Novel integration of non-invasive imaging techniques for the analysis of an egg tempera painting by Pietro Lorenzetti

The identification of an artist's palette through the application of non-invasive techniques is a challenging goal due to the huge variety of artistic materials that constitutes a painting. An effective approach is to combine several techniques providing complementary information in order to minimise the risk of misinterpreting the data. In this paper, we propose a multi-analytical method comprising three non-invasive mapping techniques, namely Reflectance Imaging Spectroscopy (RIS), Macro-X-Ray Fluorescence (MA-XRF) and Fluorescence Lifetime Imaging (FLI), for the study of a fourteenth-century painting by Pietro Lorenzetti from the Uffizi Gallery collection. For the low-cost and time-saving interpretation and integration of the data provided by the different techniques, a purposely developed software for multivariate statistical analysis was used. FLI data were acquired with a prototype applied for the first time on a work of art, and the data were processed with a method based on phasor analysis. The information obtained was discussed within a multidisciplinary team of experts on painting materials and data processing belonging to both the scientific and the conservation community.


Introduction
Pigments identification and mapping from spectral imaging datasets is an ongoing challenge in cultural heritage studies of painted artworks due to their complexity and variety of the microstructure and chemical composition. The traditional approach relies on the non-automated interpretation of data by visual comparison with spectral data found in the literature followed by the application of classification algorithms. It is common practice to combine complementary techniques, such as Macro-Scanning X-Ray Fluorescence (MA-XRF) and Reflectance Imaging Spectroscopy (RIS), in order to overcome ambiguities in the interpretation of data acquired by the individual techniques [1,2]. MA-XRF and RIS are both macro-scale spectral imaging methods providing image data cubes defined by two spatial and one spectral dimensions.
In MA-XRF, elemental spectra have discrete and known emission lines that are used to reconstruct 2D elemental distribution maps, in principle for all the elements with Z ≥ 11. Depending on the elements present in the sample and on the X-ray source characteristic lines, the occurrence of overlapping emission peaks and the deep penetration of the primary X-Ray radiation may lead to uncertainties in the accuracy of the distribution maps. In addition, no information can be obtained on compounds and organic material.
Pigment identification with RIS can be achieved based on characteristic features determined by electronic or vibrational transitions and detected in the Vis-IR spectral range. However, univocal pigment identification is not always possible because of both signal overlapping due to non-pure materials and light scattering effects by pigment's particles. Typically, the analysis of RIS data cubes follows a two-step approach: (1) reduction of data dimensionality and identification of characteristic reflectance spectra in the image cube using Multivariate Statistical (MVS) methods [3,4] for the creation of unlabelled maps; (2) examination of spectral features to identify known pigments or mixtures or stratifications of pigments to obtain labelled maps. Spectral Angle Mapping (SAM) [5], Spectral Correlation Mapping (SCM) [6,7], Maximum Likelihood (ML) estimation [8], and subspace projection methods-e.g. Minimum Noise Fraction (MNF) and Principal Component Analysis (PCA)-are among the most widely used MVS methods. The main issue with these approaches is that they consider the spectrum as a linear combination of two or more endmembers, which ideally represent the pure pigments, and thus are based on linear spectral unmixing models. However, the reflectance from pigment mixtures commonly found in paintings produces a nonlinear response as a result of the scattering between either the pigments mixed in different concentrations or laid as superimposing layer [9]. Therefore, any attempt to provide the evaluation of pigment a e-mail: alice.dalfovo@ino.cnr.it (corresponding author) concentration on painted surfaces by conventional MVS methods results inaccurate. Moreover, since hyperspectral image cubes are characterised by a spectral resolution comparable to that of point reflectance spectroscopy but distributed over large areas, the processing of such huge amounts of data is often time-consuming. Many researchers rely on the hyperspectral classification tool Spectral Hourglass Wizard (SHW) in the Environment for Visualizing Images (ENVI, Exelis VIS, Boulder, CO) [10], which utilises MNF to determine the spectral diversity and the Pixel Purity Index (PPI) convex geometry algorithm [11] to find the unique spectral components. The endmembers are manually clustered in the principal component space and image pixels are grouped based on their spectral similarity. However, the whole process typically takes hours and requires an experienced user to apply the algorithms for selecting the subpixel target. Automatic deterministic algorithms, e.g. Sequential Maximum Angle Convex Cone (SMACC) [12] and Maximum Distance (MaxD) [13], extract and quantify endmembers faster than ENVI, but with less accuracy. Limitations imposed by linear unmixing can be overcome by approximating the reflectance from mixed pigments through a simplified Kubelka-Munk (KM) model for opaque and infinitely thick samples, based on the ratio between the absorption (s) and the scattering (r) coefficients [14]. In this approach, the pure pigments best explaining the spectral features are combined following KM theory to create libraries of computed spectra to match the experimental data using linear regression [15]. To this aim, the values of s and r must be known for all the pigments, as well as the expected layering of the painting, which is extremely complex by means of non-invasive methods. Moreover, since KM modelling on a pixel basis requires intensive computational operations, the use of chromatic segmentation and clustering methods has been explored as a preliminary step to reduce the total number of spectra to fit [16].
Deep Learning (DL) approaches, especially Neural Network (NN), have recently entered cultural heritage studies for a priori pigment selection in hyperspectral data cubes, showing promising results. For instance, Rohani et al. used deep neural network to perform supervised classification to identify pigments present in a given spectrum. The fractions of the individual components in any unknown spectrum were estimated pixel-by-pixel based on a prepared library of relevant s and r values [16]. However, to create accurate material maps with NN, large training datasets of labelled reflectance spectra are required. Since spectral signature libraries of pure materials typically do not contain sufficient sample diversity to describe intimate mixtures, datasets from regions of well-characterised paintings have been recently built for training a supervised one-dimensional convolutional neural network [17]. The suitability of this data-driven approach was tested by Kleynans et al. on well-studied illuminations from a fourteenth century book, with encouraging results. This solution gets around the need to know a priori the optical properties of pigments and paint layering, but requires the creation of huge labelled datasets representative of pigments and mixtures used by a specific artist or artistic school.
In this paper, we propose a non-invasive, multi-analytical approach for the identification and mapping of pigments in a fourteenth century painting by Pietro Lorenzetti belonging to the Uffizi Gallery collection. The painting was examined with three macroscale imaging techniques, namely multispectral RIS in the Vis-NIR [2], MA-XRF mapping [18], and Fluorescence Lifetime Imaging (FLI) with a novel portable fibre-based system used to discriminate different not luminescent pigments (or compounds) based on their different effect on the luminescence lifetime of the binder [19,20]. All measurements were compared with a reference pigments dataset, taking into account the effect of pigment mixtures or paint layering. The results were discussed within a team of professionals involved in the conservation and diagnostics of the painting. RIS data processing was performed with a purposely developed software for time-reduced and low-cost multivariate statistical analysis based on linear spectral unmixing. To overcome the aforementioned limitations imposed by MVS methods, we took advantage of the complementarity of the applied techniques and the collaboration of a multidisciplinary group of experts in painting materials and data interpretation.
Information on the composing materials were obtained following a systematic approach. First, we applied RIS scanning and performed Principal Component Analysis on the RIS image data cube to reduce the dimensionality of the spectral datasets and to facilitate its interpretation, while minimising information loss. The main spectral differences over the painting surface were highlighted with False-Colour (FC) imaging using both PC and near-infrared images to guide the extraction of the endmembers corresponding to the most spectrally diverse pixels in the image cube. We performed Spectral Correlation Mapping (SCM) using the endmembers to define the distribution of the main pigments. Spectra recorded from previously prepared egg-tempera samples allowed for the identification of the pigments, based on the characteristic spectral features. We analysed the pigments distribution with SCM using the respective reference spectra and compared the results with the SC maps previously obtained with the endmembers. The elemental analysis provided by XRF confirmed the identification and mapping of the pigments, which were discussed with the restorers and art historians involved in the conservation of the painting. Finally, we used the FLI prototype to create a fluorescence lifetime dataset of pictorial reproduction containing the same binder/pigment combinations identified in the analysed artwork. The prototype is used here for the first time on an artwork. Preliminary studies conducted by this research group have demonstrated the versatility and suitability of the new FLI system, originally designed for biomedical applications, for the analysis of materials commonly found in works of art, such as protective varnishes, binders, and pictorial layers on laboratory samples [19,20]. Here, the emission detected both in the reference paints and in the painting is generated mostly by the binder, but the measured phasors are representative of the whole paint system-i.e. reference sample or painting area-as a result of the interaction between the fluorophores and the non-fluorescent species.

Case study
In this work, we analysed one of the stories of the Altarpiece of the Blessed Humility (Florence, Uffizi Gallery) by Pietro Lorenzetti, a well-known fourteenth-century Sienese painter. This panel painting, currently under restoration at the Opificio delle Pietre Dure, takes up the archaic typology of the frontal representation of the Saint in the centre, with the story of his life narrated in small compartments on the sides.
The artistic production of Pietro Lorenzetti, inspired by the masters Duccio da Buoninsegna and Giotto, is characterised by an innovative sensitivity to everyday details. The scarce and controversial historical information on the artist's life, as well as the still limited availability of scientific data on the pictorial materials used, has influenced the critical evaluation of his paintings and technique. The combined non-invasive analysis proposed in this study is aimed at a broader understanding of Lorenzetti's artistic production. In this study, we focused on the analysis of the story entitled "Vallombrosian monk refuses to have a sick foot amputated" (Fig. 1a).

Reference samples
The set of reference samples was prepared by the Opificio delle Pietre Dure, following late medieval and Renaissance artistic recipes describing practices and materials for the production of paintings in Tuscany. Therefore, samples' composition and stratigraphy are fully representative of the fourteenth-century painting under analysis. Pure powdered pigments (by Zecchi™, Florence), whose chemical composition was previously verified by FT-IR, were dispersed in protein binder (2/3 yolk, 1/3 egg white, 1/3 white vinegar) to form the egg tempera. The pictorial layers were applied on a wooden support covered with a preparation made of gypsum and natural glue and finished with a rabbit glue primer.

Reflectance imaging spectroscopy (RIS)
The multispectral scanner used in this work was developed at the National Research Council-National Institute of Optics (CNR-INO). It combines whiskbroom scanning with filtering to acquire simultaneously 32 narrow-band images (16 VIS + 16 NIR), comprising pointwise spectral information in the range 395-2550 nm [2,21]. The lighting system comprises two low-voltage current-stabilised halogen lamps equipped with an aluminium backreflector (beam divergence ± 5°) and two narrow-spot high-power white LEDs (1 W, beam divergence ± 4.5°). Each source impinges at 45°on the painting surface irradiating uniformly an area of about 5 cm 2 .
A square-shaped fibre bundle collects the reflected light from a single point of the scanned surface and delivers it to a set of Si and InGaAs photodiodes, each of them equipped with an interferential filter. The optical head, composed of the lighting system and the collecting optics (catoptric lens with a field of view (FOV) of 0.29°), is placed in a 45°/0°illumination/detection geometry and is moved by an XY scanning system with a 250 μm step (4 points/mm) and 500 mm/s speed, resulting in 3 h acquisition time for the maximum scanning area of 1 m 2 . Proper calibration procedure was performed by measuring a certified 100% reflectance Spectralon reference standard and background noise, following CIE indications for non-contact spectrophotometric measurements. The scanner is equipped with an autofocus system based on a high-speed triangulation distance metre, which ensures optimal target-lens distance during scanning. The whole system is computer controlled via a custom-made software. The instrument output is a set of perfectly superimposing monochromatic images, metrically correct and free from aberrations.

FC, PCA, SCM
The monochromatic images were processed with a suite of home-made software tailored for the analysis of paintings. In specific, the false colour method [22], performed with the traditional NRG → RGB mapping, with "N" the near-infrared image, and "R", "G", and "B" being the red, the green, and the blue channels of the colour image, respectively, was used. Colour composite images were elaborated by combining three PCs in the trichromatic RGB space.
RIS data cube was analysed with PCA, an unsupervised exploratory method that looks for directions of maximum variance within a multivariate data space, where high variance (i.e. high variability) means large amount of information. The directions of variance (PCs) are orthogonal and thus uncorrelated, i.e. information contained is never redundant [23].
Spectral mapping was performed with a SCM algorithm, which considers spectra as vectors in N-dimensional space, where N corresponds to the number of spectral bands. The angle between the reference (R) and target spectra measures their similarity. The angle is used to build the similarity maps, where a smaller value, expressed in radians < 0, π > , suggests a higher degree of similarity. In SCM images, the pixel intensity is proportional to the angle between the vector representing the spectrum of each pixel and the reference (or endmember) being mapped. A small angle means a close match and a high intensity value in the image plot. SCM represents an improvement of SAM, as it centralises the data in its mean considering also the negative correlation. In fact, SCM relies on the calculation of similarity (− 1 < R < 1), where 1 means total correlation, through the Pearson's correlation coefficient, yielding more accurate classification results. The instrument, which is described in details in [18], is composed of a measuring head mounted on three linear stages, all placed on top of a carbon fibre box containing the power supplies, the signal digitiser and all the auxiliary elements. Supports, holders and other mechanical aids are 3D-printed. The measuring head is composed of an X-Ray tube (Moxtek©, 40 kV maximum voltage, 0.1 mA maximum anode current, with Mo anode), a Silicon Drift Detector (Amptek© XR100 SDD, 50 mm 2 effective active surface, 500 μm thickness) and a telemeter (Keyence IA-100) for keeping the sample-to-instrument distance constant when scanning. The motor stages (Physik Instrumente©, 300 mm travel range in the x and y direction) allow the scan-plus a 50 mm stage along the z perpendicular direction. The acquisition and data analysis is controlled by a PC, via a software entirely developed by CHNet and based on the open-source Qt platform.
The reconstruction of an elemental distribution map is obtained by selecting a region of interest (ROI) in the spectrum that is an energy interval usually corresponding to the characteristic X line of an element. The software assigns to each pixel a greyscale level corresponding to the X-ray counts of the selected peak: white is assigned to the maximum, black to the minimum. Thus, in the elemental maps obtained from different ROI of the same spectrum, the same grey tone may correspond to different counts and/or different tones may correspond to the same X-ray counts.
The experimental conditions for this campaign were: 40 kV anode voltage, 60 μA filament current, 10 mm/s scanning speed, 1 mm pixel size, beam diameter~1 mm on sample, no helium flow. With these conditions, the acquisition time of the whole area analysed by MA-XRF, which is evidenced in Fig. 5 by the two white adjacent rectangles (size: 11.5 × 16 cm 2 , 28× 16.5 cm 2 ), is about 1 h 50 min.

Fluorescence lifetime imaging (FLI)
The setup used for Time-Resolved Fluorescence Imaging (TRFI) measurements was developed by CNR-INO and is described in detail in [25]. The excitation source is a picosecond pulsed laser diode at 375 nm (BDL-SMN-375, Becker & Hickl GmbH, Berlin, Germany) operating at 20 MHz. A 660 nm aiming beam from a fibred light emitting diode (LED, M660FP1, Thorlabs, Newton, NJ, USA) is delivered with the excitation beam to the sample surface by two 200 μm fibres (0.22 NA). The fibre bundle is handheld and can be freely moved over the sample surface at a distance ranging from 2 to 10 mm, resulting in an irradiated spot diameter of 0.9-4.5 mm. The fluorescence emitted from the specimen is collected by a third 200 μm fibre and spectrally narrowed by an emission (2023)  . For all measurements, the average power on the sample surface was kept below 10 μW. Real-time fluorescence lifetime maps are acquired at a macroscopic scale and, remarkably, under bright illumination to allow the visual control of the examined area during measurements, as described in [19].

Phasor analysis
Fluorescence intensity decay measured on the samples' surfaces was analysed using the phasor method, following the approach presented in our previous works [19,20]. A comprehensive description of this method can be found elsewhere [26]. Briefly, the measured fluorescence intensity decay is decomposed by the Fourier transform into real and imaginary components, namely g and s, from which the phase (τ-phase) lifetime is calculated following the expression τ-phase s/(2πfg), where f is the repetition rate of the laser at 20 MHz. Hence, for each fluorescence decay curve, a single point (phasor) defined by the coordinates g and s can be plotted in the so-called universal semicircle (phasor plot), i.e. a semicircle curve with a 0.5 radius centring at (g 0.5; s 0). For single-lifetime species, the Fluorescence Lifetime (FL) can be directly determined by the coordinate values of the phasor, whereas for multiple-lifetime species, the phasor (g, s) is a linear combination of multiple phasors, each (g i , s i ) representing an individual species. In this analysis, we calculated the mean phasor value of each reference pigments from the τ-phase measured by FL mapping on the samples surface. Then, FL mapping was performed on the painting in the regions previously identified by SCM and XRF maps, obtaining the phasor distributions of the main pigment mixtures. The τ-phase was calculated as an average over 3000 acquisition points with 15 ms integration time in the imaging mode. The relative abundance of the individual pigments within the measured areas was determined based on the Euclidean distance between the phasors of the reference pigments and the probability density peak of the phasor plots of the mixtures.

Results
The false-colour processing of NIR images and colour composite with PCs allowed highlighting differences in the spectral response of the painting surface and details in the distribution of pigments not perceivable to the naked eye. The FC-NIR image at 1292 nm ( Fig. 1b) clearly shows that the robes of the four figures, with apparently similar hue in the visible, are composed of different pigments with different reflectance intensity in the NIR, generating the reddish and greenish colouring of the figure on the right and of the group in the middle, respectively. The PC composite (Fig. 1c) from the first three PCs, besides differentiating the four figures, evidences the overpainting areas on top of the tree crowns and in the architecture in the background, which have not been analysed in the present work. Based on the preliminary examination of the painting by FC processing, the most spectrally diverse areas were selected for pigments identification (see Fig. SM1 in the Supplementary Material). For each area, the prevalent spectral response was assessed by comparing at least ten spectra, providing a good representation of the targeted regions. The selected spectra were used to spectrally map the painting and confirm the presence of possible pigments on the analysed surface ( Fig. SM2 in Supplementary Material). Overlapping areas identified by multiple SC maps were attributed to the presence of pigment mixtures or layering, while areas identified by single maps were interpreted as indicative of the use of pure pigments. The SCM analysis led to the identification of twelve most representative spectra, which were compared with the reference spectra (Fig. 2) showing similar spectral features, in order to facilitate the identification of the pure pigments composing the mixtures.
The spectral analysis allowed for the identification of ten pure pigments, reported in Fig. 3b, c, which were used as endmembers for SCM analysis. The resulting SC maps are shown in Fig. 4 as single-colour images obtained by assigning the same colour of its endmember spectrum to the intensity scale of the image. For each pixel of the SC maps, the maximum intensity indicates the maximum spectral similarity between the reference and the painted surface, whereas black areas mean no correlation. All SC maps are combined in a single image evidencing the distribution of each identified pigment all over the painting (Fig. 3a). Based on SCM in Fig. 4a-c, the pigment mixture used for the walls of the architecture is mainly composed of red ochre and raw sienna (both iron oxide pigments), with a small amount of orpiment (arsenic sulphide). Red ochre is also found in the red stripes of the blanket on the bed, whereas the broader yellowish stripes show high spectral similarity with orpiment. Giving the greenish hue of the blanket, we hypothesised the presence of a blue pigment in a mixture with orpiment and found high spectral correspondence with indigo (nitrocinnamic acid), as shown in Fig. 4l. Due to the numerous pictorial gaps in the area of the blanket, the spectral characteristics of indigo were not immediately identified, but the presence of this pigment was deducted through comparison with another story where the artist depicted the same subject. The better preservation of the latter made it possible to understand that the blanket was originally green (orpiment + indigo) with red stripes (red ochre). The distribution of green earth (Fig. 4f), deriving from hydrous iron and magnesium alumino-silicate clay micas, and of natural umber (Fig. 4g)  Bone black is also spectrally consistent with the darker folds of the robes of the three figures in the middle. The distribution of azurite (basic copper carbonate, Fig. 4i) includes small spots on the foliage and most part of the figure on the right, where it is mixed with red ochre and lead white (basic lead carbonate). Finally, spectral similarity with lead white is found in an almost pure form in the collar of the robe and the sheet on the bed, as well as in a mixture with the other pigments in the lighter areas of the architecture and the foliage (Fig. 3i). RIS results were integrated with macro-XRF mapping (Figs. 5 and 6) to confirm the presence of the identified pigments based on the detection of the composing chemical elements. XRF distribution maps of Fe, As, Cu, Hg, Pb, Ca, and Sr are reported in comparison with the SC maps of the corresponding pigments. The presence of Fe highlighted by XRF (Fig. 5b) is consistent with the combined SC map of the Fe-based pigments (Fig. 5a), namely red ochre (depicted in red), raw sienna (orange brown), green earth (olive green), and natural umber (brown). The absence of spectral similarities in the tree foliage and in the shaded folds of the characters' clothing (black areas), where Fe is distinctly detected, is probably due to the presence in the mixture of an opaque pigment, namely bone black (see the SCM in Fig. 4d). This pigment likely covers the signal from the ferrous pigments, whose fractions in the mixture are too low to be revealed. The presence of orpiment highlighted by the SC map on the bed blanket (Fig. 5c) is consistent with the distribution of As, which can be inferred by the comparison between the As + Hg map (Fig. 5d) and the Hg map (Fig. 5h). The distribution of As is reported together with that of Hg, due to the overlapping of As_Kb (11.73 keV) and Hg_Lb (11.82 keV). It was not possible to use the Ka line of As (10.54 keV) because it overlaps the La line of Pb (10.55 keV). Hg_La (9.99 keV) has been used for the Hg distribution. Copper, which is indicative of the presence of azurite and malachite, is detected on the robe of the figure on the right and in a spot on the tree's foliage (Fig. 5f), consistently with the distribution of azurite (light blue areas in the combined SC map of azurite and malachite, in Fig. 5e). However, no matches were found between the Ma-XRF map of copper and Cu pigments in the lighter green leaves, thus contradicting the attribution of the spectral response revealed by SCM to malachite and azurite in these areas. Moreover, Hg signal was detected in correspondence of the skin tone of the four figures (Fig. 5h), which is in contrast with the SC map of the red Hg-based pigment (Fig. 5g) typically used for the skin tone, namely cinnabar, where no spectral similarity is revealed. Such inconsistency is possibly due to the loss of low-intensity backscattered radiation from cinnabar, which is likely present in low fraction in the mixture of pigments used for these areas, according to the artistic technique of that time. SC map of lead white (Fig. 6d) is only partially consistent with Pb XRF map (Fig. 6d), which highlights the presence of this element in great amount all over the analysed area, except for the darker leaves of the tree, some of the windows, the doorway, the broad strips on the blanket, the gilded area, and some lacunae. The discrepancies with the SC map are probably due to the use of lead white in mixture with other pigments that hinder its spectral contribution. The distribution of Ca (XRF map in Fig. 6e)   that of Pb, suggesting the presence of a Ca-based preparation, and more specifically gypsum, due to the concurrent presence of Sr (Fig. 6f) [27]. The signal of Ca is detected not only on the lacunae on blanket, where the preparation layer emerges, but also on the tree foliage. This is consistent with the SC maps of another Ca-based material, namely bone black (Fig. 6c), although by MA-XRF it is not possible to distinguish whether the signal of Ca comes from the paint or the underlying preparation. Based on SCM and XRF results, we calculated the phasor values of each identified reference paint (pigment + binder) from the mean τ-phase measured by FL mapping (see Table 1 in the Supplementary material and Fig. SM3). Then, we performed the same measurement on the painting where the main pigment mixtures were identified (nine areas highlighted in Fig. 7). For the analysis of each dataset, we considered only decays with > 100 peak counts in order to avoid artefacts. The uncertainty for τ-phase value, reported in Table 1, is given by the maximum value between the standard deviation of the mean and the HWHM of the instrument response function (IRF), which in this case is 0.1 ns; IRF was previously measured as instrumental response to a laser pulse. We calculated the g and s coordinates of the probability density peak in the respective phasor cluster of the selected regions.
Here we focus on the results obtained on areas 3, 7, and 8 (results on areas 1 and 2 can be found in the Supplementary Material). The main pigments composing the mixture within region 3 are raw sienna and red ochre, as shown by the combined SCM in Fig. 8b. Each pixel in the fluorescence lifetime image superimposed on the painting detail (Fig. 8c) can be traced in the phasor cluster (Fig. 8d) due to the reciprocity between the fluorescence decay and the phasor transformation. The bimodal distribution of lifetime data within Fig. 8c (green versus red-yellow region) is probably due to the presence of two mixtures within the same imaged area, as demonstrated by the corresponding bi-modal points distribution in the phasor domain (Fig. 8d).The phasors of the two reference paints composing the mixture are reported together with those of paints not present in this area, namely lead white and azurite egg tempera, for comparison. The distance between the phasor of the reference paint and the maximum density of the phasor cluster is indicative of the contribution of each pigment to the FL signal of the painting mixture, i.e. the smaller the distance the higher the contribution. In this area, the distance of red ochre and raw sienna phasors from the centre of maximum density is 0.054 and 0.061, respectively, whereas the phasors of lead white and azurite fall at distances of 0.154 and 0.094, respectively. The analysis of area 8 (Fig. 9) shows the distance between the phasors of the reference paints composing the mixture (namely azurite, red ochre and white lead) and the probability density peak of the examined area. Here, the spectral similarity revealed by SCM and strong Cu signal detected by XRF suggests azurite to be the predominant pigment in the mixture. Consistently, the phasor of azurite is located inside the phasor cluster of the painting, at a distance of 0.085, whereas red ochre and lead white seem to contribute less to the fluorescence signal of the mixture, with a distance of 0.180 and 0.250 from the probability density peak, respectively.
In the case of area 7 (Fig. 10), two pigments are used separately for the red and yellowish stripes of the blanket, where the FL mapping was performed (Fig. 10b). The presence of two unmixed fluorescent species produces a rather large zone of high-count density in the phasor cluster where two maxima can be distinguished, each corresponding to one of the two main pigments. For simplicity's sake, the average value between the two maxima was used to calculate the relative distances to the phasors of the reference pigments: the contribution of orpiment-based paint is greater (distance 0.093) than that of red ochre (distance 0.136), consistently with the extent of the areas covered by the two pigments in the analysed area. On the other hand, the lead white paint, probably present in the mixture in small quantities, shows a greater distance (0.226) from the probability density peak of the distribution, which is consistent with XRF.
Finally, we analysed the tree foliage on the painting and found correspondence with what observed by SCM. The results on area 1 and 2 are reported in Fig. SM4 in the Supplementary Material. The reference paints best matching with the phasor plot measured on  . 7 Location of the nine areas analysed with FL mapping the light green leaves (area 1) are malachite (distance 0.050) and lead white (distance 0.053). It is noteworthy that the malachite reference phasor is located in the high-count density region of the phasor graph, signifying a high contribution of the pigment to the fluorescence decay signal measured in this area of the painting, which is consistent with SCM but not with XRF. The phasors of the other pigments composing the mixture, namely natural umber, green earth, and bone black, are located at a distance of 0.110 (bone black) and 0.126 (natural umber and green earth) from the probability density peak of the cluster, meaning lower contribution to the overall fluorescence signal. By contrast, in the darkest region of the tree canopy (area 2), the main pigments in the mixture, i.e. green earth, bone black and natural umber, show a distance of 0.059, 0.066, 0.087 from the maximum of the distribution, respectively. The phasors of malachite and lead white are further away from the maximum density of the distribution, with distances of 0.112 and 0.200, respectively. These results are consistent with both SCM and XRF.

Discussion and conclusions
The combined application of complementary imaging techniques enabled to define the palette of the artist and to map the distribution of each pigment with a good level of confidence, confirming the advantages of using a multi-analytical approach. The set of egg tempera samples, specifically prepared according to historical recipes, proved suitable as a reference for direct comparison with the painting under investigation. The identified pigments, as confirmed by cross-referencing the results of the three techniques, are found in mixture in different regions of the painting, as summarised in Table 2.
The proposed method also highlighted how ambiguities in the interpretation of data may arise from the use of a single technique. Specifically, possible misinterpretation of SCM maps can be related to the fact that multispectral reflectography does not offer sufficient spectral resolution to detect subtle differences between optically similar materials. The different degree of ageing between the painting and the reference sample set used for SCM can also be included among the possible causes of mismatches with XRF. On the other hand, as previously stated, XRF has limitations in the recognition of pigments, as it does not allow for the identification of chemical compounds but only individual elements, excluding those with an atomic weight lighter than sodium. Fruitful discussion with experienced conservators allowed confirming the presence of pigments identified by only one of the techniques applied, i.e. indigo identified by SCM in the blanket and cinnabar in the skin tone, as suggested by the detection of Hg by XRF. The main mismatch between XRF and SCM was found in correspondence of the light green areas in the tree foliage, where the spectral similarity with malachite revealed by SCM was disproved by the lack of Cu signal in the XRF map. FL mapping seems to confirm the presence of malachite in the light green leaves, but the high sensitivity of XRF to Cu (in the order of parts per million) leads to exclude the use of Cu-based pigments for the tree. All the other pigments identified by SCM and confirmed by XRF were proved to be consistent with FLI maps as well. In this specific case, the use of a non-invasive molecular analytical technique could identify the presence of an organic compound not detectable by XRF and spectrally similar to malachite, thus explaining the inconsistency between the results. Further analysis will be carried out to this purpose.
The phasor plot allows to represent the fluorescence lifetime measured from molecular micro-environmental heterogeneity as clusters of fluorescence lifetime distributions. In this case, spatial regions of the pictorial layer with sufficiently different fluorescence lifetimes result in spatially separated clusters within the phasor space enabling the visualisation of multiple compounds. The intensity fraction of the emitter with different surrounding microenviroments, which contribute to the fluorescence measured from a single  pixel of an image, can be calculated thanks to the linear combination properties of the phasor plot. As reported in literature, the proportion of the two fluorophores' contributions can then be computed by calculating the Euclidian or Mahalanobis [28] distance between the sample's phasor and the two phasor clusters of the contributing fluorophores. The relevance of the phasor approach is the ability to perform quantitative analysis of multiple composing species without any assumptions about their decay characteristics [29]. Based on this, we have proposed a method for the analysis of FLI data to explore the possibility to relate the phasor of the reference samples (i.e. pigment dispersed in the egg binder) to the composition of the paint mixtures on Lorenzetti's painting. The fluorescence spectra of each reference sample (Fig. SM5 in the Supplementary material) show that the emission detected both in the reference paints and in the painting should be attributed mostly to the binder, but the differences in the measured lifetime are produced by changes induced in the microenvironment by the presence of the non-fluorescent pigments dispersed in it. This a posteriori approach was tested in areas where the main pigment mixtures had already been identified by the two previously applied complementary techniques, RIS and XRF. This allowed us relating the multicomponent phasor cluster measured on the painting to the phasors of the reference paints actually present in the selected areas. We have tested the suitability of the distance-based phasor method, which is currently applied in the biological field [30], by quantifying the distance between the mean phasor of each reference paint and the probability density peak of the phasor cluster measured on the painting. The results show that the paints that are certainly not present in the mixture are found at a greater distance from the probability density peak than those present in the mixture. Further considerations on the composition of the mixture can be made by comparing the distances of the reference paints, which may in principle provide an indication of the contribution of each paint component.
The key challenge in this approach is that the molecular environment and the relative amounts of fluorescent molecular species can greatly differ from the reference samples and the painting, thus substantially affecting the measured τ-phase. Moreover, the overall fluorescence signal of a paint is strongly influenced by the binder in which the pigment particles are dispersed, especially if the pigment is a weak emitter or does not exhibit any fluorescence. On the one hand, given the strong contribution of the egg binder (fluorescence emission band centred at 510-520 nm [31]), which is present in both the reference samples and the painting, it is reasonable to expect a flattening of the differences between the respective phasors. On the other hand, the different degree of ageing between the reference paints and the examined painting should result in a nearly constant difference in the τ-phase between the painting and the reference samples. More specifically, the binder's emission properties can be affected by the oxidation of amino acids and lipids [32] resulting in a slight shift of the emission peak toward longer wavelengths [33]. However, the τ-phase values measured on the various samples, as well as on the painting, show non-negligible differences, in accordance with the mixture composition identified by RIS and XRF. Consequently, it is reasonable to assume that the measured phasors are representative of the whole paint system-i.e. reference sample or painting area-as a result of the interactions between the fluorophores and the non-fluorescent species [33].
The results presented here suggest that it is possible to arbitrarily determine a threshold distance from the probability density peak of the phasor cluster of a given mixture above which the probability of the presence of a paint is almost zero. The proposed Euclidean phasor analysis is a possible method for processing FLI data in an attempt to separate exogenous from endogenous fluorescent species, but for now, it is far from being applicable without an a priori knowledge of the paint sample. Systematic analyses of specifically created samples of pure pigments or mixtures with binders of different nature will allow assessing the effectiveness of the method.
Although it is recognised that the concept of pure material in a mixture cannot always be well defined, the abundance of individual pigments could be also estimated by constrained nonlinear mixing patterns giving the physical interpretation of the SCM endmembers [34], but this goes beyond the scope of this work and will be part of a forthcoming study.