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Abstract
In this study, we introduce a measurement method for the bidirectional reflectance distribution function (BRDF) for curved surfaces. Nearly all BRDF measurements assume that the target surface is planar. This is because if the object is non-planar, the reflection angle changes from position to position, making accurate measurement impossible. However, most real objects are curved. We have overcome this problem by applying the paraboloid reflection principle. If the curved target surface is a paraboloid, the light focused on its focal point is reflected as parallel light, as in a parabolic antenna. The BRDF can be measured as the deviation angle from this parallel light direction. Here, we demonstrate that this can significantly improve the BRDF measurement accuracy for curved surfaces.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Light reflection characteristics affect the perceived texture of an object and thus its appearance. Therefore, many measurement methods have been explored to determine the reflected light characteristics of objects. Almost all of these methods assume flat target objects. However, the shape of the object to be characterized greatly influences the results when measuring reflected light. Object shapes can be roughly divided into flat surfaces and curved surfaces. Here, we assumed that the curved surface could be modeled as a paraboloid [1]. Figure 1(a) shows a parabolic antenna, which is a paraboloid, and Fig. 1(b) shows spheres and a planar object.
Fig. 1. Photographs of objects with different surface shapes. (a) A parabolic antenna. (b) Spherical objects and a planar object. This study assumes that all curved surfaces can be modeled as a paraboloid.
We now consider the optical features of a paraboloid. When parallel light illuminates a concave paraboloid surface, it is focused at a single point (see Theory chapter), which is the principle underlying parabolic antennas. The light reflection characteristics can be represented by the bidirectional reflectance distribution function (BRDF), which represents how much light is reflected in each direction when the incident light arrives from a certain direction and strikes a certain point on the reflecting surface. The reflection intensity at different angles can be measured using a goniophotometer for different incidence angles [2], as shown in Fig. 2.
Measurement techniques for the BRDF have been proposed for various lighting environments and observation conditions [3–10]. However, most BRDF measurement methods assume that the target surface is planar. Although image-based BRDF measurement methods have been reported [11–14], the objects were convex solids with known surface shapes, and such methods cannot be used for a material with an unknown surface shape. Measurement targets for BRDF are not really limited to planar surfaces. The BRDF is used not only in the field of computer graphics (CG), but also for quality control of industrial materials. These materials could be pre-molded products or artwork, and their surfaces are not always flat. For non-planar objects, the angle of reflected light changes with the surface normal angle. As shown in Fig. 3, if the object surface is non-planar, the incident light will be reflected at a different angle from each point on the surface, and the illuminated area is not a single point, making BRDF measurements impossible.
We previously reported a method for simultaneously measuring the BRDF and the radius of curvature for a spherical curved surface [15]. This involved measuring the difference in reflection angle for two different points on the surface by parallel light beams. However, it was not possible to determine the planar BRDF using this method. In addition, in order to increase accuracy, the width of the two light beams had to be reduced, which was difficult to realize.
In the present study, we introduce a new BRDF measurement method for curved surfaces, based on the paraboloid reflection principle. The BRDF can be determined based on the deviation angle from the reflected parallel light direction. In the present study, this concept was inverted, and focused illumination was used instead of parallel illumination. We developed a measurement apparatus and conducted experiments on a number of sample materials. Our results for planar and curved surfaces are presented in Section 3. We also discuss the error in the BRDF associated with this measurement method; as described later, the BRDF measurement accuracy for curved surfaces can be significantly improved.
2. Theory
This section provides a theoretical description of the optical system used in the proposed method, followed by the general optical characteristics of a paraboloid surface, and finally a focused illumination method for BRDF measurements of curved surfaces.
2.1 Collimating optical system
The proposed apparatus uses a collimating optical system, and is based on the gonio-reflectance distribution measurement method previously reported by the present authors [3–5]. The collimating optical system, which is illustrated in Fig. 4, features a focal point on one side of a lens and parallel light on the other. The position of the focal point depends on the angle of incidence of the parallel light. The distance d of the focal point from the centerline is calculated based on the deviation of the light incidence angle, Δθ, from the centerline, and the focal length, f, as shown in Eq. (1).
Fig. 4. Schematic diagram of collimating optical system. The distance d of the focal point from the centerline can be calculated from the focal length, f, and the deviation of the light incidence angle, Δθ, from the centerline.
By setting a camera in the focal plane, the intensity of the focused light can be measured for each position. A typical image captured by a camera is shown in Fig. 5 (left). The BRDF can be obtained by converting the pixel position to the reflection angle using the characteristics of the optical system (see Fig. 5 (right)). The system functions as a goniophotometer with a narrow solid angle range.
2.2 Characteristics of paraboloid
In this study, all curved surfaces to be measured are considered paraboloids. First, the characteristics that led us to adopt a paraboloid shape will be described, and we will then explain how to represent all planes as paraboloids.
Figure 6 shows the schematic illustration of paraboloid reflection. A circular paraboloid (hereafter just paraboloid) is a surface of revolution obtained by revolving a parabola around its axis [1]. On the axis of a paraboloid, there is a point called the focus (or focal point), such that, if the paraboloid is a mirror, light (or other waves) from a point source at the focus is reflected as a beam parallel to the paraboloid axis. This also works in reverse: an incident beam of light that is parallel to the axis of the paraboloid is concentrated at the focal point. Therefore, the circular paraboloid is a widely used shape in astronomy for parabolic reflectors and parabolic antennas. The focal length of a parabola is half the radius of curvature at its vertex.
If the convex side of the paraboloid is a mirror, light directed toward the focal point is reflected as parallel light (see Fig. 7). The BRDF for a curved surface can be measured based on the deviation angle from the direction of this parallel light beam.
Fig. 6. The left figure shows a parabolic antenna. The right schematic illustrates paraboloid reflection.
In reality, objects have various shapes. When considering a convex surface, part of the curved surface is considered a paraboloid. If the radius approaches infinity, the surface can be regarded as planar.
Curved surfaces are often represented by the radius of curvature of an equivalent sphere. In the present study also, we use this sphere approximation and examine the circumstances under which it might perform adequately. Figure 8 shows cross-sections of a paraboloid and a sphere at the vertices. As shown in Fig. 8, close to the origin, the shapes of a sphere and a paraboloid closely approximate each other. The paraboloid can be represented by Eq. (2), and its focal point F is given by Eq. (3). A sphere can be expressed as a circle as in Eq. (4), and its relation a and R is determined as Eq. (5).
Fig. 8. Cross-sections of paraboloid and sphere at vertex. The focal point, F, for the paraboloid can be calculated from the radius of curvature, R, of the sphere. The center of the sphere is C.
The focal point of the parabola, F, at the origin, which is the vertex of the parabola, is half the radius of curvature, R, of the sphere, as shown in Eq. (6).
In this paper, the shape of an object is considered a convex paraboloid or a sphere, and its shape is represented by the radius of curvature.
2.3 Focused illumination method
Focused illumination can be described using optical lens theory. A convex lens has focal point on both sides of the lens. The focal length is the distance at which a collimated beam of light is focused to a single spot. In particular, a lens system that uses one side as parallel light is called a collimating optical system. Figure 9 shows the schematic of the collimating optical system used in this study. When a convex lens is used to converge light, the distance Ds from the light source to the lens, the distance Dp from the lens to the focus, and the focal length f are related by Eq. (7). Thus, the collimating optical system can generate focused light at any position by controlling the position of the light source. It can also generate parallel light by setting the position of the light source to F. This allows conventional BRDF measurements to be performed for both flat and curved surfaces.
Fig. 9. Schematic of focused illumination method. The focus, P, can be calculated from the distances Dp and Ds, and the focal length f.
3. Experiment
We developed an apparatus to measure the BRDF for a curved surface by the focused illumination method (see Figs. 10 and 11). We then performed experiments to investigate the effects of surface shape, type of material, and deliberate bending.
3.1 Apparatus
The apparatus was designed and manufactured by CHUO PRECISION Co., Ltd. Light can be focused at an arbitrary point by controlling the position of the light source. The light source was set on an x-stage, and its position was precisely controlled by a pulse motor. The focal length of the optical system was 39.0 mm, and the diameter of the light source was 0.15 mm. A ø8 mm mask was placed in front of the lens, and the maximum diameter of the luminous beam was 8 mm, which indicates parallel light. The illumination was reflected by a half mirror and struck the object to be measured. A CCD camera captured the reflected light. The camera resolution was 1960×1200 pixels, with a 12-bit output level per pixel. The size of one pixel corresponded to 0.00586 mm. The output values were equivalent to the light intensity because, as was confirmed in advance, the two were linearly proportional. The illumination axis and the optical axis of the camera were coaxial. Sample materials were set on the sample bed, and measurements were conducted in a dark room. We used a planar black glass with a refractive index of 1.567 to calibrate the process. The relative magnification in this experiment was 1.0 for the black glass. The apparatus was based on the gonio-reflectance distribution measurement method reported by the present authors [3–5]. We modified the method by incorporating the focused illumination method and zero-degree geometry measurements.
In this apparatus, the position of the light source and the in-focus position were measured experimentally. This relationship can also be calculated from Eq. (7). Figure 12 plots the measurement and calculation results.
Fig. 12. Measurement and calculation results for light source position and focal plane position, P. The horizontal axis shows the difference ΔDs from the position at which the light source distance is the same as the focal length. For ΔDs = 0, the illumination is parallel, and Dp = ∞.
3.2 Sample characteristics
Four surface shapes were measured: a planar surface and three surfaces with radii of curvature R of 515, 258, and 103 mm. They all had a mirror-like surface and were gold-leaf-pasted. The sample materials were planar black glass, a gold-leaf-pasted lens, and inkjet paper (high glossy), as shown in Fig. 13.
Figure 14 shows the basic features of the method proposed in this study. In a conventional BRDF measurement, parallel light is incident on a flat sample, and the angular distribution of the reflected light is determined. However, if the sample has a curved surface, this will disperse the reflected light, and the measured BRDF will have a large error. In the proposed method, the BRDF measurement error for a curved surface can be significantly reduced by making the reflected light parallel through the focused illumination method.
Fig. 14. Schematic diagram of proposed method. In a conventional measurement, parallel light is incident on a planar surface and provides a sharp BRDF (left column). For a curved surface, due to light dispersion, the BRDF becomes broadened (center column). In the proposed method, a curved surface is illuminated by a focused light beam, and the reflected light is collimated, thereby producing a sharp BRDF.
3.3 Effect of curvature radius
Figure 15 shows three-dimensional plots of the measured BRDF for a planar glass sample and gold-coated lenses with varying radii of curvature. The vertical axis is the relative intensity and the horizontal axes are the deviations Δθx and Δθy in the reflection angle. The left and right columns show the results for parallel and focused incident light, respectively. Although a sharp BRDF is obtained for the planar surface, it can be seen that for curved surfaces, the use of parallel light causes a large spread in the BRDF, which depends on the radius of curvature. However, using focused light and the proposed collimation method allowed sharp BRDFs to be obtained for curved surfaces.
Figure 16 shows two-dimensional slices through the center of the BRDFs in Fig. 15. On curved surfaces, the BRDF will spread when parallel light sources are used. However, using the proposed method, the curves become much sharper.
3.4 Effect of material
Figure 17 shows the measured BRDFs for samples of different materials. The samples are black glass, gold, and inkjet paper, and are either planar or have a curved surface. The left and right columns show the results for the conventional and proposed methods, respectively. This shows that the proposed method can be applied to a variety of materials. The figure also shows the BRDF for bent (non-planar) inkjet paper, which will discussed in the next section.
3.5 Bent samples
The two plots in Fig. 18 show the BRDF for bent inkjet paper measured using the conventional and proposed methods. The photograph on the left shows the apparatus used for bending. The radius of curvature can be controlled by the degree of bending, although its exact value is unknown. As the bending angle increased, the spread in the BRDF increased using the conventional method, with a correct result only being obtained for a bending angle of zero. However, it can be seen that by using the proposed method, the BRDF shape was similar regardless of the bending angle, thus giving a correct result.
Fig. 18. Instrument for bending inkjet paper (left photograph), and measured BRDFs for bent inkjet paper (right figures).
4. Discussion
4.1 To what extent is it necessary to consider the effect of curved surfaces in BRDF measurements?
We described in the introduction that an error occurs when measuring the BRDF for a curved surface using the conventional method, and we experimentally showed that this error depends on the radius of curvature. To what extent is it necessary to consider the effect of curved surfaces in BRDF measurements? Fig. 19 shows the simulated dependence of the angular error on the radius of curvature. The angular error is also affected by the width of the measured light beam. For example, if the width is 6 mm and the radius of curvature is 300 mm, the angular error expands by ± 1.0°. Even for a radius of curvature of 700 mm, the angular error expands by ± 0.5 degrees. This is not negligible, for example when measuring a sample with a diameter of 1400 mm. Correction is especially important for materials exhibiting strong specular reflection, such as metals and plastics.
Fig. 19. Simulated dependence of angular error on radius of curvature. The angular error shown is the value at the edge of the light beam width. Therefore, the angular error represents a plus or minus value on both sides at both ends.
4.2 Improvement effect and error in proposed method
As described above, the BRDF measured by the proposed focused illumination method was close to the correct value. We thus confirmed that the BRDF measurement accuracy for curved surfaces can be significantly improved. We next introduced a technique for estimating the BRDF for a planar surface based on measurements for a curved surface. Figure 16 shows the BRDFs for gold, which had a cylindrical distribution. The spread in the data depends on the curvature of the surfaces. Figure 18 plots the BRDFs for IJ-HG, which had a Gaussian-like distribution. The measured values were smaller than the correct value.
It should be noted that the proposed BRDF measurement method has some limitations that do not fit the general BRDF definition. The BRDF is defined as a function of the incident and reflection directions with respect to the surface normal. In this method, the incident and reflection directions are very close to the surface normal, and have the same angular deviations from the normal. However, the deviations differ depending on the measurement position. The center of the focused beam is along the normal direction, and the deviation angle becomes larger with increasing distance from the center or when the measurement object has a small radius of curvature. Based on trial calculations, the deviation angle is ±0.57° for R = 300 mm, ±0.25° for R = 700 mm, and ±0.12° for R = 1400 mm with the light beam width of 6.0 mm. This is the difference, for example, between a BRDF obtained for angles of incidence of 0.0° and 0.5°. Since the reflectance for most materials does not differ near 0.0°, we decided to treat the latter case as a BRDF with an angle of incidence of 0.0°.
The following reasons are considered for errors in the BRDF measurements. First, there is a limited range of radius of curvature that can be measured. When the radius of curvature is small, the incident light beam on the surface is small, and averaged measurements cannot be performed. In addition, a slight shift in the focal position of the illumination greatly affects the reflection. Next, there is the effect of the aperture size of the light source, which was 0.15 mm. The smaller the aperture size, the smaller the error.
4.3 Measurement of curved surface with unknown radius of curvature
The focused illumination method is based on the theory of parabolic reflection described in section 2.2. As shown in Fig. 7, the focal length for the measurement target has to know. Also required is the distance from the reference position of the measuring device to the measurement surface. However, in practice, it is possible to measure the BRDF for a curved surface even for a measurement target having an unknown radius of curvature. Assume a convex surface with a constant curvature. The range of reflection angles for a curved surface increases with increasing surface slope, and this angular spread must be added to the true value. Thus, the measurement result with the lowest spread is the true value. This can be achieved by controlling the focal point of the illumination to find the position with the lowest spread. This technique is described in section 3.5. This method can be employed to measure the radius of curvature of an object. First, the BRDF for an object with an unknown radius of curvature can be measured with this device. As shown in Fig. 12, the focusing position can be obtained from the light source position where the spread is minimal. If the distance from the device to the surface of the object is known, the focal length can be calculated. The target radius of curvature is twice the focal length, as expressed in Eq. (6).
4.4 Applications of measured BRDF
The first application that we consider is in the field of CG, for which the characteristics of specular reflection have been modeled [16–18]. Physics-based CG systems have been developed based on the BRDF for different materials [19]. Here, the measurement targets for BRDF are not limited to planar surfaces, but also curved surfaces. The BRDF can also be used for quality control of industrial materials, which could be pre-molded products or artwork. Again, the surfaces of such objects are not always flat. Another approach is appearance analysis. For example, we previously investigated Perlin noise [20], which was proposed as a CG drawing technique, and reported an attempt to apply the BRDF to mathematically model the shape of a paper surface [21]. The gloss of an image can be simulated using a transfer function such as the point spread function (PSF). We previously reported the simulation of gloss on curved paper using the PSF calculated from the BRDF [22].
4.5 Prototype development and contribution to industrial instrument: autocollimator
A prototype autocollimator was designed and manufactured by CHUO PRECISION Co., Ltd.. Figure 20(a) shows a photograph of the instrument. The setup is shown in Fig. 20(b). It consists of a measurement unit, a light source position controller, and a personal computer. The size of the measurement unit is 222 mm high, 124 mm wide, and 100 mm deep. The optical accuracy has been improved. The focal length of the optical system is 80.0 mm and the diameter of the light source is 0.05 mm. The maximum diameter of the light beam is 30.0 mm.
Fig. 20. Prototype instrument. (a) Overall view. (b) Measurement unit. (c) Measured BRDF for curved mirror.
We confirmed that this prototype acts as an autocollimator for curved surfaces. An autocollimator is an optical instrument that measures angles in a non-contact manner. It is generally used to measure the alignment and warp of optical and mechanical components. As industrial instruments, autocollimators are widely used throughout the world. However, they were limited in that they could only be used on flat surfaces. Figure 20(a) shows a curved mirror with a wide field of view. The measured BRDF of curved mirror is shown in Fig. 20(c). For a reflective material such as a mirror, where the reflection is not diffuse, the BRDF can be measured as a small point, an angular range, and indicates the normal direction for the surface. This is the same measurement result that an autocollimator would normally obtain for a flat surface. Thus, the prototype can be used as a curvature measurement instrument, an autocollimator for curved surfaces, and a BRDF measurement instrument.
5. Conclusion
We proposed BRDF measurements for curved surfaces using the focused illumination method based on the paraboloid reflection principle. We developed an apparatus to measure the BRDF for different curved surfaces. Our experiments revealed the effects of differences in surface curvature, material type, and whether or not the material was bent. We confirmed that the proposed method can significantly improve the BRDF measurement accuracy for curved surfaces. As a future study, we are developing a more accurate and practical version of this instrument.
Funding
Japan Society for the Promotion of Science (JP21K11954); Institute for Global Prominent Research, Chiba University.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. Eugene Hecht, “Optics, Fifth Edition”, Pearson Education Limited, England, 195–199 (2017).
2. F. B. Leloup, G. Obein, M. R. Pointer, and P. Hanselaer, “Toward the Soft Metrology of Surface Gloss: A Review,” Color Res. Appl. 39(6), 559–570 (2014). [CrossRef]
3. S. Inoue and N. Tsumura, “Point Spread Function of Specular Reflection and Gonio-Reflectance Distribution,” J. Imaging Sci. Technol. 59(1), 10501-1–10501-10 (2015). [CrossRef]
4. S. Inoue, Y. Kotori, and M. Takishiro, “Paper Gloss Analysis by Specular Reflection Point Spread Function Part I,” J. Tappi J. 66(8), 879–886 (2012). [CrossRef]
5. S. Inoue, Y. Kotori, and M. Takishiro, “Paper Gloss Analysis by Specular Reflection Point Spread Function Part II,” J. Tappi J. 66(12), 1416–1424 (2012). [CrossRef]
6. A. Gardner, C. Tchou, T. Hawkins, and P. Debevec, “Linear Light Source Reflectometry,” Proceeding SIGGRAPH ‘03 ACM SIGGRAPH 2003 Papers, 749–758 (2003).
7. Michael E. Becker, “High-Resolution Scatter Analysis of Anti-Glare Layer Reflection,” SID 2016 DIGEST, 368–371 (2016).
8. J. S. Arney, H. Heo, and P. G. Anderson, “A Micro-Goniophotometer and the Measurement of Print Gloss,” J. Imaging Sci. Technol. 48(5), 458–463 (2004).
9. J. Dupuy and W. Jakob, “An Adaptive Parameterization for Efficient Material Acquisition and Rendering,” ACM Trans. Graph. 37(6), 1–14 (2018). [CrossRef]
10. S. Inoue and N. Tsumura, “Measuring Method for Line Spread Function of Specular Reflection,” OSA Continuum 3(4), 864–877 (2020). [CrossRef]
11. S. R. Marschner, S. H. Westin, E. P. F. Lafortune, and K. E. Torrance, “Image-based BRDF Measurement,” Appl. Opt. 39(16), 2592 (2000). [CrossRef]
12. W. Matusik, H. Pfister, M. Brand, and L. McMillan, “Efficient Isotropic BRDF Measurement,” ACM Trans. Graph. 22(3), 759–769 (2003). [CrossRef]
13. A. Sole, I. Farup, and S. Tominaga, “An image-based multi-angle method for estimating reflectance geometries 0f flexible objects,” Color and Imaging Conf. November (2014).
14. B. Hu, J. Guo, Y. Chen, M. Li, and Y. Guo, “Deep BRDF: A Deep Representation for Manipulating Measured BRDF,” EUROGRAPHICS 39(2), 157–166 (2020). [CrossRef]
15. S. Inoue and N. Tsumura, “Measuring the BRDF and radius of curvature with patterned illumination,” OSA Continuum 4(3), 1113–1124 (2021). [CrossRef]
16. K. E. Torrance and E. M. Sparrow, “Theory of Off-Specular Reflection from Roughened Surfaces,” J. Opt. Soc. Am. 57(9), 1105–1112 (1967). [CrossRef]
17. B. T. Phong, “Illumination for Computer Generated Images,” Commun. ACM 18(6), 311–317 (1975). [CrossRef]
18. A. S. Glassner, “An Introduction to Ray Tracing,” Andrew Glassner, Ed., Academic Press Limited, Amsterdam (1989).
19. Brent Burley, “Physically-Based Shading at Disney”, SIGGRAPH 2012 (2012).
20. Ken Perlin, “Improving noise”, ACM Transactions on Graphics (TOG) 21(3). ACM, (2002).
21. S. Inoue, M. Maki, and N. Tsumura, “Mathematical Model of Paper Surface Topography by Perlin Noise Derived from Optical Reflection Characteristics,” J. Tappi J. 73(10), 1022–1029 (2019). [CrossRef]
22. N. Tsumura, K. Baba, and S. Inoue, “Simulating Gloss of Curved Paper by Using the Point Spread Function of Specular Reflection,” Bull. Soc. Photographic Imaging Japan 25(2), 25–30 (2015).
