Local rank constraint

Local-Rank Constraints, Selectivity, and Zero-Concentration Windows... 

Local-rank constraints are related to mathematical properties of a data set and can be applied to all data sets, regardless of their chemical nature. These types of constraints are associated with the concept of local rank, which describes how the number and distribution of components varies locally along the data set. The key constraint within this family is selectivity. Selectivity constraints can be used in concentration and spectral windows where only one component is present to completely suppress the ambiguity linked to the complementary profiles in the system. Selective concentration windows provide unique spectra of the associated components, and vice versa. The powerful effect of these type of constraints and their direct link with the corresponding... 

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. 

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 ... 

Figure 2.10 Procedure followed to incorporate local rank constraints in resolution (monolayer emulsion example).

To set local rank constraints, the local rank iirformation obtained from FSIW-EFA will be combined with reference spectral information [128] (see Figure 2.11). Potential pixels to be constrained will be those in the local rank map because they have a robust estimation of the rank and, therefore, of the number of missing components in the pixel. However, FSIW-EFA information is insufficient to identify the absent components in the constrained pixels. For identification purposes, the local rank information (Figure 2.11a) should be combined with reference spectral information (Figure 2.11b). The ideal reference would be the pure spectra of the constituents but, in most images, not all of them are known. For image components with no pure spectrum available, approximations of their pure spectrum can come from purest variable selection methods or, better, be the result of a simpler MCR-ALS analysis where only noimegativity constraints have been applied. 

Figure 2.11 Procedure followed to incorporate local rank constraints in resolution (monolayer emulsion example), (a) Partial local rank map (b) reference spectral...

Constraint Qualification For a local optimum to satisfy the KKT conditions, an additional regularity condition is required on the constraints. This can be defined in several ways. A typical condition is that the active constraints at x be linearly independent i.e., the matrix [Vh(x ) I VgA(x )] is full column rank, where gA is the vector of inequality constraints with elements that satisfy g x ) = 0. With this constraint qualification, the KKT multipliers (X, v) are guaranteed to be unique at the optimal solution. 

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