# Spectral Shape Descriptors

#### Rafael Maia, Thomas White, Hugo Gruson, John Endler, Pierre-Paul Bitton, Chad Eliason

#### 2020-02-08

This table is adapted from [1] and lists all spectral descriptors that are computed by the `summary` function in `pavo`.

Color variable | Names used | Formula | Reference
--- | --- | --- | ---
Brightness | Total brightness, total reflectance, spectral intensity | \[B_1=B_T=\int_{\lambda_{min}}^{\lambda_{max}} R_i = \sum_{\lambda_{min}}^{\lambda_{max}} R_i\] | [2],[3],[4],[5],[6],[7],[8],[9]
| Mean brightness | \[B_2=\frac{\sum_{\lambda_{min}}^{\lambda_{max}} R_i}{n_w}=\frac{B_1}{n_w}\] | [10],[11]
| Intensity | \[B_3=R_{max}\] | [2],[12],[13]
Saturation | Chroma, reflectance ratio, spectral purity | \[S_1= \frac{\sum_{\lambda_a}^{\lambda_b} R_i}{\sum_{\lambda_{min}}^{\lambda_{max}} R_i} = \frac{\sum_{\lambda_a}^{\lambda_b} R_i}{B_1}\] | [4],[5],[6],[11],[14],[9]
| Spectral saturation | \[S_2 = \frac{R_{max}}{R_{min}}\] | [2]
| Chroma | \[S_3 = \frac{\sum_{\lambda_{R_{max}}-50}^{\lambda_{R_{max}}+50} R_i}{B_1}\] | [9]
| Spectral purity | \[S_4 = \lvert b\text{max}_{neg}\lvert\] | [2]
| Chroma | \[S_5=\sqrt{(B_r-B_g)^2+(B_y-B_b)^2}\] | [7]
| Constrast, amplitude | \[S_6 = R_{max} - R_{min}\] | [12],[13]
| Spectral saturation | \[S_7 = \frac{\left(\sum_{\lambda_{320}}^{\lambda_{R_{mid}}} R_i - \sum_{\lambda_{R_{mid}}}^{\lambda_{700}} R_i\right)}{B_1}\] | [3],[8]
| Chroma | \[S_8=\frac{R_{max}-R_{min}}{B_2}\] | [3],[6]
| Carotenoid chroma | \[S_9=\frac{R_{\lambda_{700}}-R_{\lambda_{450}}}{R_{\lambda_{700}}}\] | [14]
| Peaky chroma | \[S_{10}=\lvert b\text{max}_{neg} \lvert\frac{R_{max}-R_{min}}{B_2}\] | [4]
Hue | Hue, peak wavelength, spectral location | \[H_1=\lambda_{R_{max}}\] | [2],[4],[5],[10],[6],[12],[11],[9]
| Hue | \[H_2 = \lambda_{b\text{max}_{neg}}\] | [5],[6]
| Hue | \[H_3 = \lambda_{R_{mid}}\] | [3],[6],[8]
| Hue | \[H_4 = \arctan\left(\frac{B_y-B_b}{B_r-B_g}\right)\] | [7]
| Hue | \[H_5 = \lambda_{b\text{max}_{pos}}\] | [13]

- \(R_i\): percentage (or proportional) reflectance at the \(i\)th wavelength
- \(\lambda_{max}\), \(\lambda_{min}\): upper and lower (respectively) limits of wavelengths
- \(n_w\): number of wavelength intervals used to calculate \(B_T\)
- \(R_{max}\), \(R_{min}\): maximum and minimum percent reflectances, respectively
- \(\lambda_{R_{max}}\): wavelength at maximum reflectance
- \(b\text{max}_{neg}\), \(b\text{max}_{pos}\): maximum negative and positive slopes of reflectance curve in a region of interest
- \(B_r\), \(B_y\), \(B_g\), \(B_b\): total brightness in red (\(r=625-700\,nm\)), yellow (\(y=550-625\,nm\)), green (\(g=475-550\,nm\)) and blue (\(b=400-475\,nm\)) segments of the spectrum
- \(\lambda_{R_{mid}}\): wavelength at the reflectance midpoint between \(R_{max}\) and \(R_{min}\) (i.e., \(\frac{R_{max}+R_{min}}{2}\))

# References

1. Montgomerie R. 2006 Analyzing Colors. In Bird Coloration, Volume 1: Mechanisms and Measurements (eds GE Hill, KJ McGraw), p. 640. Harvard University Press.

2. Andersson S. 1999 Morphology of UV reflectance in a whistling-thrush: Implications for the study of structural colour signalling in birds. Journal of Avian Biology 30, 193–204.

3. Andersson S, Pryke SR, Örnborg J, Lawes MJ, Andersson M. 2002 Multiple receivers, multiple ornaments, and a trade‐off between agonistic and epigamic signaling in a widowbird. The American Naturalist 160, 683–691. (doi:10.1086/342817)

4. Örnborg J, Andersson S, Griffith SC, Sheldon BC. 2002 Seasonal changes in a ultraviolet structural colour signal in blue tits, Parus caeruleus. Biological Journal of the Linnean Society 76, 237–245. (doi:10.1111/j.1095-8312.2002.tb02085.x)

5. Andersson S, Örnborg J, Andersson M. 1998 Ultraviolet sexual dimorphism and assortative mating in blue tits. Proceedings of the Royal Society of London B: Biological Sciences 265, 445–450. (doi:10.1098/rspb.1998.0315)

6. Smiseth PT, Örnborg J, Andersson S, Amundsen T. 2001 Is male plumage reflectance correlated with paternal care in bluethroats? Behav Ecol 12, 164–170. (doi:10.1093/beheco/12.2.164)

7. Saks L, McGraw K, Horak P. 2003 How feather colour reflects its carotenoid content. Functional Ecology 17, 555–561. (doi:10.1046/j.1365-2435.2003.00765.x)

8. Pryke SR, Lawes MJ, Andersson S. 2001 Agonistic carotenoid signalling in male red-collared widowbirds: Aggression related to the colour signal of both the territory owner and model intruder. Animal Behaviour 62, 695–704. (doi:10.1006/anbe.2001.1804)

9. Shawkey MD, Estes AM, Siefferman LM, Hill GE. 2003 Nanostructure predicts intraspecific variation in ultraviolet–blue plumage colour. Proceedings of the Royal Society of London B: Biological Sciences 270, 1455–1460. (doi:10.1098/rspb.2003.2390)

10. Delhey K, Johnsen A, Peters A, Andersson S, Kempenaers B. 2003 Paternity analysis reveals opposing selection pressures on crown coloration in the blue tit (Parus caeruleus). Proceedings of the Royal Society of London B: Biological Sciences 270, 2057–2063. (doi:10.1098/rspb.2003.2460)

11. Siefferman L, Hill GE. 2005 UV-blue structural coloration and competition for nestboxes in male eastern bluebirds. Animal Behaviour 69, 67–72. (doi:10.1016/j.anbehav.2003.12.026)

12. Keyser AJ, Hill GE. 2000 Structurally based plumage coloration is an honest signal of quality in male blue grosbeaks. Behav Ecol 11, 202–209. (doi:10.1093/beheco/11.2.202)

13. Keyser AJ, Hill GE. 1999 Condition–dependent variation in the blue–ultraviolet coloration of a structurally based plumage ornament. Proceedings of the Royal Society of London B: Biological Sciences 266, 771–777. (doi:10.1098/rspb.1999.0704)

14. Peters A, Denk AG, Delhey K, Kempenaers B. 2004 Carotenoid-based bill colour as an indicator of immunocompetence and sperm performance in male mallards. Journal of Evolutionary Biology 17, 1111–1120. (doi:10.1111/j.1420-9101.2004.00743.x)