### Table 4: Registration results of the Lena images presented in Figure 12 using the pseudo-polar based algorithm.

2002

### Table 1. The mean volume overlap and surface distance after the affine 4D registration,after the deformable 3D and after spatio-temporal deformable registration.

"... In PAGE 6: ... In order to calculate the overlap of the anatomical structures and the surface distance we used segmented images. Table1... ..."

Cited by 3

### TABLE I VOLUMES AND IMAGES USED IN REGISTRATION EXPERIMENTS

2004

Cited by 12

### Table 1: Simulation Results : All objects. 1: Linear planimetry, 2: 2-D cubic planimetry, 3: 3-D cubic planimetry, 4: Normal shape based, 5: Centroid shape based, 6: Maximal disc shape based. Missing values imply gt; 20 scans. The values for the cone fan scan were the result of the symmetry of the situation | odd numbers of sweeps gave more accurate volumes than even.

"... In PAGE 48: ... Object contours and surface reconstructions are shown with the minimum number of scans for which the cubic planimetry volume is within this margin. Table1 shows this number of scans for each object, and the percentage error for 20 scans. Registration and unsteady hand e ects In practice, the scan planes are generally not evenly spaced and their location and orientation is not known precisely, even with good... In PAGE 61: ...6 DISCUSSION 6 Discussion Accuracy of volume measurement Cubic planimetry gave the most accurate volume measurements with fewer segmented scans of the object in all cases, for both simulated and in-vivo experiments. In fact, typically only ten cross-sections were required to give an accuracy (due to the volume measurement technique alone) of better than 1% (see Table1 ). This was the case for both linear and complex scanning patterns, and simple or complex objects, even the sharp-edged cube in Figure 29.... In PAGE 61: ... There was little di erence in volume measurement accuracy between the three shape based methods, although in general the centroid based technique performed worst. There were some notable exceptions where the maximal disc guided technique gave signi cantly better volumes | namely for the freehand scan of the jester apos;s hat and the cube (see Table1 ). These are both complex shapes which are more di cult to represent using the simpler shape based methods.... ..."

### Table 1: Simulation Results : All objects. 1: Linear planimetry, 2: 2-D cubic planimetry, 3: 3-D cubic planimetry, 4: Normal shape based, 5: Centroid shape based, 6: Maximal disc shape based. Missing values imply gt; 20 scans. The values for the cone fan scan were the result of the symmetry of the situation | odd numbers of sweeps gave more accurate volumes than even.

"... In PAGE 48: ... Object contours and surface reconstructions are shown with the minimum number of scans for which the cubic planimetry volume is within this margin. Table1 shows this number of scans for each object, and the percentage error for 20 scans. Registration and unsteady hand e ects In practice, the scan planes are generally not evenly spaced and their location and orientation is not known precisely, even with good... In PAGE 61: ...6 DISCUSSION 6 Discussion Accuracy of volume measurement Cubic planimetry gave the most accurate volume measurements with fewer segmented scans of the object in all cases, for both simulated and in-vivo experiments. In fact, typically only ten cross-sections were required to give an accuracy (due to the volume measurement technique alone) of better than 1% (see Table1 ). This was the case for both linear and complex scanning patterns, and simple or complex objects, even the sharp-edged cube in Figure 29.... In PAGE 61: ... There was little di erence in volume measurement accuracy between the three shape based methods, although in general the centroid based technique performed worst. There were some notable exceptions where the maximal disc guided technique gave signi cantly better volumes | namely for the freehand scan of the jester apos;s hat and the cube (see Table1 ). These are both complex shapes which are more di cult to represent using the simpler shape based methods.... ..."

### Table 4: Per-processor communication volume for the Clump. Only 3-D FFT uses a signi cant number of bulk transfers. The communication-bound CON run su ers from a load imbalance in network tra c. Di erences in virtual processor layout result in markedly di erent tra c distributions for the EM3D runs. 8-way SMP NOW (8 proc.)

1997

"... In PAGE 9: ... For each run, Table 3 lists the input parameters and per-processor memory require- ment when running on the Clump. Table4 separates communication volume for each run into network and local tra c. The rst application, 3-D FFT, performs a fast Fourier transform in three dimensions and typi es regular applications that rely primarily on bulk com- munication.... ..."

Cited by 69

### Table 1. Ore algebras

1996

"... In PAGE 4: ...elds. We specify commutative ring or commutative eld when necessary. Moreover, all rings under consideration in this paper are of characteristic 0. Table1 gives examples of the type of operators we consider. All these operators share a very simple commutation rule of the variable @ with polynomials in x.... In PAGE 5: ...3 Examples of skew polynomial rings are given in Table1 . In all the cases under consideration in this table, A is of either form K[x] or K(q)[x] with K a eld.... In PAGE 6: ...lgebra F of functions, power series, sequences, distributions, etc. Then Eq. (1) extends to the following Leibniz rule for products 8f; g 2 F @i(fg) = i(f)@i(g) + i(f)g: (6)This makes F an S-algebra. The actions of the operators corresponding to important Ore algebras are given in Table1 . In the remainder of this article, we use the word \function quot; to denote any object on which the elements of an Ore algebra act.... In PAGE 8: ... Then O is left Noetherian and a non-commutative version of Buchberger apos;s algorithm terminates. As can be seen from Table1 , this theorem implies that many useful Ore algebras are left Noe-... In PAGE 12: ... Then the annihilating ideal for any product fg where f is annihilated by I and g is annihilated by K is also @- nite. As can be seen from Table1 , this hypothesis does not represent a severe restriction on the class of Ore algebras we consider. Again, f and g in this lemma need not be interpreted as functions but as generators of the O-mod-... ..."

### Table 5.1: Statistics for the synthetic 3D models used for global registration

2005

Cited by 1

### Table 1. Average time for 3D transforms (ms) volume size

"... In PAGE 2: ... For example, xz/y shear and xy/z shear can be implemented by memory shift while yz/x shear can only be done by voxel- by-voxel moving, which is time-consuming. Table1 shows the average running time for some elementary transforms and for naive rotation and naive general linear transformation on a PC (See Section 6). Table 1.... In PAGE 2: ...According to Table1 , another factor that affects the transformation speed is the usage of the CPU cache. For example, the slowest yz/x shear, the slowest z/xy shear and the slowest x resize can only be implemented by voxel-by- voxel moving and the CPU cache is not efficiently used.... In PAGE 3: ... For instance, ) , , ( ) , , ( b a y b a x S T xy S xy S P S P = . Now that some transposes are faster than some shears and resizes (see Table1 ), we can also use two transposes and a faster shear or a faster resize to accelerate another time-consuming shear or resize. For example, let ( ) x a D , ( ) y a D and ( ) z a D be the resizes in x, y and z directions and xy P be the xy transpose, we have: 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 ( , , ) ( , , ) ( , , ) ( ) ( ) ( ) ( , , ) ( , , ) ( , , ) ( ) ( ) ( ) ( , , ) ( , , ) ( , , ) S S S T z z y y x x xy xy S T xy xy S S T z z y y xy y x S xy S S x a a y b b z c c x a a y b b z c c y a a y b b z c c a a a a a a = = = = A DLU DS S S D D D P P S P P S S D D P D S P S S After an xy transpose, we avoid the inefficient yz/x shear and the slow x-resize or 3D resize.... ..."

### Table 1: Various algebras built using Albert.

1992

"... In PAGE 6: ... This command merely causes Albert to check if the polynomial expands to zero in the free algebra. Testing The results of several experiments are shown in Table1 . All times re ect computations made on a Sun SPARCstation1+, and are approximate.... In PAGE 6: ... A theorem of Hall apos;s [4] describes a basis for free Lie rings in terms of a set of \standard monomials quot;. The dimensions for Lie algebras, shown in Table1 , are precisely the numbers predicted by Hall apos;s theorem. Another test we employed was to make use of Artin apos;s theorem ([14], p 29) that states that any alternative algebra generated bytwo elements is associative.... In PAGE 10: ... In fact, these problems are beyond the scope of Albert, given the speed and memory sizes of present computers. The computations in Table1 were performed done on a computer with 12 mb memory. Albert can do useful work, however, with much less than this.... In PAGE 10: ... Albert can do useful work, however, with much less than this. As Table1 demonstrates, the system tends to do well when the number of generators is kept small, usually two or three. (For example note that the dimension of the degree 8 Lie algebra with 4 a apos;s and 4 b apos;s is only 39, and yet the degree 7 Lie algebra having 3 a apos;s, 2 b apos;s, 1 c,and1d is 232.... ..."

Cited by 7