Implicit Fitting Using Radial Basis Functions with Ellipsoid Constraint

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1 Volume 3 (4), number pp COMPUTER GRAPHICS forum Implicit Fitting Using Radial Basis Functions with Ellipsoid Constraint Q. Li, D. Wills, R. Phillips, W. J. Viant, J. G. Griffiths and J. Ward Department of Computer Science, University of Hull, Hull, HU6 7RX, UK Abstract Implicit planar curve and surface fitting to a set of scattered points plays an important role in solving a wide variety of problems occurring in computer graphics modelling, computer graphics animation, and computer assisted surgery. The fitted implicit surfaces can be either algebraic or non-algebraic. The main problem with most algebraic surface fitting algorithms is that the surface fitted to a given data set is often unbounded, multiple sheeted, and disconnected when a high degree polynomial is used, whereas a low degree polynomial is too simple to represent general shapes. Recently, there has been increasing interest in non-algebraic implicit surface fitting. In these techniques, one popular way of representing an implicit surface has been the use of radial basis functions. This type of implicit surface can represent various shapes to a high level of accuracy. In this paper, we present an implicit surface fitting algorithm using radial basis functions with an ellipsoid constraint. This method does not need to build interior and exterior layers for the given data set or to use information on surface normal but still can fit the data accurately. Furthermore, the fitted shape can still capture the main features of the object when the data sets are extremely sparse. The algorithm involves solving a simple general eigen-system and a computation of the inverse or psedo-inverse of a matrix, which is straightforward to implement. Keywords: implicit curve, implicit surface, implicit fitting, radial basis function, ellipsoid constraint. ACM CCS: I.3.5 [Computer Graphics]: Curve, surface, solid, and object representations. Introduction Fitting a curve or a surface to a set of data points is a basic task in various areas in computer science. Basically, there are two ways to represent a surface of an object, either parametrically or implicitly. Mathematically, a parametric surface is expressed as a mapping S : R R 3, and an implicit surface is expressed as the zero contour of a mapping F : R 3 R. Both types of representation have their advantages and disadvantages. In general, the advantages of one representation are just the drawbacks of the other. Parametrically represented surfaces have advantages in drawing, tessellating, subdividing and bounding. This has made the parametric surface the primary choice in computer graphics modelling, and computer aided design. However, implicitly represented shapes have advantages over parametric shapes in several aspects. First of all, when objects are modelled as implicit surfaces, one can tell directly whether a point lies inside or outside the shape and the problem of boundary detection can be easily solved [7]. Secondly, surface normals are easy to compute. Another advantage of implicitly represented shapes is that the most commonly used geometric shapes such as spheres, cylinders, ellipsoids take very simple forms. Finally, the value of an implicit function at a point can be used as an approximate measure of the distance from the point to the surface. This property of implicit surfaces is particularly useful in the problem of data matching. Fitting implicit curves and surfaces to a set of scattered points has gained increasing attention in recent years and has been widely applied in many areas of computer science including computer aided design, computer graphics modelling, computer graphics animation [,,4,,9,,,3,4], computer assisted surgery [5,5,6,37], computer vision, and artificial intelligence [7,,3,5,6,33,3,34,38,3]. c The Eurographics Association and Blackwell Publishing Ltd 4. Published by Blackwell Publishing, 96 Garsington Road, Oxford OX4 DQ, UK and 35 Main Street, Malden, MA 48, USA. 55 Submitted July Revised December Accepted November 3

2 56 Li et al / Implicit Fitting Using Radial Basis Functions with Ellipsoid Constraint Implicit surfaces can be classified either as algebraic or non-algebraic. An implicit surface is called algebraic if it can be represented as the set of all real roots of a polynomial. Significant achievements have been made in recent years for algebraic surface fitting [,3,33,34,3]. However, algebraic surfaces have drawbacks in representing general geometric shapes. In general, linear and quadratic algebraic surfaces are very limited in representing general geometric shapes. On the other hand, when high degree polynomials are used, the fitting is usually very unpredictable in shape and very uncontrollable in size, even for shapes represented by cubic algebraic equations. Secondly, it is highly likely that the fitted shape might poorly represent the actual shape from which the data have been sampled. In practice, it is not difficult to fit a set of points with an implicit shape with a high degree of accuracy, but the fitted shape might contain too many redundant parts which are not close to any data points. In addition, the shape fitted might be much more complex than the actual shape that the data represent. The fitted shape might be unbounded and multiple-sheeted. It may have holes or knots. A brief survey of the implicit polynomial fitting problem has been given by Savchenko et al. in [8] and some discussions on their merits can be found in []. In addition to algebraic surfaces, another implicit surface modelling technique using metaballs (also known as blobby objects), is also very popular. In this technique, an object to be modelled is regarded as a collection of blobs. Each blob, or more properly, metaball, is modelled as a particle with a potential field of influence, where the potential energy attributed to the particle decreases with distance from the particle s location. In Blinn s pioneer work [], a metaball was modelled as be ar, where r is the distance from the particle to a location in the field. However, there are several variations on this model. In general, the influence field is modelled by a function (r) of distance r. In addition to Blinn s model, where (r) = e ar (often referred to as a Gaussian radial basis function), some other popular choices for function are as follows:. (r) = r, (linear radial basis function);. (r) = r log r, (thin-plate splines); 3. (r) = r + c, (c > ), (multiquadrics). All these functions are commonly referred to as radial basis functions. It has been known for a long time [5] that a blend of two implicit surfaces can be simply defined by their sum. Therefore, an implicit surface defined by a set of blobs with centres {P i } i= n can be modelled by the function n λ i i (r), () i= where i (r) corresponds to the metaball with centre P i. In addition to geometric modelling, radial basis functions are also popularly applied to data fitting in neural networks and the theory of functional approximation. One primary advantage of radial basis functions is their invariance under Euclidean transformation and their adjustable local influence when used for implicit modelling. Recent research has shown an increased interest in using radial basis functions to reconstruct implicit surfaces from scattered points [5,6,4,,8,35,36]. In these techniques, the candidate functions for representing an implicit surface take the form F(P) = n λ i i ( P P i ) + P(P), () i= where P = (x, y, z) represents a general point, and P(P) is a low degree polynomial with respect to variables x, y, z. In practice, polynomials P(P) can be chosen as constant, linear, and quadratic. In general, the degree of polynomial P(P) should not be chosen larger than four. As the degree of the polynomial increases, it becomes increasingly difficult to control the fitting results []. In (), when the polynomial term is not present or chosen to be constant, the surface represented by F(P) = will be purely a blobby model. But in the problem of data interpolation, it has been suggested that a linear polynomial term should be used to obtain the smoothest interpolant for certain type of radial basis functions [6,9,8]. However, the smoothness of the function F(P) and the smoothness of the implicit shape described by F(P) = are different things. The smoothness of function F(P) is not a necessary condition of the smoothness of implicit shape represented by F(P) =, for one geometric shape, when represented implicitly, can be described by many different equations. In addition, from the point of view of geometric intuition, using a linear polynomial might not be an ideal choice in implicit fitting. In fact, the implicit surface represented in F(P) =, where F(P) is in form (), can be regarded as a blending of two implicit surfaces, one of which is modelled by the polynomial part, and the other by radial basis functions. In practice, blending two surfaces can be considered as an action of deforming one surface according to another. In the fitting model (), if we regard the implicit surface represented by the polynomial part as the partly finished product of the final fitting, then blending the two parts by summing the two functions is equivalent to deforming the partly finished shape into the desired shape using a set of blobs. With this view in mind, it appears much more reasonable to constrain the coefficients of the polynomial such that it always represents a bounded algebraic surface such as a sphere or an ellipsoid. In this paper, we consider a fitting model of form F(P) =, where F(P) takes form () and the polynomial part P(P) in F(P) is constrained in such a way that it always represents an ellipsoid. Compared with the technique developed in [8], the presented technique is a one-step fitting based on solving a general eigen-system. Unlike the technique presented in [6], c The Eurographics Association and Blackwell Publishing Ltd 4

3 Li et al / Implicit Fitting Using Radial Basis Functions with Ellipsoid Constraint 57 the implicit shape constructed using our technique is obtained directly from the given scatted surface points. Neither information on surface normal nor the building of the interior and the exterior layers of the data set are required. This technique is particularly suitable for the cases when data are known from a closed object and the data is relatively sparse and is not dense enough to construct a proper interior or exterior layer. The remaining parts of this paper are organized in the following way. In section, we introduce our one-step fitting technique. Experiments and some comparisons with the fitting model where the polynomial part is constant or linear are given in section 3. In section 4, implicit curve fitting based on the ellipse constraint is introduced in parallel with our implicit surface fitting model. In Section 5, we give a brief summary of the method and related future work.. Surface Fitting Using Radial Basis Functions with Ellipsoid Constraints In this section, we discuss some fitting techniques using radial basis functions. For the reader not familiar with these functions, we recommend Buhmann s paper [3] for a good introduction. Our investigation shows that the bounded polynomial part in the fitting model () is more suitable for fitting data from a closed surface... Implicit Fitting Using Radial Basis Functions Let {P i } i= n be a set of points from the surface of an object. We want to fit a function of form () such that F(P i ) =, i =,,...,n. (3) Based on our discussion in the first section, we are only interested in the fitting model where the polynomial part is always associated with a bounded algebraic surface. Note that the zero set of a polynomial of odd degree will never be bounded. The degree of polynomial part involved in our model should be even, say,,, 4. In practice, polynomials with degree up to 4 will be sufficient []. Let V represent the space with basis functions B ={ ( P P i ), i =,,...,n}, and V represent the space consisting of all polynomials with basis functions B ={P j (P), j =,,...,m}, where P j is a polynomial. Then all candidate fitting functions of form () can be decomposed into two parts by projecting them onto the two spaces. We assume that for any polynomial basis function P j (P) inv, n λ i P j (P i ) =, (4) i= where λ i is the projection of the candidate fitting function F(P) onthei th radial basis function ( P P i ). For the candidate function F(P), let its unknown projections onto the space V be v = (v,v,...,v n ) T, and its projection onto the space V be Then, F(P) = v = (v,v,...,v m ) T. n m v i ( P P i ) + v j P j (P). i= j= If all data points {P i } i= n lie on the implicit surface F(P) =, we will have for each k, k =,,..., n, n m F(P k ) = v i ( P k P i ) + v j P j (P k ) =. i= All these equations in conjunction with condition (4) can be rewritten in matrix form as ( )( ) ( ) A B v B T =, (5) v where matrix A = (a ij ) n n with a ij = ( P i P j ), and matrix B = (b ij ) m n with b ij = P j (P i ). Let matrix and let M = ξ = j= ( A ) B B T ( v v (6) ). (7) Then fitting the data set {P i } i= n with F(P) in () such that F(P i ) =, i =,,...,n, is equivalent to solving the linear system Mξ =. (8) To avoid the trivial solution, constraints on ξ are required. In general, the constraint can be put in the form ξ T Dξ =, (9) where D is a semi-positive definite matrix. One natural choice for matrix D is the identity matrix, which weights each term in () equally. In general, there might be no ξ satisfying both c The Eurographics Association and Blackwell Publishing Ltd 4

4 58 Li et al / Implicit Fitting Using Radial Basis Functions with Ellipsoid Constraint () and (9). Thus, it is more appropriate to put the problem in the following way: min ξ T M T Mξ = subject to ξ T Dξ =. () ξ The solution to this system can be found by solving the following generalized eigen-system: { M T Mξ = λdξ () ξ T Dξ =. When D is chosen as the identity matrix, the solution will be the unit eigenvector ξ associated with the smallest eigenvalue for the following eigen-sytem: M T Mξ = λξ. () However, in this research, we are only interested in those matrices D, such that the polynomial part always represents a bounded implicit surface under the constraint ξ T Dξ =... Implicit Fitting Using Radial Basis Functions with bounded Algebraic Surface Constraints As has been pointed out, the implicit shape represented by a linear combination af(p) + bg(p) = of two functions F(P) and G(P) (where a, b are real numbers) corresponds to the blending of the implicit shape F(P) = and the implicit shape G(P) =. This blending, from the point of view of shape deformation, can also be understood as deforming the implicit shape F(P) = by the implicit shape G(P) =, or vice versa. With this observation, we could see that, in implicit shape fitting, it would be more efficient to constrain the polynomial part in fitting model () such that it is always bounded and can roughly describe the shape the data actually represent. Except for the case where the data can be best represented explicitly by a function of form z = f (x, y), or of form h = f (x, y, z), the most obvious constraint could be to require that the polynomial always represents a bounded algebraic surface. In general, bounded algebraic surface fitting often involves an iterative optimization procedure with nonlinear constraints [,3,34]. Despite its computational cost, the procedure does not always produce a natural fitting in size, shape, and topology for high degree polynomials. It would be much more desirable just to constrain the algebraic surface defined by the polynomial part such that it is bounded and simple, say, an ellipsoid or a superellipsoid. However, there seems to be no simple mathematical description for bounded algebraic surfaces, especially when the degrees of polynomials are high. Even for the quadratic surfaces, we still need several highly nonlinear conditions to force a quadratic surface to be the best fitted ellipsoid for a given data set. To avoid a heuristic optimization procedure involving multiple nonlinear constraints, we shall consider the fitting model in form (), where the polynomial part always corresponds to an ellipsoid. The primary reason for making such a choice is the computational efficiency. As can be seen in the following section, the proposed fitting is just a one-step fitting. It is easy to implement and solve. On the other hand, from the point of view of shape deformation, the partly finished model defined by the polynomial involved in the fitting does not have to be very close to the actual shape represented by the data..3. Direct Implicit Fitting Using Radial Basis Functions with Ellipsoid Constraints Let P(x, y, z) represent a point. Consider equation P(P) = ax + by + cz + fyz+ gxz + hxy + px + qy + rz + d =. (3) It is well known that this equation represents either a central quadric or a paraboloid, including their degeneracies [3]. Let and I = a + b + c, (4) J = ab + bc + ac f g h, (5) a h g K = h b f g f c, (6) (7) It is known that I, J, K are invariant under rotation and translation and that equation (3) represents an ellipsoid if and only if J >, and IK > []. However, the use of the above conditions as the constraints for ellipsoid fitting will inevitably lead to an iterative optimization procedure with multiple nonlinear constraints. To develop a direct fitting method, we investigate whether it is possible to use one simple constraint to characterize a large family of, if not all, ellipsoids relating to the invariants I, J, K. It has been shown in [5] that when 4J I >, (8) equation (3) must represent an ellipsoid. On the other hand, for an ellipsoid, when the length of the short axis is at least half the length of the long axis, then 4J I must be positive. Although 4J I > is only a sufficient condition to guarantee that an equation of second degree in three variables represents an ellipsoid, a simple search procedure can be developed easily for ellipsoid fitting based on the fact that for any ellipsoid, there exists a real number α 4 such that α J I > since J > for all ellipsoids [5]. However, c The Eurographics Association and Blackwell Publishing Ltd 4

5 Li et al / Implicit Fitting Using Radial Basis Functions with Ellipsoid Constraint 59 in our fitting, the ellipsoid corresponding to the polynomial part only serves as a partly finished shape, we do not have to consider all ellipsoids. The family of those ellipsoids that satisfy condition 4J I > will be large enough for our fitting purpose. To understand the geometric meaning of the invariant 4J I for an ellipsoid, we transform equation (3) rigidly into the following standard form: Ax + By + Cz + Px + Qy + Rz + D =, (9) where A, B, C are the roots of the characteristic equation Set u 3 Iu + Ju K =. () 4J I ρ = {A + B + C }. Then ρ is an invariant under rotation and translation. It can be shown that ρ, and ρ = if and only if A = B = C, when the equation (3) defines a sphere. It can be further observed that the value of ρ tends to when one of the roots tends to infinity or when two of the roots tend to zero. In this case the corresponding ellipsoid will be flat shaped. Thus, we can see that the value of ρ can be used to measure the roundness of an ellipsoid. The larger the value of ρ, the more nearly spherical the quadric. Conversely, the smaller the value of ρ, the flatter or the longer and thinner the ellipsoid. Now we assume that the polynomial part in () is quadratic and has the form P(P) = d + px + qy + rz + ax + by + cz + fyz+ gxz + hxy. In this situation, the corresponding parameters v in (5) will be v = (d, p, q, r, a, b, c, f, g, h) T. Let {P i (x i, y i, z i )} i= n be the set of points to which a model of form () needs to be fitted, then the corresponding matrix B in (5) will take the form: where B = (X, X, X 3,...,X n ) T, () X T i = (, x i, y i, z i, x i, y i, z i, y i z i, x i z i, x i y i ), () corresponding to point P i (x i, y i, z i ), i =,,...,n. The fitting problem then becomes solving equation (8) subject to the condition 4J I >. That is: Solve linear equation system Mξ = subject to 4J I =. (3) If we define matrix C as: C = Then 4J I = (a, b, c, f, g, h)c (a, b, c, f, g, h) T. (4) If we further define ( ) C =, (5) C then 4J I = can be rewritten as ξ T Cξ =, (6) where ξ is defined by (7), and,, are zero matrices of size (n 6) (n 6), (n 6) 6, and 6 (n 6) respectively. Note that most elements in matrix C are zero except for some elements in matrix C. If we write matrix M defined in (6) and vector ξ defined in (7) as block matrices: ( M M M = M T M ),ξ = ( α β ), (7) where matrices M, M, M are of size (n 6) (n 6), (n 6) 6, 6 6 and vector α, β are of size n 6 and 6, then the constrained linear system (3) can be further simplified to { M α + M β = M T α + M β =, subject to β T C β =. (8) When M is nonsingular, solving equation (8) is equivalent to first solving for β from the constrained linear equation system Dβ = subject to β T C β =, (9) where D = M M T M M = M T M M as M is the zero matrix. Once β is obtained, α can be obtained immediately from equation M α + M β =. When D is nonsingular, there is no solution for such an over constrained system given in (9). Normally, it is recast as a minimization problem. Note that matrix D is symmetric, and in practice, it is often positive definite. Therefore, we can approximate β as min β T Dβ subject to β T C β =, (3) β This problem can then be recast as the following generalized eigen-system using Lagrange multipliers: { Dβ = λc β (3) β T C β =. c The Eurographics Association and Blackwell Publishing Ltd 4

6 6 Li et al / Implicit Fitting Using Radial Basis Functions with Ellipsoid Constraint Note that C in (4) has eigenvalues {,,, 4, 4, 4}. Using the same inference as given in [8], we state that equation (3) has only one solution when matrix D is positive definite, which is the general eigenvector associated with the unique positive eigenvalue of the general eigen system Dβ = λc β. In fact, when D is positive definite, it follows immediately that if β is the solution of the general eigen system (3), then we must have β T Dβ = λβ T C β = λ>. Therefore, the solution is unique. However, for general radial basis functions, matrix D in (9) is not necessarily positive definite [3]. When D is not positive definite, it can be replaced with D T D, which is at least semi-positive definite. The solution will be the eigenvector associated with the smallest nonnegative eigenvalue. In this case, the uniqueness of solution is not guaranteed, as the smallest nonnegative eigenvalues can be multiple. Remark. When M is almost singular, M can be replaced with its pseudo-inverse M and the corresponding solution for α can be replaced with M M β, which has the following properties [8]:. It is the least squares solution of M α = M βαwhen there is no solution.. It is the unique solution when there is exactly one solution. 3. It is the minimum norm solution when there are an infinite number of solutions. 3. Experimental Results In this section, we provide some experimental results to demonstrate the strengths and the characteristics of our fitting technique. Two types of data sets have been used. The first type of data consists of clouds of points from the surfaces of some actual 3D objects. The second type of data contains clouds of points from the surfaces of mathematically defined objects. There are four data sets of the first type. Three of these data sets are sampled from the surfaces of actual human bones, namely the femur, tibia, and patella. The fourth data set of this type is sampled from the head and shoulders of an actual human body (see Figures ). In all these cases, the data sets are relatively sparse with respect to the sizes of the objects they represent. Second, the information for surface normals at sample points is not known. Therefore, the technique presented in [6] cannot be directly applied. To evaluate the performance of the fitting technique presented in this paper, a cloud of points generated from a torus is also considered. To highlight our ellipsoid constrained radial basis function fitting, the fitting methods derived for the cases when the polynomial component is chosen to be linear or absent are also considered. To avoid the trial solution in the case where the coefficients are all zeros, constraints are required for these methods. In our experiments, we require that the norm of patella data tibia data femur data 3 head data 4 Figure : Surface data sampled from actual human bones and the human body. the coefficient vector in () is unit. Figure shows that when the polynomial part is not present in the fitting model, it works quite well for data sets from bone surfaces, though there are small holes present in the fitted surfaces for femur data and tibia data. However, it cannot provide a meaningful representation for the head and shoulder data. Compared with the pure blobby model, the fitting using the linear polynomial part appears even inferior to the pure blobby model. The addition of this part in the fitting model does not make any contribution to the fitting quality, but rather leads to unbounded surfaces for all four data sets, as shown by figure 3. Figures 3, 9, also show that radial basis function fitting with a linear polynomial component might be appropriate for data from a single layered open surface, but it is not suitable for fitting data from a bounded close surface. In this case, especially when the data set is relatively sparse and information on surface normal at data points is not available, ellipsoid constrained radial basis function fitting appears to be much more desirable. Figure 4 shows the fitting results using the technique presented in section.3. For data from the patella, femur and tibia, the inverse for matrix M exists in our fitting method. For the head data, matrix M is singular, its inverse in our fitting technique is approximated with its pseudo-inverse. The figure shows that the fitted results are much better than those obtained with fitting methods where the polynomial is absent or linear. There are no longer holes in the fitted surfaces for femur data and tibia data. Although there are two redundant parts occurring on the fitted surface for the head data, 3 c The Eurographics Association and Blackwell Publishing Ltd 4

7 Li et al / Implicit Fitting Using Radial Basis Functions with Ellipsoid Constraint 6 Figure : Surface data fitting using radial basis functions without polynomial part in the fitting model (). Figure 4: Surface data fitting using radial basis functions by constraining the polynomial part to be an ellipsoid in fitting model (). it is much more effective compared with the fitting results displayed in figures and figure 3. In the work of Savchenko et al. [8], a two-step implicit surface fitting technique is developed using a carrier solid object. It is similar to ours in that neither interior nor exterior layers of the data set needed to be built. However, our technique is a one-step fitting, which automatically selects an ellipsoid as the carrier. A comparison is made between the two techniques. The fitting results for the technique given by Savchenko et al. have been given in figure 5. The upper figure is the fitting result using a sphere as the carrier solid, and the bottom figure is the fitting result using an ellipsoid as the carrier solid which well approximates the fitting data. It is obvious that the fitting results depend on the choice of the carrier solid. Figure 3: Surface data fitting using radial basis functions with linear polynomial parts in the fitting model (). Figure 4 is obtained using the linear radial basis function (r) = r. In practice, there is a problem of choosing an appropriate type of basis function in the fitting. It is shown that the degree of independence of the present fitting technique from the choice of basis functions depends on the shape complexity of the object to be fitted. When the actual surface shape of an object is relatively simple, there are few differences amongst the fitting results in using different basis functions. c The Eurographics Association and Blackwell Publishing Ltd 4

8 6 Li et al / Implicit Fitting Using Radial Basis Functions with Ellipsoid Constraint Figure 6: Fitting head data using thin-plate basis function with ellipsoid constraint results in a broken surface. 3 Φ(r)=r Φ(r)=r log(r) Φ(r)=r.5 Φ(r)=r.5 Φ(r)=r 3 Figure 5: Fitting femur surface data with an implicit surface using the technique presented in [8] with different ellipsoid carriers. Left: the carrier solids used; Right: The fitting results. P P For instance, there are no significant differences in fitting the patella data with different basis functions. However, with increase in shape complexity, the fitting results can be different. Figure 6 shows that the fitting using thin-plate basis function for the head data leads to a fragmented surface. One explanation for this result could be given from the perspective of implicit shape blending. Consider an implicit function defined by two points P and P using a radial basis function (P): F(P) = ( P P ) + ( P P ), (3) where (P) = r α, (α >,α, 4, 6,...). For a fixed contour value δ, it can be observed that the shape of F(P) = δ tends to shrink towards the centre of the two points with increasing values of α. Figure 7 clearly indicates that a smaller α in (3) tends to generate a blending by uniformly expanding the shapes to be blended, while a larger α tends to produce a blending by contracting these shapes. The shape corresponding to thin-plate basis function has also been displayed in the figure. As can be seen, it also produces a contracting blending. Our experiments have shown repeatedly that the thin-plate basis function and radial basis functions (r) = Figure 7: The implicit shapes of ( P P ) + ( P P ) = 3 defined using different radial basis functions, where P = (, ), P = (, ). r α with larger α values are much more powerful than the linear radial basis function in deforming the shape represented by the polynomial part in () to fit those shapes with deep concave. However, when sample points are not uniformly distributed and noisy, they might over deform the base shape and produce surface fragments. Figure 8 shows that the number of fragments tend to be proportional to the changing rate of radial basis functions. c The Eurographics Association and Blackwell Publishing Ltd 4

9 Li et al / Implicit Fitting Using Radial Basis Functions with Ellipsoid Constraint 63 In general, the fitting results using linear basis functions and multiquadrics are roughly the same when the parameter c in multiquadrics is set to a small number. However, with the increase in value c in the multiquadrics, the value r becomes less effective in deforming the ellipsoid described by the polynomial part. The three fitting models can also be used for fitting data from some well known geometric objects, like the ellipsoid, supperellipsoid, and torus. In figure 9, the fitting results are compared with two fitting models based on linear polynomial parts and ellipsoid constrained fitting. The figure shows clearly how well the data can be fitted with the ellipsoid based fitting technique. Four data sets with different numbers of points are used in this example. They are sampled evenly from the surface of the same torus. The left column contains the fitting results using the linear polynomial part in function (), and the right column contains the fitting results with the polynomial part representing an ellipsoid. The top two graphs show the fitting results using 4 points. The remaining three pairs show the fitting results using, 5, and 33 points respectively. We have not listed the fitting results when the polynomial part is not present in the model, as in all these cases, the fitting results are poor. The power of the fitting technique using a radial basis function with ellipsoid constraint is also exhibited when it is used to fit very sparse data sets. It can blend the gaps between data points more naturally. Figure shows the fitting results using only 6% of the original tibia data. As shown in the figure, there is little similarity between the tibia shape and the fitted shapes when the polynomial part does not present or is chosen to be linear. However, the fitting model using the ellipsoid constraint can still capture the main feature of the tibia shape. It shows that the fitting can still be reasonable even when only a small proportion of data points are used. Figure 8: Implicit surface fitting with ellipsoid constraint using radial basis function (r) = r α with just % head data: (a) α =.5; (b) α = ; (c) α = 3; (d) α = 7. As far as fitting error is concerned, it is necessary to distinguish numerical error from topological error. The numerical error measures how close the fitted shape is to the given data in numerical terms, say, by calculating the RMS. The topological error, on the other hand, measures how close the fitted shape is to the actual shape represented by the given data in topological terms by considering, say, whether the fitted shape is bounded, connected, and how many holes and branches it contains. This is particularly important in implicit surface fitting. In practice, it is not difficult to find a function f (P) for a given data set such that the shape represented by f (P) = approximates each point in the data set with preset precision. But the problem is that the shape can be quite different from the shape represented by the given data set. In fact, for the fitting presented in Figures, 3, 4, the algebraic distance from any sample point to the fitted shape is less than 6. However, the fitting based on ellipsoid constraint provides much better fitting for the objects with respect to the topological error. c The Eurographics Association and Blackwell Publishing Ltd 4

10 64 Li et al / Implicit Fitting Using Radial Basis Functions with Ellipsoid Constraint Figure : Surface fitting using radial basis functions using just six percent of original tibia data. (a). without polynomial part in the fitting model; (b). Using linear polynomial part in the fitting model; (c). polynomial part to be constrained as an ellipsoid. Figure 9: Fitting torus data with Radial basis functions fitting. Left: Using linear polynomial; Right ellipsoid constrained. As for the computational time, the particular aspect that needs to be addressed is the time used to solve for the coefficients in (8). As the matrices involved in the generalized eigen-system (3) are all of size 6 6, they can be easily solved. The main issue is, in solving for the coefficients, the computation of the inverse or the pseudo-inverse of the matrix M in (7). All experiments are performed on a PC running Windows XP with a 55MHz Pentium III processor. The time to find the inverse of matrix M for the patella data, the femur data, and the tibia data are listed in table. When M is singular, the pseudo-inverse of the matrix needs to be computed, which is very time consuming. In our experiment, the matrix M corresponding to the head data, which contain 435 points, is singular. It takes about 557 seconds to compute the pseudo-inverse of matrix M. In practice, to fit a very large data set using the technique proposed in this paper is very time consuming, especially when matrix M is singular. Instead, we will use the c The Eurographics Association and Blackwell Publishing Ltd 4

11 Li et al / Implicit Fitting Using Radial Basis Functions with Ellipsoid Constraint 65 Table : The time to compute the inverse matrix M Data set Number of points Time of computing M Patella data femur data tibia data function can be constructed and used to blend implicit shapes derived from different regions. The gate functions can be defined using what has been called the smooth unit step functions. A smooth unit step is defined as a mapping µ : R [, ] satisfying following conditions: technique presented in the next section to turn the fitting into a sequence of implicit fittings involving small data sets. 4. Fitting Large Data set by Data Partitioning In the above section, we presented a one-step automatic technique for fitting scattered points using radial basis functions by constraining the polynomial part to be an ellipsoid. The algorithm involves computing the inverse or pseudo-inverse of a matrix and computing an eigenvector of a generalized eigen-system involving a matrix of size 6 6. However, with the increase in the size of the data set, the size of matrix M in (7) becomes large and the inverse (or pseudo-inverse) of matrix M becomes difficult to compute directly. On the other hand, some shapes to be fitted can be very complicated. In this situation, although a fitting could still be achieved with a high level of accuracy in the sense that all data points lie on or are very close to the fitted surface, the fitted surface may contain many redundant components and may not represent the actual shape properly. Some parts of the fitted surface might not be close to any data points. The fitting result presented in figure 4 for the head data serves as a good example of this, where two redundant parts are fitted around the neck. This phenomenon might be the most significant drawback associated with almost all implicit surface fitting techniques. To increase the fitting quality, one often qualifies the surface shape by exploiting additional information, such as using the surface normal, building new layers inside and outside the surface. However, this is not usually an easy task in general, especially when a data set is quite sparse or when the object has many thin parts. In these cases, it is almost impossible to distinguish between inside and outside and the task of building data points for interior and exterior layers becomes difficult. Since most implicit fitting techniques can fit data from simple geometric shapes very well, one feasible approach might be to partition the data into smaller data sets first. Each small data set is then fitted with an implicit surface piece by piece. The implicit surface based on all data points can then be obtained by blending these simpler implicit surfaces. Recently, such a technique has been developed by the introduction of gate functions. A gate function for a region D can be understood as a smooth approximation to the characteristic function associated with set D. Its value is or close to for points inside a region D and or nearly outside. Detailed discussion on such a technique can be found in [5]. Here we just give a brief description on how this type of. it takes the value when t <, and the value when t > ;. it is an increasing function for t [, ]; 3. it is a continuous function; 4. it is antisymmetric about point (, /), that is, µ( t) + µ(t) =. In constructing of a gate function, the fourth property given above for a smooth unit step function is not always required. This property is required only when we want to construct a set of gate functions for a balanced blending of those locally fitted implicit shapes. A smooth unit step can be regarded as a smooth approximation to the Heaviside step function, which is defined by, t > ; H(t) =, t = ;, t >. (33) There are several ways to construct smooth unit step functions. One approach is to construct smooth unit step functions as piecewise polynomials. Let H (t) be the Heaviside unit step function, and let Set f (t) = H (t), f n (t) = t ( n f n (t) + t ) f n (t ), n n =,, 3,... (34) ( ) n(t + ) H n (t) = f n, n =,, 3,... (35) Then it can be shown directly that all these H n (t), n =,,..., are continuous and nondecreasing. Furthermore, it can also be verified that the smooth unit step function H n (t) is C n -smooth. Obviously, any of these smooth unit step functions are piecewise polynomials. The smooth unit step functions H n (t) constructed in this way have a standard rising interval [, ]. More general smooth unit step functions with arbitrary rising range [ ε, ε] for any small positive number ε can be conveniently defined by H n (t/ε). c The Eurographics Association and Blackwell Publishing Ltd 4

12 66 Li et al / Implicit Fitting Using Radial Basis Functions with Ellipsoid Constraint ε = ε = (a). (b) ε =. ε = (c). (d) Figure : AC -continuous smooth unit step function with different rising range parameter ε. (a) ε = ; (b) ε =.5; (c) ε =.; (d) ε =.5. A C -continuous piecewise polynomial smooth unit step function constructed from the above procedure is displayed in figure with different rising ranges. Let D be a region in R 3 and be described implicitly as {P : Ɣ D (P) > }. Then the gate function corresponding to region D can then be represented by g(p) = H n (Ɣ D (P)). Suppose the Euclidean space R 3 is partitioned into subsets D, D,..., D n and assume that the gate function associated with set D i is g i (P), (i =,,...,n) satisfying n g i (P) =, i= for all P R 3. Let the implicit surface fitting based on the data from region D i and a small neighbourhood of the region be F i (P) =. Then the implicit surface based on all data points can then be approximated with n g i (P)F i (P) = i= As an illustration example for the proposed constructive fitting, let s consider a simple situation, where the space R 3 is partitioned into two parts using a plane which is represented implicitly as (P P ) n =, where P is a point on the plane and n is a unit vector representing the normal to the plane. Suppose the surface fitted for the sub-data set in the half-space that the normal n points at is F (P) = and the surface fitted for the sub-data set on Figure : Constructive implicit surface fitting for the head data. the other side of the plane is F (P) =. Then these two individually fitting implicit surfaces can be combined smoothly in the following way gf (P) + ( g)f (P) =, where g is a gate function defined using a smooth unit step function µ(t): g(p) = µ((p P ) n). There are many potential applications in using this technique in implicit shape fitting. In addition to reducing the shape complexity of the fitting objects, it can also be used to reduce the computation time and to increase the fitting accuracy. It has been demonstrated that this technique can be used to fit implicit surfaces of complex objects by properly partitioning the data space. Perhaps the most important application of the technique is that it can be used in direct support of parallel computing in implicit surface fitting involving a large data set. The fitting for the head data displayed in figure is obtained in this manner. Since the data is not particularly complicated we just separate the entire data into an upper part and a lower part using a horizontal plane. The plane passes approximately through the center of the data set. The top left graph is the fitting result based on upper data set and the one displayed on the top right is the fitting result for the lower c The Eurographics Association and Blackwell Publishing Ltd 4

13 Li et al / Implicit Fitting Using Radial Basis Functions with Ellipsoid Constraint 67 data set. To ensure that the blended surface is smooth enough at the joint when fitting the upper head data, some data points in the low region that are close to the upper region are used. We carry out the low part fitting in the same way. The blended surface is displayed in (c) in fig Implicit Planar Curve Fitting When D data points are considered, the fitting technique presented in section.3 becomes an implicit curve fitting. Correspondingly, we constrain the polynomial part in the fitting model to be an ellipse. It is well known that the discriminant that determines whether a D quadratic equation ax + bxy + cy + sx + ty + d = represents an ellipse is ac b >. Hence if the polynomial part in () is replaced by..5 P(P) = ax + bxy + cy + sx + ty + d,.95 and the constraint 4J I > for the ellipsoid is replaced by constraint ac b > for the ellipse, we obtain directly the implicit curve fitting algorithm. If we redefine vector X i in () as ) X T i = (, x i, y i, x i, x i y i, y i,, corresponding to point P i (x i, y i ), i =,,...,n and replace matrix C defined in (5) as a matrix C of size (n + 6) (n + 6) such that ( ) C =, C where C =, then the corresponding solution given in section.3 leads to an implicit curve fitting for D scattered points. Figure 3 demonstrates some fitting results for artificial data. The figure shows that the fitted curve will always be bounded and can fit highly complex shapes well. 6. Summary To summarize, an implicit surface fitting technique using radial basis functions is developed in this paper by constraining the polynomial part in () such that it always represents an ellipsoid. The investigation presented in this paper shows that a bounded algebraic surface constraint is much more desirable when a bounded and closed surface is expected. With this technique, the interior and exterior layers of the data set Figure 3: Implicit curve fitting using radial basis functions with ellipse constraint can properly represent complex shapes involving branches and holes. need not to be built and can still produce an agreeable fitting especially when data are sparse. The major limitation of this technique is its application to a large data set, which will involve the computing of the inverse or pseudo-inverse of a large scale matrix. A divide and conquer strategy has been proposed to cope with the problem. With this strategy, fitting an implicit surface to a large data set is simplified as fitting surfaces to some much smaller data sets. The proposed strategy can also be used to reduce the shape complexity of the objects from which the data points are sampled. Therefore the technique can be further used to improve the fitting accuracy both numerically and topologically as well. Our future research will focus on developing implicit surface fitting techniques that support parallel computing. References. C. Bajaj. Higher-Order Interpolation and Least-Squares Approximation Using Implicit Algebraic Surfaces. ACM Transactions on Graphics, (993) c The Eurographics Association and Blackwell Publishing Ltd 4

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