Landmark point

The points in A are called data points, and the points in B are called landmark points. Usually one assumes that there are many more data points than landmark ones. 

An alternative way to think of Definition 9.14 is the following. Assume that we have n data points and N landmark points. Every data point induces an order on the landmark points just sort them with respect to their distances to that point. Every such ordering can be visualized as a path in the Hasse diagram of the Boolean lattice starting from the point nearest to the chosen data point, then proceeding to the union of the two closest ones, then on to the three closest ones, and so on. Now, the witness complex W(A, B) is the maximal abstract sirnplicial complex whose face poset is contained in the union of these paths. 

In concrete successful applications of the witness complex, the crux of the matter is the choice of the landmark points, and many ingenious strategies have been devised. 

Here in 2007, we can now handle data in the form of discrete named landmark points, curves and surfaces from a wide variety of geometric data sources (photographs, solid medical images, surface scans) in two or three dimensions. We can accommodate the special cases, quite common in practice, for which landmark points arise from curves, or landmark curves from surfaces. Indeed, this approach, the simplicial decomposition of a single form, has become the most common way of building a coherent geometric representation (see Bookstein, 2004) that compromises between the enormous over-representation by surface meshes and the equally severe under-representation by conventional discrete point schemes. 

It is one implication of this approach that more care needs to be taken with the selection of landmark points than is presently the case. The allocation of a landmark halfway between two others, or in the middle of a surface path bounded by some curves, adds information to a data-set even if the landmark is located exactly where a spline driven by the remaining information would place it. It adds, precisely, the information that that location was observed with only digitizing error, so points nearby have a prediction error that is lower simply by virtue of the additional data. Where landmarks are widely spaced, the deformations predicted by, say, a thin-plate spline are far less reliable than where landmarks are precise regardless of the statistics of the landmark locations themselves. 

Angel, P.N. (1999) Multiscale Image Analysis for the Automated Localization of Taxonomic Landmark Points and the Identification of Species of Parasitic Wasp. PhD thesis. University of Glamorgan. 

The precision of the anchored circle techifique in comparison with using linear measures constrained directly by the landmark points was tested empirically by repeat measurements where the positions of the anchoring points were varied to mimic the situation in which the placement of the points is unclear on an image. Even when two of the anchoring points for a circle were uncertain, using anchored circles still produced more precise measurements than using a simple line. Thus, the use of anchored circles also reduces measurement error. 

The incising balls method is more robust at dealing with high-dimensional data than the MLE [21] however the computational cost is higher. The overall complexity of the incising balls method is 0 n ) which can be reduced to 0(nL) if an approximation method is used whereby the distances between n and L landmark points is used in Step 1. 

The formation of the projection matrix Z is a slightly more involved process. Once the landmark points have been selected, each data point can be expressed as a linear combination of nearby landmark points such that x = where... 

Zij is a local projection vector and xj is the y-th landmark point. The landmarks can be found in a number of ways, however, perhaps the most useful is to use the -landmark points that are closest to each x where q is the same for all data points [24] and q < m. Therefore, the projection matrix Z has only q non-zero elements for every column. Spectral dimensionality reduction is then performed on the set of identified landmarks and the low-dimensional embedding is found via Y = YZ. 

Silva, V.d., Tenenbaum, J.B. Sparse multidimensional scaling using landmark points. Tech. 

Quahtative description of physical behaviors require that each continuous variable space be quantized. Quantization is typically based on landmark values that are boundary points separating qualitatively distinct regions of continuous values. By using these qualitative quantity descriptions, dynamic relations between variables can be modeled as quahtative equations that represent the struc ture of the system. The... 

Tables 6 provides a summary of the predicted GLCs under worst case meteorological conditions. Note that the ground level concentrations summarized do not consider wind direction nor variation. More specifically, two points within the plant vicinity are presented to summarize wind persistence as related to known landmarks ...

Anti periplanar geometry for E2 reactions is particularly important in cyclohexane rings, where chair geometry forces a rigid relationship between the substituents on neighboring carbon atoms (Section 4.8). As pointed out by Derek Barton in a landmark 1950 paper, much of the chemical reactivity of substituted cyclohexanes is controlled by their conformation. Let s look at the E2 dehydro-halogenation of chlorocyclohexanes to see an example. 

The vegetative cell cycle of S. cerevisiae has received extensive attention. There are many justifications for this. Firstly, the cell cycle in this organism has many convenient landmarks (Hartwell 1974, 1978 Pringle 1978) which make it very easy to identify the exact point in the cell cycle at which a cell happens to be. Examples of these landmark events include bud emergence, the size of the bud, mitosis (nuclear division takes place through the neck between the mother cell and the bud), and cell... 

Another key contribution of the Schwarz group was the recognition of the dramatic influence of oxide surfaces on bulk solution pH. In a landmark 1989 paper, Noh and Schwarz [7] demonstrated the method of mass titration, in which successive additions of oxide cause stepwise shifts in solution pH. This procedure is illustrated in Figure 6.7 [7], As indicated in Figure 6.1, the protonation-deprotonation chemistry of the surface hydroxyl groups is coupled to the liquid-phase pH. In mass titration, as the mass (or more appropriately, the surface area) of oxide in solution increases, the solution pH is brought to the PZC of the oxide, at which point no driving force for proton transfer exists... 

In Proust s novel, the town of Balbec, with its slow pace of life and attractive landmarks, seems trapped in time, like an ancient ant trapped in amber. I like to think of In Search of Lost Time as a collection of chambers connected by tunnels in a dense ant colony. In one chamber is an endless party taking place at one point in the narrator s life—and if we escape through a tunnel, we are quickly whisked away to another chamber in which time congeals yet again. 

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