Exploring the independence of feasible solutions in two-component hyperspectral imaging data using mutual information map

In recent years, independent component analysis (ICA) has gained attention in chemistry as a blind-source separation technique (BSS) to find underlying components in multi- and hyper-spectral imaging (MSI/HSI) data [1]. Independence is the main constraint of ICA and it is believed that this constraint is a mathematical constraint which make ICA inappropriate for multivariate resolution purposes [2]. On the other side, multivariate curve resolution (MCR) methods are confronted with rotational ambiguities which means instead of a unique solution, a set of feasible solutions fulfilling the constraints can be obtained. The application of the strong constraint of independence, like in ICA, eliminate rotation ambiguities, but then the solution obtained is usually outside the range of MCR feasible solutions. In some implementations of ICA (like in mean-filed ICA, MFICA), the independence constraint is relaxed by the application of non-negativity constraints and the solution can be then feasible. In these cases, ICA finds the more independent solution within the range of feasible solutions [3]. Consequently, there is a major concern about ICA application for multivariate resolution purposes. On this matter, there are two main questions which need clear answers by chemometricians; (i) is there any change of independence among solutions in the feasible bands? and (ii) which ICA algorithm can give solutions inside the feasible band? Therefore, the main objective of the present contribution is answering to these two important questions for HSI data. In this regard, mutual information (MI) was used to calculate independence between different solutions [4]. Additionally, three different two-component HSI datasets were simulated with different degrees of overlap in spatial profiles. Feasible solutions were obtained by two common approaches of grid search and Lawton-Sylvester plot and error maps were calculated. Then, MI values were calculated for all solutions especially for solutions in feasible bands and MI maps were obtained for the different datasets. Inspection of the results showed that different solutions in the feasible bands have different MI values and therefore different independence. Additionally, the MI values were lower for spectral profiles (more independent) and bigger for elution profiles (more dependent) based on duality. In general, the know solution in MI maps was near to more dependent solutions for concentration profiles and near to less dependent solutions for spectral profiles which is due to applying “independence” constraints to the spectral profiles. The performance of three well-known ICA algorithms of MFICA, mutual information based least dependent component analysis (MILCA) and joint approximate diagonalization of eigenmatrices (JADE) as well as multivariate curve resolution-alternating least squares (MCR-ALS) were investigated using MI maps for simulated datasets. MI maps showed that the solutions of MFICA and MCR-ALS are in the feasible bands but the MILCA and JADE solutions which are just based on the independence maximization are outside the MI maps. Finally, a real two-component hyperspectral imaging dataset was used to confirm the applicability of the proposed method.