In order to obtain the correction filter C(λ), let us represent spectra as vectors. For a given temperature T and a Wien-Planckian E(λ, T) we denote the corresponding light vector as E̲T. If the visible spectrum is represented at 10 nanometers sampling from 400 nm to 700 nm, then E̲T is a 31-component vector. According to Eqs. (10)—(14) for a CCT of T we can calculate a corresponding daylight spectrum which we denote as an n-vector D̲T (where n is typically 31). Also, representing the function C(λ) as the n-vector C̲, we rewrite Eq. (15) in the discrete domain: where diag() takes a n-vector and places its components along an n × n diagonal matrix.

Supposing that we have a set of the spectra of N daylights, Ψ=[D̲1,D̲2,…,D̲N], having (correlated color) temperatures, [T1, T2, …, TN], respectively. For the same temperatures, we calculate Wien-Planckian lights Ω=[E̲1,E̲2,…,E̲N] (where the scaling term in Eq. (4) is assumed to be equal to 1). For this set of corresponding (matched temperature to correlated color temperature) Wien-Planckians to daylights we, simplistically, might think of finding a filter such that:

However, by the different natures of the equations that define Wien-Planckians and daylights, we admit a per temperature scaling factor k that can be applied. Let us denote the N scaling factors as the vector k. The approximation is now written as: We observe that Eq. (16) can be written as: where the symbol ⊙ denotes the component-wise multiplication of two matrices and t—here and henceforth—denotes a matrix/vector transpose.

Considering the individual components, we would like: where i∈[1,2, …, n] and j∈[1,2, …, N]. Expressing this approximation in log units, we use the superscript ′ to denote log-values. Then, recalling that multiplication becomes addition in log space, we can write: Rewriting Eq. (21) in matrix form: where each component of the respective N and n dimensional vectors u̲N and u̲n is equal to 1. Now, let Ci denote a n × N matrix that has 1s in the ith row but is zero, everywhere else. Similarly, let Kj denote a n × N matrix that has 1s in the jth columns but is zero everywhere else. Then: and: Equivalently, we can write Eqs. (23) and (24) as: and:

Here vec() is the operator that turns matrices into vectors (where columns are stacked on top of each other). Respectively, the ith and jth columns of the nN × n and nN × N matrices MC and MK are vec(Ci) and vec(Kj).

We are now in a position to write an optimization statement that we would like to minimise: where ∥. ∥F denotes the Frobenius norm [16]. Substituting (Ψ′− Ω′) with B:

Now we use the vec() function again and write:

Notice that the vec operator does not change the meaning of the equation. Rather, tables are transcribed to vectors but the same components are in correspondence. Substituting Eqs. (23) to (26) into (29):

Denoting b̲=vec (B) we would, equivalently, like to find C̲′ and k′ that minimizes

Let us define the nN × (n + N) matrix A = [MCMK] and the n + N coefficient vector x̲=[C̲′;k] (where “;” means stacking the vectors on top of each other). Effectively, we have written our minimisation in the form Ax̲≈b̲ which can be solved in closed form using the Moore-Penrose inverse (x̲=[AtA]−1Atb̲) [17]. The solution x̲ simultaneously solves for the filter and the scaling components. Since this minimisation is carried out in log space, the final correction filter C̲ is equal to exp(C̲′).

The reader will note that we have carried out the minimization in log units. Of course, we did this to arrive at a closed-form solution. To our knowledge, there does not exist a closed form solution to Eq. (19). Root Mean Squared Log Error minimization is commonly used and, in effect, minimizes relative error, e.g. [18].

Returning to Eq. (15) a daylight spectrum DC(λ, T), corrected from a Wien-Planckian light with the same temperature T can be written as:

This is opposed to (no superscript) the standard daylight calculated according to Eq. (10)—(14), denoted D(λ, T).

Now, suppose we have a locus filter FLocus(λ, Tf) such that a Wien-Planckian with temperature T1 is mapped to T2:

Applying the same locus filter to a daylight with a correlated color temperature T1 gives: Substituting Eq. (33) into Eq. (34) gives: and from Eq. (32) it follows that: Therefore, we can conclude that when a locus filter is applied to a daylight it will introduce the same shift in correlated color temperature as if it was applied on a Wien-Planckian illuminant with the same temperature.

To find the daylight filter C(λ) that brings any Wien-Planckian light to a daylight with the same color temperature, we start with a set of 21 Wien-Planckian lights that are equally distanced in terms of their mired color temperatures. The mired color temperature [19] (micro-reciprocal-degree), is often used to measure how similar one light color is to another. It is measured in mired units and can be calculated from a color temperature T as follows:

For our set of 21 Wien-Planckian lights, the mired color temperatures range from 50 to 250 with 10 mired step corresponding to color temperatures from 4000 K to 20,000 K with a non-uniform step. For each of these lights, we obtain an equivalent standard daylight with the same correlated color temperature (see start of Section 3 for equations on how this is done). Given these two sets of lights—Wien Planckians and standard daylights—we find a correction filter C(λ) as set out in Section 3.1. The correction filter—which, we interpret here as a transmittance filter—is shown in Fig. 4.

In order to validate the accuracy of the performed optimization, we calculate the difference between each original daylight and its reconstructed counterpart given the correction filter C(λ). For convenience, we represent the spectral functions of these two lights as vectors of measurements at a discrete number of sample points across the visible spectrum, which we denote D̲ and D̲C (for respectively the standard and corrected form of our daylights). Here we sample spectra at 10 nanometre samples from 400 to 700 nanometers (for the visible spectrum). We are most interested in how similar the shape of these daylights spectra are to one another. So, we use the angular error as an error metric: where “.” denotes the vector dot-product, ∥. ∥ is the vector magnitude and acos is the inverse cosine. We also calculate a percentage root mean square error measure. As there is a brightness difference between the spectra we normalise the actual daylight and the one predicted using the correction filter approach by their norm (respectively D̲∥D̲∥ and D̲c∥D̲c∥). We calculate the magnitude of the vector distance between these normalised vectors. As the vectors being compared have unit length they can be visualise as vectors whose tips lie on a unit (hyper sphere). In this visualization the largest distance between the tips of any two vectors has to be 2. Thus, we divide by 2 to return a percentage error. where ∥. ∥ is the L2-norm.

We found that the average angular error between the ground-truth daylights and the reconstructed ones using the daylight filter for the 21 lights used in the optimization is 1.1∘, and this corresponds to a 1%RMS. The maximum angular error was just 2.3∘ (2%RMS). These values indicate that the daylight filter gives very close spectra to the original daylights.

In Figure 5, we show two examples of spectral power distributions of original daylights and their reconstructed ones using the daylight filter. The first example corresponds to the case where the error between the two lights is equal to the mean error found after the optimization (1.1∘ angular error or 1%RMS). The second example corresponds to the maximum error found after optimization (2.3∘ angular error or 2%RMS). One can observe that in the case of maximum error the two spectra are still quite similar.

The method for calculating standard daylights was not designed to be used for creating daylights with a correlated color temperature less than 4000 K. Although lower values CCTs are possible at sunrise and sunset (and are found in measured daylights [20]). Creating daylights for correlated color temperatures lower than 4000 K is made possible with our method. For any CCT, we simply calculate our daylight spectrum using Eq. (15).

In the Granada set of daylights [20] there is a daylight spectrum that has a CCT of 3888 K (with respect to which we cannot calculate a standard daylight spectrum). In Figure 6 we plot the spectral power distribution of this measured daylight and the SPD calculated with our correction filter using Eq. (15) (both lights are normalized so their maximum relative power is equal to 1). The angular error and %RMS between these two SPDs are equal to 7.5∘ and 6.5%, respectively.

Additionally, using our new formula for daylights (Eq. (15)), we can calculate the chromaticities for a range of CCTs and make a new daylight locus. In Figure 7, we plot, in the u, v chromaticity space, both the standard daylight locus and our new correction-filter-based locus. Notice how similar the two loci are to one another (unsurprising given the closeness of the spectral fits). As our approach allows us to obtain daylights for color temperatures less than 4000 K, the dotted line in Fig. 7 shows the extended daylight locus for color temperatures between 2000 K and 4000 K (for which the conventional daylight locus is not defined).

We have shown in our previous work [9] that the shift of CCT introduced by a locus filter when applied to real lights follows very closely the shift seen by an equivalent Wien-Planckian illuminant (by equivalent we mean its color temperature is equal to the correlated color temperature of the real light) after applying the same locus filter. In the Section 3.2, we have, in effect, shown that we expect the shift in CCTs of daylights to follow the temperature shift in the corresponding Wien-Planckians when a locus filter is applied.

Let us now consider how the CCTs for the 99 (complete set of) measured daylights in Granada [20]—which do not lie on the daylight locus—shift when they are filtered by a locus filter. First, let us measure how far these lights are from the daylight locus built using our new daylight Eq. (15). For the ith daylight, DiG(λ) in the Granada set (superscript G for Granada), we obtain its CCT, Ti, using the method [21]. We calculate the 99 CCTs: {T1, T2, …, T99} for the 99 Granada daylights. Eq. (4) is then used to obtain a Wien-Planckian light Ei(λ, Ti) (where we interpret the CCT for the ith Granada daylight as a temperature for the Wien-Planckian equation). Applying our correction filter C(λ) to this light will give a daylight with the same CCT: DC(λ, Ti).

Let us denote the u′v′ 2-component vectors for the Granada daylight and its closest locus counterpart as u̲i and u̲TiC respectively. The average, median, maximum and 95th percentile of Δu′v′ over the 99 Granada daylights are reported in the first column of Table II. The just noticeable difference (JND) of this chromaticity distance has been found to be 0.004 [22, 23]. So, on average, the Granada daylights are so close to their daylight counterparts that their differences in color are close to being imperceptible. Though, there are lights up to 5 JNDs from the daylight locus.

Now we consider how the Granada daylights change when they are filtered by a locus filter. Clearly, by construction, a daylight that is on the daylight locus will shift as predicted by Eqs. (8) and (35). That is, given the correlated color temperature of a locus daylight and the LFT of the filter we can predict, in closed form, the CCT of the filtered light. How do we quantify what happens when off-daylight-locus illuminants are filtered by a locus filter?

As remarked earlier (after Eq. (8)) a single-color temperature and the set of locus filters generated by the set of all LFTs exactly generates all Wien-Planckian lights. The same is true for our corrected daylight formalism. A single daylight DC(λ, T) is mapped to all other daylights through the appropriate choice of LFT and locus filter. That is every Wien-Planckian or every corrected daylight itself seeds the whole (Planckian or daylight locus).

Similarly, an arbitrary daylight drawn from the Granada set will also induce its own locus of filtered lights (when all locus filters are applied). Intuitively, we would like these per Granada loci to be ‘like’ either the Planckian locus or, equivalently, the corrected daylight locus. Let us denote the chromaticity coordinates of the filtered Granada daylight and the corrected daylight (with the same CCTs) respectively as u̲if and u̲TiC,f (where f denotes dependence on a locus filter).

We can calculate Δi=∥u̲TiC−u̲i∥ and Δif=∥u̲TiC,f−u̲if∥. By our intuition of parallel locii, we’d like Δi ≈ Δif. If this distance is constant for each LFT then each Granada light would induce a locus parallel to the Planckian or corrected daylight locus and the distribution of temperatures across the locus would be the same. We additionally report in Table II the mean, median, max and 95th percentile of Δif after applying locus filters with locus-filter-temperature of 5000, 8000, 12,000 and −12,000. We find that the average chromaticity distance for locus filtered daylights is broadly equal to the unfiltered case except for the case of a negative LFT. But it is still, on average, within the range of just noticeable distance (JND = 0.004).

Figure 8 shows Granada daylights and filtered ones with two different locus filter temperatures in the u, v chromaticity diagram. As expected (predicted by the previous experiment), the Granada daylights shift along the locus. Left, a positive LFT shifts the u, v chromaticities for the Granada Daylights (blue stars) toward warmer colors (green circles). A negative LFT, see panel b, makes all the Granada daylights cooler (higher CCT).

In this section, we will consider the extent to which some Wratten filters filter light like locus filters. There is a large number of Wratten filters—which filter light in a variety of ways—and they are used in photographic, scientific and technical research and applications. These filters are organized into categories depending on the effects they have on the acquired image, and they are characterized by a coding system where each filter is referred to by a unique code.

Here, we consider four of the Kodak Wratten filters from two different categories: the 81 and the 82 which belongs to the light balancing category, and the 80A and the 85 from the color conversion category. Light balancing filters cause minor adjustments in the color temperature of an illuminant giving a cooler (the case of 82) or a warmer (the case of 81) light. Whereas color correction ones enable significant adjustments in the color temperature of an illuminant leading to a cooler one (the case of 80A) or a warmer one (the case of 85).

We would like to consider the extent that these filters change lights like locus filters. To do so, we need first to find the equivalent locus filter for each of the real filters. In Refs. [24, 25] the CCTs of initial lights and their filtered counterparts are given, see Table III for the filters in use. Then, the equivalent locus filters giving the same temperature change as the Wratten filters can be found by Eq. (6).

In Figure 9, we show the spectral transmittance of the four Wratten filters compared with their equivalent locus counterparts. The locus filters are normalized so that the maximum transmittance of each is equal to the maximum transmittance of the corresponding Wratten filter. We can see that locus filters have similar transmittance behavior compared to the Wrattens, but they tend to be smoother in shape. This is particularly apparent for the case of color balancing filters (Figs. 9a and 9b). In the case of light balancing filters, the difference between the Wratten ones and their locus counterparts is less apparent (Figs. 9c and 9d).

Let us now consider how the 4 Wratten filters and their companion 4 equivalent locus filters behave when they are used to filter lights. We will consider the filtering behavior for 3 sets of lights: the Granada daylights, the 102 illuminants compiled by Barnard et al. [26] which includes many artificial lights, and the set of 81 Wien-Planckian lights (from 2000 K to 10,000 K in 100 K steps). For the color conversion filters 80A and 85 (and their locus filter equivalents) we subdivide each illuminant set into two subsets. Filter 80A is designed only to make warm colors cooler so we consider how it changes the CCT of a light when it is applied only on illuminants that have an initial CCT that is less than 5000 K. Conversely, filter 85 is designed to make cool colors warmer. For this filter we will consider only the lights that have a CCT higher than 5000 K. All lights are considered for the light balancing filters 81 and 82 since they are designed only to make small alterations in the CCT.

In order to compare the behavior of Wratten filters to their equivalent locus ones we use three metrics: the change of CCT after applying the filters, the distance in u, v space to the Planckian locus, Duv, and the color rendering index (CRI Ra) of the filtered light. For the latter, we adopt the CIE 1995 method for its calculation, which is based on the color difference between standard samples rendered with the light source and the same samples rendered using an ideal light source with the same color temperature as the first one. Tables IV—VII report the mean CCT shift, the mean Duv and the mean CRI Ra when applying the Wratten filters, 80A, 85, 82, and 81, respectively, and their equivalent locus filters for the 3 sets of light. As expected, the shifts in CCT are small for the light balancing filters 81 and 82. In addition, the magnitude of the mean shift is consistent for these two Wratten filters and their equivalent locus ones. Higher mean CCT shifts are reported in the color conversion filters: 80A and 85. Here we see that the filter 85 (which makes light warmer in appearance), on average, performs like a locus filter regarding CCT shifts. However, the filter 80A, which makes lights cooler, on average, does not perform like a locus filter.

In terms of Duv, one can observe that these distances are relatively small and coherent between Kodak and locus cases for light balancing filters (81 and 82). However, in the case of color conversion filters, 80A performs slightly better than the locus filter in maintaining the distance between the filtered light and the locus when applied on Granada daylights. But it works less well for the Planckians and the Barnard set. The locus filter performs better than filter 85 for the 3 sets of illuminants.

Regarding color rendering indices, the light balancing filters 81 and 82, unsurprisingly given their small CCT change, make lights with similar indices as the unfiltered lights and this is also true for the equivalent locus filters. However, for the larger CCT changes induced by filters 80A and 85 the means values of color rendering indices change markedly and are generally significantly worse and yet for the equivalent locus filters the—on average high—color rendering indices are maintained.

To conclude, we have found that the light balancing filters, in terms of their filtering behavior when applied to Wien-Planckian, filter lights like locus filters. However, color conversion filters filter lights somewhat differently from locus filters. Indeed, when conversion filters are applied to both Wien-Planckian and real lights they do not maintain, as well as locus filters, the distance to the Planckian locus.

Of course, looking at Fig. 9 we see that the light-balancing filters are quite neutral in color, very smooth and very close, spectrally, to their locus filter counterparts. So, we expect them to filter lights like locus filters. In contrast, the color conversion filters are designed to make larger changes in correlated color temperatures and are designed to be used under specific lighting conditions. Moreover, that they do not filter lights like a locus filter must be related to their relative non-smoothness, again see Fig. 9.