In a general sense, linear discriminant analysis (LDA) is based on the construction of a sub-space of the space of descriptors which maximizes the separation of classes to be discriminated. The first contribution aiming to do this, the LDA of Fisher [1], is based on Huygens theorem which states that the total variance of a matrix X is the sum of the inter-class variance B and the intra-class variance W. Thus, it is sufficient to find the directions u that maximize the ratio u'Bu / u'Wu, that is to say the eigenvectors of W-1B. In very multivariate situations, such as spectrometry, the calculation of W-1 is unstable, if not impossible. Various strategies have been proposed to address this issue. The first is to apply PCA or PLS to the data in order to change to a space where the inversion of W is feasible. The second is to build a continuum between the PLS and the LDA and then search for the most suitable point in the continuum. The present study is based on a third mechanism, using orthogonal projections. It is shown that the eigenvectors of W-1B are identical to the eigenvectors of the matrix of inter class variance based on W-1X. Thus the LDA can be seen as the succession of two operations: (i) cleaning of the space of descriptors, to remove noise responsible for much of the intra-class variance; (ii) projection onto the space describing the inter-class variance. In the highly multivariate framework, we propose to perform the first operation by a projection orthogonal to the first k eigenvectors of W. The value of k determines the degree of cleaning of the space. If it is too high, there is a risk of removing useful information. If it is too low, the separation of classes is not complete. The optimal value can therefore be adjusted by (cross) validation.