SUPERSHAPE RECOVERY FROM 3D DATA SETS. Y. D. Fougerolle. 1 ... as computer graphics, computer vision, and medical imagery. A recent extension with ...
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Y. D. Fougerolle, A. Gribok, S. Foufou, F. Truchetet, and M. A. Abidi, “Supershape Recovery from 3D Data Sets”, in Proc. IEEE International Conference on Image Processing ICIP2006, Atlanta, GA, pp. 2193-2196, October 2006.
SUPERSHAPE RECOVERY FROM 3D DATA SETS Y. D. Fougerolle1 , A. Gribok2 , S. Foufou1 , F. Truchetet1 , and M. A. Abidi2 1
Le2i Lab. UMR CNRS 5158, Université de Bourgogne, Le Creusot, FRANCE 2 IRIS Lab. The University of Tennessee, Knoxville, USA ABSTRACT
2. SUPERSHAPES
In this paper, we apply supershapes and R-functions to surface recovery from 3D data sets. Individual supershapes are separately recovered from a segmented mesh. R-functions are used to perform Boolean operations between the reconstructed parts to obtain a single implicit equation of the reconstructed object that is used to define a global error reconstruction function. We present surface recovery results ranging from single synthetic data to real complex objects involving the composition of several supershapes and holes.
Derived from the original superellipsoids formulation , a term mφ + 4 , m ∈ R∗ controls the number of symmetries. Instead of one shape coefficients, three shape coefficients noted n1 , n2 et n3 are now used. In polar coordinates, the radius of a supershape is defined as:
Index Terms— Geometric modeling, signal reconstruction, boolean functions
with a, b, and ni ∈ R+ . Parameters a > 0 and b > 0 control the size of the supershape. Extension to 3D is performed by the spherical product of two 2D supershapes similarly to superquadrics. As a notation convention, we use capital letters for the parameters of the second 2D supershape : N1 , N2 , and N3 for the shape coefficients, and M for the symmetry number. In [7, 8], for a unit supershape and m and M natural numbers, we have defined two implicit functions as:
1. INTRODUCTION Superquadrics have been widely studied in various fields such as computer graphics, computer vision, and medical imagery. A recent extension with rational and irrational degrees of symmetry, namely the supershapes, has been proposed by Gielis [1, 2]. Using R-functions allows more complex blends as in [3], the detection and the reconstruction of complex objects with holes, and topology changes during the process. Similarly to the techniques proposed in [4–6], we apply the Levenberg Marquardt algorithm to recover most of the parameters. We use a simple heuristic to determine the number of symmetries. The rest of the paper is as follows: the second section presents the formulations of the supershapes and a cost function to be minimized, the third deals with individual supershape reconstruction of synthetic data, the fourth section is dedicated to multiple supershape and holes reconstruction, the fifth section deals with R-functions and the global error measure. We then present our conclusions and future work. Compared to previous work in [7, 8], where we applied supershape and R-functions for solid modeling purposes and surface polygonization, the focus of this paper is surface reconstruction of 3D data sets. The original contributions of this paper are an extension of superquadrics recovery techniques to supershapes and the development of a reconstruction method based on the combination of individual supershape recovery and R-functions.
,(((
1 r (θ) = mθ n2 1 , n1 1 + sin mθ n3 a cos 4 b 4
F1 (x, y, z) = 1 −
x2 +y 2 +r12 (θ)z 2 , r12 (θ)r22 (φ)
1 F2 (x, y, z) = 1 − r2 (φ)
cos2
(1)
(2)
x2 + y 2 + z 2 , φ (r12 (θ) − 1) + 1
(3)
with angles θ and φ defined by: ⎧ θ = θ (x, y) = arctan xy ⎪ ⎪ ⎨ φ = φ (x, y, z, n1 , n2 , n3 ) = arctan zr1 (θ)ysin(θ) .
⎪ ⎪ ⎩ = arctan zr1 (θ) cos(θ) x
Both equations generate a scalar field that is positive within the supershape and negative outside. The surface corresponds to the zero-set of the scalar field. The second equation, based on the notion of radial distance, is isotropic. Figure 1 presents examples of supershapes for various shape coefficients. 2.1. Cost functions The method introduced by Solina and Bajcsy [4] has been used in literature as a standard for superquadrics recovery.
,&,3
8.2
55
8
50 Rayon(mm)
Rayon(mm)
7.8 7.6 7.4 7.2 7 6.8
45 40 35 30 25
6.6 6.4
20 -3
-2
-1
0
1
2
3
-3
Angle(rad)
(a) (a)
(b)
(c)
The problem is stated as an mean square optimization problem of a well chosen cost function. During an iterative process, the recovered supershape evolves to obtain a minimal value of the cost function. Another cost function, based on the notion of radial euclidian distance, has been introduced by Gross and Boult [5]. In an experimental comparison, Zhang et al. [6] conclude that the functions introduced by Solina are efficient to precisely capture the size of the object but are not as efficient as radial functions to capture the shape. Following these conclusions, we define a supershape cost function by: Err (Λ) =
-1
0
1
2
3
Angle(rad)
(c)
(d)
(d)
Fig. 1. Supershapes with various shape coefficients and symmetry numbers. a) and b) m = 8, c) and d) m = 5.
n
(b)
-2
2
(F2 (Pi )) ,
(4)
i=1
where F2 (x, y, z) is the equation defined in Eq. 3, Pi is a 3D point Pi = (xi , yi , zi ), and n the number of 3D points. 2.2. Symmetry detection Since the implicit equations for supershapes require parameters m and M to be natural numbers, the Levenberg method cannot be applied to estimate those parameters. Among recent symmetry detection methods, Sun et al. [9,10] determine reflexion symmetries by using gradient information and Shen and Perry [11] use generalized complex moments to detect and locate symmetries. In our case, we deal with objects that may be noisy or incomplete. So these sophisticated methods are not well adapted to our goals, because we only need the number of symmetries. Localization and orientation will be adjusted during the optimization process. Fortunately, the nature of 3D supershapes can be used for a simpler symmetry number detection: the axis (Oz) is a rotation symmetry axis, which implies for all orthogonal planar sections to this axis, that the symmetry number remains constant. For each planar section, in polar coordinates, the data can be represented as a curve r(θ), as illustrated on figure 2 and symmetry numbers correspond to the number of extrema of r(θ). For incomplete data, some data are missing and the curve is not continuous. We then work by intervals to locate potential extrema: we keep detected extrema on continuous intervals and we seek in other sections if there is any additional information to reconstruct the complete curve. After a full sweep of the data, the curve may still be incomplete and the symmetry numbers can not be precisely determined. In this case, we set m and M to 4 to make them transparent in the formulation in Eq 1.
Fig. 2. Symmetry detection for incomplete data sets. a) and c) Initial data. b) and d) Curves r = f (θ) after a sweep along the data main inertia axis.
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 3. Synthetic data reconstruction. Numerical values are presented in table 1. 3. RECONSTRUCTION OF SYNTHETIC DATA We have to verify that Levenberg-Marquardt method makes the shape of an initial supershape to evolve to represent a given synthetic cloud of points. The initial guess is an ellipse because it is a common shape configuration for all symmetry numbers. Figure 3 illustrates the result of the recovery of synthetic supershape data sets. Initial and recovered shape coefficients are presented in table 1 and several results require further attention: • Recovered shape parameters are not necessarily close to the initial parameters as illustrated by figures 3(a), 3(b), and 3(d). • Reconstruction error significatively increases as shape parameters tend to zero. For incomplete data, an infinity of shapes can represent the data. To obtain a unique set of parameters, the term a1 a2 a3 is introduced to recover minimal volume supershapes. The set of parameters is extended by translation, rotation, and scale parameters. Figure 4 illustrates reconstruction results for partially incomplete data. The color coding represents the value of the cost function: in blue the error is null and maximum in red. We see the recovered supershape fits the data and keeps a minimal volume. The cost function defined in Eq. 4 becomes:
(a) 100%
(b) ∼ 80%
(c) ∼ 40%
(d) ∼ 25%
Fig. 4. Incomplete data reconstruction. a), b), c), and d) Partial data (in % of initial data).
figure 3(a) 3(b) 3(c) 3(d) 3(e) 3(f)
n1 50/1300 2/1880 1/0.8 100/0.79 100/86.87 0.5/0.44
n2 50/1300 2/1.24 1/1.25 25/2.90 100/86.85 0.5/0.77
n3 50/1300 2/1.22 1/1.25 100/1.69 100/86.85 0.5/0.77
N1 50/1400 50/49 1/1.03 5/4.24 100/153.2 0.5/0.48
N2 50/1300 50/49 1/1.03 5/4.60 100/153.6 0.5/0.48
N3 50/1300 50/49 1/0.92 5/4.41 100/152.2 0.5/0.44
Error 0.04 0.001 1.35 3.54 0.02 32.5
Table 1. Recovered shape parameters for objects in figure 3. Notation: Recovered / Initial values.
Err (Λ) = a1 a2 a3
n
2
(F2 (Pi )) .
(5)
i=1
(a)
4. MULTIPLE SUPERSHAPE RECONSTRUCTION We present a method to represent complex objects with several supershapes. Results using real data are presented in figures 6 and 8 for a waterneck model. Our algorithms proceeds in three steps, as illustrated in figure 5. The input is a segmented mesh by the technique used by Zhang for superquadrics [12]. The output is a single implicit representation. To globally evaluate our reconstruction error, we use R-functions [13]. Since segmented data can be seen as the union of several parts, the union of the reconstructed parts can be transcribed into a single analytical equation using Rfunctions. In this paper we consider the RP -function Rp=2 defined by: 1
Rp : x + y ± (xp + y p ) p ,
(6)
for all even positive p.
(e)
(b)
(c)
(f)
(g)
fj (P ) = (Fj ∧ Bj ) (P ) .
F (P ) =
p
fj (P )
(7)
j=1
Terms fj correspond to the recovered supershapes implicit equations. unfortunately, the union of reconstructed supershapes does not necessarily correspond to the data: in some cases, some parts of the recovered supershapes have to be removed. To solve this problem, we define truncated supershapes as the intersection between a reconstructed supershape and the oriented bounding box of the data by:
(8)
The term Bj corresponds to the equation of the oriented bounding box and is represented by a supershape, the reconstructed object is defined by the union of all the reconstructed truncated supershapes by: n i=1
The final object corresponds to the union of all the truncated supershapes and is defined by:
(h)
Fig. 6. Supershapes union. a), b), c), and d) reconstructed supershapes. e), f) Corresponding truncated supershapes, g) result without truncation, h) correct result.
Err(Λ) =
Fig. 5. Reconstruction flowchart
(d)
⎛ ⎝
p
⎞2 fj (Pi )⎠ .
(9)
j=1
Truncated supershapes are necessary to avoid non desired parts to appear when the union of the supershapes is performed as illustrated in figure 6. Approaches proposed by Solina and Bajcsy [4], Jakli˘c et al. [14], and Chevalier et al. [15] do not deal with this issue because the reconstructed superquadrics are contained within the cloud of points. 4.1. Reconstruction of objects with holes Traditional approaches iteratively insert objects until a desired accuracy is reached, which often requires large number of superquadrics that are fused using heuristics to simplify the final representation [4, 15]. Similarly to the approach proposed by DeCarlo et al. [3], we handle negative objects to represent holes and topological changes of the reconstructed object. A hole can be seen as the intersection between an envelop and the outside of another object. To detect holes, we
6. REFERENCES
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(b)
(c)
(d)
[1] J.Gielis, “A generic geometric transformation that unifies a wide range of natural and abstract shapes,” American Journal of Botany, vol. 90, pp. 333–338, 2003.
(e)
Fig. 7. Hole detection and reconstruction. a) Initial data. b) Reconstruction error using one supershape. c) Reconstructed hole. d) Reconstructed envelop. e) Final result.
[2] J. Gielis, B. Beirinckx, and E. Bastiaens, “Superquadrics with rational and irrational symmetry,” in Proceedings of the Eighth ACM Symposium on Solid Modeling and Applications 2003, Seattle, Washington, USA, June 2003, pp. 262–265. [3] D. DeCarlo and D. Metaxas, “Shape evolution with structural and topological changes using blending,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 20, pp. 1186–1205, 1998.
(a)
(b)
(c)
[4] F. Solina and R. Bajcsy, “Recovery of parametric models from range images: The case for superquadrics with global deformations,” The Visual Computer, vol. 12, no. 2, pp. 131–1476, 1990.
(d)
Fig. 8. Waterneck model reconstruction: a) Initial mesh, b) Segmented mesh, c) Reconstructed mesh, d) Reconstruction error. use the reconstruction error: points belonging to the borders of the cavity will introduce an important reconstruction error and can be clustered into a secondary cloud of points. We then deal with two data sets: the first corresponds to the envelop of the data, and the second to the hole that has to be reconstructed, as illustrated in figure 7. 4.2. Complex object reconstruction On figure 8,we present results on a real waterneck model composed of eight truncated supershapes. We can see that hexagonal objects are correctly reconstructed. On figure 8(d), we see two error regions. The first area corresponds to data that are within the reconstructed object. The second area presents a smaller but more diffuse error, and corresponds to the borders of the reconstructed supershape that cannot represent precisely the data. Although the symmetry numbers are correctly detected, the algorithm converges to a local minimum: some shapes cannot be accurately represented by one supershape. 5. CONCLUSIONS AND FUTURE WORK The method presented in this paper is inspired from recovery techniques for superquadrics proposed by Solina and Bajcsy [4], Gross and Boult [5], Zhang et al. [12], Jakli˘c et al. [14], and Chevalier et al. [15]. We propose a cost function based on a radial implicit function of supershapes and a simple heuristic to detect symmetry numbers to reconstruct truncated individual supershape with minimal volume. Using R-functions, we obtain one global reconstruction error to characterize the final result. Future work includes the development of cost functions, the comparison with other optimization techniques, such as Nelder-Mead, genetic, or simulated annealing algorithms.
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