Local rank pixel

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. 

In this context, a number of local rank maps equal to the number of layers in the image is obtained. It is important to note that the information in the local rank map of a particular layer has been obtained taking into consideration the pixels of that layer and those in the neighboring layers in the depth direction. 

The introduction of local rank constraints in image analysis requires one to determine the number and identity of the missing components in the pixels to be constrained. When recalling the flexibility in the introduction of constraints, it is important to stress that not all the pixels need to be constrained. Thus, pixels with an ambiguous estimation of the local rank, or a dubious identification of the missing components, should be left unconstrained. 

An exploratory analysis performed by FSIW-EFA provides an estimate of the number of components in each pixel. For resolution purposes, only those pixels in the partial local rank map will be potentially constrained, because these are the pixels for which a robust estimation of the number of missing components can be obtained. However, the FSIW-EFA information is not sufficient to identify which components are absent from the constrained pixels. For identification purposes, the local rank information should be combined with reference spectral information, the ideal reference being the pure spectra of the constituents, although in most images not all of these are known. For the image components with no pure spectrum available, the reference taken is an approximation of this pure spectrum. These approximate pure spectra can be obtained by pure variable selection methods, or they may be the result of a simpler MCR-ALS analysis where only non-negativity constraints have been applied. 

Thus, to set the local rank constraints, the two necessary inputs are (i) a partial local rank map, with the rank of pixels that can be potentially constrained (see Figure 2.10a) and (ii) a pure spectrum (or good estimate) per each constituent (see Figure 2.10b). 

The incorporation of local rank constraints in any pixel I starts by estimating the number of missing components as follows ... 

Although these pills were supposed to be formed by two constituents, a FSIW-EFA analysis detected the presence of a third compound (impurity) in some cases [65]. According to the theoretical composition of the piU and to the local rank analysis, information on the presence/absence of constituents in the different images could be introduced in the multi-image resolution process. From the pure spectra resolved and the distribution maps in Figure 2.17, the positive influence of the pill with the largest amount of impurity was noticeable when this compound was modeled in pills where it was present in very few pixels only. 

Fixed-size image window-evolving factor analysis (FSIW-EFA) is an evolution of the local rank algorithm fixed size moving window-EFA [111], particularly designed for the study of the local pixel complexity in images [112]. To do so, two main ideas are taken into account the need to divide the image into small areas to get local information and the need to preserve the 2D or 3D spatial... 

Figure 2.9 FSIW-EFA modus operand (monolayer Raman emulsion image), (a) Construction of pixel windows (b) singular value plots of local PCA analyses (c) complete local rank map and (d) partial local rank map.

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