Data cube

In EDXS the so-called spectrum-image method [4.122] can also be employed. A series of spectra is taken from a scanned rectangular field resulting in a data cube with its upper plane as the scanned x-y area and the third axis as the X-ray spectrum. Comprehensive information about the chemical composition and element distribution is extractable from this data set by subsequent processing. 

This brief summary is devoted to machines, not to the science they permitted. Yet, science relies critically on experimentation, the making of which may start in a laboratory, workshop, or in a factory. Astronomy is quite peculiar in respect of experimentation. It relies almost exclusively on contemplation recording images of inaccessible objects and their spectral properties i.e., recording data cubes (two angular coordinates and a spectral one), without any capability to act on the parameters of the observed object. Few sciences have lesser means to experiment yet none, perhaps, delivers so much with so little. 

The instrument, which is placed at the telescope focal plane, consists of optics and a detector to measure the light. As depicted in Fig. 2, the instrument attempts to measure a three-dimensional data cube - intensity as a function of wavelength (A) and two spatial dimensions on the sky (right ascension and dechnation). 

Figure 2. Three-dimensional data cube that is probed by an astronomical instmment the intensity is a function of two spatial directions on the sky (right ascension and declination - analogous to longitude and latitude) and the wavelength dimension.

In the z-direction (depth-direction) the resolution is determined by the confocal instrument settings. While it is essential to be aware of the limitations of confocal measurements, as mentioned above, it is possible to create three-dimensional maps. That means probing a sample in the x, y and z-directions. In Figure 2 the data cube of a two-dimensional map is shown. One element (row) of the cube has been picked out and its content enlarged on the right side of the figure. It is evident that each row therefore contains the information of a whole spectrum. 

Figure 2 Data cube generation in mapping and imaging. The four-dimensional hyperspectral data cube contains the full spectral information, absorbance vs. wavenumber (v, cm ), for each x,y pixel from the imaged area, as is shown above. A horizontal slice through that cube contains a chemical image (e.g., band intensity at selected v for each x,y pixel of the image) as is shown below. The latter result could be obtained by Global Imaging (in which only the intensity distribution of a certain band over the imaged area would be recorded).

A CCD array collects simultaneously the spectral data emanating from an array of spatial locations on the irradiated sample surface [20]. Thus, recorded is a three-dimensional data cube, with two coordinates representing the sample and one for the spectral dimension at each (x,y) point (cf. Figure 2). The spectral dimension in this case is only the intensity of a certain Raman band, used to identify the component of interest. This band should be unique for the component of interest of the sample and its intensity should be high enough in... 

Figure 75-2 shows third-order data or a hyperspectral data cube where the spectral amplitude is measured at multiple frequencies (spectrum) with X and Y spatial dimensions included. Each plane in the figure represents the amplitude of the spectral signal at a single frequency for an X and Y coordinate spatial image. 

In this equation, whereas the same loading matrix (YT matrix) is common for the different individual data matrices Dt, k = 1, 2, 3, 4, four different score matrices Xjt, k = 1, 2, 3, 4 are considered to explain the variation in Daug. Since these four D. matrices have equal sizes (same number of rows or samples and of columns or variables) they can also be arranged in a three-way data cube, with the four data matrices in the different slabs of this cube. However, in the frame of the MCR-ALS method and of the general bilinear model in (10), it is preferable to consider them to be arranged in the column-wise augmented data matrix Daug. 

Rx Signals Data Cube - R Matched Filter - Fl(pixel, speed, Heading)... 

The MFP process is illustrated in figure 8. The received signal, for all transmit/receive pairs, over a CPI forms a data cube. The matched filter, for a particular scene pixel and target velocity, also forms a data cube. For multiple operating frequencies, additional cubes would be formed. A single MFP output, a pixel and velocity, is the inner product... 

As for all absorbance-based spectral measurements, the intensity data represented in a raw (single-beam) chemical image are a combination of the spectral response of both the instrument and the sample. In order to remove the instrument response component, it is necessary to ratio the data to a background reference. For reflectance measurements, the background is a separate data cube typically acquired from a uniform. 

Online analysis processing mainly comprises the interactive exploration of multidimensional data sets, or data cubes, which are manipulated by operations from matrix algebra, for example, slice-and-dice, roll-up, and drill-down. Computing performance is related to data warehouse size and also data quality, for example, missing data, unsharpness, and redundancy. The multidimensionality issue is critical for extracting pertinent information and selecting the results to be stored and visualized. 

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). 

The data cube combines spectral and spatial information and therefore includes the requisite statistics for spectral classifications. However, new chemometric strategies have to be applied to interpret chemical imaging results. 

Mapping Historically, mapping [11] was the first method used to acquire hyperspectral data cubes, in particular with Raman spectroscopy and infrared (IR) microscopy. The image is created pixel by pixel in a step-and-acquire mode A spectrum is measured at one point of the sample, and then the sample moves to the next measurement position and another spectrum is acquired. The process is iterative for all positions in the area that define the image. 

Fiber Bundles Fiber bundles are used for Raman imaging. Several optical fibers are grouped together, each analyzing a specific sample area [13]. A 3D data cube is... 

Preprocessing enhances chemical information and removes noise and scattering effects [14]. Its specificity resides in the fact that data cubes can be preprocessed in both the wavelength and spatial dimensions. 

Multivariate Image Analysis Strong and Weak Multiway Methods Strong and weak -way methods analyze 3D and 2D matrices, respectively. Hyperspectral data cube structure is described using chemometric vocabulary [17]. A two-way matrix, such as a classical NIR spectroscopy data set, has two modes object (matrix lines) and V variables (matrix columns). Hyperspectral data cubes possess two object modes and one variable mode and can be written as an OOV data array because of their two spatial directions. 

Even if strong A -way methods are used to reduce image noise, compress data, and improve data cube visualization, weak multiway methods are more often used as they facilitate classification using classical single-point spectra. 

Weak Multiway Methods Figure 7 shows the three steps in weak N-way analysis Unfold the data cube, perform the selected chemometric methods, and refold the matrix in order to display distribution maps. Weak N-way analysis comprises two main variants ... 

Pattern recognition can be classified according to several parameters. Below we discuss only the supervised/unsupervised dichotomy because it represents two different ways of analyzing hyperspectral data cubes. Unsupervised methods (cluster analysis) classify image pixels without calibration and with spectra only, in contrast to supervised classifications. Feature extraction methods [21] such as PCA or wavelet compression are often applied before cluster analysis. 

Reference spectra choice is critical when applying supervised pattern recognition methods. The first solution is to use pure compound spectra as references. The drawback is that mixture spectra in data cubes often differ from the reference spectra. Applying the model may therefore give wrong results. The second solution, suitable in a few studies, is to select image pixels where only one compound is present in order to obtain the calibration sets. 

The raw images were dark corrected as described previously, then background corrected using a data cube collected from Spectralon (Labsphere, Inc., North Sutton,... 

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