Graziani F. (editor) Computational Methods in Transport

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The Rayspread Canopy Reﬂectance Model 213

highlighting RT features that are speciﬁc to vegetation canopies, however,

a one-dimensional approach may be more suitable at ﬁrst since the spatial

dependency of both ˜σ

and ˜σ

will only add additional complexity to the

RT formalism. In the following we thus consider a structurally homogeneous

plane-parallel leaf canopy of depth H, that is being illuminated from direction

Ω

in a spatially uniform manner from the top (z = H).

1.1 Foliage Properties Aﬀecting the Scattering Coeﬃcient, ˜σ

The probability for radiation, travelling along some arbitrary direction Ω



to be scattered into a particular direction Ω clearly depends both on the

amount of foliage area present within the canopy volume, i.e., the leaf area

density Λ(z)[m

], the distribution of the foliage orientation in the canopy,

i.e., g

(z,Ω

), and the leaf scattering phase function f(Ω



→ Ω; Ω

,z).

Assuming that all leaf normals Ω

(θ

,φ

)–whereθ

is the zenith angle

and φ

is the azimuthal angle – point into the upper hemisphere (2π

), the

diﬀerential scattering coeﬃcient, ˜σ

at some level z in the canopy can be

written as:

˜σ

(z,Ω



→ Ω)=

Λ(z)

2π



2π

(z,Ω

) |Ω



· Ω

|f (Ω



→ Ω, Ω

,z) dΩ

(2)

where, f(Ω



→ Ω; Ω

) describes the fraction of intercepted energy (from

photons that initially travelled in the direction Ω



) that is scattered by a

leaf with outward normal Ω

into a unit solid angle about the direction Ω.

Note that the integral of f(Ω



→ Ω; Ω

) over all exiting directions yields the

single scattering albedo, ω

, which may depend on both the initial photon

direction Ω



and the leaf normal orientation Ω

[SM88].

Probability distributions of the leaf orientation also have a signiﬁcant im-

pact on the interception probability for radiation travelling within a canopy.

More speciﬁcally, the leaf-normal distribution (LND) function g

(z,Ω

), de-

notes the fraction of total leaf area in the horizontal layer of unit thickness

at height z whose normals fall within a unit solid angle around the direction

Ω

, and must satisfy the following normalization criterion [Ros81]:

2π



2π

(Ω

) dΩ

2π



2π

dφ



π/2

(θ

,φ

)sinθ

dθ

≡ 1

Although some plants are known to change their orientations in response

to solar illumination conditions, heliotropism (and other two-dimensional

LND functions) are primarily treated in theoretical studies [SGR85, Ver87]

because the assumption of azimuthally random leaf distributions signiﬁcantly

reduces the mathematical complexity of the function g

(θ

,φ

). Among the

most commonly used azimuthally invariant leaf normal distribution functions

are 1) the trigonometric functions of [Bun78]:

214 J.-L. Widlowski et al.

∗

(θ

)=(a + b cos 2θ

+ c cos 4θ

)(3)

and (2) the beta functions of [GS84]:

∗

(θ

ξ−1

(1 − x)

ν−1

B(ξ, ν)

0 <x<1(4)

where B(ξ,ν)=Γ (ξ)Γ (ν)/Γ (ξ + ν) is the beta function, Γ the gamma func-

tion, x =2θ

/π and g

∗

(θ

)=g

(θ

)sinθ

. Table 1 provides the parameter

values for (3) and (4) to generate six common LNDs, including an erectophile

(planophile) LND that features predominantly vertical (horizontal) foliage

orientations typical for most grasses (watercress). The impact of diﬀerent

LNDs on the surface-leaving radiation ﬁeld is such as to alter the observed

BRF

values anywhere from −10 % to +40 % with respect to a 1-D canopy

with directionally invariant interception probability (g

∗

(θ

)=sinθ

). On

the other hand, for a homogeneous foliage canopy with a given LND, the dif-

ferences in BRF values arising due to the divergence of (3) and (4) amounts

to only ∼ 0.5 % on average in both the red and near-infrared wavelengths.

Table 1. Parameter values for the trigonometric (a, b, c) [Bun78] and beta (ξ, ν)

function [GS84] representation of 6 diﬀerent leaf-normal distributions, g

∗

(θ

)

Uniform Spherical Planophile Erectophile Plagiophile Extremophile

a 2/π sin θ

2/π 2/π 2/π 2/π

b 002/π –2/π 00

c 0 0 0 0 –2/π 2/π

ξ 1.000 1.101 2.770 1.172 3.326 0.433

ν 1.000 1.930 1.172 2.770 3.326 0.433

1.2 Canopy Properties Aﬀecting the Extinction Coeﬃcient, ˜σ

Due to the physical size of foliage elements the extinction of solar radiation

within a given canopy volume is wavelength independent but highly sensitive

to the direction of propagation. More speciﬁcally, the probability, per unit

pathlength of travel, that photons may hit a ﬁnite-sized leaf depends not only

on the amount and orientation of the scatterers but also on the direction of

travel of the photons. As a consequence, the probability that single-scattered

solar radiation exits a “cloud of oriented ﬁnite-sized scatterers” is signiﬁcantly

The BRF [-] is the radiant ﬂux density (Φ

target

) reﬂected into some inﬁni-

tesimal solid angle divided by the radiant ﬂux density reﬂected by an ideal, perfectly

diﬀusing Lambertian reﬂector (Φ

lambert

) into the same inﬁnitesimal solid angle

under identical conditions of illumination [NRH77].

The Rayspread Canopy Reﬂectance Model 215

enhanced if its new direction lies along (or is close to) the retro-reﬂection

direction [Hap81, Ger88, Kuu91]. Thus a viewer looking at a canopy target

along the solar illumination direction will not see any shadows but a local

peak in the surface-leaving reﬂectance ﬁeld, commonly known as the hot spot

(or, opposition, or, Heiligenschein) eﬀect. To account for this retro-reﬂection

peak [PV97] have proposed to write the extinction coeﬃcient as:

˜σ

(z,Ω, Ω

)=σ

(z,Ω)

O(z,Ω, Ω

)=Λ(z) G(Ω)

O(z,Ω, Ω

)(5)

where σ

(z,Ω) is the extinction coeﬃcient for a purely turbid medium, i.e.,

an inﬁnite number of oriented point scatterers. Λ(z)[m

] is the leaf area

density (LAD) at level z,andG(Ω) is the well known function of [Ros81]

accounting for the projection of the leaf normal orientations in a particular

direction of interest. G(Ω) is an indicator of the likelihood that radiation

may be intercepted by the canopy if travelling in direction Ω, since it is the

mean projection of a unit foliage area, in the direction Ω, per unit volume of

canopy:

G(Ω)=

2π



2π

(Ω

)



Ω

· Ω



dΩ

where for azimuthally independent leaf normal distributions G(Ω) simpliﬁes

to G(θ) and the probability distribution function of the leaf normal orien-

tations g

(Ω

) reduces to g

(θ

). Note that the probability of interception

remains directionally constant (G(θ)=0.5) if the LND is perfectly spherical,

i.e., g

∗

(θ

)=sinθ

Finally, the term

O(z,Ω, Ω

) in (5) is a geometric correction that aims

at accounting for the reduced extinction coeﬃcient along the retro-reﬂection

direction in a canopy composed of ﬁnite sized leaves. [VPD90], who stud-

ied the physical mechanism behind the hot-spot eﬀect – among others,

e.g., [Hap81, Ger88, Kuu91] – idealised this free space between scatterers by

considering two cylindrical volumes V

 and V

 drawn along the in-

coming Ω

and outgoing Ω directions, respectively. In analogy to an incident

and reﬂected beam of light, V

 and V

 have a common base deﬁned

by a circular sun ﬂeck of radius r, and therefore share a common volume free

of scatterers. With the assumption that the leaf area density Λ and the Ross

function G(Ω) are constant along the coordinate z, the correction function

for the extinction coeﬃcient

O(z,Ω, Ω

)isgivenby∂(V

•

 / V

)/∂z

which is equal to [PV97]:

O(z,Ω, Ω

)=1−



cos

−1

∗

− ζ

∗

[1 − ζ

∗

]

1/2

and unity if ζ

∗

=min(1,ζ

) = 1, where ζ

= zG

/2r>1. [GPV97]

describes how to retrieve an estimate of r, and the geometric factor G

deﬁned as a combination of illumination (observation) zenith θ

(θ

)and

azimuth φ

(φ

)angles:

216 J.-L. Widlowski et al.



tan

+tan

− 2tanθ

tan θ

cos(φ

− φ

)



1/2

A direct consequence of this formulation is that the smaller the radius

r of the circular sun ﬂecks at some depth z in the canopy the narrower the

angular range to which

O(z,Ω, Ω

) will provide signiﬁcant contributions. For

homogeneous leaf canopies, one can thus expect the hot spot contribution

due to radiation that has only collided once with the underlying soil (i.e., the

black canopy contribution [PGW04]) to be slightly narrower than that due to

radiation having collided once only with the foliage elements of the canopy

overstory. This is illustrated in the main left panel of Fig. 1 which shows the

BRF contributions due to the single collided by the leaves (dashed) and the

single collided by the soil (solid) radiation components for a homogeneous

leaf canopy with an LAI of 3 in the red spectral band.

-60 -40 -20 0 20 40 60

Viewing angle [degree]

0.00

0.01

0.02

0.03

0.04

0.05

Bidirectional Reflectance Factor

single-collided by soil

single-collided

by leaves

60 40 20 0 20 40 60

V ew ng angle degree]

0 015

0 020

0 025

0 030

B d rectona Ref ectance Fac or

total BRF in X−plane

-60 -40 -20 0 20 40 60

Viewing angle [degree]

0.00

0.01

0.02

0.03

0.04

0.05

Bidirectional Reflectance Factor

single-collided

by soil

single-collided

by leaves

multiple-collided

60 40 20 0 20 40 60

Viewing angle [degree]

0 015

0 020

0 025

0 030

B d rectona Ref ectance Fac or

total BRF in X−plane

Fig. 1. Main panels: The BRF contributions in the principal plane (θ

=50

◦

) due

to the single-collided by the soil (solid), single-collided by the leaves (dashed), and

multiple collided (dotted) radiation components for structurally homogeneous (left

panel ) and heterogeneous (right panel) leaf canopies in the red spectral band. Inlaid

graphs: The total BRFs in the orthogonal (or cross) plane for the same scenes

Heterogeneous leaf canopies are characterized by a conglomerate of dif-

ferently sized (and shaped) gaps among their foliage elements, that all have

their own hot spot contribution (of diﬀerent angular widths). Consider, for

example the simple case where all foliage elements are uniformly distributed

within spherical volumes – of identical size and LAD – that are randomly lo-

cated above a perfectly ﬂat background (compare with the lower right panel

of Fig. 2). In this case, only two typical scales of voids exist among the scat-

terers in the scene: 1) gaps of the order of a few centimeters that occur amid

the foliage elements in the spherical volumes, and 2) gaps of the order of a

The Rayspread Canopy Reﬂectance Model 217

Structurally Homogeneous Vegetation Canopy

Structurally Heterogeneous Vegetation Canopy

Stochastic Representation Deterministic Representation

Fig. 2. Top panels: structurally homogeneous vegetation canopies, where the foliage

is uniformly distributed throughout the available volume. Bottom panels: struc-

turally heterogeneous vegetation architectures, where the scatterers are conﬁned

within the volumes of several non overlapping spheres. In the right panels, the dis-

crete foliage elements are represented in a deterministic manner, whereas in the left

panels, they are displayed in a stochastic manner using pseudo-turbid objects

few meters that exist between the spherical foliage clusters. As a consequence

the black canopy BRF component will feature both a narrow hot spot contri-

bution, due to radiation exiting the canopy via the spherical foliage volumes,

and another much broader contribution due to radiation exiting the canopy

from within the gaps in between the spheres. Both of these signatures are

readily discernible in the black canopy BRF contribution (solid line) in the

right panel of Fig. 1. At the same time the angular width of the hot spot

contribution due to the single collided by the leaves BRF component (dashed

line), is – just as in the homogeneous case – primarily driven by the gap sizes

among the scatterers in the foliage clusters (spheres). Note also, that the

multiple-scattering component does not feature any discernible hot spot con-

tributions. Remotely acquired hot spot observations thus consist of several

nested contributions of increasing angular widths, that arise from mutual

shading conﬁgurations amid target constituents occupying diﬀerent spatial

scales (e.g., leaf, tree, terrain) [Mer89, QX94] and [GQW97].

The black canopy component is of particular interest here since it is

conditioned by the shapes, sizes and spatial organisation of the overlying

canopy structure without, however, being aﬀected by the spectral properties

of that medium. In addition, the shape of its BRF ﬁeld is known to exhibit a

218 J.-L. Widlowski et al.

bel l-shaped pattern – ignoring the immediate hot spot region [PWG02] –

because the presence of gaps in between the foliage structures reduces the

domain averaged extinction coeﬃcient for radiation exiting the heterogeneous

canopy at small θ

(unlike at large θ

where the optical depth increases dra-

matically). Given that the RT in vegetation canopies is essentially driven by

single-scattering interactions in the red, the bell-shaped black canopy BRF

component is thus in direct competition with the remaining, predominantly

bowl-shaped BRF components to form the overall “shape” of the surface leav-

ing radiation ﬁeld (compare with the inlaid graphs in Fig. 1). Since the re-

ﬂectance values of soils tend to be larger than those of foliage in the red, the

magnitude and shape of the overall BRF ﬁeld is largely driven by the amount

and spatial organisation of the gaps, in addition to the opacity and shapes of

the foliage clusters in the canopy. In fact, there is now ample observational

evidence to link the reﬂectance anisotropy of 3-D vegetation targets (in the

red spectral band) to the heterogeneity of the canopy [WPG01,LG02,CRH03]

and [WPG04]. Furthermore, [DM04] have recently proposed a germane con-

tribution which highlights and helps identifying geophysical conditions where

3-D eﬀects have a very signiﬁcant contribution to the establishment of the

radiative transfer regimes. Their approach and results derive from statistical

analyses of various types of internal variability of the extinction coeﬃcient.

The interpretation of multi-directional remote sensing observations could

thus, in principle, provide additional information on the degree of heterogene-

ity that exists within terrestrial environments. This statement is, however,

subject to the dimensionality of the RT model at hand, since in inversion

mode only those vegetation structures can be retrieved that the model is

actually capable of simulating (in forward mode). Obviously the more free

parameters a model possesses, the larger the number of possible combinations

(and hence structural and spectral scenarios) that may be simulated. Fractal

or L-system based approaches may facilitate the generation of visually im-

pressive plant representations. Nevertheless, the construction of ecologically

(and bio-mechanically) accurate canopy architectures does require access to

dozens (or more) structural parameters that may be diﬃcult to come by.

In eﬀect, detailed ﬁeld measurements of structural and spectral parameters

suitable for inclusion in 3-D RT model simulations are still lacking for most

species and geographical locations. Recently, [WPG03] provided a compila-

tion of allometric relationships that allow the faithful reconstruction of ﬁve

major European tree species for 3-D canopy reﬂectance modelling purposes

at medium spatial resolution. For any such forest structure, variable bound-

ary conditions may have to be accounted for, that is, variable proportions

of diﬀuse radiation at the top of the canopy, variable soil scattering proper-

ties and albedo values beneath the canopy, and – depending on the spatial

resolution of the observing sensor – variable horizontal radiation ﬂuxes due

to the discrete nature of adjacent canopy structures. Luckily the net impact

of the lateral boundary conditions is certainly negligible for medium spatial

The Rayspread Canopy Reﬂectance Model 219

resolution sensors like, for example, MODIS, MERIS, and MISR. Similarly

the directional distribution of the incident radiation may be described with

formulae like those of [Ste77] for clear sky radiation. Recent developments

now also allow for a decoupling of the soil optical properties from the BRF

simulations of 3-D plant canopies [PGW04]. As such, for each one of the 3-D

canopy structure scenarios that are generated in the context of a LUT-based

inversion approach, canopy reﬂectance simulations are required to account for

the natural variability among leaf and bark optical properties, their changes

as a function of wavelength, and diﬀerent positions of the sun in the sky.

The Rayspread model has been developed speciﬁcally to perform accurate

and computationally eﬀective BRF simulations over arbitrarily complex 3-D

plant environments.

2 The Rayspread Model

Monte Carlo (MC) ray-tracing techniques lend themselves nicely to the sim-

ulation of reﬂectance ﬁelds over complex canopy representations in that they

stochastically sample the various interactions of the incident radiation within

the scene until a certain level of stability is reached (for a recent overview of

MC canopy reﬂectance models see [DLN00]). One of their main assets, how-

ever, is the capability to perform radiative transfer simulations in 3-D media

of almost arbitrary complexity (subject to the available computer memory).

On the other hand, such MC ray-tracing models are – in their simplest form

– computationally demanding, which either leads to their parallelisation and

integration into multi-processor environments [GV96], or else, to the imple-

mentation of variance reduction techniques [TG98] that speed up the simu-

lations. The Rayspread model aims at combining both of these approaches.

2.1 Description of the Model

The Rayspread model, which is an extension of the virtual laboratory of

[GV98], samples all relevant radiative processes within a 3-D scene by fol-

lowing individual primary rays from their energy source, through all rele-

vant interactions, until an absorption or exit event occurs. In addition, a

variance reduction method known as “photon spreading” [TG98] has been

implemented to achieve signiﬁcantly faster simulations of BRF ﬁelds. This

local estimator technique implies the “spreading” of secondary rays towards

a series of detectors from every physical interaction point in the path of

theprimaryray.Yet,Rayspread maintains backward compatibility with the

model syntax of [GV98], to allow for the computation of radiative quanti-

ties other than BRFs (i.e., transmission, absorption, albedo, etc.), and relies

on the Message Passing Interface (MPI) as a communicating layer within a

distributed memory, parallel processor architecture. In its current form, the

Rayspread model assumes that 1) light propagation can be described entirely

220 J.-L. Widlowski et al.

with geometrical optics, 2) incident monochromatic radiation can be simu-

lated with a ﬁnite number of mutually non-interacting rays, and 3) whenever

a ray-matter interaction occurs along the trajectory of a primary ray, the ray

is scattered in one and only one direction under elastic scattering conditions.

To set up an experiment with this model, the structural and optical prop-

erties of the medium of interest have to be deﬁned prior to the computation of

the ray trajectories. A set of 12 geometric primitives (e.g., disc, cone, sphere,

cylinder, ellipse), may be combined using Constructive Solid Geometry (CSG)

techniques to produce objects of great complexity. Every object within the

bounding volume of the scene (known as the world object) is characterized

by its position with respect to the world cartesian coordinate system, its spa-

tial extension and an interaction model that speciﬁes the object’s scattering

properties if a ray intersects its outer envelope or is to be propagated within

the spatially homogeneous media deﬁned inside its (closed) interior. Figure 2

shows some examples of structurally homogeneous (top) and heterogeneous

(bottom) vegetation canopies, that are depicted both in a deterministic man-

ner (right), where every primitive is explicitly described, and in a stochastic

manner (left), where the statistical foliage properties are represented by vol-

umes of spatially homogeneous media with uniform characteristics. A light

source is deﬁned in terms of its location, extent, intensity and directionality.

Multiple light sources may be combined to simulate complex illumination

conditions. Last but not least, a series of virtual ﬁlters (and logical combina-

tions thereof) can be applied to BRF measurements such that only certain

scattering orders, or, object-speciﬁc physical interactions may contribute to-

wards the radiance counter of a given detector.

Primary Ray Generation and Event Tracking

Upon emission from a light source, a primary ray k is tagged by a set of

parameters describing that event: 

= {r

; Ω

; T

; O

},wherer

is a po-

sition vector indicating the origin of the ray, Ω

is a unit vector describing

the travelling direction, T

marks the type of j-th interaction (emission if

j = 0, absorption, reﬂection or transmission if j ≥ 1) and O

identiﬁes the

l-th object within the scene where the interaction occurs. For every new in-

teraction between the primary ray and the scene, another event is added to

the proﬁle of the ray path. In order to determine the position of the next

point of intersection (r

j+1

) an optimized geometric-sorting algorithm, based

on the uniform subdivision of the scene into smaller volumes called ‘voxels’,

is applied.

Ray Interaction at the Interface Between Two Media

If a primary ray falls onto an open surface, or is scattered inside some

medium, the type of interaction and the outgoing direction are both de-

termined with respect to the local coordinate system (



) at the point of the

scattering event. Knowing the incoming direction of the ray (Ω



)aswellas

The Rayspread Canopy Reﬂectance Model 221

the bidirectional reﬂectance and transmission distribution functions (from

the interaction model of the object), it is possible to compute the hemi-

spherically integrated probabilities of reﬂection P

(Ω



), transmission P

(Ω



)

and absorption P

(Ω



)=1− P

(Ω



) − P

(Ω



), e.g., [AM90]. The type of

interaction T

j+1

is simulated by generating a uniform random number R

over the interval [0, 1], such that a reﬂection event occurs if R <P

(Ω



a transmission if P

(Ω



) ≤R<P

(Ω



)+P

(Ω



) and an absorption event

if P

(Ω



)+P

(Ω



) ≤R. In the ﬁrst two cases the outgoing direction after

reﬂection (transmission) is statistically simulated with respect to the scatter-

ing distribution function of the object, normalized by P

). This sampling

scheme may either use the acceptance/rejection technique [Dev86], or, more

direct approaches if available. In any case, scattering directions are ﬁrst de-

termined with respect to the local coordinate system – where the surface

normal always points to (0, 0, 1) – and then in the coordinate system of the

world object (Ω

j+1

). The soil interface is treated in exactly the same man-

ner, allowing for the deﬁnition of various scattering distribution functions,

e.g., Lambertian, Gaussian, Torrance-Sparrow [TS67], or the RPV formula-

tion [RPV93].

Ray Propagation within a Scattering Medium

Ray propagation within a spatially homogeneous medium containing a sto-

chastic representation of ﬁnite-sized or point-like scatterers is not only con-

trolled by the simulations of the interaction types and scattering angles as

before, but also by the reckoning of the actual distances between successive in-

teractions. Rays are expected to travel over distances of the order of the mean

free path of photons in the real world, with a probability of interaction with

the pseudo-turbid medium that increases exponentially with the distance

travelled since the last interaction [SG69]. The actual path length d

(Ω

)that

a ray may travel since its last point of interaction r

inside a pseudo-turbid

medium of optical depth τ (Ω

), is simulated using a uniform random number

R (bound in the interval [0, 1]) such that d

(Ω

)=− ln R/τ(Ω

). In the

case of ﬁnite size scatterers, the formulation of the optical depth τ (Ω

)makes

use of the geometric-statistical hot spot model of [VPD90] (see Sect. 1.2)

which requires some assumptions as to the shape and organisation of the

scatterers within the medium. Nevertheless, a physical interaction will occur

only if the next position of interaction r

j+1

= r

+Ω

is actually contained

within the medium. Otherwise the ray will be placed just outside the current

medium, and (accounting for its direction of propagation Ω

) the probability

of having a physical interaction in this new medium will be computed in a

likewise manner.

Secondary Rays and BRF Measurements

At every physical interaction j that characterises the random walk process

by which a primary ray samples a scene, Rayspread traces secondary rays

222 J.-L. Widlowski et al.

towards a pre-deﬁned number of detectors. Given the direction Ω

of the pri-

mary ray k prior to its collision with object O

at location r

, the probability

that a secondary ray s actually reaches a detector d (located at r

) without

any physical interactions in between is given by:

(Ω

, r

; Ω

, r

)=P (O

; Ω

→ Ω

) P (r

→ r

)

where P (O

; Ω

→ Ω

) is the probability of being scattered by object O

into

the direction of a detector Ω

,andP (r

→ r

) accounts for the probability of

reaching that detector without further physical interactions. P (O

; Ω

→ Ω

)

is computed using the optical properties and scattering distribution functions

of the object where the interception of the primary ray occurred. If a detector

is located at inﬁnity, then all secondary rays travel along identical directions

Ω

irrespective of their origin in the scene. P (r

→ r

), on the other hand

is evaluated using the geometric-sorting algorithm (mentioned previously) to

verify if there are any physical interactions with other objects along the direc-

tion of propagation of the secondary ray. If such an event takes place, then

the probability that the secondary ray reaches the detector is set to zero.

Alternatively, if the secondary ray enters a spatially homogeneous medium

containing a stochastic representation of oriented scatterers, then P (r

→ r

)

has to be modulated by the probability of traversing that medium without

interception since entering and exiting a medium does not constitute a physi-

cal interaction per se [TG98]. In the case of a detector d located at inﬁnity its

radiant ﬂux density counter, R

is then updated for all contributions due to

secondary rays reaching it from physical interactions (of primary rays) which

satisfy the speciﬁed measurement conditions. In the case where all physical

interactions of the primary rays are accepted by the measure:



k=1



j=1

(Ω

, r

; Ω

, r

)

where N

is the number of physical interactions occurring within the trajec-

tory of primary ray k,andN

is the total number of primary rays entering

the world object’s top area A

. The resulting BRF value for that detector is

then obtained by:

BRF

lambert



cos θ

where the approximate sign arises from the fact that the contributions due

to secondary rays with P (Ω

, r

; Ω

, r

) ≤ 10

−9

are currently ignored. Obvi-

ously, it is possible to perform local BRF measurements within a larger 3-D

scene, by restricting the number of primary and secondary rays that may

participate to only those that enter and exit the scene, respectively, via a

predeﬁned reference area A

located at the top of the world object. These

reference areas can be of square, circular or elliptical shapes.