Proper-orthogonal Decomposition

A great tool to analyze large data sets, compare data sets, for commonality. 
Caution needs to be used to not fall victim to the temptation of interpreting POD modes as physical structures that can be seen as driving forces or manifestations of processes. While this may be the case, this is a coincidence and not a fundamental aspect of the decomposition.
As we have demonstrated, each POD mode inherently contains weighted contributions of all original input images.

Chen, H., Reuss, D. L., Hung, D. L. S. and Sick, V., "A practical guide for using proper orthogonal decomposition in engine research," International Journal of Engine Research  14 (4), 307-319 (2013)    10.1177/1468087412455748

Chen, H., Reuss, D. L. and Sick, V., "On the use and interpretation of proper orthogonal decomposition of in-cylinder engine flows," Measurement Science & Technology 23, 085302 (2012) 10.1088/0957-0233/23/8/085302

Independent Component Analysis

This approach can extract image features that are clearly physically independent and therefore is in contrast to POD an option to extract detail information that might point to physically sig ificant features, e. g. in flow field.

The image sequences below illustrate this behavior and emphasize the difference to the POD analysis.  See text after the figure for some details on ICA.


The aim of ICA is to find the underlying independent source flow-structures s1 ,…, sn of which every flow field is composed. Every snapshot x1 ,…, xm contains different proportions, given by a mixture matrix A, of these sources s. The coefficients in the mixture matrix consist of the absolute value and the sign. While the sign indicates relative direction, absolute normalized coefficient values represent kinetic energy, as described by Chen et al. for the case of POD.

This model can be written as x = As ó s=Wx, where W = A-1. In this equation x is known from the measurements, while A and s are unknown. Hence, the equation is solved iteratively to find the best estimate y of the independent components s while maximizing the statistical independence of these estimated components y

The table below shows an overview that illustrates how well ICA can extract the original sources compared to POD. The metric used is the relevance index  (see Liu and Haworth SAE 2001-01-0830, DOI 10.4271/2011-01-0830). This index Rp equals 1 or -1 for perfect structural match. Since both ICA and POD have multiplicative coefficients to assign direction and magnitude to components or modes, the sign of Rp in the table below is of no consequence and only the magnitude should be considered to evaluate how well a component or mode matches an original source field  si