# 0.3 Methods and results

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This module discusses the procedure of our research project and outcomes.

## Step 1: obtain the raw data

We received x-ray images of paintings from the Van Gogh Museum of Amsterdam. Images usually sampled at 600. Figure 1 shows the x-ray image of Van Gogh’s `Backyards of Old Houses in Antwerp in the Snow` (F260) provided by the Van Gogh Museum.

## Step 2: short time fourier analysis

We apply shot time Fourier analysis for each 0.5” × 0.5” swatch. We discarded the outrange frequency peaks and set the value as NaN for the corresponding swatches. For multiple peaks in the frequency region of interest, we accepted the peak that is closest to the median value.

## Step 3: spectra of whole-painting

We sampled the short-space spectrum every 1/4” in both directions (horizontal and vertical) for the whole-painting by choosing swatches overlap each other by half in each direction. Thus the spectra of whole-painting were obtained. And we could determine the warp and weft direction of the canvas according to the spread of measurements. Calculations were made in Matlab and took about three hours to analyze F205 on a laptop computer.

Figure 2 shows the resulting spectra of F205.

## Step 4: deviations matching analysis

From the spectra of the whole-painting, we obtained thread count deviations spectra by calculating the distributions of frequencies and subtracting the averages.

Figure 3 and 4 show the vertical thread count deviations of F205 and F260 respectively. Horizontal deviations spectra are not shown here.

Visually, we can see the matching strips between these two painting. But how well do they match?

## Step 5: 2d to 1d conversion

We first computed the 1D thread count deviations from the 2D data. The 1D thread count deviations are obtained by summing the column deviations of 2D data while discarding all the NaNs if any.

Figure 5 and 6 are the corresponding 1D plot of F205 and F260.

We can clearly see the similarities now. But how do they correlate then?

## Step 6: correlation determination

We then computed unbiased correlation coefficient between 1D vertical thread count deviations of F205 and F260 along the x-axis. You can clearly see a peak (0.7479) appears at the 55th alignment as two paintings are mapped to the matching alignment, or visually “best fit” together. The correlation mapping plot is shown in figure 7.

The results indicate that these two paintings (F205 and F260) were likely cut from the same canvas roll, sharing the more variable weft direction.

## Orientation issues in matching

When we computed the correlation for the paintings, we actually did it four times in our Matlab programs. This is because that any two paintings could possibly match in either direction and each painting could be rotated 0, 90 or 180 degree.

In plotting the correlation mapping, we choose the one that gave the best fit among all possible orientations.

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