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Abstract
In this study, we propose a method to measure simultaneously the bidirectional reflectance distribution function (BRDF) and the radius of curvature of a curved surface. The surfaces of pre-molded products or artworks are not always flat. The measurement targets were a planar surface and convex surfaces with constant radiuses of curvature. We developed and experimented with an apparatus that uses patterned illumination and a collimator optical system. The results show that BRDF and radius of curvature can be measured on a curved surface. In future work, we are developing a method to accurately estimate the BRDF of a planar surface from the measured BRDF and radius of curvature.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Product texture affect light reflection characteristics and, thus, appearance. Figure 1 shows a photorealistic computer-graphics image (CG). This CG was generated by the ray tracing method. Ray tracing calculates the reflection of light from an object. Since the balls in CG have different light reflection characteristics, they appear to be made of different materials.
Fig. 1. Photorealistic computer-graphics image. Four balls are drawn. Since they have different light reflection characteristics, they appear to be made of different materials.
The light reflection characteristics can be represented by the bidirectional reflection distribution function (BRDF). The BRDF represents how much light is reflected in each direction when the incident light enters from a certain direction at a certain point on the reflecting surface. Various types of BRDF models have been proposed. They include the Lambertian [1] and Oren-Nayar [2] models for diffuse reflection, and the Torrance-Sparrow [3,4], Phong [5], Blinn [6], Ward [7] and Cook-Torrance [8] models for specular reflections. The models are often used by fitting parameters to the measured reflectance data of real surfaces [9]. The measured BRDF data can also be used directly for realistic CG reproduction [10,11].
Measurement techniques for BRDF have been proposed for various lighting environments and observation conditions [12–17]. A typical measurement technique that is close to the principle is to measure the gonio reflectance distribution. For example, this reflectance at a deviation angle can be measured using a goniophotometer [12]. As shown in Fig. 2, the goniophotometer has a movable detector for measuring the reflectance at various angles for a given angle of incident light. Gardner et al. have presented a technique to estimate the spatially varying reflectance properties by using measurement equipment with a linear light source [13]. The authors of this study have reported a technique for measuring the BRDF used a collimator optical system [14]. These BRDF measurements assume that the target surface is planar.
Many other BRDF measurements assume that the target surface is a known curved surface [11,15–17]. Matusik et al. have built a BRDF measurement device suitable for rapidly acquiring high-quality BRDFs [11]. Image-based BRDF measurement methods have been reported [15,16]. Those measurement objects were convex solids with known curved surfaces. The acquisition system requires a spherically homogenous sample of the material. Arney et al. have reported a micro-goniophotometer using a cylinder solid with known curved surface [17].
On the other hand, the measurement targets of BRDF are not limited to planar surfaces and known curved surfaces. The BRDF is used not only for CGs, but also for quality control of actual industrial materials. These materials could be pre-molded products or artwork. The surfaces of these objects are not always flat. For non-planar objects, the angle of reflected light changes according to the surface normal angle. As shown in Fig. 3, if the object surface is non-planar, the incident light will reflect at a different angle at each point on the surface. The reflection angle on the surface is measured in addition to the BRDF of the material. In an actual BRDF measurement, however, the area illuminated by the incident light is not a point but an area. Therefore, if the object is a non-planar surface, the reflection angle will change at each position of the measurement area and accurate measurement is not possible.
In this study, we introduce a method for the simultaneous measurement of BRDF and radius of curvature. For the purposes of this study, we developed a measurement apparatus, which uses a collimator optical system and patterned illumination, to measure the BRDF of curved objects. The results of our experiment on four sample materials, each with a different BRDF, are presented for a planar surface and two curved surfaces with different radiuses of curvature. We discuss how to estimate the BRDF and the radius of curvature simultaneously.
2. Theory
We focused on two techniques to develop a method for measuring BRDF and radius of curvature simultaneously. One is a collimator optical system to measure the reflection angle distribution of light. The other is a patterned illumination method to measure the radius of curvature.
2.1 Collimator optical system
We used a collimator optical system for the measurement apparatus. This apparatus is based on the gonio-reflectance distribution measurement method reported by the authors of this study [14]. A schematic diagram of the imaging optical system with the collimator optical system is shown in Fig. 4. The collimator optical system has a focus on one side of the lens and parallel light on the other. The position of focal point is determined by the angle of the parallel light. The distance from the center, d, is calculated from the incident light angle, Δθ, and the focal length, f, as follows:
Fig. 4. Schematic diagram of the imaging optical system with the collimator optical system. Focal point distance, d, can be calculated from focal length, f, and incident light angle, Δθ.
2.2 Measuring the BRDF using collimator optical system
The authors have reported the BRDF measurement method using a collimator optical system previously [14]. The BRDF represents how much light is reflected in each direction when the incident light enters from a certain direction at a certain point on the reflecting surface. The BRDF can be calculated from the gonio-reflectance distribution. The proposed system is constructed with a parallel light incident light device and the imaging optical system with the collimator optical system as shown in Fig. 5.
By setting the camera in the focal plane of the collimator optical system, the intensity of light for each position can be measured. For example, an image captured by the camera is shown in Fig. 6 (left). The BRDF can be obtained by converting the pixel position to the reflection angle using the characteristics of the collimator optical system (see Fig. 6 (right)). The system functions as a goniophotometer.
Fig. 6. Schematic diagram of the captured image (left) and the BRDF (right) using the collimator optical system.
2.3 Patterned illumination method
We used patterned illumination to measure the radius of curvature. The reflection angle can be measured at each position by separating the incident light beams shone on the curved surface. For this purpose, a mask with two circular holes was installed on the incident light pass to make the patterned illumination.
Object surfaces with constant radiuses of curvature were assumed. Figure 7 shows the schematic diagram for radius of curvature. The radius of curvature can be estimated by knowing two positions on the surface and the normal for each position. The distance of positions, Δx, is the distance between the incident lights on the surface of the object. The radius of curvature, R, is calculated from the normal angle difference, Δθc (measured from the reflection angle distribution), and the distance of positions, Δx, to the following:
Fig. 7. Schematic diagram of patterned illumination method on a curved surface. The radius of curvature, R, can be calculated from position distance, Δx, and the normal angle difference, Δθc.
3. Experiments
3.1 Apparatus
We developed a BRDF measurement apparatus based on the patterned illumination method. Figures 8 and 9 show the apparatus used to measure the BRDF and radius of curvature in this study. The light source and the observation angles are set to 45°. The light from the light source is collimated by the collimator optical system. The camera captures the reflected light. The focal length, f, (shown in Fig. 4) was 50.1 mm. The magnification at the camera image was 0.7. The image resolution of the CCD camera was 1960×1200 pixels, with a 12-bit output level per pixel. The size of one pixel corresponds to 0.00586 mm.
Sample materials were set on the sample beds, and the measurements were done in a dark room. There were 3 sample beds: a flat sample bed, a curved sample bed (R = 125 mm in Fig. 8), and a cylindrical curved sample bed (R = 62.5 mm).
We used the output values of camera as the relative intensity of reflected light. The linearity between the light intensity and the output value of camera was confirmed in advance. However, the intensity of reflected light is wide-ranging. The intensity of the reflected light, which changes depending on the sample, was adjusted by the shutter speed of the camera. Black glass with a refractive index of 1.567 was used so that the intensities could be compared for all samples. The relative magnification for this experiment was 1.0 for the black glass.
The apparatus is based on the gonio-reflectance distribution measurement method reported by the authors of this study [14]. We modified that method with the addition of a patterned illumination mask with apertures. The measurement angle was fixed at 45 degrees. It is possible to decrease the measurement angle. However, increasing the angle causes a problem. This is because the patterned lights illuminate the surface of the object unevenly.
The patterned illumination was generated by placing an illumination mask with pattern apertures in the incident light path (see Figs.8 and 9). Figure 10 shows the mask used in this study. The pattern was two 1.0 mm holes with a center-to-center distance of 2.0 mm.
Fig. 10. Mask with pattern apertures used in this study. There were two 1.0 mm holes with a center-to-center distance of 2.0 mm. The left shows the schematic diagram, and the right is a photograph of the mask.
3.2 Samples and captured images
Four sample materials were measured: a plastic plate (Plastic), which had a mirror-like surface, and high gloss (IJ-HG), medium gloss (IJ-MG), and low gloss (IJ-LG) photo-like Inkjet paper. The sample materials were measured on the three sample beds (see Fig. 11). The sample materials were bent in the direction of the optical path because they were sheets. Therefore, the data were analyzed at the vertical center line of image in this study.
Fig. 11. Schematic diagram of the three sample beds. The left is a planar surface. The center is a curved surface, with a radius of curvature of 125 mm. The right is a curved surface with a radius of curvature of 62.5 mm.
The images captured by the camera are shown in Fig. 12. The BRDF can be obtained by converting the distribution of the reflection intensity [14].
Fig. 12. Captured images measured by the patterned illumination method. The intensity (image brightness) is the relative intensity, and the x- and y-axes are in the range of the reflection angle of 8°. The relative magnification at the time of measurement is shown in parentheses.
3.3 Estimating the radius of curvature
Figures 13–15 show the profiles of the reflected light intensities in the directions of the incident light centers. Here, the vertical axes are the relative intensities and the horizontal axes are the deviations of the reflection angle, Δθ.
On the planar surface, the reflected light is measured with one distribution (see Fig. 12). These distributions are the BRDFs of each sample. Even if illuminated by two incident positions, the BRDF at each position is the same, so there was only one angular distribution of reflected light measured.
On curved surfaces, the reflected light was measured in two distributions. The BRDF at each position is measured by shifting by the reflection angle because there is an inclination on the surface where the two incident lights illuminate. The radius of curvature was calculated using Eq. (2). Figure 14 shows the profile of the reflected light intensities of the curved surface with a radius of curvature of 125 mm. In this experiment, we fixed the Δx as 2.0×1.4142 mm. The measured Δθ was 2.2° in the peak-to-peak difference of IJ-HG. The normal angle difference, Δθc, was 1.1°. In this case, the estimated R was 147.3 mm. This result was larger than the radius of the sample bed. The theoretical Δθ is 2.6° at R = 125 mm.
Figure 15 shows the profile of the reflected light intensity on the curved surface with a radius of curvature of 62.5 mm. The measured Δθ was 4.1° at the peak-to-peak distance for IJ-HG. In this case, the estimated R was 79.1 mm. This result was bigger than the radius of the sample bed. The theoretical Δθ is 5.2° at R = 62.5 mm.
The estimation error of the radius of curvature in this study was large. We analyzed the possible error sources. For example, the estimation error for R = 62.5mm was 26.6%. This error is probably due to the sample settings. As the edge of sample rises above the surface of the sample bed, the curve on the sample surface becomes flatter and the radius of curvature increases. This effect is large, and it is estimated that R increases by 26.6% when 0.013 mm is lifted under this experimental condition, which is the same as the experimental result. In the calculation process of the radius of curvature, we fixed the Δx as 2.0 × 1.4142 mm. Δx = 2.0 × 1.4142 mm is the value for a flat surface. This hypothesis works well when the surface is smooth, the radius of curvature is large. This effect is not so large in case of R = 62.5 mm, the error that can be caused by this hypothesis was calculated to be 0.1% of Δx.
4. Discussion
4.1 Simultaneous measurement of the BRDF and radius of curvature
On the planar surface, the reflected light was measured with one distribution. On the curved surfaces, the reflected light was measured with two distributions. Furthermore, the difference in angle distributions was dependent on the surface curvatures. These distributions are BRDFs. These show that BRDF and radius of curvature can be measured simultaneously by the patterned illumination method.
The purpose of this study was to develop a method for measuring the BRDF of curved objects. The results of our experiments show that this method can be applied to measure the BRDF of curved objects with high specular reflection. Applicable materials are, for example, plastics, metals, and glossy paper. It was found that materials with a specular reflection angle spread within several degrees can be measured. In addition, objects with a radius of curvature of 62.5 mm or more can be measured under experimental conditions. In fact, for example, this can be applied to the measurement of car bodies, many plastic appliances, and glossy paper in the manufacturing process.
4.2 Limitations of the proposed measurement method
There are limitations to the proposed method. The system functions as a goniophotometer within a narrow solid angle range. The BRDF can only be measured in the range of −4° to 4° deviating from the specular reflection angle. Furthermore, in order to determine the radius of curvature, the two peaks of BRDF need to be separated and measured in this angular range. Therefore, this apparatus can only measure the BRDF of a relatively smooth surface material.
For example, in a BRDF with a large spread, the two distributions may not be separated (see IJ-MG in Fig. 16 and IJ-LG). In these cases, the method proposed in this paper cannot calculate the radius of curvature. Materials with strong specular reflection are suitable for this measurement method, for example, metal or plastic whose surface is close to a mirror. Materials with weak specular reflection are not suitable, for example cloth or matte paper.
In addition, technology that accurately estimates the radius of curvature is required. A possible solution is to improve the incident light pattern. In this experiment, a one-dimensional analysis was performed using two simple holes. However, a two-dimensional analysis is possible with three or more holes. Furthermore, the shape of the holes is not limited to a circle.
The BRDF measurement results for planar surface by this method were compared and verified with the results by other measurement methods, and they gave good agreements [14,18,19]. In this study, we applied this method to the BRDF measurement of curved surfaces, however the validation of the BRDF measurement results on curved surfaces is a future work.
4.3 Difference between BRDFs of the planar surface and the curved surfaces
The angular spreads of BRDF on the curved surfaces were larger than those on the planar surfaces. Figure 17 shows the BRDFs of the plastic plate. On a planar surface, the plastic BRDF had a narrow distribution, almost a point. The BRDF spread greatly on the curved surfaces. Therefore, a BRDF measured on a curved surface needs correction. Figure 18 shows the BRDFs of the IJ-HG. The IJ-HG BRDFs had a spread distribution like Gaussians. They also spread out on curved surfaces. The authors reported a technique for simulating the BRDF of a curved surface based on the measured BRDF of a planar surface [19]. In future work, it is necessary to consider developing a technique for estimating the BRDF of a flat surface from the measured BRDF of a curved surface.
5. Conclusion
We proposed a simultaneous measurement of BRDF and radius of curvature by the patterned illumination method. The patterned illumination was generated by placing an illumination mask with two holes in the incident light path. The BRDF at two illuminated positions was measured by using two incident light rays, and the radius of curvature was estimated from the difference between the two positions. The BRDF spreads on curved surfaces. Therefore, it is important to know the curvature of the surface when the BRDF is measured. In future work, we are developing a method to accurately estimate the BRDF of a planar surface from the measured BRDF and radius of curvature.
Funding
Institute for Global Prominent Research, Chiba University.
Disclosures
The authors declare no conflicts of interest.
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