BRDF

Figure 3 Example of the BRDF characteristic of a polished optical surface. Note the rapid increase in scattering at small scatter angles.

A final practical note involves instrument intensity measurement calibrations. The intensity measurement is self-calibrating relative to the incident beam from the source. However, measurements typically have a dynamic range of 10 -10 , and care must be taken to insure the detection system is linear. A method of calibrating the scatterometer is to characterize a diffuse reflector having a known scattering characteristic. For example, a surface coated with BaS04 makes a nearly Lambertian scatterer, which has a BRDF of 1/Jt at all angles. 

Figure 3.14 The bidirectional reflectance distribution function (BRDF) specifies how much of the incident light coming from direction (6l,

The BRDF specifies how much of the incoming light is reflected for any given combination of light source direction N/. and viewer direction Ny. If we fix the position of the light source relative to the patch, then the BRDF specifies how the reflected light is distributed around the patch. All BRDFs must fulfill the following requirement. 

That is, if we exchange the positions of the light source and the viewer, then the BRDF must have the same form. It must be symmetric in this respect, i.e. bidirectional. 

Ideal matte and specular surfaces have a very simple BRDF. In general, however, the BRDF does not have a simple structure. For example, cloth or brushed metal has a BRDF which reflects most of the light along a preferred direction. Sample BRDFs for aluminum, magnesium oxide ceramic, and sandpaper can be found in (He et al. 1991). If we know the BRDF for a particular material, then we can compute the radiance given off by an infinitesimal small patch of this material. The radiance leaving the patch is given by... 

This is the total irradiance falling onto the patch. The amount of light reflected in each direction depending on the direction of the irradiance is described by the BRDF. Therefore, the radiance given off by the patch into the direction of the viewer is given by... 

If we know the BRDF of the material of the viewed object, then we can compute the radiance using the preceding equation. We will now have a look at the BRDF for two idealized surfaces, a Lambertian surface and a perfect mirror. Let us address the BRDF of a perfect mirror first. If the incident light is coming from direction (9l, [Pg.54]

Let us now turn to a Lambertian surface. A Lambertian surface reflects the incident light equally in all directions. The radiance given off by matte objects can be approximated using the BRDF of a Lambertian surface. If the irradiance is reflected equally in all directions,... 

This BRDF allows us to compute the radiance given off by a Lambertian surface illuminated by a point light source. 

Macbeth 5000 K fluorescent, a Philips Ultralume fluorescent, and a Sylvania Cool White fluorescent tube. Some color constancy algorithms try to find out what type of illuminant produced the particular color sensation, which was measured by the sensor. If we measure the entire power spectrum for a particular patch of our object and also know the type of illuminant used, then it is easy to compute the BRDF. Let L(X) be the measured power spectrum and let E A.) be the power spectrum of the illuminant. Assuming a Lambertian surface, where the BRDF is independent of the normal vector of the patch Nobj, and also independent of the normal vector that points to the direction of the light source Ni, we... 

Sometimes it is important to consider the direction of reflected irradiation exitent from a surface. A property called the bidirectional reflectance distribution function (BRDF) is used to specify the directional distribution of the reflected intensity for a specified direction of incident radiation [2-4]. A specular surface is a mirrorlike surface for which the incidence angle is equal to the reflection angle. For a diffusely reflecting surface, the reflected intensity is the same in all directions, and if perfectly reflective, the BRDF is l/n sr. ... 

Thus, the reflected radiance is the integral of the energy in each incident direction times the BRDF for that particular combination of incidence and observation angles under consideration. The BRDF, or in short the reflectance, plays a central role in the remote sensing of planetary surfaces and is important for the correct assessment of their albedo. 

Purely diffuse reflection occurs at microscopically irregular surfaces, while purely specular reflection occurs when the surface is perfectly smooth, like a minor. If the reflected radiance from a surface is completely uniform with angle of observation, it is called a Lambert surface. The BRDF for a Lambert surface is independent of both the direction of incidence and the direction of observation. Then, the reflectance simplifies to p(v, —h, = where pi is the Lambert reflectance. Specular reflection from and transmission throngh a smooth dielectric surface can be calculated from Snell s law and Fresnel s equations, given the optical constants of air and the dielectric material. 

Bidirectional reflectance distribution function (BRDF) n. The ratio of radiance per unit irradiance, used for describing the... 

The reflectance properties of a surface are characterized by its bidirectional reflectance distribution function (BRDF). The BRDF is the ratio of the scene radiance in the direction of the observer to the irradiance due to a Hght source from a given direction. It captures how bright a surface will appear when viewed from a given direction and illuminated by another. For example, for a flat Lambertian surface illuminated by a distant point light source, the BRDF is constant hence, the surface appears equally bright from all directions. For a flat specular (mirrorHke) surface, the BRDF is an impulse function as determined by the laws of reflection. 

The image size on the detector (0/2jr), and hence the magnification of the mirror, depends on the BRDF of the sample. For Lambertian samples, the magnification will be a maximum. Knowledge of the maximum linear magnification of a hemispheroidal or dual-paraboloidal mirror is required to... 

For samples with an arbitrary BRDF, Coblentz (14) determined the size of this correction experimentally by placing a blackened disk near the beam port. The size of the beam optics loss can also be measured for 0/2ro mode instruments by using a beamsplitter and a second detector (56). For 2n/d... 

Interreflections between the detector and sample in the 0/2ro mode or the source and sample in the 2nlQ mode can introduce a systematic error that is dependent on the sample reflectance value as well as the sample BRDF. Additionally, the sizes of the sample, detector, and input spot in the Bjln mode or the sample, source, and viewed spot in the 2ji/0 mode, and the magnification properties of the conic mirrors, are all factors that can affect the size of the error. For InfO instruments, interreflections can increase the source temperature. In the following analysis, it is assumed that the source temperature is stabilized (57). 

Previously calibrated standards of similar reflectance and BRDF as the samples to be measured can be used to estimate the size of the interreflection uncertainty. However, this approach is limited by the a priori knowledge of the sample s reflectance and BRDF, as well as the availability of standards. The interreflection error can be directly minimized by using detectors (0/2ti) or cavity radiation sources 2nlB) having a low reflectance. 

Here, we neglect the interreflections between the reflectometer and the external optical system, which will depend on the details of the external system and need to be dealt with separately. Note that the factor by which the reflectance is enhanced due to multiple reflections can easily exceed 1.1 if the detector reflectance is greater than 20%. Clarke and Larkin (59) have suggested a separate correction for the interreflection error associated with the specular and diffuse components of samples with an arbitrary BRDF. Their technique involves measuring standards of known reflectance, such as pure barium sulfate powder and a first-surface aluminum mirror. For... 

This chapter explains how to characterize the angular distribution of optical scatter from an opaque surface. In particular it focuses on measurement of the bidirectional reflectance distribution function (BRDF). BRDF is a convenient and well-accepted means of expressing optical scatter levels for many purposes (1, 2). 

---