Background to PCA: SVD Sort

 

Image Processing Steps is CasaXPS

 

Thickness Measurements and Images

 

Image Processing using PCA and Peak Synthesis

 

 

Modern imaging XPS instruments offer a means of collecting data both spatially and energy resolved [1]. These data sets may be viewed as a stack of images acquired from the same location on the sample over a range of energies, or alternatively the data set can be interpreted as an image, where each pixel contains an energy spectrum. Typically these data sets are large and while many images, so acquired, are lacking in features, their existence is useful in the sense these images include information about the noise in the data and are therefore valuable beyond merely defining the background to images taken at peak intensities.

 

One means of utilizing the full set of data is Principal Component Analysis (PCA), where the set of images are decomposed into an orthogonal set of images, from which the maximum variation in the data is partitioned into the abstract images with the largest eigenvalues. The abstract factor images without features can be attributed to noise. If the set of original images are reconstructed from only those abstract images containing significant information, the result is a new data set where the influence of noise is reduced in magnitude. Data smoothing in the spatial domain is the net result. What is more, spectra determined from these reconstructed images are modified by the procedure without moving data between bins in the energy sense and therefore the influence of noise on each energy bin is reduced without need of averaging in the energy domain. If the data is now viewed as a set of spectra recorded at each pixel, the result is a set of spectra with counting statistics determined from the entire acquisition time used to record the image set. That is, smoothed spectra, without damage to the line-shapes. Figure 1 shows both PCA reconstructed spectra and the spectrum from the same pixel without processing.

 

Figure 1: Spectrum at an image pixel. The PCA reconstructed spectrum is fitted with three synthetic components: Cu 3p, Al 2p Oxide and Al 2p Metal

 

The result of processing the entire image set into spectra following a PCA reconstruction is remarkable and at first hard to believe. The spectra seen in Figure 1 show both the poor signal to noise in the original raw data and the smoothed data following the PCA and subsequent projection onto the significant abstract factors in the spatial domain. The PCA processed spectrum clearly shows the shapes of three underlying peaks, namely, Cu 3p, Al 2p Oxide and Al 2p Metal. Fitting the peaks using synthetic lineshapes allows the intensities attributed to these three state to be partitioned into separate images (Figure 2), which when overlaid supports the spectral information in each pixel by reproducing images consistent with the expected result seen in Figure 2.

 

Figure 2: Cu grid on Al foil after peak-fitting spectra at each pixel using the model shown in Figure 1. Note the two Al 2p peaks result in positive visual information in the background to the Cu grid. No such information was apparent in the original set of images. Data was acquired on a Kratos Axis Ultra upgraded with a pulse counting delay line detector.

 

The image of the copper grid seen in Figure 2 should be compared to the original data displayed in Figure 3. The two bands of images are displayed using the same 10%/90% threshold rule used in Figure 2, where the images are taken from the energy interval around the maximum for the Cu 3p and Al 2p Oxide peak positions. While it is true that images of similar definition can be obtained by summing the appropriate raw data, the real advantage of the procedure lies in the ability to perform proper quantification at each pixel within the image and therefore obtain quantifiable results for features with size related to the spatial dimensions of the pixels in the image set.

 

Figure 3: Unprocessed images near the peak maximum for the Cu 3p and Al 2p Oxide transitions.

 

[1] D. E. Peebles et al, Multivariate statistical analysis for XPS spectral imaging: effect of analyzer resolution, submitted JVST (2003).