Counting Vermeer


5.3 Parameters of the Fourier spectrum

To see how the behavior of the Fourier transform depends on the size or scale of the chosen region, consider figure [1], which shows the vertical thread density map of the canvas of The Milkmaid (L07) when calculated with regions that measure (a) 0.25 cm per side, (b) 0.33 cm, (c) 0.5 cm, (d) 0.75 cm, (e) 1 cm, and (f) 2 cm per side. All the images are calculated at the same set of grid points, but using the differently sized regions. The larger regions provide unreliable estimates because there is too much variability within the regions. The smaller regions show spurious (and unreliable) detail caused by the local variations. In between ‘too large’ and ‘too small’ is ‘just right’.


Using different-sized regions examines the canvas at different scales and can lead to different results for both angle and thread density maps.


Perhaps the easiest way to understand this behavior is to recognize that any object will appear different at different scales. For example, when looked at from far away, a piece of wood may appear smooth, yet may look rough when magnified. With the Fourier-derived thread density images, this sensitivity to scale can be controlled by carefully selecting the size of the regions for analysis.

Another parameter that effects the appearance of the weave map is the ‘overlap factor’. This specifies how much each region overlaps its neighbors. Say the region is 1 x 1 cm. An overlap of 2 would mean that succeeding regions begin 0.5 cm apart. An overlap of 4 would mean succeeding regions begin 0.25 cm apart. A consequence of this is that larger overlap values imply more points in the weave map. Figure [2] illustrates the same canvas X-ray calculated with the same region size, but with different overlaps.

Perhaps the most dramatic change in the appearance of weave maps is caused by different false colors used to specify the values within the weave map. Figures [3] and [4] show the same data colored using six different false color schemes. Some features may be easier to see in certain color schemes, and some of the authors of this monograph prefer looking at particular colors.


A portion of Woman Writing a Letter, with Her Maid is calculated using regions of 0.5 x 0.5 cm, but with different overlaps: 1, 2, 3, and 4. Here the same data is displayed in two ways. In the top version, each weave map is displayed proportionally to its size (to the number of pixels it contains). In the bottom version, all four weave maps are displayed at the same physical size, which emphasizes the poorer resolution of the versions with smaller overlap.


The vertical thread density map of The Geographer is colored using six different schemes: ThermometerColors, SunsetColors, Rainbow, RedGreenSplit, StarryNightColors, and FruitPunchColors. Some features of the weave map may be more clearly visible in some color maps, but color map preferences are often personal.


The angle maps of The Milkmaid are colored using the six color schemes of figure 3. Some features may be more apparent in some color schemes.

As mentioned in chapter 4, it is important to have a reasonably good initial estimate of the thread count when using the Fourier Transform. This is because the algorithm must search for the peaks (for instance, the black areas of figure 4 in § 5.2), and there may be many peaks. For example, if the initializing hand count was too small by a factor of 2, the peaks that occur in the small blue circles (in the top and bottom spectra in figure 4, § 5.2) would be the detected local peaks, and this would ‘confirm’ the existence of the thread count at this erroneous value. Therefore, it is important to have an estimate of the approximate value of the peaks that correspond to the actual thread count. Fortunately, this is easily obtained by taking a modest number of accurate hand counts.

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