (1904) 59{100. 2. J. G. Ramsay, Folding and fracturing of rocks, (McGraw Hill, New York, 1967). 3. G. G. Gordon, \Face recognition from depth maps and surface curvature", in Proceedings of SPIE Conference on Geometric Methods in Computer Vision, San-Diego, CA (July 1991) pp. 234{247. 4. A. Gueziec, \Large deformable splines, crest lines and matching", Rapports de Recherche, No 1782, INRIA, Le Chesnay, France (1992). 5. J.-P. Thirion and A. Gourdon, `The Matching Lines Algorithm: new results and proofs", Rapports de Recherche, No 1881, INRIA, Le Chesnay, France (1993). 6. I. R. Porteous, Geometric Dierentiation for the intellegence of curves and surfaces, (Cambridge, Cambridge University Press, 1994). 7. M. Hosaka, Modeling of Curves and Surfaces in CAD/CAM (Springer, Berlin, 1992). 8. H. Blum, A transformation for extracting new descriptors of shape, in Symposium on Models for the Perception of Speech and Visual Form (M.I.T. Press, 1964). 9. T. Pavlidis, Structural Pattern Recognition (Springer, Berlin, 1977). 10. L. R. Nackman, Curvature relations in three-dimensional symmetric axes, Computer Graphics and Image Processing, 20 (1982) 43{57. 11. V.I. Arnold, A. N. Varchenko, and S. M. Gusein-Zade Singularities of Dierentiable Maps vols. I, II (Birkhauser, Boston, 1985). 12. V.I. Arnold, Catastrophe Theory (Springer, Berlin, 1986). 13. V. I. Arnold, V. V. Goryunov, O. V. Lyashko, and V. A. Vasil ev, Singularity Theory II, Classi cation and Applications, Dynamical Systems VIII, Encyclopedia of Mathe0

10

matical Sciences, vol. 39, ed. V. I. Arnold (Springer, Berlin, 1993). 14. A. T. Fomenko, Visual Geometry and Topology (Springer, Berlin, 1994). 15. A. T. Fomenko and T. L. Kunii, Topological Models for Visualization (in preparation). 16. E. V. Anoshkina, A. G. Belyaev, O. G. Okunev, and T. L. Kunii, \Garment Wrinkles Modeling, Mountains Shape Coding and Singularities in Variational Calculus", Technical Report 94-1-18 The University of Aizu, Japan. 17. E. V. Anoshkina, A. G. Belyaev, and T. L. Kunii, \Edges and Caustic Singularities", Technical Report 94-1-023, The University of Aizu, Japan. 18. T. L. Kunii and H. Gotoda, Singularity Theoretical Modeling and Animation of Garment Wrinkle Formation Processes, The Visual Computer 6 (1990) 326{336. 19. B. K. P. Horn, Robot Vision (M.I.T. Press, Boston, 1986). 20. Y. L. Kergosien, H. Gotoda, and T. L. Kunii, Bending and Creasing Virtual Paper, IEEE Computer Graphics and Applications 1 (1994) 40{48. 21. J. Milnor, Morse Theory (Princeton University Press, New Jersey, 1963) 22. Y. Shinagawa, T. L. Kunii and Y. L. Kergosien, Surface coding based on Morse theory, IEEE Computer Graphics and Applications 11(5) (1991) 66{78. 23. Y. Shinagawa, T. L. Kunii, A. T. Fomenko, and S. Takahashi, Coding of Object Surfaces Using Atoms, in Frontiers in Scienti c Visualization, ed. Larry Rosenblum (book in press). 24. J. Canny, A computational approach to edge detection IEEE Trans. Pattern Analysis and Machine Intelligence 15 (1986) 679{678. 25. S. Mallat and S. Zhong, Wavelet Transform Maxima and Multiscale Edges, in Wavelets and Their Applications, eds. Mary Beth Ruskai et al. (Jones and Barlett Publishers, Boston, 1992) 69{86. 26. R. M. Haralick, L. T. Watson, and T. J. Laey, The Topographic Primal Sketch, International Journal of Robotics Research 1 (1983) 50{72.

11