### TABLE 4.8: Dropping directions corresponding to small Eigenvalues of C, i.e. dropping less important principal components (cf. (4.14)), for the translation invariant RBF kernel (see text). All results given are for the case = 0:4 (cf. Table 4.6).

1997

### Table 1: Topological invariants in the perturbative and the non-perturbative regimes for d = 3 and d = 4. The resemblance between the two pictures is very appealing. Nevertheless, there are important di erences which rise some important questions. In the case of knot theory, Vassiliev invariants constitute an in nite set. However, in Donaldson theory, for the cases studied so far, only a nite set of invariants seems to play a relevant role. One would like to know if this is general or if this fact is just a peculiarity of the only two cases (gauge group SU(2) without matter and with one multiplet of matter in the fundamental representation) which have been studied so far. The general picture of non-perturbative 2

1997

"... In PAGE 3: ... Therefore, it might happen that no new topological information is gained. The situation for three and four-dimensional TQFTs is shown in Table1 . These the- ories seem to share a common structure.... ..."

Cited by 1

### Table 1. Some affine Hu invariants up to 4-th order mo- ments (not complete)

2003

"... In PAGE 2: ... These invariants can be easily derived using the so called complex moments, see [4]. Some of the clas- sical Hu invariants and our new Hu invariants are given in Table1 . It is very important that these features are invariant against rotations and reflections for all three types of mo- ments (DM,LM,AM).... In PAGE 2: ...1 without the normalization of the translation by the centroid. Using these normalized moments, the features Hk of Table1 can be calculated which are now invariant to homogeneous affine transformations. Table 1 shows that it can be used low order invariants, e.... In PAGE 2: ... Using these normalized moments, the features Hk of Table 1 can be calculated which are now invariant to homogeneous affine transformations. Table1 shows that it can be used low order invariants, e.g.... ..."

Cited by 2

### Table 2: Classi cation performance (% correct) of Invariance Signature Neural Network Classi ers (with perfect Local Orientation Extraction) trained for 1000 iterations with the data shown in Figure 13 and tested with the perfect data set in Figure 14. The ISNNCs generalize perfectly to the the test set. The network architecture constrains the system to be shift-, rotation-, scale- and re ection-invariant in the absence of quantization noise, so this is no surprise. Importantly, the result indicates that su cient information is retained in the 5 bin Invariance Signatures for all 22 unambiguous letters of the alphabet to be distinguished. Inspection of the sum squared error values after each iteration indicated that the error on the test set was indeed identical to that on the training set: for perfect data, the ISNNC produces perfect results.

"... In PAGE 21: ...The results for the ten ISNNCs which used perfect Local Orientation Extraction are summarized in Table2 . The average number of iterations for 100% correct classi cation to be achieved was 220.... ..."

### Table 2.1: Some Desirable Properties of Time-Frequency Distributions distributions, they should be real and positive functions. Also, the time marginal of a time- frequency distribution is the integral of the time-frequency distribution over frequency, and the time marginal should be identical to the distribution of signal energy over time. These properties and several others are listed in Table 2.1. For a more complete listing see [35]. 2.5 Covariance and Invariance The notion of a density function being covariant or invariant to an operator plays an important role in time-frequency analysis. Three commonly used operators are the time- shift operator, the frequency-shift operator, and the scale operator de ned, respectively as:

1999

Cited by 11

### Table 5: System of three particles: chart adapted to the constraint structure and to the Poincar e group (p1 and p2 are invariants of E(3) in the case of a realization with no xed invariants). k1 k2 k3

"... In PAGE 5: ...e. the total invariant mass and angular momentum) as canonical variables (see for example Table5 ) is the only choice of Poincar e invariants that can survive in the interacting case where individual particle Poincar e invariants are no longer constants of the motion. Finally, these `spin bases apos; seem likely to provide an important tool in the study of the Nambu string [4] and in classical eld theory.... ..."

### Table 1: Features and rankings from down-selection. The numbering indicates the feature ranking for each segmentation method with 1 being the most important. The top three features for each segmentation method and, hence, salient features are shown in italic. These features include Histogram Equalised Features (HE), Laws textures (C4) and image invariants (AW).

"... In PAGE 5: ... subsequent stages was obtained by removing uninformative features while system perfor- mance increased to a maximum. Figure 2 shows the variation in performance during the SBS process while Table1 summarises the rankings for those features that were retained in the reduced feature sets. Classifiers: Three typical forms of classifiers were used for this study.... ..."

### Table 2. A ne Invariants The ratio of lengths of parallel lines measured on the re-

1996

"... In PAGE 12: ...s the image apex at (1430,4675). The horizon line passes through (0,182). camera centres are computed from the camera projection matrices. Table2 shows the computed angles for the 6 image triplets from sequence I, while gure 7 shows a plan view of the recovered metric structure from the rst image triplet. 5 Conclusions and extensions Wehave demonstrated the geometric importance of xed points and lines in an image sequence as calibration tools.... ..."

Cited by 99

### Table 2. A ne Invariants The ratio of lengths of parallel lines measured on the re-

1996

"... In PAGE 12: ...s the image apex at (1430,4675). The horizon line passes through (0,182). camera centres are computed from the camera projection matrices. Table2 shows the computed angles for the 6 image triplets from sequence I, while gure 7 shows a plan view of the recovered metric structure from the rst image triplet. 5 Conclusions and extensions Wehave demonstrated the geometric importance of xed points and lines in an image sequence as calibration tools.... ..."

Cited by 99