# Spatial Frequency Analysis of Perforated Board Designs

## A Comparison of Design Motifs

# Introduction

Techniques for analyzing the spatial frequencies in perforated board designs were discussed earlier [1]. This article applies these methods to characterize several perforated board design motifs.

# Data Set

The previous set of analyzed Usonian perforated boards are denoted “FLW”[2]. Several other design motifs are investigated here. These images were found on Pinterest[3] and then processed for analysis. The category “ranma” comes from images of Japanese carved wooden transom screens. The “geometric”, “leaves”, and “organic” categories contain images from laser-cut panel designs. The “southwest” category images were derived from stencils and other patterns.

# Analysis Method

Besides the spectral analysis techniques described in [1], characteristic feature size was investigated. The perforated board design from the Seth Peterson cottage is used to illustrate:

The 2D spectrum of this design is:

The spatial frequency for the vertical and horizontal directions is shown. The units correspond to frequency cycles that would cover the horizontal axis of the image.

The red circle denotes the 95% power radius. It is calculated by first performing an azimuthal integration to get power versus spatial frequency radius:

The cumulative power is then calculated by summing the spectral power contained from 0 to frequency f:

This cumulative spectral power (y-axis) is plotted using a ‘logit’ scale. This scale plots the fraction of total power and makes it easier to see fractions close to unity. In this case, 95% of the total power is below a spatial frequency of 21.

The characteristic feature size was taken as the corresponding dimension to the 95% frequency radius:

Normalized to the design width W gives a unitless version:

A red dot with the diameter d is shown overlaid on the perforated board design showing this is a reasonable definition:

While the selection of a 95% power is somewhat arbitrary, it does capture most of the information in the image. Applying a 4th order 2D Butterworth filter with this cutoff frequency removes the image information that resides in higher frequencies:

The resulting perforated board image with this information removed shows the effect of losing high frequencies:

The design is still quite recognizable despite this degradation. If the image is thresholded using the Python library function `skimage.filters.threshold_otsu`

, it is recovered fairly well:

The same process applied to a more detailed design such as a Japanese ranma gives:

The corresponding filtered image is:

After thresholding, the recovered image is:

Experimentation with power cut-offs below 95% caused a greater loss of image detail. For example, with a 90% cut-off, the above thresholded ranma image becomes:

# Results

A correlation scatterplot of the power within 95% versus 68% spatial frequencies indicates some features to investigate:

Most notably, some perforated board designs have lower spatial frequencies while some have more high frequency content. To better see this, the normalized characteristic feature dimensions are summarized for each category in a box plot:

The roll-off of 1D Power Spectral Density were fit to a power law, as described earlier. A lower value indicates more high frequency content, which can be seen in the box plot of power law exponents:

The anisotropy metric [1] `A`

varies between 0 (isotropic) and 1 (maximum anisotropy). Values for the design motif categories are:

Finally, note that the “ranma” designs have a higher aspect ratio in general:

Aspect ratio was calculated with panel dimensions (not design window dimensions as in earlier work) for convenience.

# Conclusion

Usonian perforated board designs generally have the highest spectral anisotropy and largest characteristic feature size for the data set in this analysis. Across individual designs, there is overlap with the ‘southwest’ designs in the data set on these parameters. The power law exponents between these categories are also close.

The higher anisotropy of the Usonian designs is due to lower symmetry of their oblique lines compared with “southwest” designs. For example, the Usonian rhomboidal shapes create spectral power in the vertical and one oblique direction. The triangular shapes of “southwest” designs will have energy in two oblique directions as well as a horizontal or vertical direction. This is shown below for a rhomboid and triangle of equal area and similar angles:

Japanese ranma designs contain more detail which is quantified by smaller characteristic feature sizes and lower power law exponents. These designs offer closer depictions of natural scenes. The higher aspect ratios of these designs (notice the correlation for “ranma”) is responsible for slightly higher anisotropy:

The “leaves” and “organic” categories can be viewed as having a level of abstraction mid-way between natural scenes and Usonian designs. The parameters in this study are aligned with this interpretation.

# References

**[1]** J. van Saders. “2D Spectrum Characterization: A note on analyzing spectra of perforated board designs”. https://medium.com/tangibit-studios/2d-spectrum-characterization-e288f255cc59

**[2]** J. van Saders. “Feature Analysis of Usonian Perforated Board Designs: Characterizing Frank Lloyd Wright’s Light Screens”. https://medium.com/tangibit-studios/feature-analysis-of-usonian-perforated-board-designs-9fa97c8aa383

**[3]** J. van Saders. Pinterest Board “Organic Design”. https://www.pinterest.com/15ny67vwqklu218/organic-design/