Components: the principal component calculation

If stack .stk and $I_{0}$ files are both loaded, the Components button of Fig. 17 becomes sensitive. Within that menu, the button Calculate components launches the calculation of principal components. At the end of the calculation, one is asked to save the results as a .pca file.

Figure 17: The Component menu in pca_gui. After pressing the Calculate button, you must decide the number of significant components based on observations of the eigenvalues, eigenimages, and eigenspectra.

The Component widget has three graphic areas which display eigenimages, eigenspectra, and eigenvalues. Eigevalues are depicted with white ``x's'' in the plot on lower left corner. The current eigenimage is shown on the upper right corner in a red-white intensity scale. The scale is such that negative values are scaled red, positive values white, and zero is black. The current eigenspectrum is shown in the lower right corner. The Component slider is used to select which eigenimage and eigenvalue are shown. The Move component up and Move component down buttons are used to change the order of the components. Displayed component can be moved up or down the list, in which components are sorted with decreasing eigenvalue.

Your goal in examining eigenspectra and eigenimages is to select the number of significant components present in the sample. As discussed in our first paper [2], you can generally tell when components become insigificant by observing a ``salt and pepper'' pattern in eigenimages (indicating that the component is showing only random rather than systematic variations of signal from pixel-to-pixel), and random fluctuations in the eigenspectra. Based on this examination, you should adjust the number of Significant components to be used in further analyses of the data.

The Powerspectra of eigenimages button displays the spatial frequency power spectrum for each eigenimage, all scaled on the same color scale (see Fig. 18). If an eigenimage consists only of noise, its power spectrum will be flat; that is, it will have the same values at all frequencies (because shot noise is uncorrelated pixel-to-pixel, and the Fourier transform of a delta function is a flat function). This is sometimes helpful for judging the number of significant components. The Significant components button is used to select the number of significant components. In order to proceed with the analysis, it is required to choose this number to be smaller than the number of photon energies in the stack.

Figure 18: Spatial power spectra of component images. Because insigificant components are expected to mainly show noise, one expects them to have spatial powerspectra with nearly constant signal at all spatial frequencies. Significant components should have power spectra more like a typical image, with a peak at low frequencies that declines at higher spatial frequencies.

There are also different options of saving PCA results which become available only after the number of significant components has been selected with the scroll-bar at the upper left corner of the widget. Save all displayed eigenvals, eigenspectra as ``.eps'' saves eigenvalues and eigenspectra of significant components in .eps Encapsulated PostScript files. Save all displayed eigenvals, eigenspectra, eigenimages as ``.png'' saves .png portable network graphics images of all significant components. Save all displayed eigenvals, eigenspectra, eigenimages as ``.csv'', ''.nc'' saves eigenvalues and eigenimages in Excel-readable .csv text files, together with NetCDF .nc files.