Spectral data is of increasing importance in color reproduction workflows, for instance, in brand identity colours and spot colour reproduction [1], colour characterisation or separation [2–5], gamut mapping [6], scene-referred image reproduction [7–9], and soft proofing [10, 11], where it may aid in increasing accuracy and reducing metameric mismatches. When measured spectral data is not available, it can be estimated from colorimetric data using a spectral estimation method in conjunction with suitable training spectral data.

Spectral estimation methods are widely used in colour imaging. Some of the most common uses of spectral estimation methods in this domain are for estimating reflectances of material surface or objects in a scene or estimating spectral sensitivity functions of a camera sensor. In this paper, we focus on spectral estimation of reflectance from tristimulus values such that the estimated reflectance is representative of the material object, the colorants and substrate that combine to form the spectral stimulus, in particular. There is a plethora of spectral estimation methods in the literature. The pseudo inverse is the simplest least squared solution for spectral estimation [12, 13], while the most commonly used method is principal component analysis (PCA) [14–17]. Fairman and Brill, proposed an efficient PCA-based method to estimate reflectance from tristimulus values, which was further modified by Agahian et al. by weighting the training reflectances based on minimizing the colorimetric difference from the test value. Using Cohen’s Matrix R theory to calculate fundamental stimuli and metameric black, Zhao and Berns showed how both colorimetric and spectral transformations can be used to estimate accurate spectra from a captured image [18]. The Wiener estimation method has been used to estimate spectra from RGB images [19], and there have been methods developed to impose a non-negativity constraint on estimated spectra [20, 21]. Lopez-Alvarez et al. demonstrated the effectiveness of the Wiener estimation method in accurately estimating skylight spectral curves using a limited set of training reflectances [22]. They highlighted that this method eliminates the requirement for measuring spectral sensitivity or calculating linear bases and is robust to noise. Dupont studied spectral estimation from colorimetric values using different optimization methods including genetic algorithms and used a metric that minimized colour difference under two different lights [23]. Other spectral estimation algorithms use optimization to minimize spectral and colorimetric errors [24–26], and there are methods which use interpolation [27–29]. Spectral estimation methods can also be applied to achieve colour-constant estimated spectra which is a desirable property [30]. van Trigt stated that reflectance curves must preserve some amount of colour constancy, making observed colour shift more or less a characteristic of the metameric set [31]. Based on that, van Trigt proposed a spectral estimation method, which creates a generic reflectance curve for a tristimulus value and achieved this by generating the least variation or the smoothest reflectance for a tristimulus value [31, 32]. Preserving colour constancy is a different task from estimating reflectances that match the spectral characteristics of a given material.

Estimated spectra from colorimetric values are essentially metameric reflectances, such that they match in colour under the colorimetry of the illuminant used to train the model. If a spectral estimation method produces a reflectance that is an exact match to the measured reflectance of an object then there will be no metameric mismatch under changing lighting conditions. It can be argued that these two reflectances have the same characteristics and represent a perfect match of the optical properties of the material. Based on these considerations, in the case of reflectance estimation from tristimulus values, we can select training datasets such that they have similar material components to those of the test datasets. For example, for print datasets matching printing conditions could be a way to match important components that comprise a printed material, such as ink pigments and substrate with or without concerning the printing process. It will also be important to find out what kind of similarities are enough for standardisation of training datasets in spectral estimation workflow can be considered in the future.

The perception of colour in humans is a function of the colour stimulus and the cone sensitivities. Illuminant metamerism (i.e. when two objects with different reflectance functions are perceived similar in colour under one light but different under another light because of the difference in the spectral power distribution between the two lights) can be an issue in cross media colour reproduction. When we do not have spectral reflectance of an object, its tristimulus value under different illuminant can be predicted by a sensor adjustment transform, which may be a chromatic adaptation transform (CAT) or a material adjustment transform (MAT).

CATs predict corresponding colours. Corresponding colours are two tristimulus values that result in the same perceived colour when the two samples are seen in test and reference adapting conditions [33], and they are derived from experimental data where observers match colours under different illuminants [34–36]. Bradford CAT, CAT02 and CAT16 are examples of CATs, that have used corresponding colour data to optimize transformation matrices used to convert tristimulus values to and from a sharpened cone space. The adaptation to the destination colorimetry is performed in cone space before converting to the adapted tristimulus value. These CATs draw their inspiration from Von Kries hypothesis on chromatic adaptation that states that there is a linear relationship between the stimulus and the visual response of the three cone types [37, 38]. The Linear Bradford transform, without the non-linear exponent, is recommended by ICC v4 specification [39]. CAT02 and CAT16 are part of CIECAM02 [40] and CIECAM16 [41] colour appearance models, respectively. CAT16 and CAT02 are analogous but they differ in their transformation matrices [36].

In the case of MATs, a material match of the object (i.e. how tristimulus values of an object change due to changes in either illuminant and/or observer [42]) is predicted rather than a perceived colour match [43]. Logvinenko defined a metameric mismatch volume, i.e., for a given colour under one illuminant, corresponding volume that contains all the possible colours under another illuminant [44]. This theory was evaluated by Zhang et al. by choosing all metameric reflectances under one illuminant from a large collection of measured reflectances and creating corresponding metameric mismatch volumes by predicting colours under different illuminants [45]. It was found that the centroid value of this volume can be used to predict colours with less metameric mismatch than well known CATs [45]. Logvinenko also defined an object-colour manifold which is a six-stimulus value (light colour and object colour together) and can be split into three dimensional material colour which shifts when the illumination changes [46]. Extending this concept to include a change in observer colour matching functions, Derhak developed a material adjustment transform that uses a colour equivalency representation called Wpt (Waypoint) [43]. Oleari, developed CATs by optimizing the conversion of tristimulus values under different viewing conditions to an ABC colour space, such that it preserves colour constancy [47]. Derhak argues that Oleari’s CATs are actually MATs because they optimize cone excitations [43]. van Trigt’s reflectance estimation method generates smooth reflectances with constant values near the endpoints of the visible range which improves colour constancy especially under illuminant A [31]. This method was modified by Burns and used to estimate reflectances in order to predict colour under a destination illuminant, and perform the final step towards chromatic adaptation by scaling the relative luminance Y of the destination tristimulus value to match the Y value of the source tristimulus value [48]. This modified spectral estimation method produces strictly positive and smooth reflectances. Although this procedure does not use corresponding colour data, it is considered to be a CAT rather than a MAT since it does not aim to match material attributes.

As discussed previously, estimated spectra from colorimetric values are basically metameric reflectances such that they match in tristimulus values under the illuminant used to train the model. Thus, it is interesting to evaluate if with careful selection of training data i.e. a good material match with the test data, can a spectral estimation method then be considered for material adjustment transform. However, it is easier to find training datasets for some applications than others, e.g. for spectral estimation of natural objects it would be important to have a training dataset with a wide range of representative reflectances but for a narrow application like colour chips or print datasets, the different material components such as pigments, substrate, fluorescent whitening agents etc. can be employed to categorise training reflectance datasets into groups of similar material types. It will be important to understand what kind of material component similarities between the training and test data are sufficient to obtain an acceptable metameric match. This will pave the way to have one training or reference spectral dataset to predict spectral reflectances for a group of print data that can be used in a colour management pipeline.

Colour management is built upon communicating relevant data for precise interpretation of colour content and utilizing colour conversion techniques to achieve a desired reproduction between imaging devices. The profile connection space (PCS) facilitates the accurate conversion between colour data from a source device to a destination device. In ICC.1 i.e. version 4 colour management, the PCS is colorimetric (either XYZ or CIELAB) which limits the integration of spectral reproduction workflows directly into colour management. The ICC.2 i.e. version 5, the new colour management architecture, allows spectral connection and processing through programmable calculator elements inside an ICC profile. This makes it possible to integrate spectral data into colour management workflows. The need for spectral estimation arises when the source profile is based on colorimetric PCS data, or when intermediate processing steps require spectral data. When converting from colorimetric to spectral in such a workflow, the spectral estimation method has to be simple, accurate and fast. The spectral estimation methods evaluated here are least squares fit, i.e. linear combination of bases that can be easily encoded using matrices into a colour management pipeline or other practical applications.

In this study, we evaluate the performance of different spectral estimation methods along with the training and test datasets whose material characteristics are known. We also evaluate the colorimetric performance of the two best performing spectral estimation methods under varying illuminants together with different training and test datasets and compare the results with other well known CATs and MATs. With the results obtained, we aim to answer the following questions:

In a least squares fit, the task is to find a solution x to an overdetermined set of equations Ax ≃ b by minimizing the residuals ||Ax − b||, where A ∈ Rnxm and b ∈ Rn are known [49]. This equation can be rewritten for the purpose of obtaining spectral estimates from tristimulus values as shown in Eq. (1), where S ∈ Rn is spectral reflectance and n is the number of wavelengths, T ∈ R3 is tristimulus value and M ∈ Rnx3 is the linear mapping or the basis that minimizes the residuals. In this case, for a training dataset with lnumber of samples, S and T are known; we first determine the matrix M that will be used to estimate spectral reflectance from a test tristimulus value. Depending on the number of training samples, matrix S will then be of the size nxl and, the matrix T will be of the size 3xl.

A set of XY Z tristimulus values T is the summation of the product of the illuminant spectral power distribution E, colour matching functions O and the surface reflectance S at each wavelength. Let AT be the weighted colour matching functions given by AT = kOTdiag(E) where O ∈ Rnx3, E ∈ Rn and k is the normalizing constant. The tristimulus values can be represented in the matrix form as shown in Eq. (2).

The simplest spectral estimation method is the pseudo-inverse method that takes advantage of Eq. (1) such that given a spectral reflectance training dataset and the corresponding tristimulus values from Eq. (2), M can be calculated using Eq. (3). Since, T is not a square matrix, to calculate its inverse, the Moore-Penrose inverse or the pseudo-inverse method is used. M can then be used to recover spectral reflectance Ŝ from any test tristimulus value T as shown in Eq. (4).

Babaei et al. argues that instead of creating a general basis M by giving equal weights to all the reflectance samples in the training dataset; to increase accuracy it is important to weight the training tristimulus values in proportion to its similarity or dissimilarity with respect to the test tristimulus values. This is achieved by calculating the inverse of the colour difference Eab∗ between the training tristimulus values and the test tristimulus value for which spectral reflectance is to be estimated [12]. The weights matrix W is a diagonal matrix as shown in Eq. (5), where l is the total number of samples in the training dataset and e = 0.01 is a small constant to avoid division by 0. The new weighted basis M is calculated by replacing T with weighted tristimulus values T′ = TW and replacing S with weighted reflectances S′ = SW in Eq. (3). We refer to this as the weighted pseudo-inverse spectral estimation method.

The pseudo-inverse spectral estimation method can also be modified to a polynomial fit. In order to apply a polynomial, the terms of a tristimulus value i.e. T = (X, Y, Z) have to be expanded according to the chosen polynomial order as shown in Table I. These expanded tristimulus values are used in Eq. (3) to obtain the basis matrix M. M can then be used to estimate spectral reflectance, where the terms of the test tristimulus value also have to be similarly expanded according to the respective polynomial order in Eq. (4). Increasing the order can lead to overfitting, and hence, it is important that to maintain the number of terms significantly lower than the number of samples. Also, in an optimisation problem, overfitting is handled by adding a term to the cost function in order to force the coefficient estimates toward zero and achieve a better generalisation. This is called regularisation. In this work with spectral data of print datasets, the general solution to least squares is considered without any regularisation or iterative optimisation.

Another classical spectral estimation method is Wiener Inverse, where the correlation matrix μ of the spectral reflectance of training dataset S is used to generate basis M; thus, M is given by Eq. (6). By employing a simple way to determine the spectral similarity, this method produces spectral reflectance estimates that are more or less insensitive to the illuminant [50].

Principal component analysis (PCA) determines a projection matrix which maximizes the variance in the dataset. Fairman and Brill introduced a classical mean centered PCA on spectral reflectances. The mean reflectance V o of the training dataset S is determined. The first 3 eigenvectors V are calculated from mean centered spectral reflectance of training dataset (S − V o). Then, a column vector C ∈ R3 that contains the principal component coordinates has the relationship as shown in Eq. (7) [15].

The tristimulus constrained principal component coordinates can be calculated by substituting Eq. (7) in Eq. (2) and rearranging for C as shown in Eq. (8), where T is the test tristimulus value.

Once C is synthesised, spectral reflectance estimate Ŝ can be obtained by Eq. (9).

Agahian et al. proposed a weighted version of the PCA spectral estimation that uses the weights matrix W in Eq. (5) to calculate weighted reflectance S′ = SW. The weighted coordinates C are obtained by calculating principal components V and mean reflectance V 0 from S′ in Eq. (8) [17]. This weighted PCA spectral estimation method increases accuracy of the reflectance estimates compared to classical PCA.

The Waypoint (Wpt) method uses spectral reflectance decomposition defined by Chau that represents spectral reflectance as the sum of scaled non-selective reflectance and a characteristic reflectance [42, 51]. The non-selective reflectance is a reflectance vector that reflects the same amount of light at every wavelength, making it invariant to wavelength. The characteristic reflectance, which is the wavelength selective component, is obtained by normalizing a reflectance vector such that the minimum value becomes 0 and the maximum value becomes 1; this was called the primary reflectance by Chau. Wpt coordinates are represented by (W, p, t), where W represents perceptive lightness, p and t represent perceptive chromaticness, that is, a combination of perceptive chroma and hue at constant perceptive lightness [43]. Wpt coordinates in polar form are represented by (W, c, h) where c is Wpt chroma and h is Wpt hue and are given by c =p2+t2 and h = arctan2(t, p), respectively [42]. The tristimulus values in this case are first converted to polar Wpt coordinates (W, c, h) by using Wpt normalizing matrix determined for source observing conditions. Wpt hue is used to determine its corresponding characteristic reflectance. This method uses Munsell reflectances to determine characteristic reflectances but the latter can also be determined from other measured reflectances. These characteristic reflectances are divided into groups having constant hues that form a hue-plane. One of the characteristic reflectances is selected from this group to represent that hue-plane. The Wpt coordinates (W, c, h) of the characteristic reflectances and non-selective reflectance are calculated. Using W and c coordinates of the characteristic reflectance and W coordinate of the non-selective reflectance, the scalar of spectral whiteness g and the scalar of spectral saturation s are determined. A reflectance vector is then estimated by scaling non-selective reflectance with g, scaling the characteristic reflectance with s, and then finally combining them as described in Ref. [52].

Burns altered van Trigt’s method [31] for estimating spectral reflectance from tristimulus values, which involves using optimization to identify the unique reflectance curve with the smallest slope squared integrated over the visible wavelengths such that this curve matches the tristimulus values of the source [48]. Burns implemented this optimization in log reflectance, which ensures that the resulting curve is strictly positive [48]. This spectral estimation method is also evaluated and the results are compared to the performance of weighted pseudo inverse and third order polynomial spectral estimation methods in achieving least metameric mismatch.

Chromatic adaptation is the mechanism by which the human visual system tries to preserve an object’s perceived colour under different viewing conditions [53]. This colour adaptation mechanism has to be applied in a colour reproduction workflow to reproduce consistent colour appearance when the illuminants of the input and output colorimetry differ. In ICC.1 colour management, the reference intermediate colour space or Profile Connection Space (PCS) is defined as CIE colorimetry based on illuminant D50 and CIE 1931 2∘ Standard Observer. Hence if illuminant or observer of the input or output encodings differ then they have to be transformed to or from PCS. If the transform is by a CAT, the result is a corresponding colour match and not an exact colorimetric match [54]. These transforms can be represented as a 3 × 3 linear matrix transforms. CAT matrices are calculated by mapping corresponding colours which were obtained by carrying out memory based or haploscopic experiments. These corresponding colours are colour pairs, i.e., colour of a stimuli under a source illuminant that match in appearance to the colour of another stimuli under a destination illuminant. The foundation for this specific way of modelling chromatic adaptation with scaling factors applied to cone excitations was laid down by Johannes von Kries and most modern CATs are built upon it. Some current popular CATs are Linear Bradford, CAT02 and CAT16 that are discussed in this section.

Linear Bradford CAT is recommended by ICC v4 specification, i.e., based on Bradford CAT derived by Lam, consisting a non-linear correction in the blue region which is dropped in Linear Bradford CAT [35]. In Linear Bradford CAT, the transformation matrix that converts tristimulus values to cone excitation can be found in ICC v4 specification [39].

CIE TC8-01 [55] proposed CIECAM02 as a colour appearance model which uses CAT02 for chromatic adaptation. Transformation matrix (M) of CAT02 is used to convert tristimulus values to sharpened cone responses (RGB). Considering full adaptation, CAT02 then can be calculated similar to von Kries transformation as shown in Eq. (10), where, (Rw1, Gw1, Bw1) and (Rw2, Gw2, Bw2) are the sharpened cone responses of the source and destination white points, respectively; (X1, Y 1, Z1) is source tristimulus value and (X2, Y 2, Z2) is destination tristimulus value.

CAT16 was developed by Li et al. to solve the computational issues that arose due to CAT02 [41]. CAT16 transformation matrix was optimized using several corresponding colour datasets. A chromatically adapted tristimulus value considering full adaptation and when luminance of the source and destination illuminants are equal, is obtained in a similar manner as shown in Eq. (10) where the transformation matrix of CAT02 is replaced by the transformation matrix of CAT16 [41].

Based on CAM02 model where illuminant E, corresponding to an equi-energy stimulus, was used as a reference or intermediate illuminant and a two-step CAT16 transformation was also developed [36] where, an intermediate transformation to and from illuminant E is included.

Derhak et al. defines a generalized term, sensor adjustment transforms (SAT), that includes both CAT and MAT. A CAT uses mapping of corresponding colour datasets to achieve appearance match under changing illuminants as discussed above, while a MAT uses least dissimilar color matching proposed by Logvinenko to achieve material constancy under changing illuminants. Therefore, a MAT is based on the concept that when illuminant varies, the appearance of a stimuli changes but due to some secondary mechanism the human visual system can still to some degree identify the material [43]. Based on this Derhak, et al. proposes Wpt transform, where the color of an object is mapped to coordinates (W, p, t), which is a colour equivalency representation and forms a waypoint to navigate between the source and destination colour viewing conditions. The optimized Wpt based MAT matrices from source colorimetry to Wpt representation are given in the Eq. (11) and Eq. (12), where M2∘,D65, M2∘,D50, M2∘,A, and M2∘,F11 are transformation from CIE 2∘ Standard Observer and illuminants D65, D50, A and F11, respectively, to Wpt coordinates [56]. The destination tristimulus values can be obtained from the inverse of these matrices applied accordingly to the Wpt coordinates.

Oleari’s MAT are mathematical transformation matrices that are used to convert colors from one lighting condition to another, so that they appear the same to the human eye [47]. These matrices were obtained by optimizing the conversion of tristimulus values under different viewing conditions to an ABC colour space, such that it preserves colour constancy and, hence, produces the same perceived colour for different cone excitations under different illuminants and observers. Oleari’s transformation matrices from a source colorimetry to a destination colorimetry can be found in the publication [47].

Burns defines a CAT that is performed using spectral estimation [48]. This method is developed as an improvement to other traditional CATs, to make sure that the tristimulus values predicted under a destination illuminant does not fall outside of the spectral locus and does not depend on corresponding colour datasets for training or fine-tuning it [48]. Burns modified the approach to spectral estimation proposed by van Trigt [31] that uses optimization to find the unique reflectance curve with minimum slope squared integrated over the wavelengths of the visible range with the constraint that the reflectance curve produces the source tristimulus value. Burns carried out this optimization in log space of the reflectance curve so that the conversion from the log space creates a strictly positive reflectance curve. Two spectral power distributions are also estimated that lead to source and destination whitepoints respectively. Using these two illuminants, estimated reflectance curve and colour matching functions, the source tristimulus value and destination tristimulus value are calculated. The destination tristimulus value is further adjusted such that its chromaticity is preserved and its relative luminance Y matches the Y value of the source tristimulus value. This adjusted tristimulus value is the predicted corresponding colour. Although, this method doesn’t use any corresponding colour datasets, it also doesn’t try to achieve material constancy. It is regarded as a CAT because Burns motive was to use it to predict corresponding colours well.

The metameric mismatch results obtained from the CATs and MATs discussed in this section are compared to the results obtained from the two spectral estimation methods.

Three sets of tests are carried out with different aims of using spectral estimation methods. Eight spectral estimation methods discussed in Section 2, namely, Wiener method (Wiener), Classical PCA (cPCA), Weighted PCA (wPCA), Pseudo Inverse (PInverse), Weighted Pseudo Inverse (wPInverse), Second Order Polynomial (Polynomial 2), Third Order Polynomial (Polynomial 3) and Waypoint-based (Waypoint-R) methods were applied to model a reflectance or emission spectrum recovery from tristimulus values obtained under D50 illuminant and CIE 1931 2∘ Standard Observer colorimetry. The datasets used and the tests carried out are elucidated.

Two spectral estimation methods, weighted pseudo inverse (wPInverse) and third order polynomial (Polynomial 3) using least square fit were evaluated using various training and test datasets and their performance in predicting colour appearance under different illuminants were assessed. These two methods were chosen for their simplicity and computational efficiency, as well as their performance in terms of least root RMSE and metameric differences compared to other classical methods, like Wiener method, methods using PCA and Wpt based spectral estimation demonstrated in Section 6.

Seven datasets, as shown in Table II were selected for training and test purposes. These datasets contain reflectances of Munsell colour chips, and different print datasets with varying material components such as substrates, inks and processes. These datasets are be categorised according to their material components. The main purpose is to understand, how well such spectral estimation models perform in generating estimated reflectances that are a close match to those of the object of interest, and whether these estimates predict colours under different illuminants more accurately than colour predictions using alternative sensor adjustment transforms. A Munsell dataset (M1), a newsprint dataset (N2) and a textile print dataset (T2) were employed in the estimation using k = 5 cross validation method, where a dataset is divided into 5 sets, and one set i.e. 20% is used for testing and the rest 80% is used for training and this is repeated for each set. Six other cases where the training dataset does not belong to the test dataset were also used. N2 is used as a test dataset, while the N1, W1 and M1 datasets were used for training the spectral estimation models. Similarly, T2 was used as a test dataset and T1, F1 and M1 datasets were used for training.

The results from the two estimation methods were compared with predictions from different CATs and MATs. These colour predictions were evaluated under five CIE illuminants; two standard daylight illuminants D50 and D65, illuminant A representing tungsten filament lighting, F11 representing fluorescent lamp and LED-V1 representing violet pumped phosphor-type LED lamp. LED-V1 is part of the illuminants representing typical LED lamps published by the CIE in 2018 [60]. Fifteen combinations of these illuminants are used where each illuminant is the source and for each source, three other illuminants are used as the destination. These chosen combinations of (Source - Destination) illuminants are (D50-D65), (D50-A), (D50-F11), (D65-D50), (D65-A), (D65-F11), (A-D50), (A-D65), (A-F11), (F11-D50), (F11-D65), (F11-A), (LED-V1-D50), (LED-V1-D65) and (LED-V1-A). For each pair of source and destination illuminants, the tristimulus values of the test datasets were calculated by using their measured reflectances, source illuminant and the CIE 1931 2∘ Standard Observer. They were then chromatically adapted to the destination illuminant using different CATs; the Linear Bradford transform (L-Bradford), CAT02 and CAT16 with one step and two step via equi-energy transforms (with full adaptation), Oleari MAT, Waypoint MAT, and Burns CAT by spectral estimation (Burns-CAT). Colour predictions using estimated reflectances from the intermediate spectral estimation step (Burns-R) in Burns CAT were also compared. For comparisons, colour difference ΔE00 was calculated between the destination tristimulus values obtained from the measured reflectances and the adapted or estimated tristimulus values under the destination illuminant. This gives us a measure of metameric mismatch between the reference and adapted or estimated colours with changing lighting conditions.

It is important to note that the spectral estimation methods, estimated reflectances from tristimulus values obtained under the respective source illuminant colorimetry, so the estimated reflectances were metameric matches under the source illuminant, i.e., the ΔE00, in this case is 0. This allows for a more accurate assessment of the performance of the spectral estimation methods, as the estimated reflectances are obtained from the same source colorimetry as the adapted colours.

Moreover, tristimulus values were calculated from spectral data with 10 nm interval using CIE recommendations [70], where optimum weights tables [65] were created from CIE 1 nm data of illuminants and colour matching functions, as explained previously.

The estimated reflectances from tristimulus values under the source illuminant obtained using the two spectral estimation methods were then used to find tristimulus values under the destination illuminant as described in Section 3 above. Burns spectral estimation method (Burns-R) and different CATs and MATs are also used to predict destination tristimulus values from source tristimulus values. As stated in Section 3, pre-computed transformation matrices were used for Waypoint and Oleari MATs, but since transformation matrices to and from LED-V1 illuminant are not available, we leave these computations blank. The reference tristimulus values are XYZ values obtained using measured spectral reflectances and the respective destination illuminant. The 1931 2∘ Standard Observer was used as the colour matching function throughout. When full adaptation is used, the one-step and two-step CAT02 and CAT16 transforms produce identical tristimulus values at the precision shown, and hence the results for the two-step CAT02 and CAT16 are not shown. The test and training data used to obtain the estimated reflectances, in this case, are denoted by their identifiers as (test-training) e.g.: (M1-M1) where M1 is both test and training data or (N2-M1) where N2 is test data and M1 is training data. Their results as a MAT are discussed next.

Table XVI summarises the mean and 95th percentile of ΔE00 results for dataset M1, where part of the same dataset was used for training denoted by (M1-M1) i.e. (test data-training data). Between the two spectral estimation methods, wPInverse and Polynomial 3, their mean and 95th percentile of ΔE00 are comparable. They produce the least mean and 95th percentile of ΔE00 compared to other CATs and MATs. Burns-R and Waypoint-MAT mean results are comparable and they produce less metameric mismatch results compared to other CATs and Oleari-MAT, although they are higher than Polynomial 3 and wPInverse. Burns-CAT metameric mismatch results are comparable to CAT02.

Zhang et al. [45], evaluated colour predictions under changing lights obtained using centroid of the metameric mismatch volume across Munsell dataset (M1) and found mean ΔE00 to be 1.27 and 95th percentile of ΔE00 to be 3.42 when destination illuminant is D65 and source illuminant is A as shown in Table XVII. While mean ΔE00 was 1.82 and 95th percentile of ΔE00 was 4.99 when destination illuminant is D65 and source illuminant is F11 as shown in Table XVII. For the same source and destination illuminant pairs, wPInverse and Polynomial 3 spectral estimation methods produce mean ΔE00 less than 1.03 and 95th percentile of ΔE00 less than 2.8. The two spectral estimation methods thus minimise metameric mismatch when the training data is similar to the test data, as is the case in M1-M1 in Table XVI.

Tables XVIII and XIX summarise mean and maximum ΔE00, respectively, obtained from different CATs and MATs, and estimated reflectances for newsprint dataset N2, under different combinations of source illuminants (first row) and destination illuminants (second row). The two spectral estimation methods wPInverse and Ploynomial 3 were used to perform spectral estimation using training datasets that were same (N2) or different (M1, W1, N1) from the test dataset N2 as described in Section 5. Both spectral estimation methods produce the smallest mean and maximum ΔE00 for the cases where the training dataset is either from the same dataset (N2-N2) or another newsprint dataset (N2-N1). The grand mean ΔE00 across all combinations of source and destination illuminant pairs is less than 0.5 while the average of the maximum ΔE00 values is less than 2. When the training dataset is different in material components from newsprint, such as (N2-W1), the mean and maximum ΔE00 of the two spectral estimation methods are still significantly better than other CATs and MATs. Their grand mean ΔE00 is less than 0.8 and the average of the maximum ΔE00 values is less than 3. When the training dataset is colour chips with higher variability in reflectances i.e. (N2- M1), then the metameric mismatch of estimated reflectances for the newsprint dataset increases. The mean and maximum ΔE00 in this case are comparable to the results obtained from other CATs, MATs and Burns-R. The overall good estimation results with less metameric mismatch might be because all the training and test datasets in this case have smooth reflectances with broader peaks.

Tables XX and XXI summarise mean and maximum ΔE00 obtained from different CATs and MATs and estimated reflectances for textile dataset T2, under different combinations of source illuminants (first row) and destination illuminants (second row). The two spectral estimation methods wPInverse and Polynomial 3 were used to perform spectral estimation using training datasets that were same (T2) or different (M1, F1, T1) from the test dataset T2 as described in Section 5. Both spectral estimation methods produce the smallest mean and maximum ΔE00 when the training dataset is from the same dataset (T2-T2) with a grand mean ΔE00 of less than 0.8 and an average maximum ΔE00 of less than 3.4 across all combinations of illuminants. When a different textile dataset is used as training data i.e. (T2-T1), the mean and maximum ΔE00 are still better than other CATs, MATs and Burns-R. The grand mean ΔE00 is less than 1.2 and average of the maximum ΔE00 values is less than 4.3 across all combinations of illuminants. When the training dataset has some difference in material components such as (T2-F1) which has an optically brightened substrate, such as premium coated paper rather than textile, the mean and maximum ΔE00 values are still significantly better than other CATs, MATs and Burns-R, with results that are comparable to those obtained from (T2-T1). In this case, the grand mean ΔE00 is less than 1.3 and the average maximum value of ΔE00 being less than 4.3 across all combination of illuminants. The training datasets T1 and F1 have similar spectral peaks in the blue region as in the T2 reflectances which makes them a better choice for training compared to M1 that has reflectances with smoother and broader peaks. Therefore, when the training dataset used is M1 i.e. for (T2-M1), the mean and maximum ΔE00 are worse than the results from datasets with fluorescence that has close reflectance characteristics. (T2-M1) results are similar to the other results from CATs and MATs and especially comparable to the results obtained from Burns-R. This may be due to a generalisation achieved with smoothness and higher variability present in M1 reflectances.

Table XXII summarises the mean, median, 95th percentile and maximum of ΔE00 obtained across all datasets and combination of source and destination illuminants for the two spectral estimation methods wPInverse and Polynomial 3. The grand mean ΔE00 is below 1 and average value of the maximum ΔE00 is below 4. This indicates that overall metameric mismatch will be even lower if the results from only those cases where training data is similar to the test data are considered.

A MAT is expected to perform well when there is a good degree of material similarity of training reflectances with test reflectances. Based on this, five categories of MATs are possible, they are, (1) spectral estimation based MAT to predict similar reflectance characteristics, (2) spectral estimation based MAT to predict less similar reflectance characteristics, (3) Matrix based MAT optimised to predict similar reflectance characteristics, (4) Matrix based MAT optimised to predict less similar reflectance characteristics and (5) Matrix based CAT optimised to predict corresponding colours. wPInverse and Plynomial 3 under category 1 or 2 based on the material similarity of the training data used. Burns-R falls under category 2. Oleari-MAT and Waypoint-MAT fall under category 3 or 4 based on how they are optimised. Burns-CAT falls under category 4. CAT02, CAT16 and L-Bradford fall under category 5.

Table XXIII shows mean colour differences (in ΔE94 for comparison with previous work) obtained for different corresponding colour datasets (CSAJ, Helson, Lam & Rigg and LUTCHI) using different CATs and MATs. The mean ΔE94 is also obtained for the spectral estimation methods wPInverse and Polynomial 3 where the reflectances were first estimated from the source corresponding colour using three different training datasets M1, N2 and T2, respectively. The datasets selected for this analysis and their source and destination illuminants are CSAJ (D65, A), Helson (C, A), Lam & Rigg (D65, A) and LUTCHI (D65, D50). A small inaccuracy arises from the use of standard illuminants in the spectral estimation, while the measured illuminants in the corresponding colour datasets are slightly different from the standard illuminants. The spectral estimation methods using all the different training datasets perform similar or better than Oleari-MAT and Wpt-MAT when it comes to matching corresponding colour data. All CATs perform better in this case including Burns-CAT. Similar, tests were also carried out by Derhak et al. using Wpt-based spectral reconstruction CAT which also utilises Burns spectral estimation and lightness scaling of the final tristimulus value to the source tristimulus value [42] and the results obtained were comparable to the results obtained in this work. This kind of CAT, using characterisation reflectance for print results will fall in category 1 MAT.

In the previous sections, we have established that spectral estimation methods wPInverse and Polynomial 3 perform very well in predicting colour under changing illuminants when training datasets are carefully chosen by matching the material characteristics. However, weight matrix W in the case of wPInverse has to be calculated for each pixel value which is both computationally time consuming and complicated to implement inside an ICC profile. Therefore, due to the ease of implementation and low computation time, Polynomial 3 spectral estimation method using matrices and stack-based calculator element programming of ICC.2, we propose a spectral estimation workflow to integrate it into colour management. We define an absolute rendering transform from colorimetric PCS XYZ values to the corresponding reflectances in a BToD3Tag. BToD3Tag defines a colour transform from a spectrally-based PCS, determined by the spectralPCS and PCS fields in the header, to device colours [72]. DataColourSpace tag in the header is used to define the number of wavelengths. A multiProcessElementType is defined in the BToD3Tag which incorporates a CalculatorElement tag. The multiProcessElementType defines and processes a sequence of processing elements and they are processed in the order of their position. Each processing element passes its results to the next element, with the final processing element providing the final result for the multiProcessElementType. Basis matrix M from Eq. (4) is stored as a matrix using matrixElement tag in an external file which is imported when the ICC XML file is compiled to create the ICC profile. A macro is defined to expand XYZ input to the third order polynomial terms in Table I. The spectral estimation model is encoded in the main function; the input channels are expanded using the polynomial expansion macro and then a matrix multiplication mtx is called to multiply matrix M by the polynomial terms of XYZ. The code inside the main function under CalculatorElement tag is shown in Figure 6. DToB3Tag defines a colour transform from device to a spectrally-based PCS to achieve an absolute rendering. Therefore, a transform from reflectance values to XYZ using illuminant and observer functions can be defined in DToB3Tag, if colorimetric output is required. The average roundtrip error in ΔE00 between the starting colorimetry and the output XYZ values after applying the spectral estimation and conversion back to colorimetry across datasets M1, N2 and T2 obtained from applying the profile back to the estimated reflectance values is 0.026 at 32-bit precision using the ICC.2 workflow, and 4.15E-10. when computed at double precision in MATLAB.

This study presents the estimated reflectances using different spectral estimation methods and training datasets with known material components or printing conditions. It can be seen from the results that if the training dataset used has similar material components to those of the test dataset, the estimated reflectances have reduced metameric mismatch under different illuminants. It is also observed that when training reflectances have some dissimilarity (such as using training reflectances of web-offset on lightweight coated paper for reflectance estimation of newsprint data), the results still lead to acceptable metameric differences because their reflectances are smooth and have similar peak wavelengths with some difference in amplitudes. Similar acceptable metameric differences are also obtained when training reflectances of offset-litho on premium coated paper are used to estimate reflectances of textile dataset where both exhibit fluorescence and, therefore, have reflectance curves with similar sharp peaks around the blue region. This suggests that based on similarities in material components, many spectral datasets or printing conditions can be grouped together and a single reflectance dataset can represent a common training dataset for reflectance estimation of that group. The weighted pseudo inverse (wPInverse) and third order polynomial (Polynomial 3) spectral estimation methods performed the best in all the tests, while Polynomial 2 and wPCA are not significantly different. These methods are also not computationally costly or complex to implement.

This study also compares the sensor adjustment performance of spectral estimation with chromatic adaptation transforms (CATs) and material adjustment transforms (MATs). The two spectral estimation methods, wPInverse and Polynomial 3 with good training data selection performed significantly better in minimizing the metameric mismatch compared to other CATs and MATs. When the training dataset is not a good material match or has high variability such as using Munsell colour chip reflectances to estimate reflectances for newsprint or textile datasets then the metameric mismatch increases although the performance is similar to other CATs. This paper has shown the importance of the training dataset, and that by considering spectral similarities between datasets, simple spectral estimation methods like weighted pseudo inverse or third order polynomial can be used as a sensor adjustment transform that minimizes metameric mismatches.

We show that even with training datasets with different material components match to the test data i.e. when they do not belong to the same dataset but have some spectral similarities, spectral estimation produces reflectances with least metameric mismatch and such methods can be a good alternative to using traditional CATs and MATs for predicting tristimulus values under different illuminants, when material constancy is the aim. When spectral data is not required, a more general MAT i.e. a 3 × 3 transform can be used. Also based on the application, such specific training data can also be chosen to optimise an existing MAT and improve material constancy.

Simple spectral estimation methods were evaluated such that they can be easily integrated into a colour management workflow. Among the best performing spectral estimation methods, Polynomial 3 can be easily and efficiently encoded in an ICC profile as matrices using calculator elements programming of ICC.2. In the future, regularisation of these methods can be considered. Regularisation require proper selection of the regularisation term to achieve both good generalisation and accuracy and can be considered to assess if it can help achieve spectral estimation preserve material constancy or colour constancy. The final matrices after regularisation can then be encoded into ICC profiles.

This study provides an insight into the need for standardising spectral estimation usage in colour management, where every step is essential. These steps include selecting a reference training dataset that can represent spectral data for a group of print data, determining the spectral estimation method, identifying the application’s goal (e.g. material constancy or colour constancy), and understanding the workflow to encode it in an ICC profile.