1. Introduction

The bidirectional reflectance distribution function (BRDF) describes how incident radiance is redirected in all directions across a hemisphere above the surface. Since the notion of BRDF was first raised by Nicodemus [1] in 1970, it has been treated as the ultimate quantity for non-translucent materials under uniform lighting conditions. Given that the reflectance distribution determines the surface appearance of reflective materials and plays a key role in material discrimination, BRDF finds applications in various domains including computer graphics [2–6], remote sensing [7–11], color science [12–16], and many other fields [17–20]. Given its wide usage, the measurement methodology for rapidly obtaining high-accuracy BRDF data has been explored over the past few decades.

In terms of BRDF definition, multiple goniometric arms should be employed to position a light source and a detector at different angles relative to a sample, typically a flat piece of material. This specific method of BRDF measurement is widely known as goniospectrophotometer [21–24], renowned for its complexity and high degree of freedom. Measurement institutes from different countries have made contributions to the development of goniospectrophotometers [25–29]. Based on the measurement principle, data must be acquired for each angular position of either the source or the detector, leading to lengthy measurement cycles that often span several hours. Consequently, the goniospectrophotometer is often bulky and high-cost.

To accelerate the measurement process, an imaging detector can be used to capture various angles of illumination and reflection from the surface in one picture. This approach necessitates the homogeneousness of the sample surface and the fabrication of samples made in specific shapes [30], such as spherical [31] or cylindrical [32–34], to enable an imaging sensor to acquire reflection information from different directions. However, the limited field of view of the imaging detector constrains the measurement range of the reflection angle. To ensure that the collected data covers the full angle range from 0° to 180°, the detector should capture several images with the reflection information at different positions corresponding to various ranges of reflection angle. Consequently, for these image-based methodologies, the multi-position measurements can still cause the lengthiness of the measuring process. Additionally, the requirement for uniformly curved surfaces in the measured sample can present challenges in the manufacturing process. Additionally, collecting reflection data from different points on the curved surface may lead to a reduction in data accuracy.

Many approaches have been proposed to improve the compactness and measurement efficiency of image-based BRDF measurement system. In terms of avoiding the rotational movement of light source components, illumination aperture translation [35] has been used to achieve variation in illumination direction, albeit requiring a translation structure. Mukaigawa et al. have proposed a prototype incorporating a projector [36] as an optical component to alter the angle of incident light, avoiding the necessity for mechanical rotation and translation. Additionally, adopting fixed light sources [37] at specific lighting directions also improves the compactness of the system to a certain extent. To achieve a substantial reduction in measurement time, catoptric components have been integrated in image-based BRDF measurement systems. In 1992, a hemispherical mirror along with a fish-eye lens was first employed to achieve an image-based BRDF measurement device [38]. In 2006, Sipke Wadman (at Philips) firstly employed a dome mirror and a convex mirror in image-based BRDF device and named such device as “Parousiameter” [39,40]. Remarkable progress has been made in system development and optimization, involving different types of catoptric components, such as ellipsoidal mirror [41], parabolic mirror [35], cylindrical mirror [42], plane mirror [43], freeform dome [44] and kaleidoscope [45]. Systems like Parousiameter have been implemented in the commercial devices such as IS-SATM [46]. Still, the incorporation of a large 20-inch-wide hemisphere and a movable light source robotic arm in the IS-SA system has resulted in increased physical dimensions and overall weight, which may impose constraints on the practical application and deployment of the system. Methods mentioned above require relative measurement calibration based on response measurements of a standard diffuse plate under the same illumination conditions prior to formal measurement. Therefore, the fluctuation of light source illuminance during formal measurement can cause intuitive errors in the BRDF data. Moreover, in these current image-based measurement systems, RGB sensors are commonly utilized rather than a multispectral or hyperspectral one, ignoring the need for multi-wavelength BRDF data in areas such as computer graphics rendering.

Therefore, the goal of this paper is to accomplish the rapid acquisition of BRDF data in multiple visible bands with high accuracy and resolution. Building upon a parousiameter-based design, an image-based BRDF measurement method has been proposed using a high-resolution multispectral imaging sensor and a Lambertian sample which allows for real-time calibration. Compared to existing similar methods, our method offers notable advantages regarding the compactness, extended spectral range and real-time calibration. The proposed method enables obtaining BRDF data with a large FOV (up to 160°), a high angular resolution (less than 0.6°), and a high color measurement accuracy (the ΔE*_ab color difference below 2.5). Notably, the inclusion of the Lambertian sample into the system enables drift correction during the measurement process, leading to a substantial improvement in data accuracy.

2. Image-based BRDF measurement method using a catadioptric multispectral capture and a real-time Lambert calibration

The proposed method consists of four key elements: a catadioptric multispectral acquisition unit, illumination units, a Lambertian calibration reference, and a signal processing unit. The acquisition unit collects light reflected from various directions of the test sample, consisting of a semi-ellipsoidal mirror (the primary reflector), a hyperboloid mirror (the secondary reflector), a refracting lens and a multispectral camera. The illumination units comprise multiple LED light sources with stable spectra, which are controlled to emit light by turn in a predetermined order. The multispectral camera is mainly composed of objective lens, a color wheel, a step motor and a high-resolution sensor. During the measurement process, the test sample and the Lambertian calibration reference are illuminated by LED light sources lit in turn. As illustrated in Fig. 1, the reflection distribution of the test sample is captured by the objective lens and the CMOS sensor. The primary mirror and secondary mirror redirect the light reflected from the test sample and focus it onto the CMOS sensor of the multispectral camera that captures a series of images using different filters distributed on a step-motor-controlled color wheel. The camera captures both the Lambertian sample and the test sample in each image, enabling real-time calibration of the multispectral camera. Before the formal measurement, white and black boards are used to correct the dark current and obtain the sensor's physical response value. The electronic driver and USB cable enable the personal computer (PC) to send control signals to the illumination units and multispectral camera, regulating seamless integration and synchronization of these functional units during the measurement process.

 

Fig. 1. Schematic diagram of acquisition system in the proposed image-based BRDF measurement device.

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2.1 Optical system design

The imaging performance and the optical structure for the catadioptric capture subsystem play decisive roles in achieving accurate BRDF measurements. Our design opted for a structure of two mirrors along with a refractive objective lens. It allows for efficient collection of the scattered illumination from the sample, offering a large field of view and high-quality imaging. The sample well and the entrance pupil of the objective lens were established in an object-image conjugate relationship, achieved through the combined function of the catadioptric system (comprising the two mirrors) and the refractive objective lens. By leveraging this optical setup, the reflected light from the sample could be accurately collected and focused onto the imaging sensor. A semi-ellipsoidal mirror was selected as the primary reflector (M1), and a hyperboloid mirror was selected as the secondary reflector (M2), as depicted in Fig. 2(a). In this configuration, the center of the sample well and one focal point of M2 are well aligned with the two focal points of M1, respectively. Based on the optical properties of conic sections, the light scattered from the sample at a wide range of directions could all be focused at the other focal point of M2. This initial structure allows the proposed method to efficiently focus scattered light from the test sample onto the camera, thus enabling the measurement of BRDF across a wide range of reflection directions.

 

Fig. 2. (a) System setup with an off-axis conic surface showing the conjugate pair of the aperture stop and image plane. (b) Final system for BRDF measurement. (c) MTF curve of the final system.

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To ensure an adequate area of the sample captured by the measurement system, the entrance pupil of the optical system was located at the sample well with a diameter of 3 mm. The radius of the primary mirror was restricted to ensure a compact system. Additionally, considering the blocking effect of the secondary mirror, it is necessary to reduce the field of view on the secondary mirror side. Another important factor is the distance between the sample well and the secondary mirror. The optimized distance allows the Lambertian calibration sample positioned near both the sample well and the secondary mirror. Consequently, the calibration standard can be effectively illuminated with the sample well and captured by the image sensor along with the secondary mirror. Note that the secondary mirror should not obstruct the incoming light. To simplify the optimization process and achieve desirable imaging characteristics, an ideal lens with a focal length of 35 mm was chosen as the objective lens in the system. The proposed measurement system can achieve improved imaging quality and streamline the optimization process by considering the design principles and incorporating the ideal lens. The actual eyepiece is opted as a readily available stock lens with a focal length of 35 mm. This choice not only meets the system's requirements but also helps cut costs compared to custom-designed alternatives. Once the entrance pupil size and the desired FOV are determined, the corresponding imaging zone can be roughly estimated based on the object-image conjugate relationship. In this case, the calculated image diameter is approximately 10 mm.

In line with the design principles outlined earlier, the performance of the optical system was optimized to determine the related parameters, including the mirror curvatures, mirror sizes, and lens positions. The optimization aims to reach a tradeoff between the desired field of view, image sharpness, and overall system compactness. A large FOV (-60° to 75° in yz direction and -80° to 80° in xz direction) from the entrance pupil was set from the entrance pupil, limited by the position of the secondary mirror. The 2D layout of the optimized optical system is shown in Fig. 2(b). Figure 2(c) depicts the MTF (Modulation Transfer Function) curve of the system, which is a critical metric for evaluating the system's capacity to accurately transfer grayscale information across various spatial resolutions. This capability directly affects the angular resolution of the system when measuring BRDFs. To investigate the imaging resolution of light rays across various reflected directions, the MTF was assessed for field angles extracted from the meridian plane, sagittal plane, and the edge of the view’s periphery. As illustrated in Fig. 2(c), the MTF value at all fields remains above 0 when the spatial frequency is below 18 lp/mm, representing the capability of energy collection of the system. Due to the large FOV and high relative aperture of the optical system, the two-mirror front group cannot fully correct the aberrations, which results in low MTF values. However, it successfully collects all the principle rays over the whole FOV to the entrance pupil of the subsequent lens.

2.2 Illumination unit

To achieve the mobility and commercial viability of the device, the illumination unit needs to maintain a compact configuration and seamlessly integrate into the overall system. Thus, the device is designed to measure BRDF at only five representative incident angles. Each lighting module consists of a reflector-based LED point source without lens. The LED incorporates a mixture of various phosphors, enabling its spectrum to cover the range from 400 nm to 780 nm. The conventional BRDF systems often employ collimated light sources. However, if we utilize collimated light sources in our system, the beam diameter of the light source needs to be at least 40 mm to ensure that the sample aperture and internal standard plate are fully covered from various angles of incidence, which results in a relatively large LED aperture. The choice of an LED point source involves a trade-off between size and collimation, which caters to meet the required lighting conditions and helps to minimize the aperture area on the hemisphere of the main mirror at the same time, reducing the impact of the mirror aperture on the measurement range. The size of the LED source is 1 mm × 0.8 mm, resulting in an incident angle of approximately ± 0.3° at the center of the sample, considering a lighting distance of 100 mm. The actual optical path diameter at the sample port is 3 mm, which results in an illumination angle deviation of ± 1° across different positions within that region.

Figure 3(a) depicts the view of the system on the illumination side where five LEDs are mounted in the ellipsoidal reflector dome. These LEDs are directed towards the center of the sample hole with illuminating angles of 0°, 20°, 40°, 45° and 60° relative to the sample, respectively. During the measurement, each LED is switched on by turns.

 

Fig. 3. Basic structure of the illumination unit. This unit consists of five LEDs embedded in the hemisphere, all pointing toward the center of the sample. During the measurement, each LED is switched on by turns.

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The top view of the system is pictured in the Fig. 3(b). There is an open area on the primary mirror directly above the secondary mirror, providing the camera with an unobstructed view of the secondary mirror and the white standard. Thus, these elements can be completely included in the captured multispectral image. The mounting axis of the light sources is positioned perpendicular to the main optical axis, thereby protecting the information integrity of the main reflected beam without any loss from the notch on the primary mirror.

During the measurements, the LED intensity may exhibit fluctuation to a certain degree. It may be caused material variances, manufacturing processes, temperature or the prolonged usage, which may introduce errors in the measurement results. Thus, appropriate corrections and compensations need to be implemented during the calibration process. The inclusion of the Lambertian calibration sample in the captured image enables real-time drift correction. Before the formal measurement, both the white and black Lambertian boards need to be measured to correct the dark current and obtain the sensor response of the reference sample. However, the luminance intensity of LEDs may temporally fluctuate due to heat dissipation during long continuous measurements, consequently changing the sensor response. In this case, illumination change causes the inaccuracy of measured BRDF values. To minimize the influence of luminance intensity drift, the built-in Lambertian sample placed next to the sample well is used as the calibration reference. During the measurement of the white board, the response of the built-in Lambertian sample is simultaneously captured as reference values. By comparing the captured responses of the calibration Lambertian sample with the known reference values, the drift in illumination can be promptly identified and corrected, which guarantees the accuracy and reliability of the collected BRDF data.

2.3 Multispectral camera

To obtain BRDF of the measured sample in the spectrum, we opted for a multispectral imaging camera equipped with a filter wheel. This wheel accommodates eight filters with peak transmittance wavelengths spanning from visible spectral band, and a black standard wafer. Each shot corresponds to a single round rotation of the color wheel, allowing us to obtain the surface scattering information of the test sample filtered through eight wavebands. The image of the black-wafer channel provides a reference for real-time denoising.

The pixel size plays a crucial role in determining the angular resolution of a system. According to the MTF of the optical system, an appropriate pixel size could be estimated. Based on the pixel size and the FOV of the optical system, the angular aperture of the pixel which determines the upper limits on the angular resolution of the system is defined as follows:

In Eq. (1), the Rang represents the angular aperture of a pixel. The d_fov and the d_r represents the pixel size and the angle that each pixel corresponds to in the field of view.

Before integrating the multispectral camera in the BRDF measurement system, the camera characterization needs to be thoroughly implemented to establish a mapping between the camera input values and the CIE XYZ tristimulus. Under the standard illuminant (e.g. D65), color charts with known spectral reflectance, such as the X-Rite ColorChecker Classic and Digital SG charts, were imaged using the multispectral camera. The datasets used to derive the transformation matrix consist of the camera's responses for each patch and the corresponding XYZ tristimulus values. The relationship between the device-dependent and XYZ color spaces could be characterized by a linear transformation, as given below:

2.4 Image processing

The geometric calibration is performed to accurately characterize the actual angular distribution of light rays on the sensor. This procedure allows for the correction of the non-uniformities in the angular distribution, resulting in a more accurate estimation of the pixel's angular aperture across the sensor. This calibration helps refine the imaging performance of the optical system, ensuring more reliable and accurate BRDF measurements. Figure 4(a) gives an illustration of the geometric calibration process. A specialized hemisphere is designed to facilitate the geometric calibration. On the inner surface of the hemisphere, black bumps are distributed at 10° intervals in the zenith direction and at 20° intervals in the azimuthal direction. These bumps serve as reference markers for orienting and aligning the hemisphere during calibration. By identifying and aligning these markers, the calibration process can establish the correct geometric relationship between the camera and the sample being captured. During calibration, the geometric calibration standard needs to be positioned directly over the sample well, ensuring that the center of the sphere aligns with the center of the sample well. The schematic diagram illustrating the information received by the camera is presented in Fig. 4(b). The pixel positions of points at different calibration angles can be recognized as equivalent to those of light rays from the center of the sample aperture with the same spatial trajectory. Recognizing and establishing this correspondence in calibration enables the accurate mapping of different angles of light onto the 2D plane.

 

Fig. 4. Schematic diagram of the system (a) and the captured image (b) while using the geometric calibration standard.

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However, the sampling points obtained from geometric calibration are discrete. Thus, the cubic spline interpolation is utilized to create smooth transitions and enhance the spatial continuity of the pixel locations of different reflected angles during the geometric calibration process. Assuming there are (n + 1) data nodes whose space coordinates are: (x₀, y₀), (x₁, y₁), (x₂, y₂)…, (x_n, y_n). Within each subinterval (x_i≤ x ≤ x_i + 1), the spline interpolation equation is expressed as given below:

The parameters ${a_i}\sim {d_i}$ in Eq. (3) are calculated using the following linear equations:

After performing geometric calibration and data interpolation, the correspondence between pixel locations and their respective reflected angles could be established. Before conducting formal measurements, the white and black boards need to be tested for dark correction and response calibration, respectively. After obtaining sensor's physical response values after calibration, the BRDF of the test samples can be calculated according to the following equation:

In Eq. (5), the rsp_sample and the rsp_standard correspond to the responses after black correction at the same pixel position of the test sample and the white Lambertian board, respectively. The t_standard and the t_sample correspond to the integration time for capturing the white Lambertian board and the test sample, respectively. The f_standard is the standard BRDF value of the white Lambertian board at the angle corresponding to the same pixel. During the measurement of the white Lambertian board, the responses of the built-in Lambertian sample are simultaneously captured as the reference data. The k is the response ratio of the built-in Lambertian sample during the formal test to the reference data. Pixels at different positions correspond to different combinations of incident and outgoing angles, which in turn correspond to different geometric factors of the BRDF. The geometric factor used for BRDF calculation depends on the solid angle subtended by a pixel, which is not constant over the entire image. Thus, by calculating the BRDF on a per-pixel basis according to Eq. (5) above, the impact of uneven solid angle distribution across pixels could be mitigated.

However, due to the presence of holes in the ellipsoidal dome mirror for LED installation and camera observations there may be missing reflected light information in small ranges of stereo angles. To address this issue and ensure an accurate representation of the BRDF, an enhanced Ward's BRDF model [47] was adopted to fit the data, as follows:

In Eq. (6), f_r(x, ω_i→ω_o) represents the BRDF value for the reflected light direction ω_o at position x on the sample surface, given an incident light direction of ω_i. The ρ_d and ρ_s represent the diffuse and specular reflectivity, respectively. The α and β are the parameters that control the surface roughness. The (θ_i, ϕ_i), and (θ_o, ϕ_o) are (zenith angle, azimuthal angle) of the incident and reflected light, respectively. The θ_h and ϕ_h are zenith angle and azimuthal angle of the halfway vector between the incident and reflected light. Employing a non-linear squares approach, the measured data could be well-matched to Ward’s model. By fitting the available data points and extrapolating them to fill in the missing ranges of stereo angles, the model helps to reconstruct a more comprehensive and accurate representation of the BRDF.

When measuring highly specular materials, characterized by a substantial disparity in magnitude between diffuse reflection and specular gloss of the BRDF, it is important to adopt techniques capable of accommodating a wide dynamic range of light intensities. High dynamic range (HDR) imaging proves to be a suitable approach in such cases. HDR imaging involves the capture of multiple images of the same scene with different exposure durations. By combining these images, a single composite image with an extended dynamic range is generated, allowing for the accurate representation of both brilliant specular highlights and dimmer diffuse regions. Thus, the exposure times are initially configured as 80, 1.6, 0.8, 0.16, and 0.08 in milliseconds (ms).

3. Performance of image-based BRDF measurement device

3.1 Prototype implementation

A prototype of a catoptric imaging BRDF measurement device, constructed according to the optimized parameters as mentioned above, is presented in Fig. 5. Both mirrors are made of aluminum, with their surfaces shaped through machining operation and EDM (Electrical Discharge Machining). Figure 5(a) shows the processing of the secondary mirror on a mechanical lathe. As seen in Fig. 5(b), the secondary mirror is installed directly on base plate 1, eliminating the necessity for additional support elements. During the actual fabrication process, the diameter of the sample well in the catoptric imaging BRDF measurement device was enlarged to 12 mm, allowing for the inclusion of the base plate along the thickness direction and ensuring the unimpeded light passage with a large FOV. A 10 mm × 10 mm calibration Lambertian sample made of barium sulfate material with 90% reflectivity is securely affixed next to the sample well. The machined primary mirror is shown in Fig. 5(c). Holes are installed on the primary mirror to facilitate the integration of illumination units and the multispectral camera.

 

Fig. 5. Photographs of the proposed BRDF acquisition system. (a) The processing of the secondary mirror on a mechanical lathe. (b) Distribution of components on base plate 1. (c) The mechanical structure of the primary mirror. (d) The internal construction of the system. (e) Overall appearance of the system.

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In Fig. 5(d), the components are assembled to constitute the internal structure of the prototype. The catoptric optical system supported by metal pillars is mounted on the upper part of the device. Within the hemisphere dome, five LEDs are installed, each directed toward the center of the sample hole, corresponding to five different angles of incident light with respect to the sample plane at 0°, 20°, 40°, 45° and 60°, respectively. We have selected 3030 SMD (Surface Mounted Devices) LEDs with 5000 K correlated color temperature (CCT). A monochromic 12-bit scientific CMOS sensor with a resolution of 25 Megapixels (MP) was selected as an imaging unit. The pixel size measured 2.5 µm, aligning with the specifications outlined in Section 2.1. The aperture of the camera is manually set to f/8 in advance. The focal length of chosen camera is fixed at 35 mm. To obtain the optimal image, the camera is inclined at an angle of 6° relative to the horizontal direction according to the previous optical design.

Figure 5(e) presents the operational prototype, appearing as a cylinder linked to a personal computer. The system is composed of two baseplates and a cylindrical enclosure crafted from photopolymer resin, coated in black, to form a unified and portable unit. The enclosed shell shields the inner components from ambient light interference. The dimensions of the prototype measure 350 mm in height, 250 mm in width, and 250 mm in depth.

Figures 6(a)–6(d) present the mechanical structure of the multispectral camera. The camera body (see Fig. 6(a)) consists of three primary parts: the front panel, the main panel, and the camera housing, as displayed in Fig. 6(b). A color wheel is positioned between the main panel and the front panel, as shown in Fig. 6(c). There are eight filters varying in the peak wavelength of the transmittance spectrum and a black flake installed in the wheel (see Fig. 6(d)). The eight filters have peak wavelengths of 430, 460, 490, 520, 550, 590, 620, 660 nm. The Color Checker Classic including 24 uniform color patches, shown in Fig. 6(e), was selected to characterize the multispectral camera using a linear transformation matrix as calculated in Eq. (2). The color chart was obliquely placed inside a Datacolor Tru-Vue Light Booth under D65 illumination. The reflected spectrum of each patch was measured by a Konica Minolta CS-2000 spectroradiometer. The color differences in the CIELAB color space, denoted as ΔE*_ab, of 24 color patches between the predicted and measured tristimulus are summarized in Fig. 6(f), with a mean value of 1.5 and a maximum value of less than 2.5.

 

Fig. 6. (a) The camera body (without lenses). (b) The three components of the camera body. (c) The positioning of the color wheel. (d) Filter wheel structure of the multispectral camera. (e) Color characterization process. (f) Color difference analysis of the color characterization fitting results.

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3.2 Resolution analysis

To ensure the accuracy of the data, the geometrical calibration of this instrument needs to be performed before each official use. The geometric calibration standard is positioned over the sample hole for angular calibration, as depicted in Fig. 7(a). As shown in Fig. 7(b), the captured image is composed of three key components: a grayscale image of the sampled material surface, BRDF information from the secondary mirror, and reflectance information of the calibration Lambertian standard.

 

Fig. 7. Geometric calibration (a) The shape of geometric calibration standard. (b) Valid information part of the image. (c) Angle distribution of the reflected light on the image. θ is the zenith angle of reflected light. ϕ is the azimuthal angle of reflected light.

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The specific angles represented by the black bumps on the geometric calibration hemisphere are expressed as 2D locations in the captured image, as shown in Fig. 7(b). A continuous angular distribution is obtained by interpolating the discrete coordinates of the black bumps (see Eq. (3) and Eq. (4)). Figure 8(c) visualizes the light distribution from different angles on the sensor. The distribution of iso-zenith-angle and iso-azimuthal-angle contour lines reveals that the angular resolution varies with different zenith angles. The angular aperture of a pixel approximates 0.078°/pixel at a zenith angle around 0°.

 

Fig. 8. (a) The precision aluminum coating mirror sample. (b) BRDF plots of the tested specular sample obtained under low-exposure conditions. Each plot shows the BRDF data at ϕ_o = 180°.

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The angular aperture of each pixel is considered only to represent the upper limit of the system's resolution. The actual system angular resolution may be influenced by other factors such as optical distortions, system stability, noise levels, and the angular aperture of the source. We captured images of the precision aluminum coating mirror displayed in Fig. 8(a) under low-exposure conditions and measured the half-peak width of its specular highlight to assess the system's ability to resolve subtle angular differences. Figure 8(b) depicts the specular highlight cross-sections in four incident directions at λ = 550 nm. Based on the data from Fig. 8(b), the half-peak widths, referred to as the angular resolution, are approximately 0.640°, 0.654°, 0.556°, 0.542° for 20°, 40°, 45°, and 60°incident angles, respectively. Thus, the mean angular resolution is about 0.6°. Taking the angular aperture of the pixel and the deviation of illumination angle into account, we could conclude that the angular resolution of the system is mainly influenced by the solid angle of illumination. Considering that the mechanical processing of the primary dome mirror can have an impact on the calculated angular resolution of the measured specular sample, the diamond turning techniques will be employed to fabricate the primary mirror in the updated version of this system to achieve a higher angular resolution.

3.3 Data processing

Following the geometric calibration, the recovery of reflected angles from pixel locations was accomplished. In formal measurements, for materials without obvious specular highlights, capturing the image with a single exposure time is sufficient to obtain an adequate amount of information. The Spectralon diffuse standard was selected as the test sample, as shown in Fig. 9(a). Figure 9(b) presents the BRDF data for the cross section drawn at ϕ_o= 180°and each trend line in the subfigure corresponds to one spectral band. Compared with the data of HD PTFE (high-density Polytetrafluoroethylene) measured by the National Institute of Standards and Technology (NIST) [48] under the same geometry condition, our data shows a similar trend. All plots at different illumination angles exhibit no significant specular component at 8 spectral bands, while the BRDF data decreases as the angle deviates from the specular angle.

 

Fig. 9. (a) The Spectralon diffuse standard. (b) BRDF plots of the Spectralon diffuse standard obtained through basic image processing. Each plot shows the BRDF data at ϕ_o= 180°.

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For materials with obvious highlights, the HDR technology [49] is essential to obtain the full range of brightness of reflected lights. Firstly, a series of images at multiple exposure times were captured. The base image was selected from the image where the diffuse region was properly exposed. By analyzing the response distribution, the overexposed highlight region was identified. The responses of the properly exposed region were remained. Secondly, the responses of the properly exposed region were retained and the responses of the overexposed highlight region were calculated using the image captured at a lower exposure time. This process facilitated the separation of properly exposed and overexposed areas. The responses of the properly exposed area were then adjusted by multiplying them with the exposure time ratio between the base image and the current image. These adjusted values were used to replace the overexposed values at the corresponding pixel locations in the base image. Thirdly, the second process was repeated with subsequent images taken at lower exposure times until all overexposed areas in the base image were recovered. Ultimately, a single HDR image was generated. To verify the BRDF measurement capability of our prototype, a smooth and uniform glazed ceramic sample was chosen for testing. As shown in Fig. 10, six images were captured at different exposure times to generate an HDR image which records the linear response of the reflected light within a high dynamic range.

 

Fig. 10. (a) The green ceramic sample. (b) 6 images at λ = 550 nm with different exposure times used for HDR image processing (θ_i= 60°).

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After acquiring the merged pictures, the BRDF data of the samples were calculated with respect to the physical response values of the sensor obtained through the measurement of calibration Lambertian sample before formal test. Then the BRDF data were corrected using the scaling factors obtained through monitoring the drift in the response of the built-in calibration Lambertian sample. Figure 11 presents the BRDF data for the cross section drawn at ϕ_o= 180° and each trend line in the subfigure corresponds to one spectral band. It is evident that the diffuse reflectance data exhibits variations across different spectral bands and reaches its peak at 520 nm and the specular parts increases as the incident angle increases. The shape of the specular lobe at the same incident angle remains consistent across different wavelengths. It is worth noticing that part of the specular component of BRDF data at θ_i= 0° is absent due to the obstruction caused by the illumination units, as shown in Fig. 11(a).

 

Fig. 11. BRDF plots of the tested green ceramic sample obtained through basic image processing. Each plot shows the BRDF data at ϕ_o= 180°.

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To overcome data incompleteness, the enhanced Ward’s BRDF model was employed to fit the BRDF data. The complete angular distribution of fitted data at 550 nm is shown in Fig. 12. To enhance the visualization of the diffuse reflective component, we applied a logarithmic transformation to the BRDF data for graphical representation, aimed at highlighting the subtle variations in the diffuse reflectance and offering a more visually informative depiction of the data. It can be observed that the specular component of green ceramic sample increases as the zenith angle θ_i increases.

 

Fig. 12. The complete angular distribution of BRDF data at 550 nm tested from the green ceramic sample (A base-10 logarithmic scale is used on BRDF axis).

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3.5 Accuracy analysis

After testing the glazed ceramic sample and accomplishing the data process procedure, the measurement capability of the prototype was verified. Due to the utilization of a multispectral camera in the system, which possesses the capability to detect spectral information across multiple wavelengths, the investigation of the detection proficiency of this instrument for BRDF data of different colored samples was necessary. Thus, the diffuse reflection properties of eight glazed ceramic samples with the same material but varying colors were tested and analyzed. The measurement and data processing for each sample was completed in less than 5 minutes The zenith angle of incident and reflected light were set at (0°, 45°), which is a standard configuration for the capture of the diffuse reflection characteristics. Then, using the goniospectrophotometer of National Institute of Metrology (NIM) China, the eight samples were measured under consistent illumination and measurement conditions. As a result, the standard BRDF curves as a function of wavelength were obtained after nearly four hours of testing. A comparative analysis was conducted by comparing the standard BRDF data of samples with different colors to the data obtained from our prototype under the same illumination conditions across eight wavelength bands. The specific graphical representation of the analysis is illustrated in Fig. 13. As shown in the graph, the prototype consistently demonstrates a good level of data accuracy or investigating the diffuse reflection properties of samples with different colors. The mean relative bias of measured data is 2.5%. In comparison to the lengthy measurement process of the goniospectrophotometer, our prototype is capable of acquiring BRDF data at eight spectral bands and reducing the measurement time by sixfold while ensuring the measurement accuracy for samples of different colors.

 

Fig. 13. Comparison of the BRDF data of 8 standard ceramic samples obtained from our prototype with the standard data at θ_i= 0° and θ_o= 45°.

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4. Conclusions

In this paper, we proposed an image-based method to measure BRDF quickly and accurately using an ellipsoidal dome mirror, a hyperboloid mirror, a multispectral high-resolution sensor and a real-time calibration Lambertian sample. A prototype was built to verify this method. Based on the prototype, we conducted color characterization analysis on the multispectral camera. Geometric calibration analysis was performed using the specialized hemisphere. Measurements were conducted using a glazed ceramic sample and the acquired data were processed using HDR technology and BRDF model fitting. Finally, measurements were taken on eight samples of different colors, and the BRDF data at eight spectral bands were compared to the reference data obtained from the goniospectrophotometer. The experiment results showed that our system is capable of measuring BRDF data with a large FOV (up to 160°), a high angular resolution (less than 0.6°) and a high color measurement accuracy (the ΔE*_ab color difference below 2.5) within 5 minutes. The mean relative bias of measured data is 2.5%.