### Table 2: The maximum error for 2D nonrigid image registration by cubic B-spline with normalized mutual information. The unit for the displacements and errors is mm. The column of Maximum Disp. represents the maximum displacement between the target image and the object image.

2004

"... In PAGE 43: ... Since the registration algorithm can efiectively capture the deformations between the image pair, most of the true and the computed vectors overlap. Table2 shows another data set of the registration error by running cubic B-spline with normalized mutual information over the cardiac images. These cardiac images have have a size of 386 x 323 pixels and spatial resolution of 1 mm x 1 mm.... ..."

### Table 1 The basic one-dimensional quadrature rule for cubic B-splines.

2000

"... In PAGE 19: ... A too small number of quadrature points leads to instabilities, in particular, when the quadrature points are not properly spaced; a high polynomial accuracy alone does not suffice. For the tensor- product third order B-splines described at the beginning of section 2, we had good experience with the tensor-product counterpart of the one-dimensional quadrature rule given by Table1 . This quadrature formula is exact for fifth order polynomials and assigns 52 or n = 25 quadrature points to each particle in two space dimensions.... ..."

Cited by 7

### TABLE A1. Tabulated values of a cubic B-spline and of its derivative at integer arguments

2007

Cited by 2

### Table 2: Example surfaces created using an interactive editor for hierarchical b-splines.

"... In PAGE 5: ... Most were created using an interactive editor for hierarchical B-splines, one created using SoftImage, and the last from a digitized surface data created using a Cyberware laser scan- ning system. The results are illustrated in Section 8 and sum- marized in Table2 and Table 3. All of the resulting surfaces (with exceptions noted below) approximate the initial input mesh with 0.... ..."

### Table 2a: Same as Table 1a but for B-spline FEM.

"... In PAGE 25: ... In the subsequent Tables 2a to 2f, results of corresponding calculations with B-spline nite elements are shown. In Table2 a, neutron single particle eigenvalues are listed which have been calculated with the new B-spline FEM code. A comparison of the numbers with those listed in Table 1a shows that they are identical for equal mesh point numbers.... In PAGE 25: ... In Table 1b, full precision is achieved at 60 mesh points while 121 mesh points were necessary in Table 1b. In a calculation with 4th order B-spline elements, 45 mesh points are required as shown in Table2 c whereas 145 mesh points are necessary with Lagrange elements (Table 1c) to obtain a precision of 12 digits. In the 5th order B-spline FEM, 34 mesh points have been used (Table 2d) while a corresponding 5th order Langrange FEM required 76 mesh points (Table 1d).... In PAGE 25: ... In a calculation with 4th order B-spline elements, 45 mesh points are required as shown in Table 2c whereas 145 mesh points are necessary with Lagrange elements (Table 1c) to obtain a precision of 12 digits. In the 5th order B-spline FEM, 34 mesh points have been used ( Table2 d) while a corresponding 5th order Langrange FEM required 76 mesh points (Table 1d). The 6th order B-spline FEM (see results in Table 2e) leads still to a considerable relative reduction of the number of mesh points from 34 to 30 at the same level of precision while in the 7th order method still 29 mesh points were required (Table 2f).... In PAGE 25: ... In the 5th order B-spline FEM, 34 mesh points have been used (Table 2d) while a corresponding 5th order Langrange FEM required 76 mesh points (Table 1d). The 6th order B-spline FEM (see results in Table2 e) leads still to a considerable relative reduction of the number of mesh points from 34 to 30 at the same level of precision while in the 7th order method still 29 mesh points were required (Table 2f). The results shown in the Tables 1a to 1f and in the Tables 2a to 2f lead to the conclusion that the B-spline FEM has its optimum at 6th order whereas the optimal order of the Lagrange FEM is at 5th order.... In PAGE 25: ... In the 5th order B-spline FEM, 34 mesh points have been used (Table 2d) while a corresponding 5th order Langrange FEM required 76 mesh points (Table 1d). The 6th order B-spline FEM (see results in Table 2e) leads still to a considerable relative reduction of the number of mesh points from 34 to 30 at the same level of precision while in the 7th order method still 29 mesh points were required ( Table2 f). The results shown in the Tables 1a to 1f and in the Tables 2a to 2f lead to the conclusion that the B-spline FEM has its optimum at 6th order whereas the optimal order of the Lagrange FEM is at 5th order.... In PAGE 28: ...f [10?1; 10?10] are su cient in most applications. In Fig. 8b there is an indication for 6th order to become optimal order at precisions better than 10?10. This is in agreement with the conclusion that has been drawn form the data in Table2... ..."

### Table 2c: Same as Table 1c but for B-spline FEM.

"... In PAGE 25: ... In the subsequent Tables 2a to 2f, results of corresponding calculations with B-spline nite elements are shown. In Table2 a, neutron single particle eigenvalues are listed which have been calculated with the new B-spline FEM code. A comparison of the numbers with those listed in Table 1a shows that they are identical for equal mesh point numbers.... In PAGE 25: ... In Table 1b, full precision is achieved at 60 mesh points while 121 mesh points were necessary in Table 1b. In a calculation with 4th order B-spline elements, 45 mesh points are required as shown in Table2 c whereas 145 mesh points are necessary with Lagrange elements (Table 1c) to obtain a precision of 12 digits. In the 5th order B-spline FEM, 34 mesh points have been used (Table 2d) while a corresponding 5th order Langrange FEM required 76 mesh points (Table 1d).... In PAGE 25: ... In a calculation with 4th order B-spline elements, 45 mesh points are required as shown in Table 2c whereas 145 mesh points are necessary with Lagrange elements (Table 1c) to obtain a precision of 12 digits. In the 5th order B-spline FEM, 34 mesh points have been used ( Table2 d) while a corresponding 5th order Langrange FEM required 76 mesh points (Table 1d). The 6th order B-spline FEM (see results in Table 2e) leads still to a considerable relative reduction of the number of mesh points from 34 to 30 at the same level of precision while in the 7th order method still 29 mesh points were required (Table 2f).... In PAGE 25: ... In the 5th order B-spline FEM, 34 mesh points have been used (Table 2d) while a corresponding 5th order Langrange FEM required 76 mesh points (Table 1d). The 6th order B-spline FEM (see results in Table2 e) leads still to a considerable relative reduction of the number of mesh points from 34 to 30 at the same level of precision while in the 7th order method still 29 mesh points were required (Table 2f). The results shown in the Tables 1a to 1f and in the Tables 2a to 2f lead to the conclusion that the B-spline FEM has its optimum at 6th order whereas the optimal order of the Lagrange FEM is at 5th order.... In PAGE 25: ... In the 5th order B-spline FEM, 34 mesh points have been used (Table 2d) while a corresponding 5th order Langrange FEM required 76 mesh points (Table 1d). The 6th order B-spline FEM (see results in Table 2e) leads still to a considerable relative reduction of the number of mesh points from 34 to 30 at the same level of precision while in the 7th order method still 29 mesh points were required ( Table2 f). The results shown in the Tables 1a to 1f and in the Tables 2a to 2f lead to the conclusion that the B-spline FEM has its optimum at 6th order whereas the optimal order of the Lagrange FEM is at 5th order.... In PAGE 28: ...f [10?1; 10?10] are su cient in most applications. In Fig. 8b there is an indication for 6th order to become optimal order at precisions better than 10?10. This is in agreement with the conclusion that has been drawn form the data in Table2... ..."

### Table 2e: Same as Table 1e but for B-spline FEM.

"... In PAGE 25: ... In the subsequent Tables 2a to 2f, results of corresponding calculations with B-spline nite elements are shown. In Table2 a, neutron single particle eigenvalues are listed which have been calculated with the new B-spline FEM code. A comparison of the numbers with those listed in Table 1a shows that they are identical for equal mesh point numbers.... In PAGE 25: ... In Table 1b, full precision is achieved at 60 mesh points while 121 mesh points were necessary in Table 1b. In a calculation with 4th order B-spline elements, 45 mesh points are required as shown in Table2 c whereas 145 mesh points are necessary with Lagrange elements (Table 1c) to obtain a precision of 12 digits. In the 5th order B-spline FEM, 34 mesh points have been used (Table 2d) while a corresponding 5th order Langrange FEM required 76 mesh points (Table 1d).... In PAGE 25: ... In a calculation with 4th order B-spline elements, 45 mesh points are required as shown in Table 2c whereas 145 mesh points are necessary with Lagrange elements (Table 1c) to obtain a precision of 12 digits. In the 5th order B-spline FEM, 34 mesh points have been used ( Table2 d) while a corresponding 5th order Langrange FEM required 76 mesh points (Table 1d). The 6th order B-spline FEM (see results in Table 2e) leads still to a considerable relative reduction of the number of mesh points from 34 to 30 at the same level of precision while in the 7th order method still 29 mesh points were required (Table 2f).... In PAGE 25: ... In the 5th order B-spline FEM, 34 mesh points have been used (Table 2d) while a corresponding 5th order Langrange FEM required 76 mesh points (Table 1d). The 6th order B-spline FEM (see results in Table2 e) leads still to a considerable relative reduction of the number of mesh points from 34 to 30 at the same level of precision while in the 7th order method still 29 mesh points were required (Table 2f). The results shown in the Tables 1a to 1f and in the Tables 2a to 2f lead to the conclusion that the B-spline FEM has its optimum at 6th order whereas the optimal order of the Lagrange FEM is at 5th order.... In PAGE 25: ... In the 5th order B-spline FEM, 34 mesh points have been used (Table 2d) while a corresponding 5th order Langrange FEM required 76 mesh points (Table 1d). The 6th order B-spline FEM (see results in Table 2e) leads still to a considerable relative reduction of the number of mesh points from 34 to 30 at the same level of precision while in the 7th order method still 29 mesh points were required ( Table2 f). The results shown in the Tables 1a to 1f and in the Tables 2a to 2f lead to the conclusion that the B-spline FEM has its optimum at 6th order whereas the optimal order of the Lagrange FEM is at 5th order.... In PAGE 28: ...f [10?1; 10?10] are su cient in most applications. In Fig. 8b there is an indication for 6th order to become optimal order at precisions better than 10?10. This is in agreement with the conclusion that has been drawn form the data in Table2... ..."

### TABLE I RESULTS OF SHAPE CODING USING B-SPLINES AND A QP OF 20 ( KIDS FRAME 0).

1998

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### TABLE I Gain in the sampling density brought by using the minimal interpolation instead of the B-spline equivalent L 1 2 3 4 5

1998

Cited by 13

### Table 2 B-splines estimation of the Archimedean copula generator: posterior means and 90% credible intervals for the B-splines parameters associated to K = 20 equidistant knots on (0,1)

"... In PAGE 12: ...haracterized by the copula in the continuous case. Our expectations were confirmed in our example. Note that this would not be the case anymore if we were dealing with discrete data, see Denuit and Lambert (2005), Vandenhende and Lambert (2002a) and the references therein for motivating arguments. Summary measures of the posterior distributions can be found in Table 1 for the marginal skewed-Student regression models for log(SBP) and log(DBP), and in Table2 for the B-splines parameters describing the fitted Archimedean copula generator. Table 1 reveals a positive marginal association between the cholesterol level and blood pressures (see afii98261), the positive skewness of the distributions of log(DBP) and log(SBP), as well as very moderate kurtosis.... In PAGE 12: ... Note that the quality of the fits provided by the marginal parametric models was assessed and found to be excellent. The results in Table2 are summarized graphically in Fig. 5.... ..."