# A Fluorescent Material Model for Non-Spectral Editing & Rendering

Laurent Belcour, Alban Fichet, Pascal Barla

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Laurent Belcour, Alban Fichet, Pascal Barla. A Fluorescent Material Model for Non-Spectral Editing & Rendering. SIGGRAPH Conference Papers 2025 -- Special Interest Group on Computer Graphics and Interactive Techniques Conference Conference Papers, Aug 2025, Vancouver BC, Canada. pp.1-9, 10.1145/3721238.3730721. hal-05267431

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# A Fluorescent Material Model for Non-Spectral Editing & Rendering

LAURENT BELCOUR, Intel Corporation, France

ALBAN FICHET, Intel Corporation, France

PASCAL BARLA, Inria Bordeaux, France

<figure><img src="image_2.png" /><figcaption>Fig. 1. We introduce a material model for the rendering and editing of fluorescence. Our approach relies on a Gaussian approximation of the fluorescence component of the bi-spectral reradiation, which reproduces measured data accurately thanks to an energy conserving decomposition. Our model allows for analytical spectral integration, making it compatible with non-spectral and real-time rendering, and eases the editing of fluorescent material textures.</figcaption></figure>

Fluorescent materials are characterized by a spectral reradiation toward longer wavelengths. Recent work [3] has shown that the rendering of fluorescence in a non-spectral engine is possible through the use of appropriate reduced reradiation matrices. But the approach has limited expressivity, as it requires the storage of one reduced matrix per fluorescent material, and only works with measured fluorescent assets.

In this work, we introduce an analytical approach to the editing and rendering of fluorescence in a non-spectral engine. It is based on a decomposition of the reduced reradiation matrix, and an analytically-integrable Gaussian-based model of the fluorescent component. The model reproduces the appearance of fluorescent materials accurately, especially with the addition of a UV basis. Most importantly, it grants variations of fluorescent material parameters in real-time, either for the editing of fluorescent materials, or for the dynamic spatial variation of fluorescence properties across object surfaces. A simplified one-Gaussian fluorescence model even allows for the artist-friendly creation of plausible fluorescent materials from scratch, requiring only a few reflectance colors as input.

CCS Concepts: • **Computing methodologies** → **Rendering; Reflectance modeling.**

## ACM Reference Format:

Laurent Belcour, Alban Fichet, and Pascal Barla. 2025. A Fluorescent Material Model for Non-Spectral Editing & Rendering. In *Special Interest Group on Computer Graphics and Interactive Techniques Conference Conference Papers (SIGGRAPH Conference Papers '25)*, August 10–14, 2025, Vancouver, BC, Canada. ACM, New York, NY, USA, 9 pages. https://doi.org/10.1145/3721238.3730721

## 1 INTRODUCTION & RELATED WORK

*Context.* Fluorescence is widespread in plants [16], animals [13] and minerals [12]. It is visually characterized by an apparent change of reflectance color depending on the environment lighting. This may result in a change of hue (e.g., for a same material when going from daylight to indoor lighting), and/or in an increase of reflected intensity compared to the surrounding (e.g., for materials that reflect non-visible ultraviolet light in the human visible range). Formally, fluorescence occurs when light of an incoming wavelength $λ_i$ is reradiated at another outgoing wavelength $λ_o$ [5]. This is characterized by spectral reradiation matrices $P(λ_i, λ_o)$. Such matrices may be assumed triangular (i.e., $P(λ_i, λ_o) = 0 ∀ λ_o < λ_i$) since in the vast majority of cases, reradiation only occurs toward smaller energies, or larger wavelengths. In diffuse fluorescent materials, $P$ replaces and augments the conventional spectral albedo, which then corresponds to values along its diagonal: $ρ(λ) = P(λ, λ)$.

*Related work.* In Computer Graphics and Optics, most previous methods (e.g., [7, 9, 10, 14, 15]) render fluorescence using a single measured reradiation matrix $P$ per material, assuming fluorescence and scattering properties are uncorrelated. There are two notable exceptions in the literature. The work of Hullin et al. [8] projects measured bispectral BSDFs (Bidirectional Scattering Distribution Functions) on an angularly-dependent linear combination of a few reradiation matrices. Unfortunately, measured data for that model have not been made available. The method of Benamira et al. [2] suggests a fully analytical bispectral BSDF inspired from Quantum Dots (nano-particles distributed on a microsurface). Such a structural assumption is very specific and unlikely to correspond to common fluorescent materials; in particular, the angularly-varying reradiation matrices produced by their model differ significantly from the reradiation matrices measured in diffuse fluorescent materials. In the absence of reliable angularly-varying references for common materials, we chose to focus in this paper on diffuse fluorescence.

SIGGRAPH Conference Papers '25, August 10–14, 2025, Vancouver, BC, Canada
© 2025 Association for Computing Machinery.
This is the author's version of the work. It is posted here for your personal use. Not for redistribution. The definitive Version of Record was published in Special Interest Group on Computer Graphics and Interactive Techniques Conference Conference Papers (SIGGRAPH Conference Papers '25), August 10–14, 2025, Vancouver, BC, Canada, https://doi.org/10.1145/3721238.3730721.
**Problem.** All of the aforementioned methods require a spectral rendering engine. The rendering of fluorescence in non-spectral engines was first suggested by Hulin et al. [8]. It has been recently improved by Fichet et al. [3], who construct a reduced reradiation matrix P from the spectral reradiation matrix P using:

$$P = S^{\mathrm{T}} \mathcal{P} \tilde{S}, \qquad (1)$$

where S is the $N \times K$ matrix of sensitivity functions (K functions discretized over N wavelengths), and $\tilde{S} = S(S^{\mathrm{T}}S)^{-1}$ is the $N \times K$ matrix of dual sensitivity functions (i.e., verifying $S^{\mathrm{T}}\tilde{S} = I_K$, the $K \times K$ identity matrix). An outgoing color $c_o$ is then approximated by a matrix-vector product with an incoming color $c_i$:

$$c_o \approx P \times c_i. \qquad (2)$$

The method yields satisfying approximations when rendering measured fluorescent materials using the CIE XYZ Color Matching Functions as bases for S (i.e., using $K = 3$), and is even more accurate with the addition of an UV basis function (i.e., using $K = 4$).

However, the method also presents two shortcomings: 1) for each new fluorescent material in a 3D scene, one must compute and store an additional reduced reradiation matrix P; 2) the method takes as input measured spectral reradiation matrices $\mathcal{P}$, for which there exist only a few databases [6, 9]. As a result, the approach is limited in terms of expressivity: it does not provide an artist-friendly fluorescence creation or editing solution; and spatially-varying fluorescence is restricted to interpolations of measured materials, with no independent control over reflectance and fluorescence components.

**Contributions.** In this paper, we address these two limitations through the introduction of an analytical fluorescent material model tailored to non-spectral rendering. In Section 2 we first explain how we decompose a reduced reradiation matrix into fluorescent and non-fluorescent components with explicit constraints on energy conservation. The fluorescent component is then modeled in Section 3 as a sum of Gaussians, similarly to previous work [9]. Our main contribution then lies in an analytical method to integrate this model over Gaussian-based sensitivity functions, granting on-the-fly editing of fluorescent material properties similarly to previous work on iridescence (e.g., [1, 4]). Lastly, we show in Section 4 that a simplified, one-Gaussian version of the model allows for the creation of spectral fluorescent materials with artist-friendly controls.

## 2 REDUCED RERADIATION MATRIX DECOMPOSITION

Before we introduce a fluorescence model that grants analytic, on-the-fly integration in the next section, we first discuss the decomposition of spectral and reduced reradiation matrices.

**Spectral decomposition.** Spectral reradiation may be additively decomposed into reflectance and fluorescent components:

$$\mathcal{P}(\lambda_i, \lambda_o) = \mathcal{R}(\lambda_i, \lambda_o) + \mathcal{F}(\lambda_i, \lambda_o), \qquad (3)$$

where $\mathcal{R}(\lambda_i, \lambda_o) = \delta(\lambda_o - \lambda_i)\rho(\lambda_o)$ is a diagonal spectral reflectance matrix with $\delta$ the Kronecker delta, and $\mathcal{F}(\lambda_i, \lambda_o)$ holds the remaining off-diagonal (fluorescent) spectral reradiation.

In contrast to methods that rely on measured fluorescence data that are naturally energy-conserving, our model has to remain

<figure><img src="image_3.png" /><figcaption>**Fig. 2. Model overview.** With our decomposition, our model faithfully reproduces a measured reradiation matrix (HERPIPIN) with as low as one Gaussian for the reradiation. Our matrix decomposition works both on full spectral reradiation matrices (Equation 5) and with reduced matrices (Equation 7). This ensures energy conservation during the edition without the need to recompute the reduction for all components.</figcaption></figure>

physically-plausible when fluorescence properties are modified by artists. The energy conservation constraint is defined with respect to spectral reradiation by:

$$\int_{0}^{\infty} \mathcal{P}(\lambda_i, \lambda_o) d\lambda_o \le 1 \text{ for all } \lambda_i. \qquad (4)$$

Enforcing such a constraint is problematic in our context as it depends both on the reflectance $\mathcal{R}$ and fluorescence $\mathcal{F}$ in Equation 3. In order to grant independent control over fluorescence while conserving energy (assuming $\rho(\lambda_i) \le 1 \forall \lambda_i$), we introduce a normalized fluorescent term $\tilde{\mathcal{F}}$ verifying:

$$\mathcal{F}(\lambda_i, \lambda_o) = \tilde{\mathcal{F}}(\lambda_i, \lambda_o)(1 - \mathcal{R}(\lambda_i, \lambda_o)). \qquad (5)$$

Using Equation 5 in Equation 3 and performing the integration of Equation 4 yields the following energy conservation constraint:

$$\int_{0}^{\infty} \tilde{\mathcal{F}}(\lambda_i, \lambda_o) d\lambda_o \le 1 \text{ for all } \lambda_i, \qquad (6)$$

which may be enforced regardless of reflectance, as desired. The decomposition of the spectral reradiation in terms of reflectance and normalized fluorescence is illustrated in the top row of Figure 2. In spite of the simplicity of the normalized fluorescence component $\tilde{\mathcal{F}}$, the measured reradiation is accurately reproduced, thanks to multiplication by the $[1 - \mathcal{R}]$ term.

**Reduced decomposition.** The reduced reradiation matrix is decomposed by applying Equation 1 to each component of Equation 3:

$$P = R + F = S^{\mathrm{T}} \mathcal{R} \tilde{S} + S^{\mathrm{T}} \tilde{\mathcal{F}} \tilde{S}, \qquad (7)$$

with R and F denoting reduced reflectance and fluorescence matrices respectively. The latter may be re-expressed in terms of the reduced normalized fluorescence matrix $\tilde{F} = S^{\mathrm{T}} \tilde{\mathcal{F}} \tilde{S}$ similarly to Equation 5:

$$F = \tilde{F} [I - R]. \qquad (8)$$

**PROOF.** We start from $F = S^{\mathrm{T}} \tilde{\mathcal{F}} \tilde{S} = S^{\mathrm{T}} \tilde{\mathcal{F}} \tilde{S} - S^{\mathrm{T}} \tilde{\mathcal{F}} \mathcal{R} \tilde{S}$ using Equation 5. The first term corresponds to $\tilde{F}$. We next observe that $\tilde{S} \tilde{S}^{\mathrm{T}} = (\tilde{S} \tilde{S}^{\mathrm{T}})^{-1} \tilde{S} \tilde{S}^{\mathrm{T}} = I$ from the definition of dual sensitivity functions. We may thus write $S^{\mathrm{T}} \tilde{\mathcal{F}} \mathcal{R} \tilde{S} = S^{\mathrm{T}} \tilde{\mathcal{F}} \tilde{S} \tilde{S}^{\mathrm{T}} \mathcal{R} \tilde{S} = \tilde{F} \mathcal{R}$ for the second term. As a result, we obtain $F = \tilde{F} - \tilde{F} \mathcal{R} = \tilde{F} [I - R]$. $\square$
<figure><img src="image_4.png" /><figcaption>Fig. 3. Fitting of measured materials with our model. Our reradiation rescaling approach not only ensures energy conservation but also better reproduces the shape of the reradiative part for most of the measurements, even when using a single Gaussian.</figcaption></figure>

The reduced reradiation matrix of Equation 7 then becomes:

$$P = R + \tilde{F} [I - R], \quad (9)$$

which is illustrated in the bottom row of Figure 2, with positive (resp. negative) coefficients shown in blue (resp. red).

The outgoing color $c_o$ may itself be decomposed by plugging the reduced reradiation matrix of Equation 9 into Equation 2:

$$c_o = c_{o,r} + c_{o,f} \approx R c_i + \tilde{F} [I - R] c_i, \quad (10)$$

with $c_{o,r}$ and $c_{o,f}$ denoting reflected and fluorescent outgoing colors respectively. The term $[I - R] c_i$ may then be interpreted as the incoming color available to fluorescent reradiation.

## 3 ANALYTIC FLUORESCENCE MODEL

Our focus in this paper is on the editing and rendering of fluorescence. In the following, we show that a Gaussian-based model of the normalized spectral fluorescence $\tilde{F}$ has two key advantages: a small number of Gaussians is sufficient to accurately reproduce the reflected colors of measured fluorescent materials under various illuminants (Section 3.1); and it may be analytically integrated over arbitrary Gaussian-based sensitivity functions to yield a reduced normalized fluorescence matrix $\tilde{F}$ for use in non-spectral rendering (Section 3.2). For completeness, we provide Gaussian fits of XYZ sensitivity functions, and suggest an optional Gaussian basis for ultraviolet (UV) reradiation (Section 3.3).

### 3.1 Gaussian-based normalized fluorescence

Iser et al. [9] have shown that an axis-aligned Gaussian Mixture Model (GMM) of the fluorescent component $\tilde{F}$ may accurately fit spectral measured data. We take inspiration from their work, with a few key differences: we use a GMM model to fit the normalized fluorescent component $\tilde{F}$; and we evaluate its accuracy in terms of reradiated colors under various illuminants.

Formally, we assume that the normalized spectral fluorescence $\tilde{F}$ may be approximated by a combination of Q axis-aligned 2D Gaussian functions:

$$\tilde{F}(\lambda) \approx \sum_{q=1}^{Q} \mathcal{G}_q(\lambda; \alpha_q, \mu_q, \Sigma_q) \mathcal{H}(\lambda_o - \lambda_i) \quad \text{with} \quad \lambda = \begin{bmatrix} \lambda_i \\ \lambda_o \end{bmatrix}, \quad (11)$$

where the $\mathcal{G}_q$ are 2D Gaussians of intensity $\alpha_q$, mean $\mu_q$ and diagonal co-variance matrix $\Sigma_q$, while $\mathcal{H}$ is a heaviside function used to

<figure><img src="image_5.png" /><figcaption>Fig. 4. Accuracy of our model. We compare the reradiation of standard illuminants on the measured reradiation matrices from Gonzalez and Fairchild [6] with the fitting of our model in the spectral domain. We use a different number of Gaussians for the reradiation and six Gaussians for the diagonal. The whiskers bottom and top represent resp. the minimum and the maximum, the box bottom and top represent resp. the first and third quartiles and the central line represent the average $\Delta E_{2000}^*$ distribution.</figcaption></figure>

enforce reradiation toward smaller energies (see Figure 2). As shown in Figure 3, this decomposition reproduces details of the reradiation, even for a small $Q = 1$ number of Gaussians.

In order to evaluate the accuracy of this spectral fluorescence model, we study how colors reflected by measured materials under various spectral illuminants differ from the ground-truth when using our GMM approximation. Formally, we compute:

$$c_o = \iint \mathcal{P}(\lambda_i, \lambda_o) \mathcal{L}(\lambda_i) \mathcal{s}(\lambda_o) d\lambda_i d\lambda_o, \quad (12)$$

where $\mathcal{s} = [\bar{x}, \bar{y}, \bar{z}]^\top$ is the vector of XYZ CMFs, $\mathcal{L}$ is a spectral illuminant, and $\mathcal{P} = \mathcal{R} + (1 - \mathcal{R})\tilde{\mathcal{F}}$ with $\tilde{\mathcal{F}}$ coming either from the measured material or from the approximation of Equation 11. Note that the spectral reflectance $\mathcal{R}$ remains unaltered.

As shown in Figure 4, using $Q = 1$ Gaussian function is enough to capture the effect of fluorescence on reflected colors of the measured materials we have tested. We obtain almost unnoticeable differences (< 2 in $\Delta E_{2000}^*$) between most of reflected colors obtained with the ground truth and the GMM approximations for a large number of illuminants. We refer the reader to Supplemental materials for a thorough evaluation on 143 measured materials from [6].

### 3.2 Gaussian-based spectral integration

For non-spectral rendering, we rely on the reduced reradiation approach of Fichet et al. [3], where the outgoing color $c_o$ is approximated by the matrix-vector product of Equation 2. In our case the reduced reradiation matrix P is further decomposed according to Equation 9, yielding Equation 10 for $c_o$. Since our main objective is the control of fluorescence, we next assume that the R matrix is known, and focus on the $\tilde{F}$ matrix. Its coefficients are given by:

$$\tilde{F}_{jk} = \iint \tilde{\mathcal{F}}(\lambda_i, \lambda_o) \tilde{s}_j(\lambda_i) s_k(\lambda_o) d\lambda_i d\lambda_o, \quad (13)$$

where $s_k (k \in [1, K])$ and $\tilde{s}_j (j \in [1, K])$ are arbitrary sensitivity and dual sensitivity functions respectively. When these functions are discretized, they correspond to the columns of S and $\tilde{S}$ respectively.

We now show that Equation 13 may be integrated analytically provided sensitivity functions are given in Gaussian form. Specific examples of such Gaussian bases will be given later, in Section 3.3.
<figure><img src="image_6.png" /><figcaption>Fig. 5. Analytical integration of Gaussian-based Fluorescence. We show that we can analytically integrate our 2D model (a) by changing the shape of the integration domain (in red) with a serie of shears (b-c) that produce an axis aligned Gaussian and map the Heaviside to the half plane $\lambda''_i > 0$. This later stage enables to integrate separately the dimensions (c). In this Figure, the 2D Gaussian is scaled for visualization purpose and will not spread in the negative domain in practical cases.</figcaption></figure>

**Gaussian sensitivity functions.** A first issue is that dual sensitivity functions are usually not Gaussian-shaped. Fortunately, since $\tilde{S}S^\top = I$ by design, we have $\tilde{S}S^\top S = S$ and thus $\tilde{S} = S(S^\top S)^{-1}$. Hence, dual sensitivity functions can be expressed as weighted sums of the sensitivity functions. We then obtain:

$$ \tilde{F} = S^\top \tilde{F} S (S^\top S)^{-1}. \quad (14) $$

Intuitively, this means that for Gaussian sensitivity functions, dual sensitivity functions are defined as a weighted sum of Gaussians.

A second issue is that a single Gaussian may not be sufficient to approximate a sensitivity function. Let's assume that $M \ge K$ Gaussians are required in the general case. We then rewrite the matrix of sensitivity functions as $S \approx GT_G$, with $G$ a $N \times M$ matrix whose columns are discretized 1D Gaussian functions, and $T_G$ a $M \times K$ transfer matrix that linearly combines 1D Gaussians contributing to a same sensitivity function.

The reduced normalized fluorescence matrix may now be written:

$$ \tilde{F} = T_G^\top \tilde{F}^\circ T_G C, \quad (15) $$

where we have introduced $C = (S^\top S)^{-1}$, a $K \times K$ matrix that is precomputed once for a given choice of sensitivity functions; and $\tilde{F}^\circ = G^\top \tilde{F} G$, a $M \times M$ matrix whose coefficients are given by:

$$ \tilde{F}^\circ_{jk} = \iint \tilde{F}(\lambda_i, \lambda_o) g_j(\lambda_i) g_k(\lambda_o) d\lambda_i d\lambda_o, \quad (16) $$

where $g_m$ ($m \in [1, M]$) are 1D Gaussian functions, which when discretized correspond to the columns of $G$. Our next goal is to find a method to integrate Equation 16 analytically.

**Analytical integration.** For simplicity and without loss of generality, we will now consider a single 2D Gaussian ($Q = 1$) in Equation 11, dropping the $q$ subscript: $\tilde{F}(\lambda_i, \lambda_o) = \mathcal{G}(\lambda; \alpha, \mu, \Sigma)\mathcal{H}(\lambda_o - \lambda_i)$.

Equation 16 may then be rewritten as:

$$ \tilde{F}^\circ_{jk} = \int \mathcal{G}_{jk}(\lambda; \alpha_{jk}, \mu_{jk}, \Sigma_{jk}) \mathcal{H}(\lambda_o - \lambda_i) d\lambda, \quad (17) $$

with $\mathcal{G}_{jk}$ a 2D Gaussian function defined by:

$$ \mathcal{G}_{jk}(\lambda; \alpha_{jk}, \mu_{jk}, \Sigma_{jk}) = \mathcal{G}(\lambda; \alpha, \mu, \Sigma) g_j(\lambda_i) g_k(\lambda_o), \quad (18) $$

where Gaussian product rules are applied to the $\lambda_i$ and $\lambda_o$ dimensions separately to yield $\alpha_{jk}, \mu_{jk} = [\mu_j, \mu_k]^\top$ and $\Sigma_{jk} = \text{diag}(\sigma_j^2, \sigma_k^2)$.

The key idea of our analytical method is then illustrated in Figure 5: instead of integrating the 2D Gaussian $\mathcal{G}_{jk}$ over a triangular domain as in Equation 17, we perform two changes of variable

through a pair of skew transforms. The first skew $S_i$ along $\lambda_i$ converts the diagonal boundary into a vertical boundary, which turns $\mathcal{G}_{jk}$ into an oblique 2D Gaussian (Fig. 5 (b)); the second skew $S_o$ along $\lambda_o$ leaves the vertical boundary unchanged, but produces a novel axis-aligned 2D Gaussian (Fig. 5 (c)) that is analytically integrated (Fig. 5 (d)).

The first change of variable $\lambda' = S_i\lambda$ maps the $\lambda_o - \lambda_i = 0$ boundary to $\lambda'_i = 0$, which is achieved by a horizontal shear:

$$ S_i = \begin{bmatrix} 1 & -1 \\ 0 & 1 \end{bmatrix}. \quad (19) $$

Since shearing does not change the space measure, this yields:

$$ \tilde{F}_{jk}^\circ = \int \mathcal{G}_{jk}(S_i^{-1}\lambda'; \alpha_{jk}, \mu_{jk}, \Sigma_{jk}) \mathcal{H}(-\lambda_i') d\lambda', \quad (20) $$

$$ = \int \mathcal{G}_{jk}(\lambda'; \alpha_{jk}, S_i \mu_{jk}, S_i \Sigma_{jk} S_i^\top) \mathcal{H}(-\lambda_i') d\lambda', \quad (21) $$

where we have used an equivalent 2D Gaussian with sheared parameters in the second line, with the covariance matrix given by:

$$ S_i \Sigma_{jk} S_i^\top = \begin{bmatrix} \sigma_j^2 + \sigma_k^2 & -\sigma_k^2 \\ -\sigma_k^2 & \sigma_k^2 \end{bmatrix}. \quad (22) $$

The goal of the second change of variable $\lambda'' = S_o\lambda'$ is to turn the oblique 2D Gaussian into an axis-aligned 2D Gaussian. $S_o$ should thus transform the matrix of Equation 22 into a diagonal matrix:

$$ S_o = \begin{bmatrix} \frac{1}{\sigma_k^2} & 0 \\ \frac{\sigma_j^2 + \sigma_k^2}{\sigma_j^2} & 1 \end{bmatrix}. \quad (23) $$

Applying the second change of variables then yields:

$$ \tilde{F}_{jk}^\circ = \int \mathcal{G}_{jk}(\lambda''; \alpha_{jk}, \hat{\mu}_{jk}, \hat{\Sigma}_{jk}) \mathcal{H}(-\lambda_i'') d\lambda'', \quad (24) $$

where the new covariance matrix $\hat{\Sigma}_{jk}$ is diagonal as expected:

$$ \hat{\Sigma}_{jk} = S_o S_i \Sigma_{jk} S_i^T S_o^T = \begin{bmatrix} \sigma_j^2 + \sigma_k^2 & 0 \\ 0 & \frac{\sigma_j^2 \sigma_k^2}{\sigma_j^2 + \sigma_k^2} \end{bmatrix} := \begin{bmatrix} \hat{\sigma}_j^2 & 0 \\ 0 & \hat{\sigma}_k^2 \end{bmatrix}; \quad (25) $$

and the new mean vector $\hat{\mu}_{jk}$ is given by:

$$ \hat{\mu}_{jk} = S_o S_i \mu_{jk} = \begin{bmatrix} \mu_j - \mu_k \\ \frac{\sigma_j^2 \mu_k + \sigma_k^2 \mu_j}{\sigma_j^2 + \sigma_k^2} \end{bmatrix} := \begin{bmatrix} \hat{\mu}_j \\ \hat{\mu}_k \end{bmatrix}. \quad (26) $$

Equation 24 now has a simple analytical solution since the Heaviside term $\mathcal{H}$ only applies to the $\lambda''_i$ dimension and the integrated 2D Gaussian is axis-aligned. Carrying integration, we obtain:

$$ \tilde{F}_{jk}^\circ = \alpha_{jk} \times \frac{1}{2} \sqrt{2\pi} \hat{\sigma}_j \left[ 1 - \text{erf} \left( \frac{\hat{\mu}_j}{\sqrt{2}\hat{\sigma}_j} \right) \right] \times \sqrt{2\pi} \hat{\sigma}_k, \quad (27) $$

where the second (resp. third) factor corresponds to bounded (resp. unbounded) integration along the $\lambda''_i$ (resp. $\lambda'_o$) dimension.

After a few simplifications using Equations 25 and 26, we obtain:

$$ \tilde{F}_{jk}^\circ = \pi \alpha_{jk} \sigma_j \sigma_k \left[ 1 - \text{erf} \left( \frac{\mu_j - \mu_k}{\sqrt{2(\sigma_j^2 + \sigma_k^2)}} \right) \right]. \quad (28) $$
<figure><img src="image_7.png" /><figcaption>**Fig. 6. Gaussian basis functions.** Our analytical bispectral integration relies on the availability of Gaussian sensitivity functions. We fit the CIE XYZ 2006 2° $\bar{x}$ sensitivity function (in red) with a pair of Gaussians, while $\bar{y}$ and $\bar{z}$ only require a single Gaussian. In order to account for reradiation from non-visible to visible light, we provide an additional UV basis (in purple) whose parameters are chosen to best approximate measured materials.</figcaption></figure>

### 3.3 Gaussian basis functions

The analytical bispectral integration process described in the previous section relies on the availability of Gaussian basis functions. As shown in Figure 6, the CIE XYZ 2006 2° observer tabulated sensitivity functions are well approximated by Gaussian bases. In particular, $\bar{x}$ requires a pair of Gaussians, while a single Gaussian is sufficient for both $\bar{y}$ or $\bar{z}$. Gaussian basis parameters are provided in Table 1.

<table><caption>**Table 1. Gaussian basis parameters.** We fit the CIE XYZ 2006 2deg using Gaussians ($\bar{x}$, $\bar{y}$, $\bar{z}$) and report the magnitude, mean and standard deviation here. We further include a Gaussian sensitivity function for the UV band.</caption><thead><tr><th></th><th>intensity</th><th>mean (nm)</th><th>std (nm)</th></tr></thead><tbody><tr><td>$\bar{x}$</td><td>0.35087<br>1.141263</td><td>443.412226<br>596.813847</td><td>20.838149<br>33.276659</td></tr><tr><td>$\bar{y}$</td><td>1.024335</td><td>560.186336</td><td>43.898132</td></tr><tr><td>$\bar{z}$</td><td>1.915863</td><td>447.268188</td><td>23.542626</td></tr><tr><td>UV</td><td>1</td><td>382.535501</td><td>57.432550</td></tr></tbody></table>

**Additional UV basis.** We use an additional basis in the UV range to account for reradiation from non-visible to visible light, similarly to Fichet et al. [3]. Contrary to them, we use a single Gaussian function for this additional channel. The Gaussian parameters of this extra band (also provided in Table 1) are optimized over a set of measured materials [6], as detailed in Supplemental material. In a nutshell, we searched for the Gaussian parameters that minimize the $\Delta E^*_{2000}$ difference between the color obtained with the full spectral reradiation (e.g., using Equation 12) and the color obtained with a reduced $4 \times 4$ XYZU reradiation matrix (i.e., using Equation 2), for several illuminants.

**Transfer matrices.** Recall that the matrix of sensitivity functions is given by $S = GT_G$. When working with XYZU colors, our approach requires 5 Gaussian basis functions. Hence $G$ is the $N \times 5$ matrix of discretized 1D Gaussians, and $T_G$ is a $5 \times 4$ matrix given by:

$$T_G = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}. \qquad (29)$$

<figure><img src="image_8.png" /><figcaption>**Fig. 7. Accuracy of our reduction.** We compute $\Delta E^*_{2000}$ differences between a spectral evaluation of the original data and the XYZ evaluation of the reradiation based on three XYZU reduction methods. Two of these methods apply brute-force reduction on measured materials, using either our UV basis or the one from Fichet et al. [3]. The third reduction method applies our analytical Gaussian-based integration, using a single 2D Gaussian for the normalized fluorescence component and six 1D Gaussians for the reflectance along the diagonal. The whiskers bottom and top represent resp. the minimum and the maximum, the box bottom and top represent resp. the first and third quartiles and the central line represent the average $\Delta E^*_{2000}$ over 143 reradiation matrices from [6].</figcaption></figure>

When using a different color space for light transport than the one of the sensor – say CIE XYZ 2006 2°, we also have to project the outgoing color $c_o$ back to the sensor color space. As already mentioned in the work of Fichet et al. [3], this is done through $T_{c_o}$, where $T = S_{XYZ}^\top \hat{S}$ is a final $3 \times K$ transfer matrix. When working with XYZU colors ($K = 4$), it simply amounts to:

$$T = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix}. \qquad (30)$$

**Reduction accuracy.** The accuracy of our Gaussian-based analytic reduction method is evaluated in Figure 7 on the Gonzalez and Fairchild's dataset [6] for a range of illuminants, using $\Delta E^*_{2000}$ differences with colors obtained with the full bispectral reradiation matrices. A first difference of our approach compared to the one of Fichet et al. [3] is that we use different UV bases. We thus compare the two XYZU representations using a brute-force reduction on measured materials, which shows that our representation (red ribbons) is on par with theirs (blue ribbons). Yet the main advantage of our model is to grant analytic, on-the-fly reduction, provided the material is approximated with Gaussians: here we also fit 6 Gaussians to the reradiation diagonal to grant full analytic integration. In spite of these approximations, Figure 7 shows that the $\Delta E^*_{2000}$ differences obtained with our model (marroon ribbons) remain very close to those obtained with brute-force integration.

## 4 ARTIST-FRIENDLY FLUORESCENCE MODEL

In this section, we show that besides accurately representing measured fluorescent materials, a simplified Gaussian-based model is well adapted to the creation of fluorescent materials from scratch. We first detail our choices of model simplification (Section 4.1) and explain how energy conservation is ensured (Section 4.2), before showcasing our fluorescence palette editing system (Section 4.3).
## 4.1 Simplified model

Our first design choice is to reduce the number of material parameters to make direct artistic editing tractable. We thus use a single Gaussian for the normalized fluorescence component $\tilde{F}$ (i.e., $Q = 1$ in Equation 11), yielding five parameters: the Gaussian intensity $\alpha$, its mean vector $\mu = [\mu_a, \mu_e]^\top$ and diagonal covariance matrix $\Sigma = \text{diag}(\sigma_a^2, \sigma_e^2)$, where we have used the $a$ and $e$ subscripts to denote the absorption and emission axes respectively.

We then specify the reduced reflectance matrix $R$ using an albedo color $\rho$ (three floats). This further permits to reuse workflows for albedo editing without change. We build $R$ by first upscaling $\rho$ to a spectrum using the dual basis, $\rho \times \tilde{S}$, and then reducing the diagonal matrix $R = \text{diag}(\rho \times \tilde{S})$. Thanks to linearity of the reduction, we obtain $R_{\rho} = \sum_k \rho_k R_k$, where $R_k = S^\top \text{diag}(\tilde{S}_k) \tilde{S}$. This form permits to compute $R$ from precomputed $R_k$ matrices and user specified albedo color $\rho$. The outgoing color is obtained by using $R_{\rho}$ in Equation 10. We compare this approach with the approximation $R = \text{diag}(\rho)$ in Supplemental material and conclude the former works better at approximating multiple scattering.

For RGB rendering, we assume that the albedo color is first converted to XYZ color space to yield $\rho_{XYZ}$. As shown in the work of Fichet et al. [3] and confirmed in Section 3.3, using an additional ultraviolet channel significantly increases accuracy for some materials. When working with XYZU colors, we thus also need to decide of an ultraviolet channel value for the corresponding albedo color $\rho_{XYZU}$. One option would be to let artists specify it, but this would likely be unintuitive since its effects are indirect: it reduces (resp. contributes to) fluorescent reradiation in direct (resp. indirect) lighting. We thus make the choice to determine it automatically, using $\rho_{XYZU} = T_U \rho_{XYZ}$, with $T_U$ a 4 × 3 transfer matrix given by:

$$T_U = S_{XYZU}^\top \tilde{S}_{XYZ} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ -0.0145415 & 0.0267372 & 0.397627 \end{bmatrix}. \quad (31)$$

It amounts to a spectral upsampling through $\tilde{S}_{XYZ}$, followed by a downsampling to the XYZU color space through $S_{XYZU}$. As evident in Equation 31, the X, Y and Z channels are left unchanged, while the UV channel is obtained as a linear combination of other channels in a way proportional to the overlap of their sensitivity functions.

## 4.2 Energy conservation

Thanks to the decomposition of Equation 10, energy conservation may be enforced separately on the reflectance and fluorescence components. For the reflectance component, we enforce each coefficient of the albedo color $\rho$ to be in the $[0, 1]$ range.

Energy conservation of the normalized fluorescence component is more involved, since the corresponding reduced normalized fluorescence matrix $\tilde{F}$ in Equation 10 is obtained through the analytical integration process of Section 3.2. In other words, constraints must be found on material parameters $\{\alpha, \mu_a, \sigma_a, \mu_e, \sigma_e\}$ to ensure the energy conservation constraint of Equation 6 is met.

For consision, we rewrite that constraint as $\tilde{\mathcal{A}}(\lambda_i) \le 1 \forall \lambda_i$, with $\tilde{\mathcal{A}}$ a reduced absorption spectrum shown in Figure 8. Energy conservation may then be ensured if the intensity parameter $\alpha \le \alpha_{max}$,

<figure><img src="image_9.png" /><figcaption>Fig. 8. Energy conservation. We evaluate our conservative bound $\tilde{\alpha}_{max}$ for energy conservation by showing $\tilde{\alpha}_{max} \tilde{\mathcal{A}}(\lambda_i)$ (red solid line), which should reach 1 at its maximum for perfect energy conservation. The reduced fluorescence component $\tilde{F}$ is shown in the background. We examine three absorption configurations, using $\mu_e = 650$nm and $\sigma_e = 60$nm in all cases. Our computation is based on a Gaussian approximation to $\tilde{\mathcal{A}}$ (black dotted line), whose maximum departs from the true maximum when a significant portion of the Gaussian lies below the bispectral diagonal, as in c).</figcaption></figure>

where the upperbound is a function of other material parameters:

$$\alpha_{max}(\mu_a, \sigma_a, \mu_e, \sigma_e) = \frac{1}{\max_{\lambda_i} \tilde{\mathcal{A}}(\lambda_i)}. \quad (32)$$

The intensity parameter is then computed as $\alpha = \tilde{\alpha} \alpha_{max}$, where $\tilde{\alpha} \in [0, 1]$ controls the material fluorescence strength. Even though $\tilde{\mathcal{A}}$ has an analytical form, we have not found an analytical solution to Equation 32 due to the maximum in the denominator.

A conservative analytical solution is obtained by ignoring the Heaviside term in Equation 11, leaving only the single Gaussian term $\mathcal{G}$. Since the peak of $\mathcal{G}$ is located at $[\mu_a, \mu_e]^\top$, we only need to perform integration in $\tilde{\mathcal{A}}$ at $\lambda_i = \mu_a$, yielding:

$$\max_{\lambda_i} \tilde{\mathcal{A}}(\lambda_i) \approx \int_0^\infty \mathcal{G}(\mu_a, \lambda_o) d\lambda_o = \sqrt{\frac{\pi}{2}} \sigma_e \left(1 + \text{erf}\left(\frac{\mu_e}{\sqrt{2}\sigma_e}\right)\right). \quad (33)$$

When used in Equation 32, we obtain our approximate bound: $\hat{\alpha}_{max}(\mu_e, \sigma_e)$. This bound only departs from the true bound when a significant portion of $\mathcal{G}$ lies below the bispectral diagonal (Figure 8).

## 4.3 Editing with fluorescence palettes

In order to let artists choose values for the mean and standard deviation paramers, we present them with color palettes that show a range of achievable colors (see Figures 9 and 10). For each palette, we assume that the albedo color $\rho$ is known, and that the incoming color $c_i$ is determined by a choice of spectral illuminant integrated over Gaussian basis functions in pre-process. In addition, we assume that the maximum fluorescent strength is achieved (i.e., $\alpha = \alpha_{max}$ or $\tilde{\alpha} = 1$) when displaying fluorescence palettes.

We let artists control the parameters $(\mu_a, \sigma_a)$, which light absorption for fluorescent reradiation. Intuitively, a large value of $\sigma_a$ reradiates from as many wavelengths as possible, while a smaller value permits to select a subset of wavelengths around $\mu_a$. A fluorescence palette is then obtained by computing $c_o$ using Equation 10 for every pair $(\mu_e, \sigma_e)$. When an artist picks a color in the palette, this determines the parameter pair $(\mu_e, \sigma_e)$, from which the final intensity is computed through $\alpha = \tilde{\alpha} \hat{\alpha}_{max}(\mu_e, \sigma_e)$.
<figure><img src="image_10.png" /><figcaption>**Fig. 9. Fluorescence palettes with varying absorption.** We show fluorescence palettes for three types of illuminants (daylight D65, incandescent A, and a Gaussian illumination spectrum in the UV-blue range), and three absorption configurations as seen in $\tilde{F}$ (first column): narrow absorption in the UV range (first row), in the visible range (second row) or wide absorption (third row). We use an achromatic albedo $\rho = 0.1$ throughout. In the last three columns, we show renderings of spheres, where the left and right sides use a non-fluorescent VS fluorescent material, obtained by setting $\alpha_n = 0$ and $\alpha_n = 1$ respectively. The fluorescence color has been chosen by picking a point in the corresponding palette (white dots), in this case yielding $\mu_e = 560$nm and $\sigma_e = 14$nm. This shows that adjusting the absorption parameters allows for selective fluorescence behaviors: a narrow absorption in the UV (resp. in the visual) range significantly reduces fluorescence under the A (resp. UV) illuminant, while a wide absorption produces fluorescence in all lighting conditions.</figcaption></figure>

**Editing absorption.** In Figure 9, we show several palettes in different absorption configurations and for different illuminants, keeping the $(\mu_e, \sigma_e)$ parameter pair constant throughout. We choose to visualize the corresponding normalized fluorescence matrix $\tilde{F}$ instead of the full reradiation matrix $\mathcal{P}$ in the first column as it directly conveys the manipulated Gaussian. We also show rendered spheres, with lighting environments made to emit a particular illuminant spectrum by first desaturating them then multiplying them by the XYZU color version of the illuminant. As seen in the figure, focusing absorption in the visible (resp. UV) range has a selective effect in incandescent (resp. UV) lighting, while a daylight illuminant is much less affected since it has large values over both visible and UV wavelength ranges. In contrast, with a wide absorption, the material exhibits marked fluorescent behaviors with all three types of illuminant.

**Editing re-emission.** In Figure 10, we keep the $(\mu_a, \sigma_a)$ parameter pair constant but vary both $\rho$ and $(\mu_e, \sigma_e)$, using a D65 illuminant throughout. Here we show that fluorescence may serve different purposes: it may increase the brightness of an achromatic material, increase its saturation, or change its hue. Thanks to fluorescence palettes, finding the relevant $(\mu_e, \sigma_e)$ pair for a choice of $\rho$ is immediate. We observe in particular that the hue of the fluorescence component varies with $\mu_e$, while its saturation varies with $\sigma_e$. This is to be expected since for smaller $\sigma_e$ values, reradiated chromaticities should tend to the spectral locii.

<figure><img src="image_11.png" /><figcaption>**Fig. 10. Fluorescence palettes with varying emission.** We show fluorescence palettes for a single illuminant (D65) and fixed absorption parameters ($\mu_a = 420$nm, $\sigma_a = 60$nm), but with varying albedo and emission parameters. From first to last row, we use fluorescence to boost the outgoing luminance, saturate the albedo color, and modify the albedo's hue.</figcaption></figure>
# 5 ADDITIONAL RESULTS

In this section, we explore more complex uses of our fluorescent material model: the edition of measured fluorescent materials, and spatial variations of fluorescence parameters. The supplemental video demonstrates fluorescent material manipulation in real-time, along with an animated version of Figure 1. For Figures 9, 10 and 12 to 14, we used a real-time OpenGL prototype. On a laptop with an Intel Arc 140V integrated GPU, a 1920 × 1080 frame with our model takes 8.3ms. Figures 1 and 11 use a CPU path tracer that we compare to a spectral reference following the state-of-the-art method of Mojzik et al. [11].

*Measured material editing.* In Figure 11, we fit our single-Gaussian model to a pair of measured materials [6], then interpolate all fluorescence parameters while keeping the albedo unchanged. As shown in the reradiation plots (first column), the blending of Gaussian parameters displaces the mass of the first sample towards the second one. Since the input measured materials fluoresce different colors (orange in the first row, purple in the third row), the resulting interpolated material exhibits yet another color (green in the second row). As shown in Figure 11, this editing method may be used either in spectral or non-spectral rendering, using full reradiation matrices or reduced matrices respectively. We further show that we can boost the contribution of the fluorescence component in post-process by increasing the fluorescence strength.

*Spatial variations.* In the following we rely on the model of Section 4. The simplest way to vary fluorescence across a surface is through modulation of the fluorescence strength $\alpha_n$. This is shown in Figure 12 with renderings in two environments, for a fixed $(\mu_a, \sigma_a)$

<figure><img src="image_12.png" /><figcaption>Fig. 11. **Interpolation of Fitted Parameters.** Our parametric model permits to explore a space of Fluorescent material from fits of the Gonzalez and Fairchild's [6] database. In this example we take the fluorescence $\bar{F}$ of HERPIMAG and PHP8HPWH samples, and blend their fitted parameters (strength, mean and variance) with half/half weights. Using the diagonal of HERPIMAG in each three case, we can build new plausible fluorescent materials. By changing the strength (last column), we can alter the appearance of all materials. All images were rendered using a D65 illuminant.</figcaption></figure>

pair and several parameter choices of $(\mu_e, \sigma_e)$. We also vary $\alpha_n$ spatially in Figure 13, but this time $(\mu_e, \sigma_e)$ is constant, and we modulate $\mu_a$ spatially using a fixed small absorption spread $\sigma_a$. Altering the range of variations of the $\mu_a$ parameter then results in different behaviors under different illuminants. Our last example varies the $\mu_e, \sigma_e$ and $\alpha_n$ parameters based on an input color map. More specifically, we have first converted the color map to HSV, then mapped hue to $\mu_e$, saturation to $\sigma_e$ and value to $\alpha_n$. We show in Figure 14 different results obtained with different choices of mapping for $\mu_e$.

# 6 DISCUSSION

We have introduced a new analytical model for fluorescent materials that brings several contributions. It is based on a decomposition that not only eases energy conservation, but also grants accurate reproduction of measured fluorescent materials. We model the normalized fluorescence component with axis-aligned Gaussians, and introduce an analytical bispectral integration over arbitrary sensitivity functions. This makes our approach particularly well suited to non-spectral rendering since fluorescence parameters may then be edited on the fly with real-time feedback. Yet our model is also useful in spectral rendering contexts as it opens new avenues for the editing of spectral reradiation matrices. Observing that a single Gaussian is enough to provide faithful fluorescence appearance, we use it to define a simplified model that enables artist-friendly controls through fluorescence palettes and spatial parameter variations.

*Limitations.* In this work, we have assumed that reflectance and fluorescence are only bound by energy conservation. From a physical standpoint, the two components might be more tightly coupled, and a more accurate model would need to take such coupling into account. However, we are not aware of any development in the Optics domain on that topic. We have also considered Gaussian basis functions of infinite support in our model. Using Gaussians clamped at the border of the visible range would complicate the derivation of Section 3.2 as we would need to consider explicit integration bounds. We did not find this to be necessary, as the fitted Gaussian have minimal energy outside of the visible range by construction.

*Future work.* Even though we have demonstrated our approach on diffuse fluorescent materials, it might be adapted to represent angularly varying fluorescence (e.g., as seen in *Troides Magellanus* butterflies). However, this will require the availability of a database of measured bispectral BSDFs. Our model might be useful in this respect to reduce memory footprint or to regularize measurement noise. We would also like to investigate inverse design approaches on our simplified model, whereby artist provide reradiated colors under various illuminants and model parameters are retrieved. This would in particular allow one to explore metameric configurations that arise between fluorescent materials. Finally, we have focused our approach on fluorescent surfaces, but we believe it could be readily adapted to the modeling and rendering of fluorescent volumes (e.g., as seen in the white fur of polar bears).
<figure><img src="image_13.png" /><figcaption>Fig. 12. **Spatial variations of fluorescence strength.** The leftmost images show the textures used to vary the achromatic albedo $\rho \in [0, 0.2]$ and the fluorescence strength $\alpha_n \in [0, 1]$ respectively. Rendering results are shown in the remaining four images, each under two illuminants (D65 at left, UV at right), with fixed $\mu_a = 420$nm and $\sigma_a = 100$nm, and $(\mu_e, \sigma_e) \in \{(419, 69), (593, 48), (515, 21), (645, 10)\}$ from left to right.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 13. **Spatial variations of fluorescence strength and absorption.** The leftmost images here show the textures used to vary the fluorescence strength $\alpha_n \in [0, 1]$ and $\mu_a$ respectively. In the second image, $\mu_a$ is fixed to 400nm, while in the remaining images, it varies in different intervals: from left to right, $\mu_a \in [400, 440]$, $\mu_a \in [400, 480]$, $\mu_a \in [400, 520]$. The wider the interval, the more pronounced the difference in appearance between the D65 and UV illuminants (left and right halves respectively). All other parameters are kept fixed: $\sigma_a = 33$nm, $\mu_e = 419$nm, $\sigma_e = 69$nm, and $\rho = (0.14, 0.14, 0.2)$.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 14. **Spatial variations of fluorescence color.** We use the color texture in the leftmost image to control the fluorescence parameters that govern color appearance. This is done by first converting the color texture to HSV, and then mapping hue to $\mu_e$, saturation to $\sigma_e$ and value to $\alpha_n$. The absorption parameters are set to $(\mu_a, \sigma_a) = (420, 100)$ to ensure consistent appearance among illuminants, while the albedo is set to $\rho = (0.14, 0.14, 0.2)$. We manually set the mapping in the second image to approximately reproduce the color texture; while in the remaining images, we explore fluorescent color appearance freely.</figcaption></figure>

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