Reflectance models

Let Nobj be the normal vector of the patch, let N/, be the normal vector that points to the direction of the light source, and let Ny be the normal vector that points to the direction of the viewer. Let / (N/.) be the irradiance falling onto the patch, and L(Ny) be the radiance leaving the patch in the direction of the viewer. The fraction of the radiance relative to the irradiance falling onto the patch is called the bidirectional reflectance distribution function 

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 

Let us first consider a point light source and let L be the radiance given off by this light source. If we assume that the light source is far away from the patch, then the irradiance E(Nl) is simply LNohjN/.. In this case, we obtain for the radiance L(Ny) given off by the patch in the direction of the viewer 

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Zeno paradox. On the other hand, recovering these interferences from a single path leads to excessive correlation, as evidenced by the highly oscillatory results obtained with TSH for Tully s third, extended coupling with reflection, model. This is remedied effortlessly in FMS, and one may speculate that FMS will tend to the opposite behavior Interferences that are truly present will tend to be damped if insufficient basis functions are available. This is probably preferable to the behavior seen in TSH, where there is a tendency to accentuate phase interferences and it is often unclear whether the interference effects are treated correctly. This last point can be seen in the results of the second, dual avoided crossing, model, where the TSH results exhibit oscillation, but with the wrong structure at low energies. The correct behavior can be reproduced by the FMS calculations with only ten basis functions [38]. 

For the example data, the RMSEC is calculated for models containing 1 -22 variables (adding the variables in the order listed in Table 5.9). The RMSEC versus the number of variables included in the model is plotted in Figure 5.66 for component A. nte fit improves as variables are added to the model (RMSEC decreases). However, knowing these results reflect model fit, there is a concern about overfitting (i.e., fitting noise from the calibration data). It is known that the error in the concentration values is 0.010 (la). The RMSEC drops below this level after the fifth variable is included, and therefore the tentative conclusion is that a four-variable model is appropriate. 

Figure 20. Estimated precipitation weighted hypsometric mean elevations of paleo-drainages sampled by bivalves in the Siwaliks based on estimated A(8lsO, ) shown in Figure 19 as a function of age. Error bars reflect model uncertainty at 2c based on A(6180 m) of the estimated precipitation weighted hypsometric mean elevations.

Color Constancy Using a Dichromatic Reflection Model... 

Figure 6.29 The dichromatic reflection model assumes a matte reflection in combination with a specular reflection. Part of the light is reflected at the outer surface. The remainder enters the transparent coating. The second reflection is modeled as being Lambertian. (Reproduced by permission of Pearson Education from 3D Computer Graphics Third Edition, Alan Watt, Pearson Education Limited, Pearson Education Limited 2000.)...

Figure 6.32 Algorithm of Risson (2003). First, noise is removed by prefiltering the image. Next, the image is segmented. Regions that do not satisfy the assumptions of the dichromatic reflection model are removed. The dichromatic line is computed for the remaining regions. Finally, the most probable point of intersection is computed.

Algorithms based on the dichromatic reflection model are not applicable as the samples are considered to be matte. 

Color constancy using a dichromatic reflection model. 

Finlayson GD and Schaefer G 2001 Solving for colour constancy using a constrained dichromatic reflection model. International Journal of Computer Vision 42(3), 127-144. 

Tominaga S 1991 Surface identification using the dichromatic reflection model. IEEE Transactions on Pattern Analysis and Machine Intelligence 13(7), 658-670. 

Measurements of the polarized reflectance in the NIR have frequently been used to obtain estimates for the transfer integrals. The method consists in fitting a reflectance model based on the Drude expression [Eq. (1)] to the experimental data. The Drude expression should be considered as a tool in estimating the plasmon frequency, ftp the background dielectric constants, e0 plasma frequency, (op and so on. The validity of the Drude analysis is limited to the conducting organic materials, with the electrical conductivity not less than a few S cm-1. 

Currently only Monte Carlo approaches can handle the wide range of surface geometries, reflection models and support complex atomic and molecular processes that occur in real fusion edge plasmas. Therefore the neutral particle transport (ionization, dissociation, etc.) as well as impurity ion transport in the edge region of fusion plasmas is often treated by Monte Carlo simulation on a kinetic level. 

A simple engineering approach to specular reflection is the so-called diffuse plus specular reflection model. Here the total reflectivity p2 = 1 — e = ps + pa is represented as the sum of a diffuse component... 

When applied to isotope effects, the main weakness of the reflection method is the assumption that the transition dipole moment is constant for all isotopologues. This weakness remains in the improved model presented below. Only ab initio calculations are able to go beyond this approximation. However, the dependence of the transition dipole moment along the nuclear coordinates can be introduced (numerically or analytically) in the model below, even if a less compact analytic form is expected. This paper is organized as follows in Section 2 the "standard" reflection model is improved by taking into account the curvature of the upper state potential (in addition to its slope). In Section 3, the quantum character of the final state is taken into account by replacing the Dirac function by an Airy function. In Section 4 the model is applied to the CI2 molecule. In Section 5 the model is adapted and applied to the O3, SO2 and CO2 triatomic molecules. Conclusions and perspectives are presented in Section 6. 

In this section we apply the model presented in Section 3 to the XS of triatomic molecules such as O3, 2/ CO2. In line with Heller [5], Schinke, on pages 115-116 of his book [6], proposed a 2D version of the simple reflection model but without the curvature and quantum effects taken into account in Sections 2 and 3. For triatomic molecules the quantities corresponding to Vq/P and V / or r (see Formula (12)) defined in Section 2 and corresponding to t (see Formula (27)) defined... 

Alternatively, Formulae (12 ), (27 ) or (29) can be used numerically to fit the ct(E)/E of triatomic molecules, even if the interpretation of the fitted parameters is not yet possible. The results presented below show that the numerical improvement obtained by using Formulae (12), (27) or (29) (all have 4 parameters and are able to describe the asymmetry of a a E)/E) is comparable with the improvement observed for CI2 (the Chi /DoF is reduced by typically up two orders of magnitude see Section 4). Here, it is essential to note that the reflection models are only able to describe the envelope of the XS, (corresponding to very short time evolution (f < 10 fs (femtosecond)) of the wavepacket after the photon absorption) and not the vibronic structures which are specific to each molecule and correspond to some vibrational (and or vibronic) oscillations at a time scale of several hiuidred femtoseconds. 

Bricaud A. and Morel A. (1987) Atmospheric corrections and interpretation of marine radiances in CZCS imagery use of a reflectance model. In Oceanography From Space Proceedings Of The Atp Symposium On Remote Sensing, Brest, France vol. 7, pp. 33-50. 

To this group belong several models of very different kinds such as the image model (RE-IE), the charge transfer (RE-CT) model, the electron-hole excitation model (RE-EH), and the Raman reflectivity model (RE-RF). These models have very little in common except that they all lead to enhancements by virtue of a resonance scattering mechanism. The validity of the last statement is not always realized by people, but it will be shown below to hold true. 

To summarize this part, one might say that the Raman reflectivity models, though possible in principle, still need more theoretical substantiation before one can decide regarding their importance in SERS. Certainly, these mechanisms cannot be the whole story. 

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