## 2 Measurement Approach
### Principal Component Analysis (PCA)  
PCA [23] is a multivariate statistical procedure with the goal of extracting the important information of a dataset and representing them as uncorrelated variables, called principal components (PCs) [24]. The PCs are characterized by their orthogonal orientation to each other and are sorted according to their variance. Therefore, the first PC inherits the most distinct information of the data set, hence representing the highest variety of the complete data set. As a result, the first \(N\) PC can be linearly combined to approximate each individual input of the data set sufficiently, while \(N\) remains much lower than the number of individual input data. This can be interpreted as a dimensionality reduction while most of the information to distinguish between the individual input remains.  
If the complete data set is arranged in a \(I\times J\) matrix \(X\) with \(I\) observations of \(J\) individual variables, the PCs are obtained with a matrix factorization, the singular value decomposition. The singular value decomposition of \(X\) is defined as  
\[X=P\Delta Q^{T}, \tag{1}\]  
with the columns in \(P\) being the left singular vectors of \(X\) and also the eigenvectors of the matrix \(XX^{T}\). The columns of \(Q\) being the right singular vectors of \(X\) and the eigenvectors of the matrix \(X^{T}X\). \(\Delta\) is the diagonal matrix of the singular values, also being the diagonal matrix of the eigenvalues of the matrix \(XX^{T}\). The PC \(F\) are afterwards calculated by \(F=P\Delta\)[24]. By obtaining only the first \(L\) largest singular values and their singular vectors, a truncated matrix \(F_{L}\) with rank \(L\) can be calculated, while the Frobenious norm between \(F\) and \(F_{L}\) is minimal. With \(L\) being smaller than \(J\) the dimension of the data set has been reduced while retaining most of its variance. In the case of three-dimensional data, a dimensionality reducing PCA gives the orthogonal base under which the maximum variance of the data is maintained, i.e., the data are projected on a two-dimensional plane with a maximal spread.  
The PCA was applied early on to a variety of problems to handle multivariant data in different fields of science to improve the differentiation between measurements [25]. Since the calculated PCs are sorted with descending variances concerning the data, the PCA can be used for de-noising through simply ignoring the PCs with low variance and reconstructing the data set based only on the high-variance basis [26; 27].  
Another application of the PCA that fueled the motivation for this work is the calculation of eigenfaces [28], a term for the PC of a data set of face images usually used in face recognition tasks. By projecting the face images to a feature space, the significant variations among those faces are extracted. Reducing a large data set of many images to a smaller set of images representing features led to the hypothesis of a useful application in the thermographic flow visualization. If applied to thermographic raw images with multiple flow regimes, flow feature images with an increased distinguishability between the flow regimes arise. The flow characteristics of different flow states contribute to different amounts of variance throughout the image series in the respective image regions and, therefore, are extracted as different features. In the resulting PC, these emphasized features might have an increased distinguishability between flow regimes if compared to the average of the raw image series.