# colorSpec User Guide

#### Glenn Davis

#### 2020-04-01

colorSpec is an R package providing an S3 class with methods for color spectra. It supports the standard calculations with spectral properties of light sources, materials, cameras, eyes, scanners, etc.. And it works well with the more general action spectra. Many ideas are taken from packages hsdar [16], hyperSpec [4], pavo [17], photobiology [3], and zoo [30].

Some features:

- a clear classification of the common color spectra into 4 types
- flexible organization for the spectra in memory, using an S3 class - colorSpec
- a product algebra for the colorSpec objects
- uniform handling of biological eyes, electronic cameras, and general action spectra
- a few advanced calculations, such as computing optimal colors (aka MacAdam Limits)
- inverse colorimetry, e.g. reflectance recovery from response
- built-in essential tables, such as the CIE illuminants and color matching functions
- a package logging system with log levels taken from the popular Log4J
- support for reading a few spectrum file types, including CGATS
- bonus files containing some other interesting spectra
- minimal dependencies on other R packages

Regarding the dependencies, the following packages are Imports:

- spacesXYZ [6], required for computing CCT
- MASS [26], required for camera emulation, which uses the matrix pseudoinverse `MASS::ginv()`

The following packages are Suggests:

- rootSolve [23], required for inverse colorimetry
- rgl [5], required for plotting optimal colors in 3D
- knitr [29], rmarkdown [1], and spacesRGB [7], required for building the vignettes
- microbenchmark [18], for a timer with higher precision than R’s built-in timer
- arrangements [13], makes the zonohedron calculations for optimal colors a little bit faster
- quadprog [24], may be used in `responsivityMetrics()` to determine whether the generators lie in an open linear halfspace

Some non-features:

- There is no support for 3D colors spaces other than XYZ and RGB; see packages colorspace [10], colorscience [8], spacesRGB [7], and spacesXYZ [6] for more 3D spaces.
- there are few non-linear operations. The only such operations are conversion of absorbance to transmittance, and the reparameterized wavelength \(\omega\) from \(\lambda\) in `computeADL()`. The electronic camera model is purely linear with no dark current offset or other deviations.
- there is little support for scientific units; for these see packages photobiology [3] and colorscience [8]
- photons are parameterized by wavelength in nanometers (nm); other wavelength units (such as Ångstrom and micron) and alternative parameterizations (such as wavenumber and electronvolt) are not available.

# 1 Spectrum Types

Pick up any book on color physics (e.g. [28], [20], [19], or [12]) or color management (e.g. [9]) and you will see plots of many spectra. Let’s start with a simple division of these spectra into 4 basic types:

For the infinite-dimensional spaces, the interval [380,780] is used for illustration; in specific calculations it can vary. Note that of the 4 vector spaces, only \(L^*\) and \(M\) are isomorphic, but we take the mathematical point of view that although they are isomorphic, they are not the same. For a proof of this isomorphism, see Appendix D. Multiplication operators are the infinite-dimensional generalization of diagonal matrices. For more background on this functional analysis, see [27] and [14].

For the finite-dimensional spaces, it takes the full sequence of wavelengths and not just the endpoints. The wavelength sequence is typically regular not always. In this case all 4 vector spaces are isomorphic (since they are the same dimension), but we still take the mathematical point of view that they are not the same space.

The last `type='responsivity.material'` is the least common. There is an example in [9] (Figure 10.11a, page 141) of a scanner, where the 3 spectra are called the effective spectral responsivities of that scanner. They are the pointwise product of the scanner light source and the responsitivies of the scanner’s RGB sensor; see also pages 146-147. On page 335, in equation (B.4), there is a more general product. In colorSpec, both of these products and more are peformed in the function `product()`. Another example of effective spectral responsivities are those of a reference scanner from SMPTE, see [22].

Every colorSpec object has one of these types, but it is not stored with the object. The object stores a `quantity` which then determines the `type`; see the next section for more discussion. A synonym for `type` might be `space`, but this could be confused with color space.

colorSpec does not actually use the finite-dimensional representations in Table 1.1; the organization is flexible. And it would not be efficient memory use to store a diagonal matrix as such. For discussion of the organization, see section 4.

Given 2 finite-dimensional spectra of types `'light'` and `'responsivity.light'` the response (a real number) is their dot product multiplied by the step between wavelengths.

All materials in this document are non-fluorescent; i.e. the outgoing photons reflected (or transmitted) only come from incoming photons of the same wavelength. A transparent material transmits an incoming light spectrum and a new spectrum emerges on the other side. If the material is not fluorescent, the outgoing spectrum is the same as the incoming, except there is a reduction of power that depends only on the wavelength (and the material). If the light power were divided into N bins, the transmitted power spectrum would be a diagonal NxN matrix times the incoming spectrum.

A reflectance spectrum is mathematically the same as a transmittance spectrum, except we compare the outgoing light spectrum to that of a perfect reflecting diffuser. Such a material does not exist, like many concepts in physics, but it is a very useful idealization.

# 2 Spectrum Quantities

Unfortunately there are two common metrics for quantifying spectra with `type='light'` - energy of photons and number of photons. The former - radiometric - is the oldest, being used in the 19th century. The latter - actinometric - was not used until the 20th century (after the modern concept of photons was proposed in 1905). So colorimetry uses radiometric quantities by convention and actinometric ones are converted to radiometric automatically for calculations. The conversion is easy; see the function `radiometric()`, [20] pp. 93-94, and [19] p. 12.

Similarly, `'responsivity.light'` can be radiometric (e.g. the CIE color matching functions) or actinometric (e.g. the quantum efficiency of a CMOS sensor). These actinometric spectra are also converted to radiometric on the fly.

For responsivity, we distinguish between 3 types of response: electrical, neural, and action. In colorSpec this 3-way distinction is only used in a few places:

- for the y label of the spectrum in `plot()`
- to determine the default adaption method in `calibrate()`
- for the conversion equations in `radiometric()` and `actinometric()`

Note that the `action` response is really a grab-bag for responses that are neither `electrical` (a modern solid-state photosensor) nor `neural` (a biological eye).

Here are the valid types and their quantities:

The colorSpec quantities are typically not the same as the SI quantities; they are more general.

First consider light sources (`type='light'`).

The colorSpec `quantity='energy'` includes all 5 of these power-based SI quantities: radiant power (radiant flux), irradiance, radiant exitance, radiant intensity, and radiance. And it also includes these energy-based quantities: radiant energy, radiant exposure, and the time integrals of radiant exitance, radiant intensity, and radiance. Thus `quantity='energy'` includes 10 true physical SI quantities, which all include energy and optionally include area, solid angle, and time.

Similary, the colorSpec `quantity='photons'` includes all 5 of these SI quantities: photon flux, photon irradiance, photon exitance, photon intensity, and photon radiance. It also includes these 5 quantities integrated over time, e.g. photon fluence.

Versions of colorSpec before 0.7-1 used `power` in place of `energy`. But now we have switched to `energy`; see Appendix E for the reasons why. `power` and `power->*` are still supported, but deprecated, and will eventually be phased out.

For `type='light' and type='responsivity.light'`, each radiometric quantity has a corresponding actinometric quantity. The following table shows the correspondences:

The colorSpec functions `radiometric()` and `actinometric()` convert back and forth between the two metrics. For `energy` and `energy->electrical` and `energy->action` the functions actually do assume the example units. For `energy->electrical` the example units in the table are in common use for electronic cameras. Note that for `energy->action`, the action we have in mind is photosynthesis. A quick internet search shows that the maximum theoretical photosynthesis response is between 1/16 and 1/8 of an \(\text{O}_2\) molecule per photon. For `energy->neural` see the functions man pages for more discussion.
Since these are spectra parameterized by nm, the example units should all add \(\text{nm}^{-1}\) at the end, but this is suppressed for simplicity.

Now consider materials (`type='material'`). The situation here is simpler. The colorSpec `quantity='reflectance'`, `'transmittance'`, and `'absorbance'` correspond directly to the SI quantities. All reflecting materials are Lambertian and opaque, and all transmitting materials have only direct transmission with no scatter.

# 3 Construction of colorSpec objects

The user constructs a `colorSpec` object `x` using the function `colorSpec()`:

    x <- colorSpec( data, wavelength, quantity='auto', organization='auto', specnames=NULL )

The arguments are:

`data`
a vector or matrix of the spectrum values. In case `data` is a vector, `x` has a single spectrum and the number of points in that spectrum is the length of `data`. In case `data` is a matrix, the spectra are stored in the columns, so the number of points in each spectrum is the number of rows in `data`. It is OK for the matrix to have only 0 or 1 column.

`wavelength`
a numeric vector of wavelengths for all the spectra in `x`. The length of this vector must be equal to `NROW(data)`, and the unit must be nanometers. The sequence must be increasing. The `wavelength` of `x` can be changed after construction.

`quantity`
a character string giving the quantity of all spectra; see Table 2.1 for a list of valid values. In case `quantity='auto'`, a guess is made from the `specnames`. The `quantity` of `x` can be changed later.

`organization`
a character string giving the desired organization of the returned colorSpec object. In case `organization='auto'`, the organization is `'vector'` or `'matrix'` depending on `data`. The `organization` of `x` can be changed after construction. See the next section for discussion of all 4 possible organizations.

`specnames`
a character vector with length equal to the number of spectra in `data`, and with no duplicates. If `specnames=NULL` and `data` is a vector, then `specnames` is set to `deparse(substitute(data))`. If `specnames=NULL` and `data` is a matrix, then `specnames` is set to `colnames(data)`. If `specnames` is still not a character vector with the right length, or if there are duplicate names, then `specnames` is set to `'S1', 'S2', ...` with a warning message. Names can be changed after construction.

Compare `colorSpec()` with the function `stats::ts()`.

# 4 colorSpec object organization

A spectrum is similar to a time-series (with time replaced by wavelength), and so the organization of a `colorSpec` object is similar to that of the time-series objects in package stats. A single time-series is organized as a vector with class `ts`, and a multiple time series is organized as a matrix (with the series in the columns) with class `mts`. We decided to use a single class name `colorSpec`, continue the idea of different organizations, and allow 2 more organizations. Here are the 4 possible organizations, in order of increasing complexity:

`'vector'`
The object is a numeric vector with attributes but no dimensions, like a time-series `ts`. This organization works for a single spectrum only, which is very common. The common arithmetic operations work well with this organization. The length of the vector is the number of wavelengths. The class of the object is `c('colorSpec','numeric')`.

`'matrix'`
The object is a matrix with attributes, like a multiple time-series `mts`. This is probably the most suitable organization in most cases, but it does not support extra data (see `'df.row'` below). The common arithmetic and subsetting operations work well; and even `round()` works. The number of columns is the number of spectra, and the spectrum names are stored as the column names. This organization can be used for any number of spectra, including 0 or 1. The class of the object is `c('colorSpec', 'matrix')`.

`'df.col'`
The object is a data frame with attributes. The spectra are stored in the columns. But the first column is always the wavelength sequence, so the spectra are in columns 2:(M+1), where M is the number of spectra. This organization mirrors the most common organization in text files and spreadsheets. The common arithmetic operations do not work, and the initial wavelength column is awkward to handle. The spectrum names are stored as the column names of the data frame. This organization can be used for any number of spectra, including 0 or 1. This organization imitates the “long” format in package hyperSpec. The class of the object is `c('colorSpec', 'data.frame')`.

`'df.row'`
The object is a data frame with attributes. The last (right-most) column is a matrix with spectra in the rows. This matrix is the transpose of the matrix used when the organization is `'matrix'`. The common arithmetic operations do not work. The spectrum names are stored as the row names of the data frame. This organization can be used for any number of spectra, including 0 or 1. This organization imitates the “tall” format in package hyperSpec. This is the only organization that supports extra data associated with each spectrum, such as physical parameters, time parameters, descriptive strings, or whatever. This extra data occupies the initial columns of the data frame that come before the spectra, and can be any data frame with the right number of rows. This extra data can be assigned to any spectrum with the `'df.row'` organization. The class of the object is `c('colorSpec', 'data.frame')`.

# 5 colorSpec object attributes

The attribute list is kept as small as possible. Here it is:
The user should never have to modify these using the function `attr()`.

# 6 Spectrum File Import

There are 5 text file formats that can be imported; no binary formats are supported yet. The function `readSpectra()` reads a few lines from the top of the file to try and determine the type. If successful, it then calls the appropriate read function; see the colorSpec reference guide for details. The file formats are:

`XYY`
There is a line matching `'^(wave|wv?l)'` (not case sensitive) followed by the the names of the spectra. This is the column header line. All lines above this one are taken to be metadata. This is probably the most common file format; see the sample file `ciexyz31_1.csv`.

`spreadsheet`
There is a line matching `'^(ID|SAMPLE|Time)'`. This line and lines below must be tab-separated. Fields matching `'^[A-Z]+([0-9.]+)nm$'` are taken to be spectral data and other fields are taken to be extradata. All lines above this one are taken to be metadata. The organization of the returned object is `'df.row'`. This is a good format for automated acquisition of many spectra, using a spectrometer. See the sample file `E131102.txt`.

`scope`
This is a file format used by Ocean Optics spectrometer software. There is a line `>>>>>Begin Processed Spectral Data<<<<<`. The following lines contain wavelength and energy separated by a tab. There is only 1 spectrum per file. The organization of the returned object is `'vector'`. See the sample file `pos1-20x.scope`.

`CGATS`
This is a standardized format for exchange of color data, covered by both ANSI and ISO standards, see [2] and [11]. It might be best understood by looking at some samples, such as `inst/extdata/objects/Rosco.txt`. Unfortunately these standards do not give a standard way to name the spectral data. The function `readSpectra()` considers field names that match the pattern `"^(nm|SPEC_|SPECTRAL_)[_A-Z]*([0-9.]+)$"` to be spectral data and other fields are considered extra data. The organization of the returned object is `'df.row'`.

`Control`
This is a personal format used for digitizing images of plots from manufacturer datasheets and academic papers. It is structured like a Microsoft `.INI` file. There is a `[Control]` section establishing a simple linear map from the image pixels in the file to the wavelength and spectrum quantities. Only 3 points are really necessary. It is OK for there to be a little rotation of the plot axes relative to the image. This is followed by a section for each spectrum, in XY pixel units only. Conversion to wavelength and spectral quantities happens during on-the-fly after read. The organization of the returned object is `'vector'`.

# 7 Package Options

There is a function `cs.options()` for setting options private to the package. There are 3 such options, and all are related to the package logging mechanism. All messages go to the console.

There is an option for setting the logging level. The levels are the 6 standard ones taken from `Log4J`: `FATAL`, `ERROR`, `WARN`, `INFO`, `DEBUG`, and `TRACE`. One can set higher levels to see more info.

By default, when an `ERROR` event occurs, execution stops. But there is a colorSpec option to continue. The logging level `FATAL` is reserved for internal errors, when execution always stops.

Finally, there is an option for how the message is formatted - a layout option. For details see the help page for the function `cs.options()`.

# 8 Future Work

Here are a few possible improvements and additions.

wavelength
handling the wavelength sequence, e.g. for `product()` and `resample()`, is an annoyance. We might consider adding a global wavelength option that all spectra are automatically resampled to.

fluorescent materials
Recall that a non-fluorescent material corresponds to a diagonal matrix, which operates in a trivial way on light spectra. A diagonal matrix can be stored much more compactly as a plain vector, and multiplication of a diagonal matrix by a vector simplifies to entrywise (Hadamard) multiplication. A fluorescent material corresponds to a non-diagonal matrix – called the Excitation Emission Matrix or Donaldson Matrix. The product in Appendix C is still multilinear, but the material product in the middle is no longer symmetric, so enhancements to the product computations must be made. This is a new level of complexity and memory usage, and may require a new type of memory organization.

comparisons
There should a metric of some kind that compares two material spectra. There should be a way to compare 2 colorSpec objects of the same type, especially `'responsivity.light'`. For example, there would then be a way to evaluate how close an electronic camera comes to satisying the Maxwell-Ives Criterion. Possible metrics would be the principal angles between subspaces.

`plot()`
the `product()` function saves the terms with the product object, but the `plot()` function ignores them. It may be useful to have an option to plot the individual terms too.

# 9 References

[1] ALLAIRE, JJ, XIE, Yihui, MCPHERSON, Jonathan, LURASCHI, Javier, USHEY, Kevin, ATKINS, Aron, WICKHAM, Hadley, CHENG, Joe, CHANG, Winston and IANNONE, Richard. rmarkdown: Dynamic Documents for R [online]. 2018. Available at: https://rmarkdown.rstudio.com

[2] ANSI/CGATS.17. Graphic technology — Exchange format for colour and process control data using XML or ASCII text [online]. Standard. New York City, USA: American National Standards Institute. 2009. Available at: https://webstore.ansi.org/RecordDetail.aspx?sku=ANSI%2fCGATS.17%3a2009+(R2015)

[3] APHALO, Pedro J. The r4photobiology suite. UV4Plants Bulletin [online]. 2015, 2015(1), 21–29. Available at: doi:10.19232/uv4pb.2015.1.14

[4] BELEITES, Claudia and SERGO, Valter. hyperSpec: a package to handle hyperspectral data sets in R [online]. 2017. Available at: http://hyperspec.r-forge.r-project.org

[5] DANIEL ADLER, Duncan Murdoch and OTHERS. rgl: 3D Visualization Using OpenGL [online]. 2018. Available at: https://CRAN.R-project.org/package=rgl

[6] DAVIS, Glenn. spacesXYZ: CIE XYZ and some of Its Derived Color Spaces [online]. 2018. Available at: https://cran.r-project.org/package=spacesXYZ

[7] DAVIS, Glenn. spacesRGB: Standard and User-Defined RGB Color Spaces, with Conversion Between RGB and CIE XYZ [online]. 2018. Available at: https://cran.r-project.org/package=spacesRGB

[8] GAMA, Jose. colorscience: Color Science Methods and Data [online]. 2016. Available at: https://CRAN.R-project.org/package=colorscience

[9] GIORGIANNI, E.J., MADDEN, T.E. and KRISS, M. Digital Color Management: Encoding Solutions. B.m.: Wiley, 2009. The Wiley-IS&T Series in Imaging Science and Technology. ISBN 9780470512449.

[10] IHAKA, Ross, MURRELL, Paul, HORNIK, Kurt, FISHER, Jason C. and ZEILEIS, Achim. colorspace: Color Space Manipulation [online]. 2016. Available at: https://CRAN.R-project.org/package=colorspace

[11] ISO/28178. Graphic technology — Exchange format for colour and process control data using XML or ASCII text [online]. Standard. Geneva, CH: International Organization for Standardization. 2009. Available at: https://www.iso.org/standard/44527.html

[12] KOENDERINK, J.J. Color for the Sciences. B.m.: MIT Press, 2010. ISBN 9780262014281.

[13] LAI, Randy. arrangements: Fast Generators and Iterators for Permutations, Combinations and Partitions [online]. 2018. Available at: https://cran.r-project.org/package=arrangements

[14] LANG, Serge. Real Analysis. Reading, Massachusetts: Addison-Wesley Pub. Co., 1969. Addison-wesley series in mathematics. ISBN 0-201-04179-0.

[15] LANG, Serge. Linear Algebra. B.m.: Springer New York, 2013. Undergraduate texts in mathematics. ISBN 9781475719499.

[16] LEHNERT, Lukas W., MEYER, Hanna and BENDIX, Jörg. hsdar: Manage, analyse and simulate hyperspectral data in R [online]. 2017. Available at: https://CRAN.R-project.org/package=hsdar

[17] MAIA, Rafael, ELIASON, Chad M., BITTON, Pierre-Paul, DOUCET, Stephanie M. and SHAWKEY, Matthew D. pavo: an R Package for the analysis, visualization and organization of spectral data. Methods in Ecology and Evolution [online]. 2013, 4(10), 609–613. Available at: doi:10.1111/2041-210X.12069

[18] OLAF MERSMANN AND CLAUDIA BELEITES AND RAINER HURLING AND ARI FRIEDMAN AND JOSHUA M. ULRICH. microbenchmark: Accurate Timing Functions [online]. 2018. Available at: https://CRAN.R-project.org/package=microbenchmark

[19] OLEARI, Claudio. Standard Colorimetry: Definitions, Algorithms and Software. B.m.: Wiley, 2016. SDC-society of dyers and colourists. ISBN 9781118894446.

[20] PACKER, Orin and WILLIAMS, David R. Light, the Retinal Image, and Photoreceptors. In: Steven K. SHEVELL, ed. The Science of Color. B.m.: Optical Society of America, 2003.

[21] RUDIN, Walter. Real and Complex Analysis. Second. New York: McGraw-Hill, 1974. McGraw-hill series in higher mathematics. ISBN 0-07-054233-3.

[22] SMPTE/2065-2. ST 2065-2:2012 - SMPTE Standard - Academy Printing Density (APD) #x2014; Spectral Responsivities, Reference Measurement Device and Spectral Calculation. ST 2065-2:2012 [online]. 2012, 1–14. Available at: doi:10.5594/SMPTE.ST2065-2.2012

[23] SOETAERT, Karline. rootSolve: Nonlinear root finding, equilibrium and steady-state analysis of ordinary differential equations [online]. 2009. Available at: https://CRAN.R-project.org/package=rootSolve

[24] TURLACH, Berwin A. quadprog: Functions to Solve Quadratic Programming Problems [online]. 2019. Available at: https://cran.r-project.org/package=quadprog

[25] USER163644. Multiplication operator on \(L^1\) [online]. B.m.: Mathematics Stack Exchange. Available at: https://math.stackexchange.com/q/1378498. URL:https://math.stackexchange.com/q/1378498 (version: 2015-07-29)

[26] VENABLES, W. N. and RIPLEY, B. D. Modern Applied Statistics with S [online]. Fourth. New York: Springer, 2002. Available at: http://www.stats.ox.ac.uk/pub/MASS4

[27] WIKIPEDIA. Multiplication operator — wikipedia, the free encyclopedia [online]. 2016. Available at: https://en.wikipedia.org/w/index.php?title=Multiplication_operator&oldid=733677418. [Online; accessed 3-November-2017]

[28] WYSZECKI, G. and STILES, W.S. Color Science: Concepts and Methods, Quantitative Data and Formulae. B.m.: Wiley, 2000. Wiley series in pure and applied optics. ISBN 9780471399186.

[29] XIE, Yihui. knitr: A General-Purpose Package for Dynamic Report Generation in R [online]. 2018. Available at: https://yihui.name/knitr/

[30] ZEILEIS, Achim and GROTHENDIECK, Gabor. zoo: S3 Infrastructure for Regular and Irregular Time Series. Journal of Statistical Software [online]. 2005, 14(6), 1–27. Available at: doi:10.18637/jss.v014.i06

# 10 Appendix A - Built-in colorSpec Objects

The following are built-in colorSpec objects that are commonly used. They are global objects that are automatically available when colorSpec is loaded. For more details on each see the corresponding help topic.

# 11 Appendix B - Bonus Spectral Data

Each built-in colorSpec object in Appendix A takes time to fully document in `.Rd` help files. Here are some bonus spectra files under folder `extdata` that users may find interesting and useful. Use the function `readSpectra()` to construct a colorSpec object from the file, for example:

    sunlight = readSpectra( system.file( 'extdata/illuminants/sunlight.txt', package='colorSpec' ) )
    sunlight

    ##
    ## colorSpec object.   The organization is 'df.col'.  Object size is 4024 bytes.
    ## the object describes a single source of light, and the quantity is 'energy' (energy of photons, which is radiometric).
    ## Wavelength range: 300 to 830 nm.  Step size is 10 nm.
    ##
    ## 1 spectra
    ## 54 data points / spectrum
    ##
    ##            Source Min  Max LambdaMax Integral
    ## 1 sunlight.Energy   0 1167       480   472390

See the top of each file for sources, attribution, and other information. Alternatively, one can run `summary()` on the imported object. Some of the files in `Control` format have associated `JPG` or `PNG` images of plots.

# 12 Appendix C - Spectrum Products

This Appendix is a very formal mathematical treatment of spectra. In infinite dimensions we use the terminology of functional analysis. In finite dimensions we use the terminology of linear algebra. For easier reference here is a repeat of Table 1.1:

There are 5 natural binary products on these spaces:

An equivalent way to handle these material diagonal matrices is to represent them instead as simple vectors – the entries along the diagonal. The above products with diagonal matrices then become the much simpler entrywise or Hadamard product. This is how it is done in colorSpec, using R’s built-in entrywise product operation.

The first 4 products can be strung together to get a product: \[L \times M_1 \times ... \times M_m \times L^* \to R\] It is not hard to show that this product is multilinear. This means that if one fixes all terms except the \(i^{th}\) material location, then the composition: \[M \to L \times M_1 \times ... \times \bullet \times ... \times M_m \times L^* \to R\] is linear, see [15]. The first inclusion map means to place the material spectrum in \(M\) at the ith variable slot \(\bullet\) in the product. The composition map is a functional on \(M\) which is an element of \(M^*\), i.e. a material responder. This special method of creating a material responder - a spectrum in \(M^*\) - plus all the products in the above table, are available in the function `product()` in colorSpec. See that help page for examples. Compare the previous equation with equation (B.4) (page 335) in [9].

The right-hand term \(R\) can be thought of as standing for Response or Real numbers. In colorSpec the light responders can have multiple channels, e.g. R, G, and B, and so there are conventions on the admissible numbers of spectra for each term in these products. See the help page for `colorSpec::product()` for details.

# 13 Appendix D - Proofs

This appendix gives some proofs of some earlier statements about infinite dimensional function spaces. It is not relevant to the software in any way, and is likely of interest only to mathematicians and physicists. This proof is not original and is largely an expanded version of a discussion on math.stackexchange.com, see [25].

Throughout this appendix, \(L^1\) denotes \(L^1[0,1]\), which is isomorphic to \(L^1[ \lambda_{min}, \lambda_{max}]\) where \([ \lambda_{min}, \lambda_{max}]\) is an arbitrary interval of wavelengths. Furthermore, \(L^\infty\) denotes \(L^\infty[0,1]\), and \(\mu\) denotes Lebesgue measure on \([0,1]\).

Proposition: Suppose \(\phi :[0,1] \to \mathbb{R}\) is a measurable function, and that \(\phi f \in L^1\) whenever \(f \in L^1\). Define the multiplication operator \(M_{\phi} : L^1 \to L^1\) by \(M_{\phi}(f) = \phi f\). Then

1. \(M_{\phi}\) is a continuous linear operator on \(L^1\)
2. \(\phi \in L^\infty[0,1]\)
3. \(\left\lVert M_\phi \right\rVert ~=~ \left\lVert \phi \right\rVert _ \infty\)

Lemma: Given \(f,g \in L^1\) and a sequence \({f_n} \in L^1\) and \(\phi\) as above. Suppose \[a) ~ f_n \to f ~~~~~~~\text{and} ~~~~~~~~ b) ~ \phi f_n \to g\]

where both convergences are in \(L^1\). Then \(\phi f = g\) almost everywhere.

Proof: From \(a)\), and Theorem 3.12 in [21], p. 70, \(f_n\) has a subsequence that converges to \(f\) a.e. Replace \(f_n\) by this subsequence and \(a)\) and \(b)\) are still true. From \(b)\), \(\phi f_n\) has a subsequence that converges to \(g\) a.e. Replace \(\phi f_n\) and \(f_n\) by this subsequence and \(a)\) and \(b)\) are still true. So we have \[a') ~ f_n \to f ~~~~~~~\text{and}~~~~~~~~ a'') ~ \phi f_n \to \phi f ~~~~~~~\text{and}~~~~~~~~ b') ~ \phi f_n \to g\] where all convergences are almost everywhere. From \(a'')\) and \(b')\) we conclude that \(\phi f = g\) a.e. \(\square\).

Proof of Proposition: Parts \(a)\) and \(b)\) of the Lemma state that \((f_n,\phi f_n) \to (f,g)\) in \(L^1 \times L^1\). Define the graph of \(M_{\phi}\) in \(L^1 \times L^1\) to be the set of all pairs \((f,\phi f)\), \(f \in L^1\). The conclusion of the Lemma states that this graph is closed. So by the closed graph theorem ([21] p. 122), \(M_\phi\) is continuous. This shows part \(1.\)

Consider the functional \(f \mapsto \int \phi f \, d\mu\) on \(L^1\). It is the composition of \(M_\phi\) and a trivially continuous functional, and is therefore continous. Since \(L^1\) is \(\sigma\)-finite, the standard duality theorem ([21] p. 136), implies that there is a unique \(g \in L^\infty\) so that \(\int \phi f \, d\mu = \int g f \, d\mu\) for all \(f \in L^1\). Therefore \(\phi = g\), and this shows part \(2.\)

If \(\left\lVert \phi \right\rVert_\infty = 0\) then \(\phi=0\) and \(\left\lVert M_\phi \right\rVert = 0\), so part \(3.\) is trivially true. Assume now that \(\left\lVert \phi \right\rVert_\infty > 0\). Let \(f \in L^1\) with \(\left\lVert f \right\rVert = 1\). Then \[\left\lVert M_\phi (f) \right\rVert_1 ~=~ \int_0^1 \left\lvert \phi f \right\rvert \, d\mu ~=~ \int_0^1 \left\lvert \phi \right\rvert \left\lvert f \right\rvert \, d\mu ~\le~ \left\lVert \phi \right\rVert_\infty \int_0^1 \left\lvert f \right\rvert \, d\mu ~=~ \left\lVert \phi \right\rVert_\infty \] This shows \(\left\lVert M_\phi \right\rVert \le \left\lVert \phi \right\rVert_\infty\). For the other direction, let \(\alpha\) be any number with \(0 < \alpha < \left\lVert \phi \right\rVert_\infty\), and let \(E_\alpha := \left\lvert \phi \right\rvert ^{-1} ( [\alpha,\infty] )\). Then by the definition of \(\left\lVert \phi \right\rVert_\infty\), \(\mu( E_\alpha) > 0\). Let \(f_\alpha := \chi_{E_\alpha} / \mu( E_\alpha)\) (the \(L^1\)-normalized indicator function of \(E_\alpha\)). Then \[ \left\lVert M_\phi (f_\alpha) \right\rVert_1 ~:=~ \left\lVert \phi f_\alpha \right\rVert_1 ~:=~ \int_0^1 \left\lvert \phi \right\rvert f_\alpha \, d\mu ~\ge~ \int_0^1 \alpha f_\alpha \, d\mu ~=~ \alpha \int_0^1 f_\alpha \, d\mu ~=~ \alpha \left\lVert f_\alpha \right\rVert_1 ~=~ \alpha \] So \(\left\lVert M_\phi \right\rVert \ge \alpha\) for every \(\alpha < \left\lVert \phi \right\rVert_\infty\), which implies \(\left\lVert M_\phi \right\rVert \ge \left\lVert \phi \right\rVert_\infty\). This shows part \(3.\) \(\square\).

Corollary: Let \(M\) be the vector space of all multiplication operators on \(L^1\). Then the mapping \(L^\infty \to M\) given by \(\phi \mapsto M_\phi\) is a norm-preserving isomorphism.
Proof:The mapping is clearly injective. The Proposition shows that it is surjective and norm-preserving. \(\square\)

# 14 Appendix E - Energy vs Power

Consider these subtle differences in the way light sources and responders (detectors) are appropriately measured:

- Power is an appropriate way to measure constant light sources, such as the lighting in an office, or a standard illuminant. But energy is appropriate for variable sources, such as a pulsed light source.
- The response to power is an appropriate way to measure non-integrating responders, such as a biological eye (`'power->neural'`), a photovoltaic cell (`'power->electrical'`), or photosynthesis (`'power->action'`). All of these respond (almost) instantaneously.
  The response to energy is an appropriate way to measure integrating responders, such as an electronic camera (`'energy->electrical'`), or erythemal exposure (`'energy->action'`). For these responders there is a well-defined integration time.

Since color science emphasizes constant light sources and biological eyes, power has always seemed more appropriate to me than energy. But starting with colorSpec version 0.7-1 I decided to switched to `energy` for these reasons:

- Energy is more fundamental than power. Power is defined from energy by the messy process of differentiation. The conversion between energy of photons and number of photons is straightforward, without a messy integration time. The terms energy-based and photon-based are well-established in vision science, and in software packages like photobiology [3].
- In flash-based photography (`energy->electrical`) what matters to the color of the photograph is the integral of the spectrum (the energy) of the flash bulb over the exposure interval of the camera. This is a case when the light spectrum is not constant; it can vary over that interval. Similarly, in photosynthesis (`energy->action`) what matters to the plant is the integral of daylight from sunrise to sunset. Think of the daytime as a very long pulse. For an example, see the file `solar.exposure.txt` in Appendix B.

I also considered allowing both `energy` and `power`, and both `photons` and `photons/time`. But this would force the user to decide whether a light source is constant or variable, and whether a responder/detector is integrating or non-integrating. So things quickly got complicated. These common radiant SI quantities - radiant power, irradiance, radiant exitance, radiant intensity, radiance - differ only in area and steradian. Time is now grouped with these 2 geometric units.

# 15 Appendix F - Continuous vs Discrete

In physics, wavelengths are in some interval of real numbers - an uncountable set. But in engineering, one is forced to use wavelengths taken from a finite table of values. Given a table of wavelengths and values, a software package must make some sort of choice of what the physical interpretation of this table really is. In colorSpec the choice is schizophrenic - there are multiple interpretations.

With few exceptions, a table of wavelengths and values is interpreted as a step function. Such functions are sometimes called piecewise-constant. This requires a lengthy explanation. Suppose `X` is a colorSpec object with \(N\) wavelengths: \(\lambda_1 < \lambda_2 < \ldots < \lambda_N\). Define \(N\) intervals \(I_i := [\beta_{i-1},\beta_i]\) where \[\begin{equation} \beta_0 := \tfrac{3}{2}\lambda_1 - \tfrac{1}{2}\lambda_2 ~~~~~ \beta_i := (\lambda_i + \lambda_{i+1})/2, ~ i{=}1,\ldots,N-1 ~~~~~ \beta_N := \tfrac{3}{2}\lambda_N - \tfrac{1}{2}\lambda_{N-1} \end{equation}\]

The intervals \(I_i\) are a partition of \([\beta_0,\beta_N]\). Note that \([\beta_0,\beta_N]\) is slightly bigger than \([\lambda_1,\lambda_N]\) because the endpoints are extended. Define the \(i'th\) step \(\mu_i := \operatorname{length}(I_i), ~ i{=}1,\ldots,N\). If the sequence \(\{\lambda_i\}\) is regular (\(\lambda_{i+1}-\lambda_i\) is constant), then \(\mu_i\) is constant with the same value, and each \(\lambda_i\) is the center of \(I_i\). Now suppose `X` has \(m\) spectra (channels) with vector values \(\mathbf{y}_i \in \mathbb{R}^m\). Then the physical function realization of `X` is a function \(\mathbf{y}(\lambda) : [\beta_0,\beta_N] \to \mathbb{R}^m\) that takes the constant value \(\mathbf{y}_i\) on \(I_i\). If the sequence \(\{\lambda_i\}\) is regular, then all \(\mathbf{y}_i\) have the same weight, including the first and last. This is the step function interpretation used in `product()`, `interpolate()`, `bandSpectra()`, and many other places.

The exceptions are `resample()` and `plot()`. In `resample()` the physical functions are piecewise-linear, piecewise-cubic, or piecewise-quintic, depending on the argument `method` (a smoothing method is also available). In `plot()` the spectra are plotted as piecewise-linear (using `lines()`) by default, but an option to plot as step functions (using `segments()`) was added in v. 1.2-0.

The function `product()` often takes the product of spectra. Note that the product of piecewise-linear functions is not piecewise-linear, but the product of step functions is still a step function.

In lengthy calculations using both interpretations, there are inevitable numerical errors, which are certainly larger than the usual numerical roundoff. But we do not attempt carry the error analysis any further than that.

# 16 Session Information

    R version 3.6.3 (2020-02-29)
    Platform: i386-w64-mingw32/i386 (32-bit)
    Running under: Windows 7 (build 7601) Service Pack 1

    Matrix products: default

    locale:
    [1] LC_COLLATE=C
    [2] LC_CTYPE=English_United States.1252
    [3] LC_MONETARY=English_United States.1252
    [4] LC_NUMERIC=C
    [5] LC_TIME=English_United States.1252

    attached base packages:
    [1] stats     graphics  grDevices utils     datasets  methods   base

    other attached packages:
    [1] colorSpec_1.2-1

    loaded via a namespace (and not attached):
     [1] Rcpp_1.0.3           digest_0.6.25        spacesXYZ_1.1-1
     [4] MASS_7.3-51.5        magrittr_1.5         evaluate_0.14
     [7] rlang_0.4.4          stringi_1.4.6        rmarkdown_2.1
    [10] tools_3.6.3          stringr_1.4.0        xfun_0.12
    [13] yaml_2.2.1           compiler_3.6.3       microbenchmark_1.4-7
    [16] htmltools_0.4.0      knitr_1.28