Q Hua

V Tázlar

A Fichet

A Wilkie

Efficient Storage and Importance Sampling for Fluorescent Reflectance

Keywords:

: In this scene, we use a near-UV light for illumination. While non-fluorescent materials only reflect shades of the illumination colour (left), fluorescent surfaces also reemit a portion of the absorbed energy from incident light as a light at additional wavelengths, leading to the colourful appearance of other objects in the scene. To represent fluorescent materials in a renderer, we typically use reradiation matrices (centre), which have a significant memory overhead. Instead, we propose a more efficient representation of these matrices using Gaussian mixture models (right -eight Gaussians). Although compact, this representation also provides compelling results even with such challenging illumination.

Introduction

Traditionally, rendering pipelines mimic the human visual system and rely on RGB or tristimulus values to compute light transport. This approach, however, is insufficient if high colour accuracy or reproduction of spectral effects is required. Resolving the wavelength domain with additional bands, and thus performing spectral rendering, allows handling of such cases at the expense of increased computational demands. However, due to the increase in computation power and algorithmic efficiency [WND * 14], spectral rendering is becoming increasingly popular even in production [PBC * 18]. In this paper, we focus on the reproduction difficulties of one particular spectral effect: fluorescence.

Fluorescence is a phenomenon often found in natural elements and chemical compounds. It has a significant impact on how we perceive an object's colour in terms of hue, saturation, and luminance because it transfers a part of non-visible or near-visible light into a different part of the visible range. Therefore, the inclusion of fluorescence can enhance material brightness, making the colour appear more vivid compared to a pure reflection. While the importance of this effect for appearance modelling in computer graphics has been known for years, it has not been used much in practice, as adding it to graphics workflows -both on the side of user interfaces, as well as within rendering software proper -is a demanding problem.

An issue preventing a more widespread adoption of fluorescence is the significant memory footprint of its representation. To fully describe fluorescent spectra, one typically uses reradiation matrices [Don54], also called Donaldson matrices, representing reemission distributions for a given set of wavelengths. Although explicitly storing these data is manageable for a few fluorescent elements, this does not scale for fine-grained material definitions such as fluorescent textures.

In recent years, there has been an increasing interest in fluorescence in the research community. For example, Jung et al. [JWH * 19] used fluorescence to enlarge the colour gamut during spectral uplifting in a fashion analogous to the use of optical brighteners in real-world materials. However, due to the lack of better alternatives, their method had to store a full reradiation matrix for every texel, which in practice limited its usage to low-resolution input images.

We have proposed a method for efficient storage and importance sampling of reradiation matrices using a Gaussian mixture model in [HFW21]. This preliminary work was evaluated on three measured fluorescent materials. In this revision, we present an extended evaluation using an additional database of measured materials [START_REF] Gonzalez | Evaluation of bispectral spectrophotometry for accurate colorimetry of printing materials[END_REF]. Further, we compare the previously proposed weighted Expectation-Maximisation method with the weighted Bayesian approach of Gaussian mixture fitting. Finally, we provide additional evaluation metrics to assess the fidelity of the fitted reradiation matrices.

Previous work

Bispectral rendering Although not widely used, the concept of fluorescence in computer graphics is not new. Glassner [START_REF] Glassner | A model for fluorescence and phosphorescence[END_REF] in-troduced fluorescence and phosphorescence to the computer graphics community and proposed a method to support those effects in a Whitted ray tracer. Wilkie et al. [START_REF] Wilkie | Combined rendering of polarization and fluorescence effects[END_REF] proposed a path tracing system capable of rendering both fluorescence and polarisation effects, but they relied on a defined number of spectral bands, the wavelength not being part of the Monte Carlo integration. Mojzík et al. [START_REF] Mojzík | Handling Fluorescence in a Uni-directional Spectral Path Tracer[END_REF] made the wavelength domain part of the integrand and adapted Hero Wavelength Spectral Sampling [WND * 14] to their method capable of handling fluorescent media in a Monte Carlo path tracer.

Colour gamut enlargement One of the challenges when using a spectral renderer is the requirement of spectrally defined assets. Assets are commonly defined in a tristimulus space (e.g., RGB, XYZ) and cannot be used directly as the input of a spectral renderer. To allow the usage of RGB assets in a spectral renderer, the original RGB data needs to be 'uplifted' to a spectral form. A wide range of work addresses nonfluorescent spectral uplifting, assuming smooth spectra [Smi99, JH19], or data-driven from measured spectra [OYH18, TWF21]. Recently, Jung et al. [JWH * 19] proposed an uplifting pipeline that additionally uses fluorescent spectra to enhance the colour gamut. A significant drawback of this bispectral uplifting method is its memory requirement. In the worst case, when dealing with texture uplifting, each pixel needs to be stored as a full reradiation matrix. However, this is not an intrinsic drawback of the uplifting method, but an indicator that the representation of fluorescent data in rendering technology required improvement.

Bispectral measurements Bispectral reflectance measurements are not as widespread as BRDF measurements because their acquisition is a lengthy process requiring expensive equipment. Additionally, with the limited number of renderers supporting fluorescence, there is little incentive to take bispectral material measurements for computer graphics applications. However, few exceptions are noteworthy. [START_REF] Gonzalez | Evaluation of bispectral spectrophotometry for accurate colorimetry of printing materials[END_REF] offers a database of reradiation matrices of numerous materials, mainly papers and inks. They used a Labsphere BFC-450 bispectral fluorescence colorimeter for the measurements. We use this database to consolidate our earlier published evaluation on the three reradiation matrices from Labsphere distributed in ART [START_REF]The Advanced Rendering Toolkit[END_REF] measured with the same device. Gaussian representations in rendering Gaussian distributions are often used in rendering practice, as their sufficiently large combination can approximate distributions of any shape. submitted to COMPUTER GRAPHICS Forum (8/2022).

Bispectral material models

Background

In this section, we review the core concepts used in the remainder of this article. We explain the fluorescent effect, its properties, and the typical representation of fluorescent data in a bispectral renderer. We also review the basic notation of Gaussian mixture models, which we employ to represent reradiation matrices.

Fluorescence

Fluorescence is an effect where a fluorophore absorbs energy from incident photons, causing excitation of its molecules which, during their return to the ground state, may reemit part of this energy as photons with lower energy. In contrast to phosphorescence, where emission can transpire over a long time period, fluorescence occurs in a few nanoseconds (10 -9 to 10 -7 s) and, as such, is considered to be instantaneous in the context of computer graphics.

Fluorescence produces noticeable visual effects and is commonly used to increase the brightness and saturation of pigments and dyes. This works by shifting light from the barely visible ultraviolet region to longer wavelengths, where the human eye is more sensitive.

A fluorophore can be characterised by its absorption and reemission spectrum (Fig. 2a). The first defines which incident wavelengths are absorbed and lead to reemission events, with magnitudes representing the total intensity of the reemissions. The second describes the amount of reemission across all incident wavelengths.

According to Kasha's rule, the reemission spectrum shape can be considered independent of the specific incident wavelength that caused the excitation (reemission intensity scales). The difference between the spectral positions of the band maxima of absorption and reemission arising from the same electronic transition is called a Stokes shift [START_REF] Mcnaught | IUPAC. Compendium of Chemical Terminology[END_REF]. In practice, this shift usually corresponds to a difference between the rightmost local absorption maximum and the reemission maximum.

However, there are exceptions to Kasha's rule, leading to the need for reradiation matrices (Fig. 2b), which characterise, for each incident wavelength (λ i ), a distribution of reemission wavelengths (λ i ̸ = λo) and surface reflectance (λ i = λo).

In the context of path tracing, a normalised reemission spectrum (column excluding λ i = λo) can be seen as a PDF of the wavelength shift of an irradiation λ i and a normalised absorption spectrum (row excluding λ i = λo) can be seen as a PDF of absorption of a wavelength λ i for a given outgoing radiation λo.

To preserve energy conservation, the sum of each column of the reradiation matrix never exceeds 1. The resulting value corresponds to the total albedo of the material at a given incident wavelength, that is, the sum of surface reflectance and fluorescence reemission. But, while reradiation matrix is more flexible and suited for computer graphics applications, it has a significant memory footprint.

Gaussian Mixture Model

A Gaussian mixture model (GMM) is a parametric probabilistic model representing a dataset as a linear superposition of Gaussian distributions. This linear superposition is represented by:

p(x) = K ∑ k=1 π k N (x | µ k , Σ k ) (1)
where:

p(x)
is the density of mixture of Gaussians,

N (x | µ k , Σ k ) is a single Gaussian density, π k
is the mixing coefficient (weight) of each Gaussian, K is the number of Gaussians.

In this article, we model reradiation matrices as mixtures of 2dimensional multivariate Gaussian distributions:

N (x | µ, Σ) = 1 2π |Σ| e -1 2 (x-µ) T Σ -1 (x-µ) (2)
where: µ is the mean, Σ is the covariance matrix.

Method overview

Our technique consists of two related but distinct operations that form an efficient fluorescence rendering pipeline (Fig. 3):

1. Fitting, where the reradiation matrices are fitted to GMMs (5). 2. Rendering, where the GMM representation is used to importance sample wavelength-shifting events (6). We model a reradiation with a GMM in the fitting phase, and then during the rendering phase, we importance sample wavelength shifting events according to the GMM.

The pipeline starts with a fitting preprocessing step. Here, the reradiation matrices representing fluorophores are filtered to remove invalid values and fitted to GMMs. During the rendering phase, we use these GMM representations for fluorescent wavelength-shifting events, where a conditional GMM is importance sampled. Depending on the path direction, we either sample based on absorption or reemission wavelengths.

Using a GMM to represent a bispectral reflectance drastically reduces its memory footprint in the renderer, as instead of relying on a tabulated reradiation matrix, we only need to store a few parameters of the mixture. This reduction is even more pronounced when the rendering technique relies on random sampling, as additional tabulated CDFs are necessary for efficient sampling of reradiation matrices.

Fitting phase

In the fitting phase, we apply parametric density estimation to fit a reradiation matrix into a GMM. Later, we show that we can accurately reconstruct the reradiation matrix information from the GMM on the fly.

Dataset filtering

Some measurements in the dataset include a significant amount of noise, potentially lowering the accuracy of the fit. To mitigate this problem, we filtered the reradiation matrices before the fitting phase.

Although not ideal, we zeroed values under the diagonal (λ i > λo) and negative values above it, since these values in our dataset correspond to measurement errors. We do not apply any further data filtering during the fitting phase. A further improvement would be possible with a proper characterisation of the acquisition device's noise ratio.

Fitting algorithms

In our previous publication, we used the implementation of weighted Expectation-Maximisation (EM) provided by the Pomegranate library [Sch]. In this revision, we instead use weighted EM in the better-behaving scikit-learn [PVG * 11, sci21] library and compare its fitting accuracy with the weighted Bayesian fitting algorithm.

Weighted Expectation-Maximisation Expectation Maximisation (EM) is a two-stage iterative algorithm suitable for the finding of mixture model parameters. The E-step (expectation) performs an estimation of the expected log-likelihood for a complete dataset. The M-step (maximisation) maximises the expected complete loglikelihood. The E-step and M-step are run iteratively until the expectation converges to the target. As a detailed introduction of EM is beyond the scope of our work, we review only a sketch of EM. For a detailed study on EM, please refer to [START_REF] Bishop | Pattern Recognition and Machine Learning (In-formation Science and Statistics)[END_REF]. Vorba et al. [VKŠ * 14] also showcase an adaptation of EM to the rendering context.

The EM algorithm works with a set of observation samples and computes a mixture matching its distribution. But, the reradiation matrix instead provides weights of samples of pairs (λ i , λo). We could use a large set of samples distributed according to these weights, but then the accuracy of the representation would depend on the size of the sample set, whereas bigger sample sets would lead to higher memory consumption and a slower fitting process.

A more suitable approach is to use the weighted EM algorithm [GAPFH16, VKŠ * 14] where all non-zero samples are used as observation outcomes and weighted according to their respective values.

Variational Bayesian inference Variational inference follows the principle of the EM algorithm. It is a two-stage iterative algorithm that maximizes a posterior probability (MAP) instead of maximising local likelihood (ML) as in the EM. Apart from updating the approximations in EM, the model evidence in variational inference also includes a predefined lowest prior distribution. By maximising their posteriors using Bayesian inference in each iteration, the variational inference model gives more stable results compared to the EM, which can produce singularities and exhibit overfitting [START_REF] Bishop | Pattern Recognition and Machine Learning (In-formation Science and Statistics)[END_REF]. Implementation of weighted variational Bayesian inference in our pipeline directly follows the derivation of weighted EM [HFW21].

Optimal number of Gaussians Choosing an optimal number of Gaussians for the EM is a non-trivial problem, as with a higher number of Gaussians, we may face overfitting issues. Bayesian information criterion (BIC) can mitigate overfitting by the introduction of a penalty for the number of parameters of the model.

Variational Bayesian inference can take advantage of the predefined prior distribution, such as the Dirichlet process (used in the scikit-learn implementation as default) to select the optimal number of clusters based on the fitted data.

We evaluate our fitting on different numbers of Gaussians for EM as the selection may depend on the use case. Conversely, we let Bayesian inference use a lower number of Gaussians than the specified maximum if it opts to do so for improved accuracy of the fit.

submitted to COMPUTER GRAPHICS Forum (8/2022).

Reradiation reconstruction

We can reconstruct the reradiation matrix from the GMM found previously. Gaussian mixture gives a probability density function, and so its integral is unity. Because the reradiation integral does not hold this property, the resulting Gaussian mixture has to be scaled to reconstruct the reradiation later on:

Φ (λ i , λo) = S • p (λ i , λo) (3)
where:

S is the scaling factor, Φ (λ i , λo) is the reradiation function, p (λ i , λo)
is the probability density function of the GMM defined in Eqn. 1, λ i , λo are the wavelengths of a given absorption and reemission event.

There are different strategies to retrieve the scaling factor S. We originally proposed two approaches:

Error minimisation The first strategy consists of a minimisation process. S is set as a parameter to minimise the error between the measured reradiation and the reconstructed mixture.

S = arg min S ∑ Λi ∑ Λo ∥Φ measured (λ i , λo) -S • p (λ i , λo) ∥ (4)
The choice of a norm function has little influence on the result when dealing with fluorescence -the data have a low dynamic range. By definition, this strategy reduces the average reconstruction error.

Integral ratio

The second strategy consists of defining S as the ratio between the integral of the original dataset and the GMM.

S = Λ Φ measured (λ i , λo)dλ i dλo Λ p(λ i , λo)dλ i dλo (5)
This method ensures that the reconstructed reradiation has the same albedo as the original measured one. Although this technique does not reduce the measurable reconstruction error, its application provides better rendering results because our eyes are more sensitive to brightness variation than to a slight chromaticity shift.

Strategy selection

The strategy for scaling factor computation depends on the target application. The integral ratio is better suited for perceptual applications, such as rendering with tone mapping. If the accuracy of the spectral reconstruction of the reradiation is needed, then error minimisation is a better choice.

In this publication, we apply integral ratio scaling as it is more closely related to the intended application of our method.

Rendering phase

When a fluorescent event occurs during the rendering phase, we first need to sample an in-shifting or an out-shifting wavelength depending on the ray direction. Then, we need to evaluate the attenuation given a set of λ i and λo wavelengths. The sampling phase relies on conditional probability: either λ i or λo is known, and we have to sample a λo or a λ i according to the direction of the ray, that is, evaluate an in-or out-shifting event.

Conditional GMM

To sample a random wavelength shift after a fluorescent event, we need to 'sample a slice' on the multivariate Gaussian mixture. With a fixed input wavelength λ i , we construct the conditional parameters for the resulting Gaussian mixture and generate the random variable as the output wavelength λo.

Our model is made of:

µ k = µ ka µ k b , Σ k = σ kaa σ k ab σ k ba σ k bb (6)
where µ k and Σ k are the GMM parameters of the k th Gaussian.

We define T, the inversion of the k th covariance matrix as:

T k = Σ -1 k = τ kaa τ k ab τ k ba τ k bb . (7)
The conditional probability that, given λ i , we shift to λo in component k is:

p k (λ i → λo) = N λo | µ k b|a , τ -1 k bb (8) with : µ k b|a = µ k b -τ -1 k bb τ k ba λ i -µ ka . (9)
Then, the probability for the entire conditional GMM from a given λ i to λo is:

p(λ i → λo) = K ∑ k=1 π (λi) k p k (λ i → λo) (10)
with : π

(λi) k = π k N λo | µ ka , σ kaa ∑ k N λo | µ ka , σ kaa . (11)
Finally, we generate the random variable based on the new conditional GMM model. First, we choose which Gaussian component we will sample using the conditional mixing coefficients π (λi) k , and then we generate the sample following the normal distribution of the chosen k th Gaussian:

µ (λi) k = µ k b|a , σ (λi) k = τ -1 k bb . (12)
We only discuss sampling on the light path, but the same approach can be symmetrically applied on an eye path.

Importance sampling

When a ray hits a fluorophore, two events can occur:

• the ray is reflected with the same wavelength as the incident ray -no fluorescent event, submitted to COMPUTER GRAPHICS Forum (8/2022). When a fluorescent event occurs, the outgoing wavelength λo is sampled according to this conditional PDF. Notice the presence of measurement noise lowering the highest values of the PDF derived from the raw reradiation matrix after normalisation compared to the PDF derived from the fitted GMM.

• the ray is absorbed and its energy reradiated to a different wavelength -fluorescent event.

We first determine if a fluorescent event occurs based on the ratio between reflectance and fluorescence for a given incident wavelength. If a fluorescent event occurs, we use importance sampling to select a Gaussian distribution from the mixture and a specific wavelength from the distribution (Alg. 1). Fig. 4 shows an example of a conditional PDF of outgoing wavelengths of a specific incident wavelength λ i . Given an incident wavelength, we can generate this distribution from GMM and use it to importance sample the reemitted wavelength λo.

The probability of transition between absorption at wavelength λ i and reemission at wavelength λo is:

p shift (λ i → λo) =    Φ(λi,λi) rt (λi) if λ i = λo 1 - Φ(λi,λi) rt (λi) • ∑ K k=1 π (λi) k N (λo | µ (λi) k , σ (λi) k rt (λ i ) = Λ Φ(λ i , λr)dλr (13)
where:

Φ(λ i , λo) is the bispectral reflectance, rt (λ i )
is the total reflected and reemitted energy given λ i .

In our renderer, we use Hero Wavelength Spectral Sampling (HWSS) [WND * 14]. The HWSS technique multiplexes multiple wavelengths in a Monte Carlo (MC) sample. It differs from a simple multiplexing approach by using a 'Hero' component, which is used to make all directional decisions. This technique drastically reduces colour noise in contexts with a wavelength dependence on the directional decisions, such as in a participating medium. In the case of fluorescence, we allow each of the wavelengths in the HWSS vector to be independently importance sampled. If done directly, we can find non-zero samples with a zero probability, which is incorrect for a MC integrator. Mojzík et al. [START_REF] Mojzík | Handling Fluorescence in a Uni-directional Spectral Path Tracer[END_REF] introduced a balance term to avoid this artefact. We use the same term in our implementation:

Algorithm 1: Random wavelength shifting. Given a λ i , we either have no fluorescent event, i.e. λ i = λo or, we sample a λo after a fluorescent event.

input : λ i output: λo and pdf := p shift (λ i → λo)

ξ 1 ← U (0, 1); p diag ← Φ(λi,λi) rt (λi) ; if ξ 1 ≤ p diag then //
← N µ (λi) k , σ (λi) k ; while λo ≤ λ i ; // Eqn. 13 pdf ← (1 -p diag ) • ∑ K k=1 π (λi) k N (λo | µ (λi) k , σ (λi) k ; end p j (λ i j → λo j ) = N ∏ k=1,k̸ = j p shift (λ i k → λo k ) (14)
where:

p j (λ i j → λo j )
is the probability of j th HWSS sample, p shift (λ i → λo) is the probability of shifting from λ i to λo, N is the size of the HWSS vector.

Implementation details

To evaluate the reradiation, we store the mean, covariance matrix, and weight for each Gaussian of the mixture. We also need to store the scaling factor S. The determinant and inverse of the covariance matrix are precomputed to speed up the evaluation of the mixture during rendering. The diagonal is stored in its original tabulated form without any alteration.

To efficiently perform conditional wavelength shifting, we also precompute two additional tabulated values to retrieve π Although additional precomputed values increase the total memory footprint during rendering, it is still much smaller than the memory required when using tabulated CDFs, which are necessary for the use of reradiation matrices.

We use rejection sampling to guarantee that absorption wavelengths do not exceed reemission wavelengths. It proved sufficient in our case, as we observed only a small number of rejected samples. Alternatively, truncated distributions could be used.

The implementation of our method is available on Gitlab.

Results and discussion

Our fluorophore representation consists of the original reradiation matrix diagonal representing the reflectance spectrum and the scaling factor S with the GMM representing the fluorescent effect. We need to store seven values for each Gaussian of the mixture:

• a mixing coefficient π (1 element),

• a mean vector µ (2 elements),

• a covariance matrix Σ (2×2 elements).

Dataset

In our previous publication [HFW21], we evaluated our method on three measured reradiation matrices distributed in the assets of the ART rendering toolkit -originally provided by Labsphere Inc. In this revision, we provide a more in-depth evaluation on a much larger bispectral dataset. We additionally use the dataset from RIT [START_REF] Gonzalez | Evaluation of bispectral spectrophotometry for accurate colorimetry of printing materials[END_REF], which contains 144 measured reradiation matrices of different materials.

All measurements in the resulting dataset come from the same device: Labsphere BFC-450. The materials were sampled at 10 nm precision with an absorption wavelength ranging from 300 to 780 nm and a reemission ranging from 380 to 780 nm, resulting in 49 × 41 reradiation matrices.

One material measurement (IXCAXORA) was discarded from our evaluations. Its reradiation matrix appears to be incorrect due to human error. We still include this sample in the supplemental for reference.

In this section, we provide the results of the evaluation on the whole dataset but directly showcase only its small subset. Detailed results of our method on all samples are available in the supplemental.

Illuminants

To evaluate the accuracy of our representation, we compare the results of our fitted model with the original tabulated data. We base our accuracy evaluation on the CIE 2000 Delta E (∆E * 00 ) [Com01] between a rendering of the reference data and the corresponding fits observed under a large set of illuminants.

We use two sets of illuminants and distinguish the results of each group:

• Monochromatic illuminants: We evaluate the bispectral reflectance of each fluorophore under monochromatic illumination from 300 to 780 nm to cover the full range of incident wavelengths in the dataset. The performance of our model is calculated as an average ∆E * 00 of all monochromatic illuminants. • CIE standard illuminants: We also evaluate the colour reproduction accuracy of our model under the CIE standard illuminants, which cover often used illumination conditions in rendering practice.

Any measured differences between our model and pure reradiation matrices are caused by our compact representation of the fluorescence, as the reflectance spectrum is not modified.

Rendering and colour reproduction

While a Monte Carlo rendering provides additional information about interreflection between different fluorescent materials and self-interreflection, the choice of the scene and its setup also limit the evaluation to particular configurations. And so, given the large number of material measurements, we instead predominantly use an analytical rendering method. Specifically, we evaluate the accuracy of the fitting on an analytically calculated interaction of light with a fluorescent material represented as a reradiation matrix:

C = Λo Λi φ(λ i , λo)L i (λ i )dλ i C(λo)dλo ( 15
)
where: C is the computed outgoing colour, λ i , λo are the incident and outgoing wavelengths, φ(λ i , λo) is the reradiation matrix value for λ i , λo,

L i (λ i ) is the illuminant radiance at λ i , C(λo)
is the colour matching function at λo.

This equation is equivalent to a single light bounce configuration on a Lambertian surface where both incident and outgoing directions are aligned with the surface normal. This configuration allows us to assess the quality of the fits without any additional parameters interfering with our evaluations.

The reradiation matrix corresponding to our fit can be constructed by evaluating the GMM at regular intervals for λ i , λo. This approach allows efficient generation of noise-free results for each material in the dataset for a large variety of illumination types.

Rendering accuracy

Due to the fluorescent nature of our dataset, illumination conditions have a significant impact on error, making an objective evaluation difficult. Therefore, we evaluate our representation of fluorophores under a wide variety of illuminants visualised as:

• Colour band: To visualise outgoing colour after interaction of a monochromatic illumination with a fluorophore, we use a 'colour band'. It consists of three strips representing from top to bottom: outgoing colour based on a reradiation matrix, outgoing colour based on our fitted model, and visualisation of ∆E * 00 between these two. The colour bands cover the entire excitation spectrum of the fluorophore (horizontal axis) and allow wavelength-specific comparisons of the measured bispectral reflectance with our fit. • Colour patches: To compare the quality of our fit with the original measured data, we use colour patches to visualise how well a fluorescent material is represented under a specific CIE standard illuminant. The fitted patch is embedded inside of a reference patch. The colour values are divided by the Y value of the used illuminant.

Fig. 5 shows the five worst fluorophore fits based on their average ∆E * 00 for monochromatic illuminants at 4 Gaussians using EM. While the colour difference is visible for some monochromatic illuminants, the difference is hardly noticeable for most CIE standard illuminants, especially with a higher number of Gaussians. While 4 Gaussians would be suitable for most rendering purposes, 8 Gaussians already provide high accuracy within a small memory footprint. This also applies to global illumination, as showcased in Fig. 1. This is supported by a comparison of the quality of the GMM fit with respect to the number of Gaussians used (Fig. 6). Even though the reconstruction of the original reradiation at 4 and 2 Gaussians is very rough, it does not translate to big observable differences as our visual system is much more sensitive to luminance discrepancies than to chromaticity variations.

Average accuracy on the whole dataset

To compare the overall accuracy of the two fitting methods with a varying number of Gaussians, we provide a ∆E * 00 averaged across the entire dataset.

Fitting methods comparison Figs. 7b & 8b show a consistent decrease of the ∆E * 00 with an increasing number of Gaussians. Bayesian inference performs better with a lower number of Gaussians, but is outperformed by EM from 8 Gaussians. This is caused by the Bayesian inference implementation opting to use a lower number of Gaussians for fitting than the allowed maximum.

Model accuracy Fig. 7a shows the average ∆E * 00 over the entire dataset for monochromatic illuminants ranging from 300 to 780 nm. The significant error observed in the lower parts of the spectrum can be attributed to limitations of the ∆E * 00 involving colours with low luminance. In our case, this part of the spectrum usually represents non-visible light or a negligible amount of reradiation, as absorption bands of fluorophores are often situated there. Therefore, the practical influence of this error on real-world rendering applications is limited. The monochromatic illumination error is visualised for a few selected fluorophores in Fig. 5 and the whole dataset in the supplemental.

In Fig. 8a, we provide an evaluation of a more practical application of our approach -rendering with CIE standard illuminants. We show an average of the whole dataset for the ten worst performing illuminants while fitting 4 Gaussians with EM. On average, we achieve a ∆E * 00 lower than 1.1 for two, 0.6 for 4 and 0.3 for 8 Gaussians.

Memory footprint

While retaining high accuracy, our method is memory efficient, requiring only a fraction of the memory necessary for use of the raw reradiation matrices.

In the most favourable case, we only have to store the upper triangular part of the reradiation matrix due to the nature of fluorescent reradiation being of lower energy than excitation caused by the incident illumination. For our dataset measured at 10 nm precision, it is approximately 1150 values. If maximal precision was required, each matrix could be interpolated to 1 nm precision, resulting in a hundredfold increase in memory footprint. Additionally, to facilitate efficient importance sampling of wavelength-shifting events, it is desirable to compute tabulated CDFs, effectively tripling the total required memory when using reradiation matrices for rendering.

With our method, we only need a small set of GMM parameters to both evaluate the reradiation on the fly and to importance sample wavelength shifting events.

In practice, for a representation using N Gaussians, we need to store 1 + 7 • N values. For importance sampling, we need two additional 1D arrays to efficiently evaluate the conditional GMM. This is much less than the two tabulated 2D CDFs required for a fluorescent material when using a reradiation matrix. Additionally, if desired, our method can be used solely for importance sampling instead of tabulated CDFs.

The memory efficiency of our method becomes especially pronounced when working with larger numbers of reradiation matrices. It makes fluorescent textures where each pixel may have to be represented by a reradiation matrix usable in rendering practice. This, in turn, facilitates the use of related techniques such as enlargement of the colour gamut by uplifting of RGB textures [JWH * 19].

Conclusion and future work

We presented a method for efficient storage of reradiation matrices via GMM. We showed that this approach drastically lowers required memory for the rendering of fluorescence while retaining colour reproduction accuracy matching that of the original data. Further, the resulting GMM representation can be directly used in Monte Carlo-based rendering pipelines, as it supports efficient importance sampling.

We evaluated weighted EM and Bayesian inference fitting approaches on an extensive dataset of measured reradiation matrices under all CIE standard illuminants and a wide range of monochromatic illuminants. Both fitting methods proved to provide a comparable and highly accurate representation of fluorescent reradiation with as little as 8 Gaussians for most practical rendering use cases.

Different models and fitting approaches could be explored, but considering the rendering accuracy and small memory footprint of the presented solution, the potential gains seem marginal.

We think that the main direction of future work is in the development of a parametric fluorescence model. Such model would not require tabulated CDFs -further lowering total memory requirements for efficient rendering of fluorophores. More importantly, it would allow a smooth integration of editable fluorescent materials into artistic workflows.

  Wilkie et al. [WWLP06] proposed one of the early bispectral models adapted to current rendering techniques. This model uses a layered BRDF with a diffuse fluorescent component. Hullin et al. [HHA * 10] proposed an efficient acquisition setup to capture bispectral bidirectional reflectance and reradiation distribution functions (BRRDF). They guided their acquisition by a Principal Component Analysis to lower the acquisition time. Based on this work, Jung et al. [JHMD18] derived a new BRRDF. They exploited Kasha's rule to represent their distribution with three 1D distributions (absorption and emission spectra and non-fluorescent reflectance) and two ratios (emitted to absorbed energy and reradiation to non-fluorescent reflectance).

  Fluorophore characterised by its absorption and reemission spectrum. The offset between the two spectra is called the Stokes shift. Fluorophore characterised by its reradiation matrix. The diagonal (λ i = λo) was removed to emphasise the fluorescent effect.

Figure 2 :

 2 Figure 2: (a) If Kasha's rule holds, the shape of a reemission spectrum is the same regardless of the incident excitation wavelength, and absorption with reemission spectrum form a sufficient representation of a fluorophore. (b) Otherwise, a reradiation matrix providing precise information about the reemission spectrum given an incident wavelength has to be used at the cost of its significant memory footprint.

Figure 3 :

 3 Figure3: Fluorescence rendering pipeline employing our technique. We model a reradiation with a GMM in the fitting phase, and then during the rendering phase, we importance sample wavelength shifting events according to the GMM.

Figure 4 :

 4 Figure 4: Comparison of conditional PDFs derived from a reradiation matrix and its corresponding GMM representation. The figure shows the expected distribution of the reemission wavelengths λo when the HERPICHA material is illuminated at λ i = 440 nm.When a fluorescent event occurs, the outgoing wavelength λo is sampled according to this conditional PDF. Notice the presence of measurement noise lowering the highest values of the PDF derived from the raw reradiation matrix after normalisation compared to the PDF derived from the fitted GMM.

  k : the sum of rows and the sum of columns of the Gaussian mixture. In our implementation, we use a 1 nm sampling rate. submitted to COMPUTER GRAPHICS Forum (8/2022).

FL10∆EFigure 5 :

 5 Figure 5: Comparison of analytical renderings based on our fitted model with EM and reradiation matrices. Five worst behaving fluorophores on the fitting of 4 Gaussians based on the average ∆E * 00 over the set of monochromatic illuminants from 300 to 780 nm are showcased. The colour bands show the bispectral reflectance of the reradiation matrix and our corresponding fit under monochromatic illumination along with ∆E * 00 for each wavelength. The colour patches show bispectral reflectance of the fitted model embedded inside of a patch rendered with a reradiation matrix. The worst performing (average ∆E * 00 ) CIE standard illuminants on the whole dataset are displayed.submitted to COMPUTER GRAPHICS Forum (8/2022).

Figure 6 :

 6 Figure 6: Worst performing fluorophores' reradiation matrices with absorption and reradiation spectra and corresponding fits of EM.

  Distribution of ∆E * 00 over the whole dataset over all monochromatic illuminants for different number of Gaussians.

Figure 7 :

 7 Figure7: Evaluation of the fitting accuracy on monochromatic illuminants. In the box plot. the box extends from first to third quartile, the lines show the median and the triangles show the average value. The whiskers extend from quartile to 1.5 inter-quartile range.

  Distribution of ∆E * 00 over the whole dataset over all CIE standard illuminants for different number of Gaussians.

Figure 8 :

 8 Figure8: Evaluation of the fitting accuracy on CIE standard illuminants. The box extends from first to third quartile, the lines show the median and the triangles show the average value. The whiskers extend from quartile to 1.5 inter-quartile range.

submitted to COMPUTER GRAPHICS Forum (8/2022).

Acknowledgements

We thank the reviewers for their extensive feedback. This work has received funding from Czech Science Foundation grant 19-07626S, the EU H2020-MSCA-ITN-2020 grant number 956585 (PRIME), and the Charles University grants number SVV-260588 and GA UK number 240422. The teaser uses the Serapis bust model from [McG17].


A Kraemer-Schafhalter

S Domenek

H Boehling

S Feichtenhofer

H Griengl

H Voss

Biohydroxylation of 2-cyclopentyl-1,3-benzoxazole with the ®lamentous fungus Cunninghamella blakesleeana DSMZ 1906 was studied in a 15-l stirred tank reactor. The aim of the work was to avoid substrate limitation through sub-optimal mixing by formation of pellets with a uniform pellet size distribution of 250±500 lm, obtained at an inoculum concentration of 10 7 spores ml )1 and an agitation rate of 390 rpm. Due to the high toxicity of the educt, 2-cyclopentyl-1,3-benzoxazole, on the fungus, the medium composition, the time of educt addition, and the educt starting concentration were optimized to reach high educt tolerance and hydroxylation activity. A good maintenance of biotransformation capacity was obtained without excessive loss of activity of the biocatalyst by addition of 30 mg 2-cyclopentyl-1,3-benzoxazole/g biomass (cell dry mass) during the stationary phase in a medium which was optimized in batch fermentations with experimental designs. An increase in product yield and quality (enantiomeric excess) was achieved by developing feeding strategies combining the educt and medium components. The resulting fermentation broth contained 450 mg l )1 of the product (1S,3S)-3-(benz-1,3-oxazol-2-yl)cyclopentan-1-ol with an enantiomeric excess of 95%, which represents a 48% increase over former reported results.

Introduction

Biohydroxylation represents a powerful method for the regioselective and stereoselective introduction of hydroxyl groups into organic compounds (Holland 1992). As such, it can be viewed as an intermediate step for any other subsequent functionalization. In this kind of biocatalytic process, ®lamentous fungi, especially the Zygomycete Cunninghamella blakesleeana DSMZ 1906, have been widely used [START_REF] Kreiner | Stereo-speci®c biohydroxylations of protected carboxylic acids with Cunninghamella blakesleeana[END_REF][START_REF] Zhang | Biotransformation of chlorpromazine and methdilazine by Cunninghamella elegans[END_REF][START_REF] Pothuluri | Metabolism of metachlor by the fungus Cunninghamella elegans[END_REF].

A method often applied for the synthesis of pharmaceuticals is the microbial oxidation of 2-cyclopentyl-1,3-benzoxazole (CPB) into 3-(benz-1,3-oxazol-2-yl)cyclopentan-1-ol (Fig. 1), a model reaction for cycloalkanecarboxylic acids. Its main advantages are a good educt availability, known reaction steps and established analytics of the main products (deRaadt et al. 1996a(deRaadt et al. , 1996b(deRaadt et al. , 1996c)). [START_REF] Kreiner | Stereo-speci®c biohydroxylations of protected carboxylic acids with Cunninghamella blakesleeana[END_REF] have performed the initial screening of the regiospeci®c and ste-reospeci®c hydroxylation activity of the fungus. Subsequently, the main product formation, (1S,3S)-3-(benz-1,3-oxazol-2-yl)cyclopentan-1-ol, by C. blakesleeana was optimized to gain high enantiomeric excess (e.e.). It was then important to control the special morphological features of the fungus in order to attain reproducible high quantities of the desired diastereomer. Furthermore, the environmental conditions need to be optimized in order to resolve the problem of educt and metabolite intolerance and to obtain a biocatalyst with high activity and stability.

C. blakesleeana grows either in ®lamentous or in pellet form. A uniform pellet size is preferred, since it provides reproducible transformation conditions in the stirred batch reactor. The optimal size of the pellets is less than 400 lm diameter, so that diusion limitations are avoided within the pellets [START_REF] Kobayashi | Oxygen transfer into mycelial pellets[END_REF][START_REF] Wittler | Investigations of oxygen transfer in Penicillium chrysogenum pellets by microprobe measurements[END_REF]). However, the pellet size may be in-¯uenced by a number of factors. Among these are the inoculum size, the growth medium, and environmental conditions, i.e. rates of agitation and aeration [START_REF] Whitaker | Fungal pelleting[END_REF][START_REF] Braun | Mycelial morphology and metabolite production[END_REF][START_REF] Znidarsic | Studies of a pelleted growth form of Rhizopus nigricans as a biocatalyst for progesterone 11 alpha hydroxylation[END_REF].

The aim of the present work is the development of a new feeding strategy, which includes a combination of nutrient feed and toxic educt addition. This should provide a stable and highly active biocatalyst enabling a high ®nal product concentration with good yield and high e.e.. A range of starting concentration levels of CPB and dierent timings of CPB addition are therefore investigated.

Materials and methods

Strain and culture conditions

C. blakesleeana DSMZ 1906 was obtained from the Deutsche Sammlung von Mikroorganismen und Zellkulturen GmbH (DSMZ), Braunschweig, Germany. Stock cultures were grown at room temperature in the dark on potato/dextrose (Biolife) agar slants and subcultured every four weeks. Each shake ¯ask or fermentation experiment was inoculated under sterile conditions with a spore suspension (®nal concentration approx. 10 7 spores l )1 ) prepared from agar-slant cultures grown for three weeks.

Medium composition and fermentation procedure

Medium

The fermentation medium, which was developed from shake ¯ask experiments, consisted of glucose 10 g l )1 ; peptone M66 5 g l )1 ; malt extract 0.3 g l )1 ; NH 4 Cl 2.9 g l )1 ; Na 2 HPO 4 2.4 g l )1 ; KH 2 PO 4 2.4 g l )1 ; MgSO 4 á 7H 2 O 1.3 g l )1 ; CaCl 2 á 2H 2 O 0.1 g l )1 ; iron(III) citrate hydrate 0.04 g l )1 ; ZnSO 4 á 7H 2 O 0.02 g l )1 ; CuSO 4 á 5H 2 O 0.03 g l )1 . All chemicals were obtained from Merck (p.a. quality). Media were prepared using distilled water and sterilized in an autoclave.

Experimental designs were performed in two parallel runs in a rotary shaker (120 rpm, 30 °C) in 300-ml baed shake ¯asks containing 100 ml medium. The factorial screening design (2^4) employed two levels with one center point. Data analysis was performed with Statgraphics 3.0 plus. Tested complex nitrogen sources in the ®rst design were yeast extract, malt extract, peptone M66 and meat extract at 0, 2.5 g l )1 and 5 g l )1 . The inorganic component calcium (0, 0.125 g l )1 and 0.25 g l )1 concentration levels) and the trace elements manganese, copper and molybdenum (0, 0.02 g l )1 and 0.04 g l )1 ) were tested in the second experimental design.

Further investigation of the medium components peptone M66, malt extract, NH 4 Cl, MgSO 4 á 7H 2 O, iron(III) citrate hydrate and ZnSO 4 á 7H 2 O was done with the ``genetic algorithm'' (12 individuals, four generations, selection method ``keep the best and roulette on the rest''), using Galop software from FZ JuÈ lich [START_REF] Beste | NaÈ hrmedienoptimierung mit dem Programm Galop[END_REF][START_REF] Karutz | Optimierung der Spurenelement-Konzentration bei der fermentativen L-Isoleucin-Produktion ± Anwendung Genetischer Algorithmen[END_REF]).

Bioreactor and fermentation procedure

Experiments were performed in a 15-l stirred tank bioreactor (Biostat E, Braun, Melsungen, Germany) with 10 l working volume. Temperature was monitored and held at 30 0.1 °C. pH was measured with a glass electrode (Ingold, Urdorf, Switzerland) and controlled at 6.0 0.1 through addition of either 15% (mass/ vol.) NaOH or 15% (mass/vol.) H 3 PO 4 . The aeration rate was kept at 1 vvm during the growth phase and at 0.5 vvm during the production phase by adjusting the gas-¯ow rate. The agitation rate was kept constant during the fermentation. The in¯uence of the agitation rate was tested at 160, 390, 550 and 730 rpm (0.61, 1.38, 2.10 and 2.79 ms )1 stirrer tip speed, respectively). CPB was added aseptically giving a starting concentration of 150, 350 or 1000 mg l )1 in the fermentation broth. CPB was provided by the Institute of Organic Chemistry, TU Graz (de Raadt et al. 1996b).

Analytical methods

Biomass was determined as cell dry mass (CDW) by vacuum ®ltration using a cellulose acetate ®lter (0.45 lm pore size) and drying with a moisture analyzer (SATORIUS MA30) at 105 °C until constant mass.

The pellet size distribution was estimated with Retsch sieves. After sieving the pellet suspension for 20 min and simultaneously gently washing with water, the masses of the dierent fractions were determined as CDW. Morphological observation was done with an Olympus phase-contrast microscope BSH2.

Glucose and organic acids were determined from the supernatant of the fermentation broth (centrifuged at 14800 g, 10 min, 4 °C) by HPLC (Merck IR detector L-7490 and ultraviolet detector L-7400, pump L-7100, column HPX-87H, eluent 0.005 M H 2 SO 4 , ¯ow rate 0.6 ml min )1 , 60 °C, ultraviolet detection at 210 nm). Concentrations of CPB and its metabolites were monitored by gas chromatography with injector and detector temperatures of 250 °C and 280 °C, respectively (Hewlett Packard HP 5890 series II, autosampler HP 6890 Series Injector, FI-detector and HP 5 column). The oven temperature was held at 200 °C for 3 min, then heated at a rate of 10 °C min )1 to 250 °C and held for 5 min. Samples were prepared as described by [START_REF] Kreiner | Stereo-speci®c biohydroxylations of protected carboxylic acids with Cunninghamella blakesleeana[END_REF].

-(benz-1,3-oxazol-2-yl)-cyclopentan-1-ol (product), 2-(benz-1,3-ox- azol-2-yl)-cyclopentan-1-ol (by-product 1) and 3-(benz-1,3-oxazol-2- yl)-cyclopentan-1-one (by-product 2)
e.e., which was de®ned as the excess of the major enantiomer in relation to both enantiomers, was determined by chiral HPLC using the equipment and methods described by deRaadt et al. (1996c).

Results

Media optimization

The investigated medium components, which were selected from the literature [START_REF] Sedlaczek | Transformation of steroids by fungal spores[END_REF][START_REF] Venkateswerlu | Morphological, ultrastructural, and chemical changes induced in Cunninghamella blakesleeana by copper and cobalt[END_REF][START_REF] Hartman | Microbial biotransformation of retinoic acid by Cunninghamella echinulata and Cunninghamella blakesleeana[END_REF][START_REF] Kreiner | Stereo-speci®c biohydroxylations of protected carboxylic acids with Cunninghamella blakesleeana[END_REF], were tested in shake ¯ask experiments in two screening designs. An average value of 4 2 g l )1 CDW was achieved in the shake ¯asks containing a combination of dierent nitrogen sources, in sum 10 g l )1 . The combination of peptone M66 (5 g l )1 ) and malt extract (5 g l )1 ) had a positive in¯uence on the biomass concentration after 48 h (7.3 g l )1 ). This combination of complex nitrogen sources was used in the second design. None of the tested inorganic media components had a signi®cant in¯uence on the biomass production with an average of the ®nal CDW of 7.5 0.8 g l )1 after 48 h.

In subsequent shake ¯ask experiments, which were planned using a genetic algorithm with the improvement of the CPB conversion rate as the target function, the resulting yield of the main product (60%) after 20 h transformation remained constant, although the concentration of malt extract was diminished from 5 g l )1 to 0.3 g l )1 .

In these experiments, the obtained pellets were hollow, with a diameter of 0.5±7 mm; the ®nal pH of the broth, which was not controlled, varied over pH 3±7.5.

Pellet formation and in¯uence of the agitation rate

Three hours after inoculation of the bioreactor, the unstructured spores started swelling and branching in the culture medium. The growth of a densely branched mycelium with a large hyphal diameter was accompanied by NaOH consumption for pH adjustment after 7 h. In contrast to the results with other fungi [START_REF] Tough | Experimental veri®cation of a mathematical model for pelleted growth of Streptomyces coelicolor A3(2) in submerged batch culture[END_REF][START_REF] Pothuluri | Metabolism of metachlor by the fungus Cunninghamella elegans[END_REF]), acids such as acetate, pyruvate, maleate, oxalate, citrate, succinate, fumarate or malate were not found by HPLC analysis. The maximum growth rate, depending on the agitation speed (Table 1), was achieved after 15 h fermentation. Pellet formation started at the end of the exponential growth phase after approximately 18 h. The resulting pellets consisted of a loose network of hyphae.

Dierent agitation speeds resulted in changes of morphological and kinetic properties (Table 1). At low agitation speed (160 rpm), the fungus showed ®lamentous growth. Impeller speeds higher than 340 rpm induced pellet formation. The main fraction of pellets in all fermentation experiments had a diameter of 250±500 lm, and the percentage of pellets ³250 lm decreased slightly with increasing agitation speed.

Biotransformation

Due to the high toxicity of CPB to the fungus, its start concentration and the time point of addition were examined in the optimized fermentation medium at 390 rpm agitation speed. CPB was added to the fermentation broth, giving start concentrations of 150, 350 or 1000 mg l )1 (30, 60, 200 mg g )1 CDW, respectively) after 26 h when the fungus was in the stationary phase. The optimal starting concentration was 350 mg l )1 regarding transformation rate, and 150 mg l )1 regarding catalyst stability (Table 2). CPB adsorbs immediately after the addition on the biomass, which explains the rapid decrease of substrate concentration in the supernatant. The main product and by-product 1 formation started immediately, and the formation of by-product 2 started with slight delay (30 min) after product formation (Fig. 2). The e.e. of the main product increased rapidly from almost racemic to approximately 80% of its ®nal value (Fig. 5). Further conversion rate was rather slow. The enhancement of the e.e. of the main product was accompanied by an increase in by-product 2 concentration (Fig. 2, Fig. 5).

CPB addition in the transition phase (from exponential growth to the stationary phase, after 19 h) and in the stationary phase (after 26 h) resulted in nearly the same conversion rates and a high ®nal e.e. of the main product. The stability of the biocatalyst against the toxic educt CPB was higher in the stationary phase than in the transition phase (Fig. 3, Table 2).

Feeding pro®le

A feeding strategy was developed in order to increase the ®nal concentration of the product and to maintain biocatalyst stability. CPB was added in three pulses starting in the stationary phase (26, 35, 44 h), and each pulse and Steiner (1975); percentage of pellets with a diameter of 250 lm and 500 lm d 250 lm , and e.e. of 1S,3S-diastereomer). ND, not determined created a ®nal concentration of 100±150 mg l )1 (30 mg g )1 CDW). The ®nal product concentration of 250 mg l )1 in the supernatant was reached without rapid cell lysis (Fig. 3). The e.e. of the product was increased from 71% in the unpulsed process to 82% in the pulsed process and reached its maximum value after 100 fermentation hours (Fig. 5). The biomass concentration remained stable during the three pulses, then the CDW concentration dropped down from 5.2 g l )1 to 3.0 g l )1 within 26 h after the last CPB pulse (Fig. 3).

U [rpm] d 250 lm [%] l¢ max [h )1 ] k ¢ deakt [h )1 ] r P [mg l )1 h )1 ] e.e
For extended, long-term experiments, a stable biocatalyst concentration with high activity is essential. Therefore, the biomass concentration was kept in the range 5.5±8.5 g l )1 by regeneration with a glucose feed between the CPB pulses. When the product formation rate rapidly slowed, glucose feeding was started, which was accompanied by increasing oxygen consumption. The feed was stopped when the oxygen demand decreased again. After 30 min recreation time for the biomass, the next CPB pulse was added (Fig. 4). The resulting product concentration of 450 mg l )1 was achieved after six CPB pulses. The e.e. of by-product 2 decreased with the fermentation time and the e.e. of the main product reached a ®nal value of 95% (Fig. 5).

Discussion

In shake ¯ask experiments, the medium was optimized but the morphology of the fungus could not be controlled. The pellets were hollow, indicating transport limitation to the inner hyphae; the pellet size and pH were not reproducible. Literature data indicate that the pH may in¯uence the rate of oxidation of the product to by-product 2 [START_REF] Schwartz | Microbial oxidation of ebastine[END_REF]. Therefore, the Table 2 In¯uence of the starting concentration and time point of adding 2-cyclopentyl-1,3-benzoxazole (CPB) on fermentation parameters (death rate k¢ deakt , formation rate r P of the main product and e.e. of the (1S,3S)-diastereomer). ND, not determined optimization of the biotransformation process was performed in a bioreactor with controlled pH, aeration, and agitation rate.

CPB [mg l )1 ] k ¢ deakt [h )1 ] r P [mg l )1 h )1 ] e.e
Fermentation at an optimal agitation rate of 390 rpm and an inoculum concentration of 10 7 spores l )1 showed pellet formation in accordance with the literature [START_REF] Whitaker | Fungal pelleting[END_REF][START_REF] Metz | The growth of molds in the form of pellets ± a literature review[END_REF][START_REF] Braun | Mycelial morphology and metabolite production[END_REF].

The pellet size distribution during the cultivation depended mainly on the agitation speed, as the other parameters were kept constant. At a low agitation rate (160 rpm), the fungus grew ®lamentous. At average and high agitation rates (390, 550 and 730 rpm), loose and ¯uy pellets were formed with a maximum diameter of 500 lm providing optimal oxygen transfer to the inner hyphae [START_REF] Kobayashi | Oxygen transfer into mycelial pellets[END_REF][START_REF] Wittler | Investigations of oxygen transfer in Penicillium chrysogenum pellets by microprobe measurements[END_REF][START_REF] Hotop | In¯uence of the preculture conditions on the pellet size distribution of Penicillium chrysogenum cultivations[END_REF]. With increasing agitation rate, the pellet diameter decreased slightly, due to the higher shear forces. However, prevention of pellet formation at high power input, as reported for strains with a coagulative pellet formation [START_REF] Metz | The growth of molds in the form of pellets ± a literature review[END_REF], was not found in the investigated range, although pellets did become smaller and more compact.

The stability of the biocatalyst towards the educt CPB decreased signi®cantly at low and at high agitation speeds (Table 1). The surface area of the fungus which was exposed to CPB probably played an important role for the stability of the biomass. Due to the high toxicity of CPB, only low concentrations were tolerated by C. blakesleeana (30 mg g )1 CDW). Although the mechanism of deactivation is not exactly known, a disintegration of cell membranes could be considered, because of the lipophilic character of CPB. The hydroxylation performed via cytochromes P-450 is a type of detoxi®cation [START_REF] Asperger | Induction and functional role of cytochromes P450 in the ®lamentous fungi Mortierella alpina ATCC 8979 and Cunnninghamella blakesleeana DSM 1906 during hydroxylation of cycloalkylbenzoxazoles[END_REF]. Filamentous growth (at 160 rpm agitation rate) enhanced membrane surface exposure. Pellet formation probably protected the cells in the pellet core due to a concentration gradient of CPB from the outer to the inner pellet regions. Presumably, CPB is quickly and strongly absorbed on the outer pellet regions, while the pellet core remains unloaded. At high agitation rates, torn hyphae were observed under the microscope due to mechanical stress. In this damaged state, equivalent amounts of CPB could cause greater harm as con®rmed by higher cell death rates (Table 1).

The strong adsorption of CPB on the biomass explains the rapid decrease of its concentration and the low product concentration in the fermentation medium [START_REF] Kreiner | Stereo-speci®c biohydroxylations of protected carboxylic acids with Cunninghamella blakesleeana[END_REF]. Extraction of the whole fermentation broth including the biomass will result in higher ®nal product yields. Since representative sampling of the fermentation broth was dicult due to the high amount of solid material, analysis during the fermentation was done from the centrifuged medium in order to obtain reproducible data over all experiments.

The stability of the fungus towards the chemical educt depended on the physiological state of C. blakesleeana. The highest stability of the biocatalyst was observed in the stationary growth phase, while addition of CPB in the lag phase or the exponential growth phase of the fungi resulted in very slow conversion rates and rapid cell lyses [START_REF] Kreiner | Stereo-speci®c biohydroxylations of protected carboxylic acids with Cunninghamella blakesleeana[END_REF]. However, addition of CPB in the transition phase from the exponential growth phase to the stationary phase did not aect the transformation rate or the enhancement of e.e., but did signi®cantly decrease the stability of the biocatalyst (Fig. 3). This phenomenon may be linked to an altering of the membrane structure due to the increasing age of the fungus [START_REF] Sedlaczek | Transformation of steroids by fungal spores[END_REF] or due to exposition to toxic agents [START_REF] Venkateswerlu | Morphological, ultrastructural, and chemical changes induced in Cunninghamella blakesleeana by copper and cobalt[END_REF].

Although the transformation rate was not in¯uenced by variation of the agitation speed, the e.e. decreased with increasing agitation speed (Table 1). Similar to reported observations, by-product 2 was formed by delayed oxidation of the main product [START_REF] Schwartz | Microbial oxidation of ebastine[END_REF]. The diastereomers of the main product are presumably oxidized at dierent rates with slightly faster oxidation of the (1R,3R)-diastereomer, which could explain the initial enantiomeric pure formation of by-product 2 and the increase of the e.e. of the main product. The regeneration of the fungus with the glucose feed provides active biomass during the whole fermentation, which also increases the e.e. of the main product. The ®nal e.e. is strongly enhanced from 71% to 95%, which represents an increase of 48% over previously reported results [START_REF] Kreiner | Stereo-speci®c biohydroxylations of protected carboxylic acids with Cunninghamella blakesleeana[END_REF].

The optimal strategy for the biotransformation of the toxic educt CPB was a fermentation in optimized medium, at an agitation rate of 390 rpm and an aeration rate of 1 vvm during growth phase and 0.5 vvm during biotransformation. Pulses of CPB started in the stationary phase with 30 mg g CDW )1 , and the biomass was regenerated with a nutrition feed. Due to this stable semi-continuous process, long-term experiments were performed and the product concentration was strongly Fig. 5 e.e. of product (1S,3S)-3-(benz-1,3-oxazol-2-yl)cyclopentan-1ol and by-product 2 (3R)-3-(benz-1,3-oxazol-2-yl)cyclopentan-1-one depending on the substrate feeding pro®le. Course of the e.e. of the product in the unpulsed process with a 2-cyclopentyl-1,3-benzoxazole (CPB) starting concentration of 350 mg l )1 (n), in the pulsed process with three pulses (r), and in the pulsed process with regeneration feed (d) and course of the e.e. of the by-product 2 (s) in the pulsed process with regeneration feed increased while the concentration of the toxic educt CPB was kept low. Further research should be done to clarify the mechanisms of enzymatic e.e enhancement in order to achieve a pure enatiomeric product.

Fig. 1

 1 Fig.1Hydroxylation of 2-cyclopentyl-1,3-benzoxazole (CPB) by Cunninghamella blakesleeana. Structural formulae of products: 3-(benz-1,3-oxazol-2-yl)-cyclopentan-1-ol (product), 2-(benz-1,3-oxazol-2-yl)-cyclopentan-1-ol (by-product 1) and 3-(benz-1,3-oxazol-2yl)-cyclopentan-1-one (by-product 2)

Fig. 2 Fig. 4

 24 Fig. 2 Fermentation characteristics when 2-cyclopentyl-1,3-benzoxazole (CPB) is added in the stationary phase. Course of glucose (d), cell dry mass (CDW; m), CPB (e), product (h), by-product 1 (´) and by-product 2 (n)

Table 1

 1 In¯uence of the agitation rate U on characteristic fermentation parameters (maximum growth rate l¢ max , death rate k¢ deakt , formation rate r P of the product, all calculated after Moser

	. [%]

Acknowlegments This work was supported by the Spezialforschungsbereich Biokatalyse (SFB). The authors thank M. Kreiner and G. Braunegg for preliminary studies.