# Spectral BSDF Measurement using Computational Imaging Spectroradiometry within a Lambertian ¾-of-a-Sphere (preprint)

Pierre Boulenguez, Samuel Carré, Nicolas Noe, Laurent Najman

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Pierre Boulenguez, Samuel Carré, Nicolas Noe, Laurent Najman. Spectral BSDF Measurement using Computational Imaging Spectroradiometry within a Lambertian ¾-of-a-Sphere (preprint). 2013. hal-05449292

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# Spectral BSDF Measurement using Computational Imaging Spectroradiometry within a Lambertian ¾-of-a-Sphere

Pierre Boulenguez, Samuel Carré, Nicolas Noé, and Laurent Najman

**Abstract**—The bidirectional optical scattering properties of surfaces are of importance to a wide range of disciplines. We classify existing measurement approaches and detail some emblematic instruments under a unified figurative formalism. We then present a new instrument, able to measure the spectral bidirectional scattering-distribution function over the visible-bidirectional domain (anisotropic spectral BRDF and BTDF). The general principle of the acquisition stage is to collect the scattering upon a hemispherical Lambertian screen. Interreflections are handled by subtracting the constant interreflections component. The detector is a novel omnidirectional spectral camera, which captures the spectral distribution of the scattering. Data processing, to recover a dense sampling of the SBSDF, is founded on a radiometric model of the image formation process within this device. Validation of the approach was performed through a round-robin procedure. Measured SBSDF are ultimately presented.

**Index Terms**—9.IV.I.IV Radiometry, 9.IV.I.V Reflectance, 9.IV.I.I Camera calibration, 9.IV.I.II Imaging geometry, 9.III.V.IX Physically based modeling, 9.III.VII.II Color, shading, shadowing, and texture, 9.II.IX.X Sensors, 7.I.IX.III Fredholm equation, 7.I.III.VII Pseudoinverses.

<figure><img src="image_2.png" /></figure>

THE efficient detection of the optical scattering by macroscopic surfaces—high-resolution, fast, and economic—is of importance to a wide range of disciplines.

The spectral-bidirectional dependence of this scattering is characterized by the spectral bidirectional scattering-distribution function (SBSDF), introduced by Nicodemus et al. [1], defined at a position $\vec{x}_M$ on a flat surface of normal $\vec{n}_M$:

$$f_s : \Omega \times \Omega^+ \times \mathbb{R}^{+,*} \rightarrow \mathbb{R}^+ \quad (1)$$

$$(\vec{\omega}_i, \vec{\omega}_o, \lambda) \rightarrow \frac{dL_{\lambda}(\vec{x}_M, \vec{\omega}_o)}{L_{\lambda}(\vec{x}_M, \vec{\omega}_i) [\vec{\omega}_i \cdot \vec{n}_M] d\vec{\omega}_i} \quad \{\text{sr}^{-1}\},$$

where $\Omega$ is the unit sphere, $\Omega^+$ the unit upper hemisphere, $\vec{\omega}_i$ and $\vec{\omega}_o$ incoming and outgoing directions, $\lambda$ the wavelength, and $L_{\lambda}$ the spectral radiance.

In this paper, we aim to show our contributions towards an efficient SBSDF measurement instrument by:

* addressing of interdisciplinary research fragmentation, with the first classification of approaches to measurement and the detail of recent instruments under a unified formalism;

* creating a new approach, that is fast, economic, and the first able to densely sample the function over the whole visible-bidirectional domain (excluding

grazing incident directions and a small solid angle around the direction of the backscattering);

* examining a computational imaging spectroradiometry framework, applied to the presentation of the data processing.

This paper is organized as follows. The classification is introduced in Section 1 and detailed in Section 8; in Section 2, the general principle of the new approach is presented; in Section 3, the acquisition stage is detailed; in Section 4, the spectral calibration aspects are discussed; in Section 5, a formal relationship between an acquired image and the SBSDF is established; the numerical resolution of this equation is studied in Section 6; in Section 7, validation, measured SBSDFs, and perspectives are discussed.

# 1 TAXONOMY OF BIDIRECTIONAL SCATTERING MEASUREMENT INSTRUMENTS

As arguably the single most important radiometric quantity, a considerable research effort has been dedicated to the measurement of the SBSDF. These wide-ranging efforts have, however led to fragmentation: interdisciplinary; and with instruments dedicated to selected aspects of the function (BRDF, BTDF, isotropic, plane of incidence, non-spectral, etc.). As an attempt to structure the field, bidirectional scattering assessment instruments have been classified according to Fig. 1.

The 'gold standard' sequential approach relies on the displacements of the source and detector along two spherical trajectories relative to the material. The imaging approach relies on a camera to instantly account for a multitude of bidirectional (source/detector) configurations. Imaging instruments are classified as direct when the camera is

* P. Boulenguez is with the Signal, Information, Communication Department of the XLIM Laboratory, Poitiers, France. E-mail: pierre.boulenguez@univ-poitiers.fr.

* S. Carré and N. Noé are with the Centre Scientifique et Technique du Bâtiment, Nantes, France. E-mail: samuel.carre@cstb.fr; E-mail: nicolas.noe@cstb.fr.

* L. Najman is with the University Paris-Est, Laboratoire d'Informatique Gaspard-Monge, Team A3SI, ESIEE Paris, France. E-mail: l.najman@esiee.fr.
<figure><img src="image_3.png" /><figcaption>Fig. 1. Taxonomy of bidirectional scattering assessment instruments, as detailed in Section 8.</figcaption></figure>

oriented towards the material<fnref n="1" />, exploiting the variation of the bidirection along its surface; and *indirect* when the camera is oriented towards an intermediary surface, or screen, upon which the scattering is projected. *Direct* instruments are further categorized according to the geometry of the material: instances of *spherical*, *cylindrical*, and *cylindrical stripes* were found. *Indirect* instruments are further categorized according to the coating of the projection screen: *specular*, *glossy*, and *Lambertian*; then to its geometry: *hemiellipsoidal*, *off-axis parabolic*, *kaleidoscopic*, *cubic*, *triangular*, and *hemispherical*.

To alleviate the bulk of this paper, the foundations of this taxonomy and details of referenced instruments are found in Section 8.

The rationale for the new approach draws on ideas and shortcomings identified in these instruments. Altough theoretically straightforward, the sequential approach is, in practice, paradoxically mechanically cumbersome, expensive, slow, and low-resolution. Yet, no imaging instrument was found able to resolve the scattering over the whole bidirectional domain. Few instruments account for its spectral dependence, moreover often producing data in an ill-defined spectral space, dependent on both the spectrum of the source and quantum efficiency of the camera. Finally, to express the data processing, a framework for imaging spectroradiometry in the formalism of Nicodemus et al. [1] was found lacking.

## 2 INDIRECT IMAGING APPROACH WITH A LAMBERTIAN ¾-OF-A-SPHERE SCREEN

The new approach outlined in this paper is classified as indirect imaging with a Lambertian ¾-of-a-sphere screen. It is spectral, fast, economic, and, to the best of our knowledge, the first imaging design able to densely sample the SBSDF over the complete visible-bidirectional domain (excluding grazing incident directions and a small solid angle around the direction of backscattering).

Its general principle is to illumine the material with a collimated beam, and to collect its scattering after another diffuse reflection on a hemispherical Lambertian screen. Interreflections are handled by subtracting the

<figure><img src="image_4.png" /><figcaption>Fig. 2. Principle of the new approach acquisition stage.</figcaption></figure>

interreflections component, which is constant in the hemispherical geometry of the screen. The detector is a novel omnidirectional spectral camera, which measures the spectral distribution of the scattering.

In more depth, the acquisition stage is illustrated in Fig. 2. The beam from the source $S$ enters the mobile Lambertian ¾-of-a-sphere screen $N$ through the circular aperture $I$. It is scattered by the material $M$ towards a hemispherical section of the screen; of which the spectral radiance is detected by the composite omnidirectional spectral camera ($C, L, E, F$); assembly of a fisheye lens, an optical adapter, a liquid crystal tunable filter (LCTF), and a radiometry-grade camera. The incident direction of the beam $\tilde{\omega}_i$ varies over the unit sphere by rotations of: (i) the material along axis $A_1$; (ii) the base along axis $A_2$; and (iii), the screen along axis $A_3$.

The data processing stage is founded on a model of the image formation process within this device, relating the level of a pixel to the SBSDF. Its numerical resolution notably involves a spectral reconstruction procedure, geometric and radiometric calibrations of the omnidirectional spectral camera, an estimation of the interreflections component, and a unit sphere discretization scheme.

<footnote n="1">1. The term 'material' is used throughout the paper to denote the surface being measured (in order to avoid confusion with the mathematical meaning of the term 'sample').</footnote>
# 3 PROTOTYPE ACQUISITION DEVICE

<figure><img src="image_5.png" /><figcaption>Fig. 3. Scale model of the acquisition device prototype.</figcaption></figure>

An acquisition device prototype was built as a proof-of-concept (see Fig. 3) of this new approach. Its technical description is focused on the aspects that seem the most profitable to the community, avoiding subjects seen as dispensable such as holder for the material and motorization.

## 3.1 Source

The illumination module is a continuing source of concern in the scattering community. Source $S$ was conceived to fulfill the usual constraints of: (i) collimation, and (ii) spectrum (i.e. energy for all visible wavelengths); in addition to a specific (iii) extensive power, as the detected light on the imaging sensor is the product of scatterings by the material and the screen and filtering by the LCTF.

<figure><img src="image_6.png" /><figcaption>(1: met. hal. lamp [23], 2: parab. luminaire [24], 3: IR-C filter [25], 4: air cooler, 5: support frame of 3).</figcaption></figure>

<caption>Fig. 4. Light source $S$.</caption>

As illustrated in Fig. 4, the high-intensity discharge metal halide lamp [23] was selected for: (iii) its 200 kilolumens of luminous flux, (ii) its continuous spectrum (see [23]), and (i), its small dimensions, allowing a precise collimating assembly through its installation at the focus of the parabolic luminaire [24]. Other implementation details are presented in Fig. 4.

<figure><img src="image_7.png" /><figcaption>Fig. 5. Mobile Lambertian ¾-of-a-sphere screen $N$.</figcaption></figure>

## 3.2 Screen

The mobile ¾-of-a-sphere Lambertian screen is obviously a critical component. The prototype in Fig. 5 was built by the assembly of six resin ⅛-of-a-sphere, molded from a wood and plaster frame. The inner surface was coated with layers of a Lambertian paint, whose spectral reflectance is noted $\rho_{\lambda,N}$ in the following².

Its diameter, noted $d_N$, is one meter. The diameter of the material and the aperture, noted $d_M$, is eight centimeters. Let $\alpha_{max}$ be the angle such that:

$$ \alpha_{max} = \arcsin \frac{d_M}{d_N} \approx 5 \cdot \frac{2\pi}{360} \{\text{rad}\}. \quad (2) $$

These dimensions limit the domain of measurable incident directions to:

$$ \bar{\omega}_i \in \Omega \setminus \{(\theta_i, \varphi_i) | \frac{\pi}{2} - \alpha_{max} \le \theta_i \le \frac{\pi}{2}\}, \quad (3) $$

where $\theta_i$ is the incident zenith angle³; and outgoing

2. The spectral distribution of $\rho_{\lambda,N}$ is precisely known; for information: $\rho_{\lambda,N} \approx 0.25\% \{1\} \forall \lambda \in [400\text{nm}; 700\text{nm}]$.

3. The base meets the aperture when axis $A_1$ is positioned at the angle $\theta_i = \frac{\pi}{2} - \alpha_{max} \{\text{rad}\}$.
directions to⁴:

$$ \vec{\omega}_o \in \Omega^+ \setminus \{\vec{\omega}_o | \arccos[\vec{\omega}_o \cdot \vec{\omega}_i] \le \alpha_{max}\}, \quad (4) $$

### 3.3 Axes

The direction of incidence varies over the former domain of (3) by rotations of axes $A_1$, $A_2$, and $A_3$. The axis $A_1$ sets the material in motion, varying the azimuthal angle $\varphi_i$. The axis $A_2$ rotates the base, varying its zenith counterpart $\theta_i$ over a single hemisphere⁵. The axis $A_3$ rotates the screen itself, switching between hemispheres: $\Omega^+$ for reflectance/SBRDF measurement, and $\Omega^-$ for transmittance/SBTDF.

In the device presented in Fig. 4, the axis $A_3$ (rotated by $\pi$ radians once per acquisition) is manual; the axes $A_1$ and $A_2$ (set for each incident direction) are fully automated.

### 3.4 Fisheye

Fisheyes are a class of photographic lenses whose fields of view is greater than or equal to $\pi$ radians⁶. Fisheyes are conceived to satisfy the stereographic, equisolid angle, orthogonal, or equidistant projection; the latter, also commonly referred as to $f\theta$, exhibits the lowest vignetting. Hence, an equidistant fisheye [28] was used for the instrument.

Traditional camera geometric calibration methods (e.g. Zhang [29]) assume the perspective projection (or pinhole camera model) and are thus unfit for fisheyes. In this study, we used the generic camera model by Kannala and Brandt [30].

### 3.5 Liquid Crystal Tunable Filter

<figure><img src="image_8.png" /><figcaption>Fig. 6. Transmittance $\tau_{\lambda,\mathcal{L}}(\mu)$ of the LCTF $\mathcal{L}$ (measure).</figcaption></figure>

A liquid crystal tunable filter (LCTF) is a recent evolution of the optical filter design by Lyot, which allows

4. The diameter of the aperture prevents measurement of outgoing directions in a cone around the direction of retroreflection.

5. The beam from the source 'matches' the material when $\theta_i = 0$ or $\theta_i = \frac{\pi}{2}$ {rad}; otherwise, it overfills the material.

6. The reader can refer to the book by Ray [26] for a timeline of their development and to the article by Kumler and Bauer [27] for a comparison of more recent designs.

the dynamic 'tuning' of the transmittance⁷. In the LCTF [32] retained, the transmittance is tuned through a real parameter noted $\mu$, transmitted via RS232.

The transmittance of the LCTF [32] was measured and reconstructed (with samples at $\Delta\mu = \Delta\lambda = 2$ nm) into the continuous function:

$$ \begin{gathered} \tau_{\lambda,\mathcal{L}}(\mu) : [400\text{nm}; 720\text{nm}] \times [390\text{nm}; 730\text{nm}] \rightarrow [0; 1] \quad (5) \\ (\mu, \lambda) \rightarrow \tau_{\lambda,\mathcal{L}}(\mu) \{1\}, \end{gathered} $$

plotted in Fig. 6.

### 3.6 Camera

<table><caption>TABLE 1
Relevant properties of the camera C.</caption><tr><td>Ref. camera, sensor:</td><td>[33], [34],</td></tr><tr><td>misc.:</td><td>ILT CCD, b&w, Si, 5°C Peltier-cooled,</td></tr><tr><td>bus:</td><td>IIDC1394 [35],</td></tr><tr><td>response function:</td><td>linear,</td></tr><tr><td>resolution:</td><td>$n_c = 1344 \times 1024$ pixels,</td></tr><tr><td>dynamic:</td><td>$b_c = 12$ bits,</td></tr><tr><td>full well capacity:</td><td>$f_c = 18000$ electrons,</td></tr><tr><td>area of a pel:</td><td>$a_c = 6.24 \times 6.24 \ \mu\text{m}^2$,</td></tr><tr><td>quantum efficiency:</td><td>$\eta_{\lambda,c} = \text{in [33]}$ ($\{1\} = \text{dimensionless}$).</td></tr></table>

A solid-state imaging sensor is a matrix of photoelements, called pels, which convert the incident light into electrical signals by photoelectric effect⁸. The relevant properties of the camera chosen to equip the composite omnidirectional spectral camera⁹ are given in TABLE 1.

### 3.7 Driver

<table><caption>TABLE 2
Acquisition driver D.</caption><tr><td colspan="2"><strong>inputs:</strong></td></tr><tr><td>$t_{\mathcal{L}}$: {1, 2, ..., $n_{\mu}$} → [400; 720] (LCTF $\mathcal{L}$ tunes)</td><td></td></tr><tr><td>$t_{\vec{\omega}_i}$: {1, 2, ..., $n_{\vec{\omega}_i}$} → [0; $2\pi$ [$\times$[0; $\alpha_{max}$ [$\times${0; $\pi$}] (incident directions $\vec{\omega}_i$ ⇔ axes $A_1-3$ angles)]</td><td></td></tr><tr><td>$D \leftarrow \emptyset$</td><td></td></tr><tr><td colspan="2"><strong>for all (a, b) in {1, 2, ..., $n_{\mu}$} × {1, 2, ..., $n_{\vec{\omega}_i}$} do</td></tr><tr><td>set $\mathcal{L}$ in $t_{\mathcal{L}}[a]$, ($A_1, A_2, A_3$) in $t_{\vec{\omega}_i}[b]$</td><td></td></tr><tr><td>$D \leftarrow D \cup \{(\text{capture } \mathcal{C}), t_{\mathcal{L}}[a], t_{\vec{\omega}_i}[b]\}$</td><td></td></tr><tr><td colspan="2"><strong>return D (images and attributes; card (D) = $n_{\mu} \cdot n_{\vec{\omega}_i}$)</strong></td></tr></table>

Insight on the data flow during the acquisition stage was given in Fig. 2: tunes and angles are sent to the LCTF

7. Insight on its principle can be found in [31].

8. A discussion on their diverse technologies can be found in the chapter by Seitz [36] and the book by Hopkinson et al. [37]

9. The LCTF [32] was installed between the fisheye [28] and the camera [33] using a custom optical adapter, made of a rod lens, a depolarizer (note that the detected light is also depolarized by the Lambertian reflection on the screen), an IR-filter, and two C-Mount threads.
and the axes and images are received from the camera. This procedure is clarified in TABLE 2$^{10,11}$.

# 4 SPECTRAL IMAGING

The relationship between the acquisition data set $D$ and the SBSDF is complex. The reason arises from the fact that the radiometric formalism of the SBSDF is not particularly well-suited to the optoelectronic context of an imaging instrument; in the present indirect imaging approach, new complications arise with the interreflections compensation and spectral reconstruction problems.

A formal relationship between the acquisition data set $D$ and the SBSDF is established in Section 4. Founded on this model, the data processing to actually recover the function is presented in Section 5.

## 4.1 Imaging Sensor

Starting from an acquired image (of the data set $D$), let $p$ be an element of $\{1, \dots, n_c\}$ (symbols subscripted by $C$ are related to the camera and defined in TABLE 1) and $E_{\lambda,p}$ be the spectral irradiance of the $p^{th}$ pel during its capture.

First consider the definition of quantum efficiency$^{12}$:

$$ \eta_{\lambda,c} = \frac{i_{\lambda,p} h c}{E_{\lambda,p} a_c \lambda e} \quad \{1\}, \qquad (6) $$

where $i_{\lambda,p}$ is the observed electrical intensity, $h$ is Planck's constant, $c$ is the speed of light in a vacuum, $a_c$ is the area of a pel, and $e$ is the elementary charge. In a radiometry-grade CCD context, quantum efficiency is assumed to be independent of $p$.

For polychromatic light, the observed electrical intensity is also:

$$ i_p = \frac{a_c e}{h c} \int_0^\infty \lambda \eta_{\lambda,c} E_{\lambda,p} d\lambda \quad \{A\}. \qquad (7) $$

Assuming stable illumination during the capture, the charge photogenerated in the $p^{th}$ pel is obtained:

$$ q_p = \frac{t_c a_c e}{h c} \int_0^\infty \lambda \eta_{\lambda,c} E_{\lambda,p} d\lambda \quad \{C\}, \qquad (8) $$

where $t_c$ is the integration time.

This charge is transported by the interline transfer charge-coupled device (ILT CCD) to the capacitor. The resulting tension is amplified, setting the gain noted $\gamma_c$, and quantified into the level of the pixel, noted $s_p$. These stages are represented as a generic discrete map:

$$ r: \begin{cases} 0, & e, 2e, \dots, f_c e \end{cases} \rightarrow \begin{cases} 0, & 1, 2, \dots, 2^{bc} - 1 \end{cases} \qquad (9) \\ q_p \rightarrow s_p \quad \{C\}, \qquad (10) $$

where $f_c$ and $b_c$ are respectively the full well capacity and the dynamic of the sensor. In an effort to maintain physical units throughout the data processing pipeline, the level $s_p$ is expressed as homogeneous to the generated charge in Coulomb. The map $r$ of the radiometry-grade linear camera [33] is assumed to be affine$^{13}$ such that:

$$ s_p = \xi_{*,p} \gamma_c q_p + \xi_{\circ,p} \gamma_c t_c + \xi_{+,p} \quad \{C\}, \qquad (10) $$

where $\xi_{*,p}$ is multiplicative noise, $\xi_{\circ,p}$ is time-dependent noise, and $\xi_{+,p}$ is additive noise$^{14}$.

The time-dependent $\xi_{\circ,p}$ and additive $\xi_{+,p}$ noises were least squares fitted from dark-frames acquired for different integration time $t_c$ and gain $\gamma_c$. Multiplicative noise $\xi_{*,p}$ remains unknown; assumed to be Gaussian, according to the central limit theorem, and of relatively low intensity, it is nonetheless attenuated by the narrow-band regularization and reprojection algorithms, presented in Subsection 6.3 and Subsection 4.3, respectively.

Finally, the level $s_p$ is related to the spectral irradiance $E_{\lambda,p}$:

$$ s_p = \begin{cases} \{C\} \\ \frac{\xi_{*,p} \gamma_c t_c a_c e}{h c} \int_0^\infty \lambda \eta_{\lambda,c} E_{\lambda,p} d\lambda + \xi_{\circ,p} \gamma_c t_c + \xi_{+,p} \end{cases} \qquad (11) $$

## 4.2 Fredholm's Integral Equation Framework

<figure><img src="image_9.png" /><figcaption>C: camera (i.e. sensor), L: LCTF, E: adapter, p, q: $p^{th}/q^{th}$ pels, E$_{\lambda,p}$, E$_{\lambda,q}$: spectral irradiance L.</figcaption></figure>

<caption>Fig. 7. Representation of the LCTF $\mathcal{L}$ and adapter $\mathcal{E}$.</caption>

Fig. 7 depicts the camera-LCTF-optical adapter assembly. The surface of the filter is subdivided into $n_c$ smaller surfaces, so that each of them is in correspondence with a pel of the sensor. The notation is slightly modified accordingly, with $E_{\lambda,p}$ now denoting the unfiltered spectral irradiance.

Let, then, $s_p(\mu)$ be the level of the $p^{th}$ pixel in an image captured with the filter tuned at $\mu$. From (11):

$$ s_p(\mu) = \begin{cases} \{C\} \\ \frac{\xi_{*,p} \gamma_c t_c a_c e}{h c} \int_0^\infty \lambda \eta_{\lambda,c} \tau_{\lambda,\mathcal{L}}(\mu) E_{\lambda,p} d\lambda + \xi_{\circ,p} \gamma_c t_c + \xi_{+,p} \end{cases} \qquad (12) $$

where $\tau_{\lambda,\mathcal{L}}(\mu)$ is the transmittance of the LCTF.

<footnote n="10">10. The 'capture (C)' function acquires high-dynamic range images, relying on the drains of the Interline Transfer sensor (ILT CCD) to reject pixels potentially altered by smear.</footnote>

<footnote n="11">11. The incident directions in input are not set on a regular zenith-azimuth grid, in order to more harmoniously span the domain of (3), the underlying unit sphere parameterization is discussed in Subsection 6.2.</footnote>

<footnote n="12">12. Adapted to account for $E_{\lambda,p}$, see (5.1) in [36], for example.</footnote>

<footnote n="13">13. For the map $r$ to be truly affine, the mapped sets in (9) should have same cardinality. From TABLE 1, that is not the case for the camera [33] and a more accurate model could represent (10) as a piecewise increasing function. Such a level of detail seems unnecessary for the vast majority of imaging radiometry applications.</footnote>

<footnote n="14">14. A discussion on the origin and the modeling of noise can be found, for instance, in the studies by Boneclet [38], Ratner and Schechner [39], and Hasinoff et al. [40].</footnote>
Hardeberg [41] advanced (multi/hyper) spectral imaging by identifying the spectral reconstruction problem as a Fredholm's integral equation of the first kind:

$$f(x) = \int_{a}^{b} g(x, y) h(y) dy, \qquad (13)$$

where $f(x)$ and $g(x, y)$ are two known continuous functions, $a$ and $b$ are constants, and $h(y)$ is the unknown. Adhering to this framework, we identify (12) as (13) by defining:

$$h(\lambda) = \xi_{*,p} E_{\lambda,p} \{\mathbf{W} \cdot \mathbf{m}^{-3}\}, \qquad (14)$$

the unknown spectral irradiance $E_{\lambda,p}$¹⁵ and multiplicative noise $\xi_{*,p}$; and:

$$f(\mu) = \frac{s_p(\mu) - \xi_{o,p} \gamma_c t_c - \xi_{+,p}}{\gamma_c t_c} \{\mathbf{A}\}, \qquad (15)$$

the pth pixel's level for the LCTF tuned at $\mu$, compensated by additive $\xi_{+,p}$ and time-dependent $\xi_{o,p}$ noises, and normalized by the integration time $t_c$ and gain $\gamma_c$; and:

$$g(\mu, \lambda) = \frac{a_c e \lambda \eta_{\lambda,c} \tau_{\lambda,\mathcal{L}}(\mu)}{h c} \{\mathbf{C} \cdot \mathbf{J}^{-1} \cdot \mathbf{m}^2\}, \qquad (16)$$

the tunable spectral sensitivity of the camera-LCTF assembly¹⁶, which takes significant values only for wavelength $\lambda$ in [390nm; 730nm] and tune $\mu$ in [400nm; 720nm] according to (5).

## 4.3 Spectral Reconstruction

Regularization of the ill-posed spectral reconstruction problem is still an active field of research with no clear optimal solution. For the instrument, four methods were assessed¹⁷.

In the presence of noise, experiments conducted on *Moore-Penrose pseudoinverse* and *non-negative least square*¹⁸ found these methods to be both unstable with respect to the discretization scheme, and relatively slow. *Wiener estimation*¹⁹ was also found to be slow, and highly dependent on the quality of the learning data set.

Until a consensual solution is found, the approach of approximating the transmittance of the LCTF by a delta function was employed. Measurements at different spectral bands are then treated completely independently. This pragmatic 'narrow-band regularization' method is robust, fast, and provides a smoothing of the reconstructed spectrum.

It is formulated within the Fredholm's integral equation framework by defining the new function:

$$k(\mu, \lambda) = \delta(\lambda - \mu) \int_{0}^{\infty} g(\mu, \lambda) d\lambda \quad \{\mathbf{C} \cdot \mathbf{J}^{-1} \cdot \mathbf{m}^{-2}\}, \quad (17)$$

15. The unit {$\mathbf{W} \cdot \mathbf{m}^{-3}$} expresses the spectral dependence of the irradiance (i.e. {$\mathbf{W} \cdot \mathbf{m}^{-2} \cdot \mathbf{m}^{-1}$}).

16. Homogeneity is verified with {$\mathbf{C} = \mathbf{A} \cdot \mathbf{s}$} and {$\mathbf{W} = \mathbf{J} \cdot \mathbf{s}^{-1}$}.

17. Details of this study can be found in the dissertation by Boulenguez [22].

18. See Hardeberg [41], for instance.

19. See Shen et al. [42], for instance.

where $\delta$ is the Dirac distribution. Approximation in (13) of $g(\mu, \lambda)$ by $k(\mu, \lambda)$ then yields²⁰:

$$h(\mu) = \frac{f(\mu)}{\int_{0}^{\infty} g(\mu, \lambda) d\lambda} \{\mathbf{W} \cdot \mathbf{m}^{-3}\}. \qquad (18)$$

# 5 EXPRESSION OF THE SBSDF

<figure><img src="image_10.png" /><figcaption>Fig. 8. Omnidirectional spectral camera (C, L, E, F).</figcaption></figure>

Fig. 8 illustrates the relationship between the reconstructed spectral irradiance of the filter and the spectral radiance of the screen.

## 5.1 Spectral Radiance of the Screen

In more depth, the irradiance $E_{\lambda,p}$ is:

$$\Phi''_{\lambda,p} = a_c E_{\lambda,p} \{\mathbf{W} \cdot \mathbf{m}^{-1}\}. \qquad (19)$$

This flux is, in turn, induced by the image formed on the optical adapter by the fisheye, so that:

$$\Phi'_{\lambda,p} = \frac{a_c E_{\lambda,p}}{\tau_{\lambda,\varepsilon}} \{\mathbf{W} \cdot \mathbf{m}^{-1}\}, \qquad (20)$$

where $\tau_{\lambda,\varepsilon}$ is the spectral transmittance of the adapter.

Let $\vec{\omega}_p$ and $\Omega_p$ be, respectively, the direction in object-space and the solid angle around that direction associated with the pth pixel. The direction was recovered $\vec{\omega}_p$ using the backward model of Kannala and Brandt [30]. The solid angle $\Omega_p$ was deduced from a flat field measurement in a large Ulbricht sphere²¹.

The spectral flux that travels in the solid angle $\Omega_p$ around the direction $\vec{\omega}_p$ is then:

$$\Phi_{\lambda,p} = \frac{a_c E_{\lambda,p}}{\tau_{\lambda,\varepsilon} \tau_{\lambda,\varepsilon}} \{\mathbf{W} \cdot \mathbf{m}^{-1}\}, \qquad (21)$$

where $\tau_{\lambda,\varepsilon}$ is the transmittance of the fisheye.

Let $\vec{x}_N$ be the intersection between the direction $\vec{\omega}_p$ and the screen, and let $a_N$ be the area of the surface on the

20. With $\int_a^b \delta(x - x_0) f(x) dx = f(x_0)$ iff $x_0 \in]a, b[$.

21. These two calibrations stages assume no chromatic aberration; this could be alleviated by making $\vec{\omega}_p$ and $\Omega_p$ both dependent on wavelength and calibrating for different transmittances of the filter.
screen that subtends the solid angle $\Omega_p$. The radiance at $\vec{x}_N$ in the direction $-\vec{\omega}_p$ of the fisheye is, by definition:

$$L_{\lambda}(\vec{x}_{N}, -\vec{\omega}_{p}) = \frac{\Phi_{\lambda;p}}{a_{N}[\vec{-\omega}_{p} \cdot \vec{n}_{N}] \Omega_{F}(\vec{x}_{N})} \{W \cdot sr^{-1} \cdot m^{-3}\}, \quad (22)$$

where $\vec{n}_N$ and $\Omega_F(\vec{x}_N)$ are, respectively, the normal vector to the screen and the solid angle subtended by the fisheye at $\vec{x}_N$.

The solid angle $\Omega_F(\vec{x}_N)$ was estimated by modelling the fisheye as a small sphere, of area noted $2a_F$, centered at a position, noted $\vec{x}_F$. This representation was made with the rationale that the solid angle subtended by the fisheye is constant at positions on a hemisphere centered on the fisheye, and bears similarities with the two-step projection via a unit-sphere for geometric calibration of Ying and Hu [43].

The relationship between the radiance of the screen and the irradiance of the filter is therefore²²:

$$L_{\lambda}(\vec{x}_{N}, -\vec{\omega}_{p}) = \frac{a_{c}}{a_{F}\tau_{\lambda,F}\tau_{\lambda,E}} \cdot \frac{E_{\lambda,p}}{\Omega_{p}} \{W \cdot sr^{-1} \cdot m^{-3}\}. \quad (23)$$

## 5.2 Spectral Radiance of the Material

Now, from the radiative transfer equation²³:

$$L_{\lambda}(\vec{x}_{N}, -\vec{\omega}_{p}) = \{W \cdot sr^{-1} \cdot m^{-3}\} \int_{\Omega} f_{s,N}(\vec{\omega}_{N}, -\vec{\omega}_{p}, \lambda) L_{\lambda}(\vec{x}_{N}, \vec{\omega}_{N}) [\vec{\omega}_{N} \cdot \vec{n}_{N}] d\vec{\omega}_{N}, \quad (24)$$

where $L_{\lambda}(\vec{x}_N, \vec{\omega}_N)$ is the incident radiance at $\vec{x}_N$ and $f_{s,N}(\vec{\omega}_N, -\vec{\omega}_p, \lambda)$ is the SBSDF of the screen. The Lamber-tian coating reduces the expression to:

$$L_{\lambda}(\vec{x}_{N}) = \{W \cdot sr^{-1} \cdot m^{-3}\} \frac{\rho_{\lambda,N}}{\pi} \int_{\Omega^{+}} L_{\lambda}(\vec{x}_{N}, \vec{\omega}_{N}) [\vec{\omega}_{N} \cdot \vec{n}_{N}] d\vec{\omega}_{N}. \quad (25)$$

where $\rho_{\lambda,N}$ is the reflectance of the screen.

Reconsider, then, Fig. 2. Components visible from a position on the screen are restricted to the aperture $\mathcal{I}$, the base $\mathcal{B}$, the fisheye $\mathcal{F}$, the screen $\mathcal{N}$ itself, and the material $\mathcal{M}$. Breaking down the domain of integration accordingly:

$$L_{\lambda}(\vec{x}_{N}) = \{W \cdot sr^{-1} \cdot m^{-3}\} \frac{\rho_{\lambda,N}}{\pi} \int_{\mathcal{I} \cup \mathcal{B} \cup \mathcal{F} \cup \mathcal{N} \cup \mathcal{M}} L_{\lambda}(\vec{x}_{N}, \vec{\omega}_{N}) [\vec{\omega}_{N} \cdot \vec{n}_{N}] d\vec{\omega}_{N}. \quad (26)$$

Contributions from the first three components are negligible. The contribution from the screen is a noise called interreflections. The contribution from the material conveys the information of its SBSDF.

This contribution is:

$$\int_{\mathcal{M}} L_{\lambda}(\vec{x}_{N}, \vec{\omega}_{N}) [\vec{\omega}_{N} \cdot \vec{n}_{N}] d\vec{\omega}_{N} = \{W \cdot sr^{-1} \cdot m^{-3}\} \lim_{\frac{d_M}{d_N} \to 0} L_{\lambda}(\vec{x}_{M}, \vec{\omega}_{O}) \Omega_{M}(\vec{x}_{N}), \quad (27)$$

22. With $\Omega_F(\vec{x}_N) = \frac{a_F}{||\vec{x}_N - \vec{x}_F||^2} \{sr\}$ and $a_N = \frac{\Omega_p ||\vec{x}_N - \vec{x}_F||^2}{[-\vec{\omega}_p \cdot \vec{n}_N]} \{m^2\}$.

23. Also called the rendering equation in computer graphics.

where $\vec{x}_M$ is a position on the material, $\vec{\omega}_o$ the direction from $\vec{x}_M$ to $\vec{x}_N$, $\Omega_M(\vec{x}_N)$ the solid angle subtended by the material at $\vec{x}_N$²⁴, and with $[\vec{\omega}_N \cdot \vec{n}_N] = 1$.

The radiance scattered by the material is therefore:

$$L_{\lambda}(\vec{x}_{M}, \vec{\omega}_{o}) = \{W \cdot sr^{-1} \cdot m^{-3}\} \lim_{\frac{d_{M}}{d_{N}} \to 0} \frac{\pi L_{\lambda}(\vec{x}_{N}) - \rho_{\lambda,N} \int_{\mathcal{N}} L_{\lambda}(\vec{x}_{N}, \vec{\omega}_{N}) [\vec{\omega}_{N} \cdot \vec{n}_{N}] d\vec{\omega}_{N}}{\rho_{\lambda,N} \Omega_{M}(\vec{x}_{N})}. \quad (28)$$

## 5.3 SBSDF of the Material

The SBSDF of the material is, by definition:

$$f_s(\vec{\omega}_I, \vec{\omega}_o, \lambda) = \lim_{\frac{d_M}{d_N} \to 0} \frac{L_{\lambda}(\vec{x}_M, \vec{\omega}_o)}{L_{\lambda}(\vec{x}_M, \vec{\omega}_I) [\vec{\omega}_I \cdot \vec{n}_M] \Omega_I(\vec{x}_M)} \{sr^{-1}\}, \quad (29)$$

where $\vec{\omega}_I$ is the direction from the position $\vec{x}_M$ to the aperture, and $\Omega_I(\vec{x}_M)$ the solid angle subtended by the aperture at $\vec{x}_M$. Taking the circle segment distance into account²⁵ leads to the following formal expression of the SBSDF measured in the instrument:

$$f_s(\vec{\omega}_I, \vec{\omega}_o, \lambda) = \{sr^{-1}\} \lim_{\frac{d_M}{d_N} \to 0} \frac{d_N^2 - d_M^2}{\pi d_M^2} \cdot \frac{\pi L_{\lambda}(\vec{x}_N) - \rho_{\lambda,N} \int_{\mathcal{N}} L_{\lambda}(\vec{x}_N, \vec{\omega}_N) [\vec{\omega}_N \cdot \vec{n}_N] d\vec{\omega}_N}{\rho_{\lambda,N} L_{\lambda}(\vec{x}_M, \vec{\omega}_I) [\vec{\omega}_I \cdot \vec{n}_M] \Omega_M(\vec{x}_N)}. \quad (30)$$

# 6 RECOVERY OF THE SBSDF

This equation²⁶ clarifies the relationship the acquisition data set $D$ and the SBSDF. Yet, it does not say in itself how to recover the function.

## 6.1 Compensation of the Interreflections

<figure><img src="image_11.png" /><figcaption>Fig. 9. Interreflections within the screen $\mathcal{N}$.</figcaption></figure>

A first difficulty arises from the interreflections term, noted $E_{\lambda,N}(\vec{x}_N)$:

$$E_{\lambda,N}(\vec{x}_N) = \int_{\mathcal{N}} L_{\lambda}(\vec{x}_N, \vec{\omega}_N) [\vec{\omega}_N \cdot \vec{n}_N] d\vec{\omega}_N \quad \{W \cdot m^{-3}\}. \quad (31)$$

24. An elliptic integral.

25. Yields $\lim_{\frac{d_M}{d_N} \to 0} \Omega_I(\vec{x}_M) = \frac{\pi d_M^2}{d_N^2 - d_M^2} \{sr\}$.

26. Together with (12), (18), and (23).
Interreflections are an infinitely recursive phenomenon, of decreasing intensity: a fraction of the light incident on an infinitesimal surface on the screen is scattered towards its whole surface that, in turn, scatters a fraction of it towards the infinitesimal surface, etc. The detected flux is therefore a dynamic quantity that has converged at the speed of light.

To estimate $E_{\lambda,\mathcal{N}}(\vec{x}_{\mathcal{N}})$, suppose that the screen is subdivided into $n_{\mathcal{N}}$ smaller surfaces and its radiance $L_{\lambda}(\vec{x}_{\mathcal{N}})$ discretized over that subdivision$^{27}$. Then, (31) can be rewritten:

$$E_{\lambda,\mathcal{N}}(\vec{x}_{\mathcal{N},l}) \approx \{\text{W} \cdot \text{m}^{-3}\} \quad (32) \\ \frac{1}{a_{\mathcal{N},l}} \sum_{\substack{m=1 \\ m \neq l}}^{n_{\mathcal{N}}} \left[ L_{\lambda,m} \int_{\mathcal{N}l} \int_{\mathcal{N}m} \frac{[\vec{\omega}_{l,m} \cdot \vec{n}_{\mathcal{N},l}] [\vec{\omega}_{m,l} \cdot \vec{n}_{\mathcal{N},m}]}{||\vec{x}_{\mathcal{N},l} - \vec{x}_{\mathcal{N},m}||^2} d\mathcal{N}_m d\mathcal{N}_l \right],$$

where $\mathcal{N}_1, \dots, \mathcal{N}_{n_{\mathcal{N}}}$ are the $n_{\mathcal{N}}$ surfaces of the subdivision, $a_{\mathcal{N},1}, \dots, a_{\mathcal{N},n_{\mathcal{N}}}$ their areas, $\vec{x}_{\mathcal{N},1}, \dots, \vec{x}_{\mathcal{N},n_{\mathcal{N}}}$ positions on those surfaces, $\vec{n}_{\mathcal{N},1}, \dots, \vec{n}_{\mathcal{N},n_{\mathcal{N}}}$ normal vectors to the screen in these positions, $\vec{\omega}_{l,m}$ the direction from a position $\vec{x}_{\mathcal{N},l}$ to a position $\vec{x}_{\mathcal{N},m}$, and $L_{\lambda,l}$ the radiance of the $l^{\text{th}}$ surface.

Now, using Fig. 9, the following exact reduction is obtained:

$$E_{\lambda,\mathcal{N}}(\vec{x}_{\mathcal{N},l}) = \frac{1}{d_{\mathcal{N}}^2} \sum_{\substack{m=1 \\ m \neq l}}^{n_{\mathcal{N}}} \left[ a_{\mathcal{N},m} L_{\lambda,m} \right] \quad \{\text{W} \cdot \text{m}^{-3}\} \quad (33) \\ = E_{\lambda,\mathcal{N}}(\vec{x}_{\mathcal{N},1}) + \frac{1}{d_{\mathcal{N}}^2} \left[ a_{\mathcal{N},1} L_{\lambda,1} - a_{\mathcal{N},l} L_{\lambda,l} \right].$$

This simplification$^{28}$ brings with it a drastic acceleration of the estimation of the interreflections, as only one value $E_{\lambda,\mathcal{N}}(\vec{x}_{\mathcal{N},l})$ needs to be computed $\mathcal{O}(n_{\mathcal{N}})$, the remaining $n_{\mathcal{N}} - 1$ being computed $\mathcal{O}(1)$.

## 6.2 Subdivision of the Hemisphere

<figure><img src="image_12.png" /><figcaption>Fig. 10. Noé and Péroche's hemisphere subdivision [44].</figcaption></figure>

Many schemes to efficiently subdivide the unit hemisphere have been put forward in the literature. For the problem at hand, the ideal subdivision would be: (i) made of spherical caps, associating a radiance sample $L_{\lambda,l}$ with

27. These two stages are discussed in the following Subsections 6.2 and 6.3

28. Thanks to the canceling of inverse-square and cosine factors, the interreflections component $E_{\lambda,\mathcal{N}}(\vec{x}_{\mathcal{N},l})$ tends towards a constant when the resolution of the subdivision tends to infinity.

the topological sphere; (ii) uniform, in order not to favor one single direction; (iii) flexible in the choice of the number $n_{\mathcal{N}}$ of faces; and (iv), implicit, so as to minimize the storage space of the SBSDF by not storing a the coordinates $(\vec{\omega}_i, \vec{\omega}_o, \lambda)$ but only the values $f_s(\vec{\omega}_i, \vec{\omega}_o, \lambda)$.

Fig. 10a depicts the elegant solution provided by Noé and Péroche [44] subdivision, where: (iv) the structure is implicit (as opposed to stochastic approaches); (iii) the choice of the number $n_{\mathcal{N}}$ of faces is flexible; (ii) the variance of their areas is low (significantly lower than in recursive platonic subdivisions, notably); and (i), faces are closely spherically square.

Consider Fig. 10b and let a central direction be identified on each face as:

$$\vec{\omega}_{\mathcal{N},l} = \left( \frac{\theta_l + \theta_{l+1}}{2}, \frac{\varphi_{l,m} + \varphi_{l,m+1}}{2}, 1 \right)^T, \quad (34)$$

where $l$ in $\{1, \dots, n_{\mathcal{N}}\}$ is the index of the face.

## 6.3 Samples of the SBSDF

For each incident direction, $n_c$ irradiances samples were reconstructed by (18), with each sample a $n_{\mu}$-valued spectral distributions. In turn, the radiance of the screen is recovered by projecting these samples on the screen, subdivided into $n_{\mathcal{N}}$ faces and $n_{\mu}$ spectral samples per face.

In more detail, let $L_{\lambda,l,i}$ be the $i^{\text{th}}$ spectral sample of the $l^{\text{th}}$ face. From (23):

$$L_{\lambda,l,i} = \frac{1}{\Omega_p} \cdot \frac{\sum_{p=1}^{n_c} w_{p,l} E_{\lambda,p,i}}{\sum_{p=1}^{n_c} w_{p,l}} \quad \{\text{W} \cdot \text{sr}^{-1} \cdot \text{m}^{-3}\}, \quad (35)$$

where $E_{\lambda,p,i}$ is the $i^{\text{th}}$ spectral sample of $E_{\lambda,p}$ and $w_{p,l}$ a weighting of its influence on $L_{\lambda,l,i}$.

Let $\alpha_{p,l}$ be the angle formed by the directions $\vec{\omega}_{\mathcal{N},l}$ and $\vec{\omega}_o$. The weighting $w_{p,l}$ was constructed as a radial basis function in spherical geometry, decreasing the influence of $E_{\lambda,p,i}$ as $\alpha_{p,l}$ increases. In practice, a centered unitary Gaussian with full width at half maximum of $\sqrt{a_{\mathcal{N},l}}$ was used, in order to provide a distance threshold dependent on the resolution of the subdivision.

The interreflections can now be estimated and compensated for. Yet, a number of unknowns still remain to be determined before the actual recovery of the SBSDF$^{29}$. These unknowns are accounted for by a reference measurement procedure with a material of known SBSDF: Spectralon for reflectance and a diffusing glass for transmittance.

These interreflections compensation and reference calibration stages are summarized by the following form:

$$\begin{gathered} f_{s,l,i}(\vec{\omega}_l) = \{\text{sr}^{-1}\} \quad (36) \\ \frac{\pi d_{\mathcal{N}}^2 L_{\lambda,l,i} - \rho_{\lambda,\mathcal{N},i} \sum_{\substack{m=1 \\ m \neq l}}^{n_{\mathcal{N}}} a_{\mathcal{N},m} L_{\lambda,m,i}}{\pi d_{\mathcal{N}}^2 L_{\lambda,l,i}^R - \rho_{\lambda,\mathcal{N},i} \sum_{\substack{m=1 \\ m \neq l}}^{n_{\mathcal{N}}} a_{\mathcal{N},m} L_{\lambda,m,i}^R} \cdot f_{\mathcal{R}}(\vec{\omega}_l, \vec{\omega}_{\mathcal{N},l}, \lambda_i), \end{gathered}$$

29. These unknowns are, from (30): radiance incident on the material $L_{\lambda}(\vec{x}_m, \vec{\omega}_l)$ and solid angle subtended by the material on the screen $\Omega_M(\vec{x}_m)$; from (23): area of the fisheye $a_{\mathcal{N}}$, its transmittance $\tau_{\lambda,\mathcal{N}}$, and that of the optical adapter $\tau_{\lambda,\mathcal{c}}$; and from (12): residual of multiplicative noise $\xi_{*,p}$.
where $L_{\lambda,m,i}^R$ is the reference measurement, $f_\lambda$ the SBSDF of the reference material, and $f_{s,l,i}(\vec{\omega}_i)$ a sample of the measured SBSDF.

# 7 RESULTS AND DISCUSSION

## 7.1 Validation

<figure><img src="image_13.png" /><figcaption>Fig. 11. Albedo as measured using the instruments REFLET, EZContrast and [this].</figcaption></figure>

A range of validation experiments were conducted during the development of the instrument. A two-axis motorized Laser was notably installed in place of the material, to validate the radiometric and geometric calibrations stages. The spectra recovered by the omnidirectional spectral camera were also validated by comparing measurements with a spectrophotometer [45].

Ultimately, a round-robin validation was conducted with the sequential REFLET 90 [46] and the direct imaging EZContrast XL88MS [47] instruments, on the basis of a 'yellow glossy-specular' automotive paint material³⁰.

The spectral coefficient of reflection of this material $\theta_i$ is plotted in Fig. 11 as a function of the incident zenith angle and at two wavelengths. The values were computed by integrating monochromatic BRDF lobes over the refected domain, rejecting samples within a small solid angle around the specular and retroreflection directions, as all three instruments behave very differently in these regions.

## 7.2 Plots of Measured SBSDFs

Five measured materials are presented in Fig. 12. Fig. 12a is a window made of transparent polycarbonate cells; Fig. 12b is a 'green glossy' metal sheet; Fig. 12c is a 'yellow glossy-diffuse' ground coating; Figs. 12d and e are common fabrics, with visible threads for Fabric II.

An acquisition image is presented in Fig. 13. Measurements were performed using $n_\mu = 80$ tunes and

<figure><img src="image_14.png" /><figcaption>Fig. 12. Five measured materials</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 13. Acquisition of Fabric II (Fig. 12e).</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 14. SBTDF ($\log_2$ of Polycarbonate (Fig. 12a).</figcaption></figure>

30. Two other materials were actually measured, but discrepancies between the REFLET 90 and EZContrast measurements led to their rejection as valid for the intended experiment.
<figure><img src="image_17.png" /><figcaption>Fig. 15. SBRDF (log₂) of Painted metal (Fig. 12b).</figcaption></figure>

<caption>Fig. 16. SBRDF of Ground coating (Fig. 12c).</caption>

$n_N = 1001.103$ faces. The acquisition time per incident direction ($n_μ \times n_N$ samples) was around one minute. Corresponding monochromatic BSDF lobes were plotted at two wavelengths in Figs. 14 to 17.

These figures show that signal-to-noise ratio (SNR) decrease with an increase in the incident zenith angle $\theta_i$, as less light is received by the material for grazing angles. An increase in the coefficient of reflection with respect to the angle $\theta_i$ is visible in Figs. 16 to 18. A specular peak is apparent in Figs. 15e and f, as well as off-specularity, of which the offset with the specular direction increases with $\theta_i$. A strong directional feature is present in Fig. 14, induced by the parallel cells of Polycarbonate. A strong anisotropic pattern is also visible in Figs. 18e and f, induced here by the thread structure of Fabric II.

A wavelength discrepancy is observable in Figs. 16 and 17, mitigating a hypothesis by Nicodemus et al. [1] that the spectral and bidirectional components may be separable for common materials.

## 7.3 Perspectives

The interreflection component may be quite significant, potentially higher than the direct reflection, thus leading to low SNR in (36). Also, the 'directional signature' of the instrument, or how sharp a peak can be measured meaningfully, is highly dependent on the ratio $\frac{d_M}{d_N}$. Improvements on these two aspects are expected from a new screen that is being built, with both a lower reflectivity $\rho_{\lambda, N}$ and a lower $\frac{d_M}{d_N}$ ratio³¹.

Besides, a realistic SBSDF model could be fitted to the Spectralon reference measurement in (36) as concordant studies, such as by Bhandari et al. [48], suggest that assuming this material to be Lambertian might lead to inaccuracies.

Finally, advance in light emitting diodes (LED) technology call for the investigation of LED-based multispectral illumination systems, such as presented by Park et al. [49] and Fauch et al. [50]. Such system would improve the ergonomy of the source module.
<figure><img src="image_18.png" /><figcaption>Fig. 17. SBRDF of Fabric I (Fig. 12d).</figcaption></figure>

<caption>Fig. 18. SBRDF of Fabric II (Fig. 12e).</caption>

## CONCLUSION

The hope for the emergence of an efficient optical scattering measurement instrument is shared by numerous communities, for such applications as appearance modelling, paint and coating development; surface defect detection; realistic image synthesis; telecommunications; food and agricultural product quality control; cosmetics; target recognition; normalization of illumination conditions on images; and road quality control and signalization, to name a few. With regard to the energy crisis, it may also seem particularly urgent to provide the lighting engineering community with such a facility.

The first contribution was a detailed survey and a classification of available approaches. It is the hope of the authors that this work can build bridges between the aforementioned disciplines.

A new approach was then presented, classified as *indirect imaging* with a *Lambertian ¾-of-a-sphere* screen, and able to densely sample the spectral-bidirectional domain.

The prototype acquisition device is composed of a metal halide lamp installed the focus of a parabolic

luminaire, a motorized Lambertian ¾-of-a-sphere screen, and a composite omnidirectional spectral camera made of a fisheye lens, a liquid crystal tunable filter, and a radiometry-grade camera.

The data processing was next presented in detail. It relies on a model of the image formation process within the acquisition device, and is expressed in the relevant radiometric formalism of the SBSDF.

Validation of this approach was carried out through a round-robin experiment with two other instruments. Finally, soundness of the approach was emphasized by displaying plots of SBSDF from five measured materials.

## 8 RELATED WORK IN DETAIL

This last section lays the foundation of the classification introduced in Section 1. Despite our best efforts, it cannot claim to be an exhaustive survey of the multitude of unique instruments found in the literature. Rather, classes of approaches were identified and ideas gathered under a unified formalism.
Also, hybrid methods, such as enriching an imaging instrument with a discrete measurement of the specular component, are not included. As such, it is by no means the wish of the authors to express that these methods are not fruitful. Instead, it was considered that the aim of structuring the field with a comprehensive categorization would be lost in a dense network of interdependency relationships.

For each instrument, shortcomings which led to the proposition of the imaging indirect with a Lambertian ¾-of-a-sphere screen approach are discussed.

In the rich variety of existing instruments, the single invariant identified was the presence of a source, a material, and a detector. This detector defines the first separation in the classification: sequential instruments rely on a monopoint detector whereas imaging instruments rely on a camera.

## 8.1 Sequential Approach

<figure><img src="image_19.png" /><figcaption>(a) Picture from Serrot et al. [2].</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>(b) Picture from Deniel [3].</figcaption></figure>

<caption>Fig. 19. Examples of sequential instruments.</caption>

The 'gold-standard' sequential approach relies on the displacement of the source and detector along two spherical trajectories relative to the material, with a mechanical axis associated with each of the angles $(\theta_i, \varphi_i, \theta_o, \varphi_o)$.

### 8.1.1 Gonioreflectometer by Serrot et al. [2]

<figure><img src="image_21.png" /><figcaption>Fig. 20. Sequential approach by Serrot et al. [2].</figcaption></figure>

A characteristic implementation by Serrot et al. [2] is depicted in Figs. 19a and 20. The displacement of the source along axis $A_1$ varies the zenith angle $\theta_i$. Axis $A_2$

varies its azimuthal counterpart $\varphi_i$. The spectrophotometer is mobile along axes $A_3$ and $A_4$, varying the outgoing angles $\theta_o$ and $\varphi_o$.

The acquisition time, for a fixed incident direction with an angular resolution of ten degrees in angles $\theta_o$ and $\varphi_o$ (108 samples over a quarter of the unit sphere) and a spectral resolution of two nanometers, is about ten minutes [2].

### 8.1.2 Gonioreflectometer by Deniel [3]

A second sequential instrument with reduced proportions<fnref n="32" /> is illustrated in Fig. 19b. The source is a halogen fibre-optic and the detector is a fibre-optic spectrophotometer. Samples are collected over the whole visible-BRDF domain, excluding a $\pi/8$ steradian solid angle around the direction of retroreflection. The measurement time for a fixed incident direction is in the order of one week [3].

## 8.2 Imaging Approaches

The mechanical displacement of the source and detector for each bidirectional configuration severely limits the speed and resolution of sequential instruments. On the contrary, emerging imaging approaches are hopeful of a much higher resolution and speed, each pixel potentially accounting for a bidirectional configuration.

However, where sequential instruments conveniently capture spectral distributions using a spectrophotometer as the detector, the level of a pixel is a scalar function of this distribution<fnref n="33" />. In addition, cameras are seldom calibrated in a standard spectral space (e.g. the $V_\lambda$-defined photometric space [51]), and measured levels are then dependent on their specific quantum efficiencies.

Also, the recovery of the SBSDF from the acquired data set is more complex with imaging instruments<fnref n="34" />.

Two main imaging approaches were identified: instruments where the camera was oriented towards the material, denoted as direct; and instruments where it was oriented towards an intermediate surface, called indirect.

### 8.2.1 Direct Imaging Approaches

The direct approach relies on the variation of the bidirectional configuration along the surface of the material.

8.2.1.1 Spherical material: Its development can be traced back to the study by Minnaert [4], relying on the sphericity of the Moon to permute bidirections associated with positions observed on its surface.

Contemporary direct instruments with a spherical material were proposed by Marschner et al. [5], Matusik et al. [6], and Kim et al. [7]. The latter, depicted in Fig. 21, is a recent embodiment that adds accounting for the spectral

<footnote n="32">32. As a general rule, bidirectional scattering measurement instruments can be scaled up or down to the limits of geometrical optics.</footnote>

<footnote n="33">33. A model of this relationship adapted to linear ILT CCDs was presented in Subsection 4.1.</footnote>

<footnote n="34">34. It was found that, despite this fact, this process was never described to its full extent in the literature. This consideration led to the devising of the framework presented in Sections 5–6.</footnote>
<figure><img src="image_22.png" /><figcaption>Fig. 21. Kim et al.'s *direct* instrument (picture from [7]).</figcaption></figure>

dimension. The detector is indeed an LCTF-based spectral camera (as in the instrument presented in this paper) coupled with a high dynamic-range CMOS camera. Despite referencing Hardeberg, the spectral reconstruction, to recover CIEXYZ color components, is performed outside of Fredhlom's integral equation framework. Hullin et al. [8] also recently presented a similar setup to assess the bispectral bidirectional reflectance and reradiation distribution function (bispectral BRRDF; for fluorescent scattering) for selected wavelengths and bidirections. Still, the spectral reconstruction problem is likewise not formalized.

Despite being fast and economic, the *direct* approach with a *spherical* material is restricted to isotropic bidirectional scattering; it makes measuring the transmittance impossible; it assumes the material to be highly malleable (i.e. paints); and the implications of using the BRDF formalism [1] for scattering by non-planar surface remains a subject of controversy.

<figure><img src="image_23.png" /><figcaption>Fig. 22. *Direct* approach by Lu et al. [9].</figcaption></figure>

### 8.2.1.2 Material in cylindrical stripes: The instrument by Lu et al. [9] and Ngan et al. [10], illustrated in Fig. 22, is classified as *direct imaging* with the material in *stripes* wrapped on a right *cylinder*. The stripes are along different orientations in the material, allowing for a selected measurement of the anisotropy.

Apart from this aspect and a relaxed malleability constraint, the approach presents the same limitations as the previous *direct spherical* design, in addition to a particularly delicate characterization of the covered measurement domain and material preparation.

## 8.2.2 *Indirect Imaging Approaches*

In *indirect imaging* instruments, the scattering is captured on an intermediate surface, or screen, towards which the camera is oriented. These instruments were further categorized according to the SBSDF of that screen, and then according to its geometry.

Specular, Lambertian and glossy instruments were found; with hemiellipsoidal, off-axis parabolic, kaleidoscopic, triangular, cubic, hemispherical, and spherical geometries. The approach proposed in this paper is classified as *indirect imaging* with a Lambertian ¾-of-a-sphere screen.

<figure><img src="image_24.png" /><figcaption>Fig. 23. *Indirect* approach by Ward [11].</figcaption></figure>

#### 8.2.2.1 Specular hemiellipsoidal screen: The semi-

The semi-indirect imaging instrument was proposed by Ward [11] and its principle is illustrated in Fig. 23. The beam from the source is transmitted by a half-silvered hemiellipsoidal screen and illuminates the material installed at one of the foci. The scattered light is then specularly reflected towards the second focal point, where an omnidirectional camera is positioned. A single capture therefore almost instantly collects a whole reflection lobe. Moreover, the associated incident direction varies over the unit hemisphere by rotations of the source and material along two axes.

However, the construction of a half-silvered hemiel-
lipsoidal screen proved too difficult at the time and a
simpler hemispheric PMMA surface was substituted for
the prototype.

Pursuing this research, Andersen et al. [12] recently managed to free-blow a half-silvered hemiellipsoidal surface, though the process proved to be delicate and error-prone. In this second implementation of the catoptrics design by Ward [11], a filter-wheel was added in order to account for the spectral dimension. Spectral reconstruction was performed using an *ad hoc* algorithm, but without reference to the spectral imaging literature. Besides, the handling of the polarization and refraction phenomena was not discussed. The authors of this paper also explored the filter-wheel spectral imaging alternative (see Fig. 24), but abandoned it in favor of the use of a
<figure><img src="image_25.png" /><figcaption>Fig. 24. Filter-wheel [22], spectral imaging alternative.</figcaption></figure>

LCTF, vastly improving the spectral resolution.

Several papers have been dedicated to the instrument by Andersen et al. [12] but little experimental evidence was actually provided demonstrating that the method is functional in practice. The white paper photometric solid for BRDF presented in [12], for instance, despite being one of the few BRDF data plot from an imaging instrument found in the literature<fnref n="35" /> displays a questionable shape for a surface acknowledged as Lambertian.

<figure><img src="image_26.png" /><figcaption>Fig. 25. *Indirect* approach by Dana and Wang [13].</figcaption></figure>

### 8.2.2.2 Specular off-axis parabolic screen: The *indirect* instrument with an *off-axis parabolic mirror* by Dana and Wang [13] is illustrated in Fig. 25.

The beam of the source is transmitted by a beamsplitter towards the *off-axis parabolic mirror* and focused on the material. The backscattered light is then detected by the camera. Displacement of the diaphragm along axes $A_{1-2}$ varies the incident direction; displacement of the mirror along axes $A_{3-5}$ 'sweeps' the beam along the material, accounting for the spatial variation of the BRDF.

This instrument popularized the concept of bidirectional texture functions (BTF), but measures the BRDF and BTF over a significantly truncated bidirectional domain.

A similar design dedicated to isotropic BRDF measurement was recently proposed by Ren and Zhao [14], measuring the function at a fixed wavelength (650 nm), using a laser as the source.

### 8.2.2.3 Specular kaleidoscopic screen: The instrument by Han and Perlin [15] depicted in Fig. 26 exploits

<figure><img src="image_27.png" /><figcaption>Fig. 26. *Indirect* approach by Han and Perlin [15].</figcaption></figure>

the optical properties of a tapered kaleidoscope to both observe and illuminate the material.

The source is a videoprojector whose beams, after multiple reflections within the kaleidoscope, illuminate the material from multiple directions. Each pel of the camera in turn 'sees' the material from a specific direction.

The three-face approach by Han and Perlin is optimal in terms of simplicity but results in a sparse bidirectional resolution. Bangay and Radloff [16] studied new geometries for the kaleidoscope. Despite this, the impact of the multiple polarizing events induced by the specular reflections is still not clear.

<figure><img src="image_28.png" /><figcaption>Fig. 27. *Indirect* approach by Deniel [3].</figcaption></figure>

### 8.2.2.4 Lambertian Cubic Screen: The indirect Lambertian instrument by Deniel [3] is depicted in Fig. 27.

The Lambertian radiance of the screen, detected by the omnidirectional camera, originates from the material, as well as from diffuse interreflections within the screen. These interreflections were compensated for by an inverse radiosity solution.

A strength of this approach is the depolarization brought about by the Lambertian reflections. However, artifacts are induced by the edges of the cubic screen, and the incident direction $\tilde{\omega}_i$ cannot be varied.

### 8.2.2.5 Lambertian hemispherical screen: Understandably, few details are available on the Imaging Sphere IS-SA™ commercialized by Radiant Zemax but from information made public by Yeo et al. [21], the instrument can still be classified as indirect Lambertian with a hemispherical screen. Its general principle is illustrated in Fig. 28.

<footnote n="35">35. The only other bidirectional scattering data plots from an imaging instrument were found in the dissertation by Andersen [19] and associated publications, dedicated to another indirect imaging instrument with a Lambertian triangular screen designed by Scartezzini et al. [18]</footnote>
<figure><img src="image_29.png" /><figcaption>*M*: material,
*S*: source,
*C*: imaging photometer,
*N1*: Lamb. hemisp.,
*N2*: curved mirror,
*A1*: rotation *M*,
*A2*: rotation *S*,
*D*: driver.</figcaption></figure>

<caption>Fig. 28. *Indirect* approach by Yeo et al. [21].</caption>

However, neither the incident beam zenith displacement principle, nor the information regarding the data processing have been made public in any great detail, and although there is a 'light trap' depicted in Fig. 28, neither its purpose or behavior when the specular direction evolves have been disclosed.

<figure><img src="image_30.png" /><figcaption>*M*: material,
*S*: Ar/Kr ion laser,
*C*: fisheye camera,
*N*: Lamb. sphere,
*A1*: galvanometer,
*D*: driver.</figcaption></figure>

<caption>Fig. 29. *Indirect* approach by Hawkins et al. [20].</caption>

### 8.2.2.6 Lambertian spherical screen
The indirect instrument with a *Lambertian spherical screen* by Hawkins et al. [20] is illustrated in Fig. 29. Conceived for the assessment of the reflectance field of a three-dimensional object, it shows an interest in the assessment of the bidirectional scattering and is presented as such.

As for Deniel's *imaging* instrument, the Lambertian coating of the screen simplifies the interreflections compensation from a full global illumination to a simpler radiosity solution, which is then further reduced in the spherical geometry of the screen, to the subtraction of a constant to each pixel<fnref n="36" />.

A notable innovation of the instrument is that, relying on Helmholtz's reciprocity principle, the source and the camera are virtually permuted, which means that it assesses the appearance of the material as observed from the source under omnidirectional illumination.

For bidirectional scattering, however, a limit is that the outgoing direction $\tilde{\omega}_o$ cannot be varied.

<figure><img src="image_31.png" /><figcaption>*M*: material,
*S*: fisheye videoprojector,
*C*: camera,
*N*: glo. hemiellipsis,
*F*: foci *N*,</figcaption></figure>

<caption>Fig. 30. *Indirect* approach by Peers et al. [17].</caption>

### 8.2.2.7 Glossy hemiellipsoidal screen
The indirect instrument with a *glossy hemiellipsoidal screen* of Peers et al. [17], illustrated in Fig. 30, was, likewise, conceived for the assessment of the reflectance field of a three dimensional object, and the same consideration applies.

The source is an omnidirectional videoprojector installed at a focus of a glossy hemiellipsoidal screen. Beams from distinct pixels are reflected towards the other focal point, where the material is installed. The glossy coating spreads the beams across the surface of the material, averaging spatial variations in the BRDF.

However, the design remains conceptual, due to the difficulty of constructing a *glossy hemiellipsoidal screen* in practice. As for the design by Ward [11], a simpler hemispherical screen was substituted in the actual implementation of the instrument.

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<figure><img src="image_32.png" /></figure>

Pierre Boulenguez received the master degree in computer science in 2002 from the University of Marne-la-Vallée and the PhD degree in computer science from the University of Paris-Est in 2010. In his dissertation, he notably presented instruments for near field photometry of light sources and optical scattering's measurement. Following a Post-Doc at the CNRS/University of Poitiers in 2010, he joined the Health and Comfort Department of the CSTB (French Scientific and Technical Centre for Building) in 2012, where he

continues his research at the interface between radiometry, photometry, computer vision, and lighting simulation.

<figure><img src="image_33.png" /></figure>

Samuel Carré received the master degree in computer science in 1993 and the PhD degree in 1998 from University of Rennes 1. In 1998, he joined the lighting engineering team of the CSTB (French Scientific and Technical Centre for Building). His current research interests include physics based lighting simulation, global illumination in building environments, GPU-based rendering, and computer vision.

<figure><img src="image_34.png" /></figure>

Nicolas Noé received the ingénieur's degree from the Ecole des Mines de Saint-Etienne in 1996, the PhD degree in computer graphics from Jean-Monnet University in 1999, working on BRDFs modelization. He then joined CSTB (French Scientific and Technical Centre for Building) in the Health and Comfort Department where he manages industrial software development for transportation industry. His current research interest is electromagnetic and acoustic waves propagation simulation in complex environments

using asymptotic techniques.
<figure><img src="image_35.png" /></figure>

Laurent Najman received the ingénieur's degree from the Ecole des Mines de Paris in 1991, the PhD degree in applied mathematics from Paris-Dauphine University in 1994 with the highest honor (Félicitations du Jury), and the Habilitation à Diriger les Recherches from the University of Marne-la-Vallée in 2006. Following an industrial career, he joined the Informatics Department of the ESIEE Paris engineering school in 2002, where he is a professor and a member of the Gaspard-Monge computer science research laboratory, Université Paris-Est Marne-la-Vallée. His current research interest is discrete mathematical morphology. He recently co-edited (with Hugues Talbot) a book on mathematical morphology published by ISTE-Wiley.