Which is a true statement about the diffusion-of-innovation theory?

2 Diffusion theory

Diffusion theory concerns with the spread of an innovation through a population. Researchers in diffusion theory have developed analytical models for explaining and forecasting the dynamics of diffusion of an innovation (an idea, practice, or object perceived as new by an individual) in a socio-technical system. Rogers suggests that adopters of innovations can be categorized as innovators, early adopters, early majority, late majority and laggards, whose process of adoption over time is based on the classical normal distribution curve. According to Rogers, innovators are the first 2.5% to adopt an innovation, early adopters, early majority, late majority and laggards are the next 13.5%, 34%, 34%, and 16%, respectively.Bass (1969) formulated a model for the diffusion of consumer durables and other products. Since its original formulation, the Bass model has been used for forecasting innovation diffusion in retail service, industrial technology, agricultural, educational, pharmaceutical, and consumer durable goods markets (Wright and Charlett, 1995). The Bass model considers two factors that affect the decision of a person of becoming an adopter of an innovation. In this paper, we adopt the general view that these two factors are advertisement and word of mouth. The model is formulated in the form of a differential equation: dN(t) dt=(p+qmN(t))(m- N(t)) where N(t) is the cumulative number of persons having adopted at time t, m is the size of the population, p is the influence of advertisement and q is the influence of word of mouth.

15.5.2 Diffusion theory

Diffusion theory provides a way of transitioning from the equation of transfer to a simpler diffusion equation, which provides a solution to the equation of transfer for the case of homogeneous, optically thick, highly scattering materials (i.e., those with relatively large albedos). For the application to subsurface scattering in pbrt, it can be derived by writing the equation of transfer using the reduced scattering and attenuation coefficients and an isotropic phase function.

Starting with the integro-differential form of the equation of transfer from Equation (15.1),

∂∂tLop+tω,ω= −σtpωLip,−ω +σspω∫S2pp,−ω′,ωLipω′dω′+Lepω ,

we assume spatially uniform material parameters and switch to an isotropic phase function p = 1/4π, making a corresponding change to the scattering and attenuation coefficient using similarity theory. We also replace Lo(p, ω) = Li(p, − ω) with a single function L(p, ω).

[15.15]∂∂tLp+tω,ω=−σt′Lpω+σs′4π∫S2Lpω′dω′+Lepω.

The key assumption of diffusion theory is that because each scattering event effectively blurs the incident illumination, high frequencies disappear from the angular radiance distribution as light propagates farther into the medium; in dense and isotropically scattering media, all directionality is eventually lost. Motivated by this observation, the radiance function is restricted to a simple two-term expansion based on spherical moments. Formally, for a function f:S2→ℝ, the n-th moment on the unit sphere is defined as3

μnfi,j,k,…=∫ S2ωiωjωk⋯⏟n factorsfωdω.

In other words, to get the i, j, k, … entry of the n-tensor μn [f ], we integrate the product of f and the i, j, k, … components of the direction ω written in Cartesian coordinates. Note that there is some notational overlap with the angle cosines μk from Section 8.6; the remainder of this chapter will exclusively refer to the definition above.

The zeroth moment, for instance, gives the function’s integral over the sphere, the first moment can be interpreted as a "center of mass” 3-vector, and the second moment is a positive definite 3 × 3 matrix. Higher order moments have many symmetries: for instance, exchanging any pair of indices leaves the value unchanged. High-order moments are useful to derive extended versions of diffusion theory that allow for more pronounced directional behavior; here, we will just focus on degrees n ≤ 1.

The moments of the radiance function have special designations: the zeroth moment ϕ is referred to as the fluence rate:

ϕp=μ0Lp⋅=∫S2Lpωdω.

Note that this expression differs from the fluence function H(p) in Equation (6.6), which was defined as the fluence rate on a surface boundary integrated over time.

The first moment is the vector irradiance:

Ep=μ1Lp⋅=∫S2ωLpωdω.

The two-term expansion mentioned before is defined as

[15.16]Ldpω=14πϕp +34πω⋅Ep,

so that the moments can be exactly recovered, i.e.,

μ0[Ldp⋅=ϕpandμ1Ldp⋅=Ep.

(Here, the "d” subscript denotes the diffusion approximation, not the direct lighting term as it did in Section 14.3.)

To derive the diffusion equation from the equation of transfer, we simply substitute the two-term radiance function Ld into Equation (15.15). The resulting expression is unfortunately not guaranteed to have a solution, but this issue can be addressed with a simple trick: by only enforcing equality of its moments, i.e., by requiring that

[15.17]μi∂∂tL dp+tω,ω=μi−σt′Ldpω+σs′4π∫S2 Ldpω′dω′+Lepω

for i = 0 and i = 1. Computing these moments is a fairly lengthy and mechanical exercise in trigonometric calculus that we skip here.4 The end result is an equation equating the zeroth moments:

divEp=−σt′+σs′ϕp=−σaϕp+Q0p,

where div Ep= ∂∂xEp+∂∂yEp+∂∂zEp is the divergence operator and

Qip=μiLep⋅

is the i-th moment of the medium emission. This equation states that the divergence of the irradiance vector field E is negative in the presence of absorption (i.e., light is being removed) and positive when light is being added by Q0.

Another similar equation for the first moments states that the irradiance vector field E, which represents the overall flow of energy, points from regions with a higher fluence rate to regions with a lower rate.

[15.18]13∇ϕp=−σt′E p+Q1p,

A reasonable simplification at this point is to assume light sources in the medium emit light uniformly in all directions, in which case Q1(p) = 0.

The next step of the traditional derivation is to solve the above equation for E and substitute it into the equation relating zeroth moments. The substitution removes E(p) and yields the diffusion equation, which now only involves the fluence rate ϕ(p):

13σt′div∇ϕp=σaϕp−Q0p+1σt′∇⋅Q1p.

Assuming that Q1(p) = 0, the diffusion equation can be written more compactly as

[15.19]D∇2ϕp−σaϕp=−Q0p,

where D = 1/(3σt′) is the classical diffusion coefficient and ∇2 is a shorter notation for div ∇, which is known as the Laplace operator.

With the diffusion equation at hand, we’ll proceed as follows: starting from a solution for a point source that is only correct in a space where the medium infinitely extends in all directions, we will consider ways of improving the solution’s accuracy in more challenging cases and introduce an approximation that can account for the effect of a refractive boundary.

We’ll initially focus on a point light source that is placed below the surface to approximate the effect of incident illumination striking the surface. Later, we switch to a more accurate light source approximation and derive the beam diffusion solution to the multiple scattering component as well as a single scattering correction that is based on the classical equation of transfer.

4 Sociology

In sociology, adoption-diffusion theory was the dominant approach through the 1960s (Rogers 1995). Diffusion theorists accepted the products of agricultural research as wholly desirable. Hence, their work focused almost exclusively on the fate of innovations designed for farm use. They employed a communications model adopted from engineering in which messages were seen to be transmitted from sender to receiver, later adding the engineering term ‘feedback’ to describe receivers' responses to the messages sent to them. They argued that adoption could be best understood as based on the social psychological characteristics of adopters and nonadopters. Early adopters were found to be more cosmopolitan, better educated, less risk-averse, and more willing to invest in new technologies than late adopters, pejoratively labeled ‘laggards.’ This perspective fitted well with the commitments of agricultural scientists to transforming agriculture, making it more efficient and more modern. However, it ignored the characteristics of the innovations. Often they were large, costly, and required considerable skill to operate and maintain. Not surprisingly, those who rejected the innovations lacked the capital and education to use them effectively.

Later studies challenged the diffusion theorists. First, critics of the Green Revolution asked questions about the appropriateness of the research undertaken (Perkins 1997). They noted that, although inexpensive in themselves, Green Revolution varieties were often parts of packages of innovations that required considerable capital investment well beyond the means of the average farmer. While acknowledging that yields increased, they documented the considerable rural upheaval created by the Green Revolution: growing farm size, displacement of both small farmers and landless laborers to the urban slums, declining status of women, declining water tables due to increased irrigation, and contamination of ground water from agricultural chemicals.

Others asked how agricultural scientists choose their research problems (Busch and Lacy 1983). They noted that science and commerce were necessarily intimately intertwined in agriculture, in the choice of research problems, in the institutional relations between the public and private sectors, and in the value commitments of scientists (often from farm backgrounds) and wealthier farmers. They challenged the engineering model of communication, seeking to substitute for it one drawn from the hermeneutic-dialectic tradition. Drawing on philosophers such as Jürgen Habermas and Hans-Georg Gadamer, they asserted that communications between scientists and the users of the products of agricultural research had to be able to debate fundamental assumptions about what constitutes a desirable future for agriculture as well as specific technical details.

Sociologists have also studied agricultural commodity chains, i.e., the entire spectrum of activities from the production of seed through to final consumption (Friedland et al. 1978). Such studies have examined the complex interaction between scientists and engineers involved in the design of new seeds and equipment and various constituent groups. Unlike the diffusion and induced innovation theorists, proponents of this approach have engaged in detailed empirical analyses of new technologies, challenging the assumptions of the designers. For example, both the tomato harvester and the hard tomato needed to withstand mechanical harvesting were built on the initiative of scientists and engineers in the public sector rather than to meet any need articulated by growers. Together, these technologies transformed tomato production in many parts of the world by reducing the number of growers and farm workers while increasing farm size. Given the limited employment opportunities of those displaced, critics question whether this was an appropriate investment of public funds.

In recent years, sociologists have devoted considerable attention to the new agricultural biotechnologies (e.g., gene transfer, plant tissue culture), especially those involving transformations of plants (see Biotechnology). It is argued that these new technologies have begun to transform the creation of new plant varieties by (a) reducing the time necessary for breeding, (b) reducing the space necessary to test for the incorporation of new traits from large fields to small laboratories, and (c) making it possible in principle to incorporate any gene into any organism. However, analysts note that the vast sums of private capital invested in this sector stem as much from changes in property rights as from any advantages claimed for the new technologies. In particular, they point to the advent of plant variety protection (a form of intellectual property right), the extension of utility patents to include plants, and the imposition of Western notions of intellectual property on much of the rest of the world.

Before these institutional changes, most plant breeding was done by the public sector. Private breeding was not profitable as seeds are both means of production and reproduction. Thus, farmers could save seed from the harvest to use the following year or even to sell to neighbors. Put differently, each farmer was potentially in competition with the seed companies (e.g., Kloppenburg 1988). In contrast, once the new intellectual property regimes began to be implemented, it became possible to prohibit the planting of purchased seed developed using the new biotechnologies. Suddenly, the once barely profitable seed industry became a potential source of profits. Agrochemical companies rapidly purchased all the seed companies capable of engaging in research in hopes of cashing in on the new opportunities. The result has been a shift of plant breeding research for major crops to the private sector and strong prohibitions on replanting saved seed.

Another line of work has examined particular agricultural scientific practices and institutions. These include approaches to irrigation and chemical pest control strategies (Dunlap 1981). Moreover, as environmental concerns have taken on greater significance for the general public, studies of agricultural science have begun to merge with environmental studies.

5.4 Radiative Transport Models

The K–M and diffusion theories mentioned in the previous sections can be seen as special cases of radiative transfer phenomena. When deterministic accurate solutions of the radiative transport equation in biological tissues are required, more robust methods need to be used, e.g., the successive scattering technique, Ambartsumian's method, the discrete ordinate method, Chandrasekhar's X and Y functions, and the adding-doubling method [201]. Their applicability, however, is usually limited to simple conditions and slab (an infinite plane parallel layer of finite thickness [201]) geometries. A comprehensive review of these methods is beyond the scope of this book, and the interested reader is referred to the texts by van de Hulst [260] and Prahl [201]. In this section, we highlight applications involving two of these methods in investigations of light interaction with human skin, namely the adding-doubling method and the discrete ordinate method.

The adding-doubling method has several advantages with respect to the other radiative transfer methods mentioned above. It permits asymmetric scattering, arbitrarily thick samples, Fresnel boundary conditions, and relatively fast computation [201]. The adding method requires that the reflectance and transmittance of two slabs be known. They are used to compute the reflectance and transmittance of another slab comprised of comprising these two individual slabs. Once the transmittance and reflectance for a thin slab are known, the reflectance and transmittance for a target slab can be computed by doubling the thickness of the thin slab until it matches the thickness of the target slab (Figure 5.7). In the original definition of this doubling method, it is assumed that both slabs are identical [260]. Later on, this method was extended to include the addition of two nonidentical slabs [201].

Prahl et al. [205] applied an inverse adding-doubling (IAD) method (“inverse” implying its use as an inversion procedure) to determine optical properties, namely scattering coefficient, absorption coefficient, and asymmetry factor, of biological tissues. The IAD is an iterative method which consists of guessing a set of optical properties, calculating the reflectance and transmittance using adding-doubling method, comparing the calculated values with the measured reflectance and transmittance, and repeating the process until a match is obtained. This method may be used when the propagation of light through the specimen can be described by the one-dimensional radiative transport equation. Its accuracy, however, depends on the criteria applied to define a “sufficiently thin slab” [201]. There are also restrictions on the sample geometry; i.e., it must be an uniformly illuminated and homogeneous slab [205]. The IAD method has also been used to process spectral data (reflectance and transmittance) measured with a spectrophotometer equipped with an integrating sphere in order to derive (in-vivo and in-vitro) optical properties of skin and subcutaneous tissues [28, 255]. For this type of application, the main drawback of this method is the possibility of scattering radiation losses at the lateral sides of the sample [205]. Such radiation losses can erroneously increase the value computed for the optical properties [28].

Nielsen et al. [182] have proposed a skin model composed of five epidermal layers of equal thickness, a dermal layer, and a subcutaneous layer. The radiative transfer equation associated with this layered model is solved using the discrete ordinate algorithm proposed by Stamnes et al. [234] for the simulation of radiative transfer in layered media. Incidentally, Stam [232] has previously applied a similar approach in his model aimed at image synthesis applications (Section 6.2). The discrete ordinate method divides the radiative transport equation into n discrete fluxes to obtain n equations with n unknowns. These equations are solved using numerical techniques. Numerical linear algebra packages, such as EISPACK [53] and LINPACK [67], are usually used for that purpose [234]. This method is feasible when the phase function can be expressed as a sum of Legendre polynomials [41]. For highly asymmetric phase functions, it is necessary to consider a large number of fluxes, which may result in a numerically ill-conditioned system of equations [201]. The subdivision of the epidermis into five layers allow Nielsen et al. [182] to simulate different contents and size distributions of melanosomes (Figure 5.8). Their model accounts for the absorption associated with the presence of blood [202] and melanosomes [129]. The absorption of keratin in the ultraviolet region [31] is also taken into account. In addition, the model accounts for the scattering by small particles using the Rayleigh phase function and for the scattering by large particles using the HGPF. The latter was selected due to the limited knowledge about the actual phase functions for these turbid media and for mathematical tractability since it can be expanded in terms of Legendre coefficients, which are employed in the discrete ordinate formulation. The reflectance curves obtained using their model showed good qualitative agreement with measured curves. They were also able to reproduce counter-intuitive empirical observations [82]. These observations indicated a higher reflectance at wavelengths below 300 nm for individuals with higher level of pigmentation as opposed to a lower reflectance for individuals with a lower lever of pigmentation. Other experimental investigations by Kölmel et al. [145], however, found a different relationship, i.e. a gradual decrease in reflectance with increasing pigmentation. According to Nielsen et al. [182], this apparent discrepancy may be explained by a putative shorter post-tanning period considered in the experiments by Köolmel et al. [145], which, in turn, may not have allowed for a fragmentation of the melanosomes. Nielsen et al. [183] later employed their model in the investigation of the role of melanin depth distribution in photobiological processes associated with the harmful effects of ultraviolet radiation on human skin. In this investigation, they employed a coupled atmosphere tissue–discrete ordinate radiative transfer (CAT–DISTORT) model [134] to account for solar light incidence as well as atmospheric and physiological conditions.

III.A.3 Calculational Methods

The neutron population of any chain reacting system is difficult to model because of the essentially continuous variation in energy and direction. The variety of reactions, some with very complex cross sections (e.g., as shown by Fig. 4) and secondary neutron emissions, increases the difficulty.

The first calculational techniques were based on simplified models. Then, as digital computer technology evolved, successively more sophisticated methods have been developed and used.

III.A.3.b Transport theory

A more complete description of the neutron chain reaction requires specification of not only general neutron flow, but of neutron energies and directions. A full model needs seven variables for:

1.

position in space r (a vector quantity requiring three coordinates, e.g., x, y, and z or r, θ, and ϕ for rectangular and cylindrical systems, respectively)

2.

velocity v (a vector quantity requiring three coordinates) usually split between energy E(=12mv2) and direction Ω (consisting of components θ and ϕ)

3.

time t

The Boltzmann neutron transport equation (of which the diffusion theory approximation may be considered a subset) for the multivariable flux Φ(r, E, Ω, t) may be written as:

(36)

where each term represents a rate (per unit parameter) involving neutrons with the specified coordinates. Terms 1 and 2 are the net rate of neutron accumulation and the leakage, respectively. The third term is the total interaction rate or the rate of removal of neutrons due to absorption and scattering interactions (since the latter “out-scatters” result in at least some change in neutron energy and direction).

The last two terms in Eq. (36) represent the production phenomena where neutrons at arbitrary energy E′ and direction Ω′ react with nuclei to produce those at reference energy E and direction Ω. The integrals sum over all initial energies and directions. Specifically, the double integral in term 4 yields the total fission rate; and its product with the neutron spectrum function χ(E) represents the fission-neutron energy distribution. (It may be recalled, for instance, that all fission neutrons are fast while in some reactors almost all fissions are caused by thermal neutrons).

The last term in Eq. (36) is based on differential scattering of neutrons from initial energy E′ to final energy E and from initial direction Ω′ to final direction Ω. The cross section Σs(r; E′ → E; Ω′ → Ω) accounts for the relative probabilities of all possible changes (recalling, for example, that fast neutrons can only lose energy in scattering collisions with stationary nuclei). This “in-scatter” term is the only source of neutrons at energies below the fission-neutron range, including the “slow” neutrons upon which thermal reactor designs are based.

The complex energy dependence of the reaction cross sections (e.g., as shown on Fig. 4) precludes closed form solution of Eq. (36). One solution approach begins by obtaining approximate fluxes and reaction cross sections by averaging over one or more parameters. The continuous energy dependence of the flux Φ(E), for example, may be divided into intervals or “groups” according to

(37)ΦΔE =∫ΔEΦ(E)dE

and the cross section developed as

(38)ΣrΔE=∫ΔEΣr(E)Φ(E)dE/∫ΔEΦ(E)dE,

for (energy-dependent) flux Φ, cross section Σr for (arbitrary) reaction r, and energy interval ΔE corresponding to group g. Multiplying Eqs. (37) and (38) shows that the formulation preserves the product of flux and cross section as the reaction rate. Cross sections in the form of Eq. (38) are said to be flux-averaged or flux-weighted.

For the special case where leakage can be approximated by diffusion theory [Eq. (29)], the multigroup approximation [Eqs. (37) and (38), with the latter also applying to diffusion coefficient D] to the Boltzmann transport equation results in the expression

(39)−▿⋅Dg▿ φg+∑φg=1kχg∑ g′=1GυΣfgφg′+∑g′=1GΣg′→gφg′

for each of G groups where summations replace the integrals, the Σg′→ g are cross sections for scattering from group g′ to group g, and the other symbols represent group formulations of previously defined functions and parameters.

It must be recognized that the formulation represented by Eqs. (37)–(39) still requires knowledge of the interrelated neutron flux and the reaction cross sections. An advantage is realized only when good first approximations are available and iterative procedures can be used to refine successively the results. In general, the finer the group structure divisions (i.e., the larger the number of groups), the better the approximation.

The diffusion theory approximation for neutron leakage assumes homogeneous, or at least smooth, variation of material properties. Thus, application to the heterogeneous geometries typical of reactors requires more sophisticated methods for complete calculations or, at least, to determine “effective” parameters that allow Eq. (39) to provide accurate answers.

Better approximations to Eq. (36) may be obtained by employing a transport theory method which adds explicit representation of the directional dependence of the neutron flux. The discrete ordinates method divides these directions into discrete “groups” analogous to the energy representation. The Monte Carlo method is capable of modeling both energy and direction in discrete groups or with essentially continuous variation.

The Monte Carlo approach tracks individual idealized neutron paths one collision at a time based on cross sections and random number generation. The random numbers are used with nuclear data to predict reaction types, directions of post-reaction neutron scatter, and neutron energy loss. Multiplication is calculated as the ratio of neutrons produced to neutrons lost [i.e., consistent with the definition in Eq. (23)].

11.2 Final Conclusions

Considering cumulatively the content of the book, it becomes evident that malware diffusion theory can be rather useful for network designers, administrators, and professionals separately. Depending on the network type and employed applications, one of the four frameworks presented in Part 2 can be utilized for properly predicting the behavior of a network under attack and design proper countermeasures. More importantly, these approaches enable designing dynamic response mechanisms, which are able to intelligently adapt to the fundamental nature and features of the threats, thus more effectively securing the underlying infrastructure. This was not possible in the past at the magnitude attained by the approaches presented in the book.

At the same time, the analyzed models and frameworks have been shown to be generic enough, so that their analytic properties cover broader application areas. Similar phenomena to malware diffusion emerge in information flow applications and the proposed frameworks could be easily extended and adapted to cover more general problems of information diffusion over complex communication networks. Other similar application domains can be identified within the areas of future wireless Internet [86] and other complex networks, while properly extrapolating the techniques presented for obtaining faster the desired outcomes.

The techniques and models presented in this book may be considered as the first steps of a broader vision to develop holistic frameworks describing the flow of information in communication networks. Starting with the diffusion of malware, similar attempts for other problems and application areas of content dissemination can be inspired. This would signify the successful potential of the content of this book and provide even more efficient mechanisms for designing infrastructures and information management mechanisms of the future.

4.2.2 Assessing fruit quality using empirical approaches

This chapter has so far focused on using the fundamental approach (i.e. the diffusion theory model) to determine the optical properties of fruits. While promising results have been obtained on predicting fruit firmness by optical properties, the results for predicting fruit SSC are far from satisfactory. Two empirical methods are presented here for describing the scattering profiles of the hyperspectral images. The methods, as presented below, are simpler and generally better than the fundamental approach in predicting fruit firmness and the SSC of apples and peaches.

In an experiment conducted in 2003, hyperspectral images were collected from 700 Golden Delicious apples that had been kept in a controlled environment (2 percent O2 and 3 percent CO2 at 0°C) for about 5 months. Mean reflectance values were calculated for each scattering profile for a total scattering distance of 25 mm (Figure 14.7) over the wavelengths of 500–950 nm. Relative mean spectra of apples were obtained by dividing the sample mean reflectance by the mean reflectance of a white Teflon standard (with the dark current subtracted). Figure 14.12 shows the relative mean reflectance spectra for 10 Golden Delicious apples over wavelengths of 500–950 nm. Similar to the absorption spectra in Figure 14.9, there are two absorption peaks for the mean relative reflectance spectra, at 680 nm and 950 nm, due to chlorophyll and water, respectively. Compared with the μa spectra in Figure 14.9, chlorophyll absorption peaks of the mean spectra in Figure 14.12 appear to be sharper and more prominent. The relative reflectance increases steadily over the wavelengths 725–900 nm.

Principal component analysis was applied to reduce the data dimensionality and extract the main features. A back-propagation feed-forward neural network with inputs of principal component scores was used to predict fruit firmness and SSC. Good predictions of fruit firmness and SSC of Golden Delicious apples are obtained using relative mean spectra, with correlation coefficients of 0.85 and 0.89, respectively, and the corresponding standard errors of prediction (or SEP) of 6.9 N and 0.72 percent (Figure 14.13). Both firmness and SSC prediction results are considerably better than those obtained using μa, μs', or μ′eff, and the firmness prediction results are also better than those using Vis/NIR spectroscopy (Lu et al., 2000; Park et al., 2003).

In another study, hyperspectral scattering images were obtained from 700 Red Haven peaches using the same imaging system shown in Figure 14.3 but with a slightly different arrangement for the light beam (1.6 mm in diameter and 17° incident angle).

A typical hyperspectral image from a peach fruit is shown in Figure 14.14a, which is somewhat different from the one for the apple in Figure 14.7. First, the peach fruit has lower reflectance at 600 nm and below, since its absorption coefficient appears to be higher than that for apples. Second, the scattering profiles for peaches are broader than those of apples. However, this does not necessarily mean that the reduced scattering coefficient of peaches is higher than that of apples, since absorption and scattering are intertwined.

Instead of using mean spectra, a two-parameter Lorentzian distribution function was proposed to describe each scattering profile over wavelengths of 600–1000 nm:

where x is the distance in mm, b represents the peak value of the scattering profile, and c is the full scattering width at half maximum (mm). For convenience, the scattering distance on the left side of the beam incident point was designated to be negative and the right side to be positive. The offset distance of 1.6 mm between the light beam center and the scanning line was not considered. As shown in Figure 14.14b, the two-parameter Lorentzian distribution function gives an excellent fit to the scattering profiles. For all 700 peach fruit, the average values of the correlation coefficient for the curve-fitting results are greater than 0.995 for wavelengths of 600–1000 nm. Spectra of Lorentzian parameters b and c for selected peach fruit are shown in Figure 14.15. The scattering width, represented by Lorentzian parameter c, is relatively flat over the entire spectral region. While the parameter b, which has not been corrected by a standard, changes dramatically over the spectral region of 600–1000 nm, the magnitude of change for the parameter c is much smaller. The values of Lorentzian parameter c vary between 2.5 mm and 3.8 mm for all peach fruit. There are two downward peaks on the parameter c spectra; one is around 675 nm and the other at 950 nm, due to chlorophyll and water. Multiple linear regression models were developed relating Lorentzian parameters b, c, and their combination b & c (set side by side according to wavelength) to fruit firmness for the calibration samples. Both Lorentzian parameters b and c are well correlated with the firmness of peach fruit; however, parameter c, the scattering width, gives worse predictions of fruit firmness than does parameter b (Figure 14.16). Better predictions of fruit firmness (r = 0.88) are obtained when parameters b and c are combined.

The above two application examples indicate that the empirical approach has better predictions of fruit firmness and SSC for apples and peaches than those obtained with absorption and scattering coefficients. Poor results from the fundamental approach could be attributed to errors in determining the absorption and reduced scattering coefficients. Moreover, the empirical approach also compares favorably with NIR spectroscopy in predicting fruit firmness. This could be due to the fact that the hyper-spectral scattering method can better characterize the scattering properties of the fruit than does NIR spectroscopy.

1.5.1 Adoption of Innovation Theories

Theories have been put forward that map out factors associated with the adoption of innovations and may offer a potentially useful lens through which to understand consumer response to novel food. Rogers’ (2003) diffusion of innovation theory, for example, describes the process by which an innovation disseminates through a societal group and focuses upon decision-making processes which lead to adoption of a new product or service. Ideally, according to this decision-making process, a consumer passes from first knowledge of an innovation to forming a positive attitude toward the innovation to an intention to adopt the innovation and, finally, to buying and consuming the innovation. However, in practice, knowledge of an innovation not always leads to a positive attitude, let alone to positive behavioral responses. Existing habits play an important role in prohibiting adoption of food innovations. As most people act on a routine basis in food-related contexts, positive consumer responses to food innovations are poor predictors of actual behavior (Van ‘t Riet et al., 2011). Consumers may also more actively reject innovations. For instance, when they are in conflict with existing beliefs or norms or when they have an unfavorable image (Kleijnen et al., 2009; Ram and Sheth, 1989). In addition, the degree of perceived risk associated with an innovation is one of the main barriers that evokes rejection of innovations, especially in relation to food (Frewer et al., 2003). Note that consumer perceptions of risk differ from expert assessments of risk (Fischer and Reinders, 2016). Consumers assess risks (and benefits) to some extent based on heuristics (i.e., rules of thumb) rather than on a full deliberation of the factual risk (Tversky and Kahneman, 1974). For example, in the debate of genetic modification of foods, this has led to the fact that mainly the uncontrollable and unobservable nature of genetic modification contributed to increased risk perception (Fife-Schaw and Rowe, 1996, 2000).

Ronteltap et al. (2007) further conceptualized acceptance of new food innovations by extending diffusion theory. More specifically, they state that a consumer’s decision to adopt or reject a food innovation is determined by their intention to do so, which in turn is a function of a number of psychological process constructs:

the trade-off between perceived costs and benefits,

perceived risk and uncertainty,

subjective norm; or, the opinion of important others such as peers, parents, or experts, and

the degree of perceived control the user may have over the technology.

In addition, these perceptions are evoked by a case-specific combination of features of the innovation, characteristics of the consumer who is confronted with the adoption decision, and characteristics of the socio-cultural system (i.e., the nature of the economic, political, cultural, and social environment in which food choices are made). Trust in the agent, producer, or government that launches or communicates about an innovation plays an important role in reducing risk perception and increasing benefit perception (Frewer et al., 2003; Siegrist, 2000). Trust has emerged as an important factor as over the past decades major technological changes have taken place in the way food is produced, prepared, and packaged (Kjærnes et al., 2007). It has been demonstrated that food innovations have reduced concerns and higher acceptance among consumers when their producers address consumers’ needs and offer tangible benefits (Bruhn, 2008). In Europe, the most trusted sources of information are health professionals, university scientists, and consumer organizations, followed by scientists working in industry, and media (European Commission, 2006; Rollin et al., 2011).

Publisher Summary

This chapter examines a class of local lighting models whose design was based on the application of the diffusion theory. These models were developed primarily to be incorporated into image-synthesis pipelines. The chapter starts with the review of the model proposed by Jensen et al., henceforth referred to as DT model. This model applies the general concept of the bi-directional scattering-surface reflectance distribution function (BSSRDF) to describe the propagation of light from one point on a surface to another. The performance of the DT model was later improved by Jensen et al. through the incorporation of a two-pass hierarchical algorithm. As these works represent the first appearance modeling efforts to use the diffusion theory within the computer graphics field, they are the focal point of the discussions presented in this chapter. However, the application of the diffusion theory in the modeling of material appearance has evolved beyond these models. Accordingly, the discussion of models based on the diffusion-approximation approach proceeds to address relevant works that built on and extended the techniques used in the DT model. Most of the models examined in this chapter have been applied to the rendering of a variety of materials, from marble to milk. In this book, they are examined with respect to theoretical and practical issues involving their use in the generation of realistic images depicting human skin.

3 A SYNTHESIS

Although pickup ions are energized by the MRI mechanism, not all the MRI accelerated ions are sufficiently energetic to be further accelerated by a second-stage diffusive shock acceleration process. For diffusion theory to be applicable at a perpendicular shock, particles downstream of the shock must be capable of diffusing upstream. This requires the cosmic ray anisotropy be small. For a perpendicular shock, this implies that the particle velocity v satisfy the condition [10] v ≫ (3Vn1/r)(1 + ηc2)1/2, where Vn1 is the upstream flow speed in the stationary shock frame and r the shock compression ratio. For the present, we assume hard-sphere scattering to describe the transport of diffusive particles. Thus, if λ|| is the parallel mean free path κ||/κ⊥ = 1 + ηc2, ηc = 3κ||/(vrg) ≡ λ||/rg. Here rg = pc/(QB) is the particle gyroradius (p is particle momentum, Q the charge, c the speed of light, and B the magnetic field strength). In the hard sphere scattering model, ηc is a measure of the strength of scattering: ηc small implies strong scattering whereas ηc large corresponds to weak scattering. For resonant scattering, λ|| ∝ R1/3, where R ≡ pc/Q is the particle rigidity. It follows then that ηc ∝ R-2/3, or ηc = ηcp(m/M)2/3, ηcp ≡ (λ||/rg)proton. The important implication is that even if scattering is weak for protons, i.e., ηcp ≫ 1, the inverse dependence of ηc on M implies that ηc can be much smaller for heavy ions. Thus, heavy ions are accelerated diffusively at a much lower threshold velocity than lighter ions. Consequently, a larger fraction of heavy MRI accelerated ions will enter a second-stage diffusive shock acceleration process than light MRI accelerated ions.

It remains to determine whether the two competing mass dependence effects conspire to satisfy the Cummings and Stone [4] ACR injection results. We consider two cases; (i) a highly peaked distribution such as a shell, and (ii) a power law distribution such as that obtained by Vasyliunas and Siscoe [11]. This gives (i) Rref = Rref (H+)[m/M]1/2, and (ii) Rref = Rref (H+)[m/M]. Simulations suggest that for (i), we use a (v/v0)-4 accelerated spectrum, and for (ii) a (v/v0)-5 accelerated spectrum. We suspect that the correct model lies between these two extremes.

The differential intensity of the ith species ji(m-2 s-1 sr-1 MeV-1) at the termination shock is given by ji = p2fia(p, M). In order to calculate the acceleration efficiency in the same way as [4], we rewrite the differential intensity in their form, i.e., ji=qεiFi4πE0iq−4/2 E−q+2/2, where Fi(cm− 2s− 1) is the flux of the ith pickup ion species at the termination shock, E0i the injection energy of 5.2 × 10-3 MeV/nuc. used by Cummings and Stone, and εi is the acceleration efficiency that is to be computed. Note our distinct use of the term “acceleration efficiency” for εi, which is referred to as “injection efficiency” by [4]. We use the former term to distinguish our use of injection efficiency in the context of MRI injection. The injection energies needed to render an ion diffusive range from ~ 330 keV for pickup H+ to about either ~ 185 (shell) or ~ 75 (power law) keV for C+, N+, O+, and Ne+. Our predicted acceleration efficiency and that inferred by Cummings and Stone can now be compared directly since the same parameters are used. In Fig. 1, we plot, following [4], the inverse acceleration efficiency εi− 1, normalized to He+, as a function of ion mass. The solid triangles correspond to an in initial pickup ion shell distribution, the open circles to the VS power law case, and the open squares to the inferred observations presented in [4]. Two points stand out from Fig. 1. The first is the obviously close agreement between the model results and the observed results. The second is that the extreme cases considered here, the initial shell and the VS distributions, yield acceleration efficiencies that are almost identical, suggesting that a more realistic pickup ion distribution at the termination shock is unlikely to alter our conclusions significantly.

In the limit of strong scattering, the injection threshold for ions to be viewed as diffusive is virtually identical for all ion species. Consequently, no difference exists between the injection efficiencies of different mass ion species in the case of strong scattering. This is an important point since it relates for the first time the differential injection efficiency of pickup ions of different masses to the particle scattering strength.

One can use the diffusive part of the MRI spectrum for pickup H + as the explicit source for ACRs at the termination shock, and hence determine the ACR spectrum at the shock and the modulated spectrum at any point within the heliosphere. The combined pickup ion, MRI accelerated and ACR spectrum at the termination shock, assumed to be located at 80 AU with parameters given by [4], is illustrated in Fig. 2. The flux of ACRs shown in Fig. 2 is in accord with the expected ACR source spectrum used to model the observed modulated ACR flux within the heliosphere.