Sampling spatial dimension

One of the more recent advances in XPS is the development of photoelectron microscopy [ ]. By either focusing the incident x-ray beam, or by using electrostatic lenses to image a small spot on the sample, spatially-resolved XPS has become feasible. The limits to the spatial resolution are currently of the order of 1 pm, but are expected to improve. This teclmique has many teclmological applications. For example, the chemical makeup of micromechanical and microelectronic devices can be monitored on the scale of the device dimensions. 

An ESRI system can be built with small modifications of commercial spectrometers by, for example, gradient coils fixed on the poles of the spectrometer magnet, regulated direct current (DC) power supplies, and required computer connections [40,53,55]. Gradients can be applied in the three spatial dimensions, and a spectral dimension can be added by the method of stepped gradients. The spectral dimension is important when the spatial variation of ESR line shapes (as a function of sample depth) is of interest this situation will be described below, in the ESRI studies of heterophasic polymers. In most systems, the software for image reconstruction in ESRI experiments must be developed in-house. 

With chemical imaging, a new type of data structure needs to be analyzed. Chemical imaging experiments yield a 3D X x Y x A, matrix or data cube, where X and Y are the spatial dimensions and A, the spectral dimension. One spectrum per pixel is recorded and selection of a wavelength will show an absorbance picture of the sample [10] (Figure 1). 

An important variant on PIXE is micro-PIXE. By using a proton beam whose spatial dimension is 0.5 pm (rather than the usual 10 mm), one can determine the trace-element content of a small portion of the sample, giving one a trace-element microscope. This application is important in probing samples of medical interest. A related technique is used in the electron microprobe where the ionization is caused by electron impact. 

Exploration of a data set before resolution is a golden rule fully applicable to image analysis. In this context, there are two important domains of information in the data set the spectral domain and the spatial domain. Using a method for the selection of pure variables like SIMPLISMA [53], we can select the pixels with the most dissimilar spectra. As in the resolution of other types of data sets, these spectra are good initial estimates to start the constrained optimization of matrices C and ST. The spatial dimension of an image is what makes these types of measurement different from other chemical data sets, since it provides local information about the sample through pixel-to-pixel spectral variations. This local character can be exploited with chemometric tools based on local-rank analysis, like FSMW-EFA [30, 31], explained in Section 11.3. 

Two different sandstone samples are used to demonstrate the methodology developed in Sections 2.1-2.3 in one spatial dimension. The first sample is a Bentheimer sandstone sample we have labeled KBE, which is saturated with oil. The second sample is a Brown sandstone sample, labeled MCD, that is saturated with water. 

Materials applications of imaging are often done without slice selection. For many studies, the sample geometry can be chosen with a suitable symmetry, so that a projection in one dimension will reveal the features in question. Because a projection corresponds to an integration along one spatial dimension, the signal-to-noise ratio of projections is usually superior to that of selected slices. 

A hyperspectral image is made up of a series of wave bands for each spatial position of an area studied. Therefore, each pixel contains a spectrum, which is a representative of the characteristic of the position. This is like a fingerprint for the characteristic of the position, and this would match any other positions within the sample with a similar characteristic. Hyperspectral images are called hypercubes, three-dimensional (3D) blocks of data, comprising of two spatial dimensions and... 

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