Radiometry _Introduction_ Perfect specular reflection is by csgirla

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Figure 1: perfect specular reﬂection

Perfect specular reﬂection is illustrated in Figure 1 where i, e, and
g denote the incident, emergent, and phase angles, respectively, and
N is a surface normal at the point of reﬂection. In perfect specular
reﬂection, i = e and the incident and reﬂected rays and surface
normal all lie in the same plane (so g = i + e).
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Figure 2: angular notation for surface reﬂection

In reality, corresponding rays of incident and reﬂected light and the
surface normal do not necessarily lie in the same plane, so further
geometric parameters are required. This is illustrated in Figure 2
where the direction of incident and reﬂected light rays is speciﬁed
in terms of a local coordinate system (X, Y, Z), with its origin at
the point of reﬂectance and the Z axis corresponding to the surface
normal. Both incident and emergent, or reﬂected, rays have zenith
(θ) and azimuth (φ) angles.
In the case of an isotropic reﬂecting surface, the rotation of the
surface about the surface normal does not introduce any changes.
In other words, we can substitute the diﬀerence φr − φi for φr and
φi.
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Using spherical trigonometry, we can obtain the cosine of the phase
angle g as follows:
cosg = cosθicosθr + sinθisinθr cos(φr − φi)
Thus, the directions of incident and reﬂected light for isotropic sur-
faces have three degrees of freedom.
In general, reﬂected light is characterized as a function of incident
light and the reﬂecting material, and not just by geometry.
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We need four parameters (θi, φi, θr , φr ) to specify the local geometry
of the incident and reﬂected rays. So, essentially there are four
degrees of freedom in the local co-ordinate system. By local, we
mean that these angles are deﬁned relative to the tangent plane at
the point under consideration.
Further, if we say that our surfaces are invariant to rotation about
the normal (they are isotropic), only the diﬀerence (φr −φi) matters—
so, we are left with just three degrees of freedom represented by the
triplet (i, e, g). The phase angle g is related to (φr − φi) by the
spherical cosine rule (see the formula above).
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The choice of a proper co-ordinate system (local or global) is an im-
portant part of the mathematical formulation of a problem—and
the choice is made for convenience. In vision, often, a global (gen-
erally, viewer-centered) co-ordinate system is the desirable choice.
Later, we would like to represent the triplet (i, e, g) in the viewer-
centered co-ordinate system using the notion of the gradient space
deﬁned earlier.
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In the subsequent discussion, the following terminology will be used:
M → Exitance [W m−2].
ω → Solid angle [sr].
Ω → Projected solid angle [sr].
Φ → Flux [W ].
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Notes:
minance and illuminance are photometric concepts. Radiance
characterizes the outgoing energy (energy radiated by the sur-
face) whereas irradiance characterizes the incoming energy. In
the deﬁnition of luminance and illuminance, a predeﬁned human
observer response curve is built in. As a consequence, luminance
is deﬁned only for the visible portion of the spectrum.
• It will be useful to keep in mind the three fundamental principles of optics:
1. Light travels along the shortest path.
2. There is no sense of direction associated with light (this is
called Helmholtz reciprocity). If you can see me, I can see
you.
3. Optics is linear (this implies scaling and superposition).
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A Thought Experiment
Li                       T

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Ei dAΦi
Figure 3: Diagram to understand the concepts of radiometry.
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To understand the concepts of radiometry, let us perform a gedanken
(thought) experiment in the following steps.
• Step 1: Consider a hemisphere made up of a perfectly absorb-
ing material (see Figure 3). Let dA denote the area [m2] of an
inﬁnitesimal surface element.
• Step 2: Cut an inﬁnitesimal hole in the hemisphere which
subtends an inﬁnitesimal solid angle dωi [sr]. Let Ωi [sr] denote
the projected solid angle of the hole at the surface element. Solid
angle and projected solid angle are related by the equation:

dΩi = cos θidωi

Intuitively, projected solid angle includes the cosine foreshort-
ening (see Figure 4). Generally, solid angle is converted to the
spherical co-ordinate system for ease of integration. Thus,
T
t     t
t      t
t       t
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t
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          θ
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θi
Figure 4: Cosine foreshortening.

dΩi = cos θi sin θidθidφi
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• Step 3: Imagine shining light through the hole with (radially
directed) radiance Li [W m−2sr−1]. Note that Li is a directional
quantity, i.e., Li(θi, φi). Let Ei denote the irradiance [W m−2],
and Φi the incident ﬂux [W ] on the surface element dA. The
incident ﬂux and the irradiance are related by the equation:
dΦi = dEi(θi, φi)dA

Then, the irradiance at the surface element is given by (intu-
itively, the bigger the hole, the larger the irradiance):

dEi(θi, φi) = Li(θi, φi)dΩi

Converting to spherical co-ordinates yields:

dEi(θi, φi) = Li(θi, φi) cos θi sin θidθidφi

or, after multiplying both sides by dA,

dΦi = Li(θi, φi) cos θi sin θidθidφidA
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• Step 4: Cut another inﬁnitesimal hole for the measurement of
reﬂected light. Let the inﬁnitesimal solid angle subtended by
this hole be dωr [sr].
Three things can happen to the incident ﬂux. It may be [i] ab-
sorbed, [ii] reﬂected, or [iii] transmitted in diﬀerent proportions
(the sum of the relative proportions should be one). Let us look
more closely at the reﬂected component of the ﬂux (this is the
one which may reach the viewer and is of importance to us). In
general, Φi would be reﬂected in all directions. Using symmetry,
we can directly write:
dΦr = Lr (θr , φr ) cos θr sin θr dθr dφr dA
= dMr (θr , φr )dA

where, Mr is the exitance of the surface [W m−2].
Now, how can we deﬁne the role that a surface plays in this pic-
ture? The answer is that we would like to get the reﬂected ﬂux
as a function of the incident ﬂux. For intuition only, let us deﬁne
“reﬂectance” as the ratio dΦr /dΦi which is a unitless quantity.
Note that there is a bug in this deﬁnition—this quantity is not
independent of the size of the viewing hole! The smaller the Ωr ,
the smaller the measurement and vice-versa. So, this idea needs
some patching—intuitively, a better quantity to use would be
dΦr /dΦidΩr . But now, this quantity is not unitless [sr−1].
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Aside:
Many standard books contain tables of reﬂectance for diﬀerent
types of surfaces. It is worth mentioning that these tables are
not very useful to us (the vision community), essentially, for two
reasons:
– The tabulated values are for a ﬁnite number of illumination
conditions whereas we would be interested in knowing the
reﬂectance properties of a surface under any illumination
condition.
– The values tabulated give the fraction of Φi reﬂected—there
is no direction attached to it. However, we would also like
to know the spatial distribution of the reﬂected ﬂux.
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The Bidirectional Reﬂectance Distribution Function (BRDF )
To characterize the reﬂectance properties of a material, we deﬁne
its BRDF (fr ) as,
dLr (θi, φi, θr , φr , Ei)
fr (θi, φi, θr , φr ) =
dEi(θi, φi)
The BRDF characterizes the intrinsic reﬂectance properties of a
material and is independent of particular illumination and view-
ing conditions. On the other hand, it allows us to compute the
reﬂectance properties under diﬀerent illumination and viewing con-
ditions. However, the above deﬁnition of BRDF has two implica-
tions:
1. The deﬁnition does not say anything about the dependence of
BRDF on the wavelength (hence color) of the light used.
2. It assumes point behavior of a surface, i.e., we can deﬁne things
based only on the properties of a point.
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It should be noted that the BRDF deﬁned above cannot be a mea-
sured. It is deﬁned in the limit when the hole size at the measuring
side tends to zero and assumes continuity in the properties of a sur-
face. Often, the surface is not homogeneous and taking this limit
does not make any sense (for example, in remote sensing, trying
to characterize a part of the surface of the earth with patches of
forests). The situation is analogous to velocity and speed—all we
can measure is speed, which is the average velocity over a ﬁnite
time interval—velocity is deﬁned at a particular time instant, and
is hence immeasurable. In both cases (BRDF and velocity), any
measurement necessarily is a function of scale while the deﬁnitions
are not.
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Suppose an oracle gives us the BRDF of a surface. What do
we do with it? Essentially, we would like to get the scene radiance
(because this is what the camera or the eye respond to) with respect
to the surface normal in a co-ordinate system we know about. The
general formula for achieving this is:
Lr (θn, φn) =   Ωi   fr · LidΩi

=   ωi   fr · Li cos θidωi

π π/2
=   −π 0  fr (θi, φi, θr , φr )   · Li(θs, φs) cos θi sin θidθidφi
where, the integration is taken over the full hemisphere of possible
incident directions (this formula is derived utilizing the fact that
optics is linear). The BRDF fr is deﬁned using the local co-
ordinate system whereas the radiance term Li is deﬁned in the
global co-ordinate system—so we have to work either in the local
or in the global co-ordinate system (for issues related to this, see
the paper by Horn and Sjoberg (§ 8.8 in the old reading list) on the
calculation of reﬂectance maps).
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Let us consider some surfaces with simple BRDF s.
1. Constant: This is the simplest BRDF that one can conceive.
Ideally, this constant should be greater than zero and less than
1/π (because greater than 1/π means that the surface ampliﬁes
light1; a negative value does not make sense). For a surface with
ideal diﬀuse reﬂectance (i.e., a Lambertian surface),
ρ
fr =           0≤ρ≤1
π
where, ρ is a reﬂectance factor.
Let us look at the scene radiance for a Lambertian surface under
two diﬀerent illumination conditions:
[a] A collimated source: Consider a collimated source with irra-
diance E0 from the direction (θ0, φ0). Then the scene radi-
ance is given by:
E0ρ
cos(i)
Lr (θn, φn) =
π
The term ρ is the fraction of light that is reﬂected (0 ≤ ρ ≤
1). The factor 1/π balances the incoming and outgoing light.
[b] A hemispherical uniform source: Consider a hemispherical light
source with:


   L0 θs < π/2
Li(θs, φs) = 
          0 θs ≥ π/2
The scene radiance then is given by:
1 + cos(e)
Lr (θn, φn) = L0
2

1Recall that light is reﬂected in all directions; the factor (1/π) is derived by integrating the scene
radiance equation. It certainly is not intuitive.
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Figure 5: A Lambertian surface element illuminated by a hemispherical uniform light source.

For an intuitive understanding of the situation, see Figure 5.
When e = 90◦, Lr = L0/2. When e = 0, Lr = L0.
Note that [a] and [b] represent the scene radiance for a Lam-
bertian surface under diﬀerent conditions of illumination—they
are not the deﬁnitions of a Lambertian surface.
Aside:
There exist two deﬁnitions of a Lambertian surface in literature:
Deﬁnition 1: It is the unique surface that appears equally
bright from all viewing directions.
Deﬁnition 2: It is a surface that reﬂects light equally in
all directions.
Correctness of the deﬁnitions depends on how the various terms
are deﬁned precisely. In our terminology, the second deﬁnition
is wrong while the ﬁrst one is correct. If a surface does reﬂect
light equally in all directions—the more obliquely it is viewed,
the brighter it will appear due to cosine foreshortening!
2. Based on Helmholtz Reciprocity: To be physically plau-
sible, an optical system should satisfy the Helmholtz reciprocity
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k           Examples

1          Lambertian
0.5   Lunar astronomy (moon)
0         SEM imagery

Figure 6: Some special cases of the general Minnaert surfaces.

which states that:

fr (θ1, φ1, θ2, φ2) = fr (θ2, φ2, θ1, φ1)
Obviously, when BRDF = k, a constant, the above relation-
ship is satisﬁed.
A Lambertian surface can be generalized by considering Lr (θn, φn)
α f (e) cosk (i). This leads to the surfaces of the type studied by
Minnaert for which the BRDF is given by:
k+1
fr =       (cos θi cos θr )k−1,                         0≤k≤1
2π
or,

Lr (θn, φn) α cosk−1(e) cosk (i)

Some special cases of these surfaces are shown in Figure 6.
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Background material: Horn Chapter 12 and FP 4
Properties of the BRDF

1. Helmholtz Reciprocity: “If I can see you, then you can see
me.”
fr (θ1, φ1; θ2, φ2) = fr (θ2, φ2; θ1, φ1)
2. “Conservation” of Energy:”

viewing directions

3. Separability (wrt λ): We expect the BRDF to be a function
of λ.
fr (θi, φi; θr , φr ; λ)
This is the spectral BRDF. If it is separable then:
fr (θi, φi; θr , φr ; λ) = fr (θi, φi; θr , φr )fr (λ)
Is it separable? Consider if it wasn’t. If an object was held
at a certain length and rotated about while ﬁxing the eye on
a particular point in the object, then relative amounts of red,
green and blue would change but this doesn’t appear to happen
in typical objects. Therefore, by and large, an object’s BRDF
is separable (an example of an exception is the neck feathers
of certain water fowl). It is not well known how separable the
BRDF of typical objects is.
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Physical Processes Aﬀecting Reﬂectance
(See Figure 7.)

Figure 7: Some of the physical processes aﬀecting reﬂectance

1. Light rays are reﬂected once specularly from a planar microfacet
whose dimension is suﬃciently greater than the λ of incident
light (ie: reduce problem to geometry and ignore diﬀraction).
2. Light rays that penetrate the surface, internally reﬂect and re-
fract multiple times and then refract once more back through
the surface. For this, we need to know properties of the material
such as the index of refraction and coeﬃcient of extinction.
3. Light rays are reﬂected twice or more specularly from planar
microfacets whose dimensions are signiﬁcantly greater than the
λ of incident light.
4. Light rays (now think of as waves) diﬀract at interfaces whose
dimensions are the same size or smaller than λ (e.g., at corners).
Most models in vision and graphics ignore diﬀraction and only use
geometry.
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Compiling the BRDF into the Reﬂectance Map

(θi, φi, θr , φr) ←− BRDF function
↓
(i, e, g) ←− Reﬂectance function

(θn, φn) ← Global Coords            (p, q) ← Reﬂectance Map
When compiling a BRDF into a reﬂectance map, we reduce the
variables involved by specifying constraints. Although we lose gen-
erality, we specialize the problem and get more useful information.
Let’s start with φ(i, e, g) and try to get (p, q)
Determining φ(i, e, g):
1. Phenomenological models (e.g., Lambertian and Minnaert Sur-
faces)
2. Analytical models based on surface microstructure (e.g., Re-
ﬂectance Spectroscopy: infer properties from “ground” powder
products)
3. Measure it (change normal to surface according to the 2 degrees
of freedom and measure it; this will change i and e for a ﬁxed
g)
Phong shading model (see Figure 8): λ cosn(s)+(1−λ) cos(i)   0≤
λ≤1
2 parameters λ, n (e.g., λ = .75, n = 10 is a good model of Al
paint)
This model follows our intuition but doesn’t obey Helmholtz reci-
procity.
22

The Scanning Electron Microscope (SEM)

Figure 9: A simple mode of the SEM

Brightness is measured by measuring the current of electrons trav-
elling to the charged plate. Current is a function of surface orien-
tation. (See Figure 9.)
More oblique ⇒ more reﬂection ⇒ more current ⇒ brighter image
1
Brightness ∝ cos(e) where e is the angle between the beam and
the surface normal
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The Reﬂectance Map
[0, 0, −1] −→ points to viewer
[p, q, −1] −→ surface normal
[ps, qs, −1] −→ points to light source
e: angle between ﬁrst and second vectors: cos(e) = √        1
p2 +q 2 +1
1+pps +qqs
i: angle between second and third vectors: cos(i) = √       2 +q 2 +1)(p2 +q 2 +1)
(p             s   s
g: angle between ﬁrst and third vectors:      cos(g) = √      1
2
p2 +qs +1
s
We can always transform φ(i, e, g) into a function of (p, q) called
the reﬂectance map R(p, q).
For an image E(x, y): E(x, y) = R(p, q)
This is the Image Irradiance Equation. It is valid in a vacuum and
most everyday cases. We recognize it as a ﬁrst order nonlinear par-
tial diﬀerential equation but we will treat it simply as an equation
in two unknowns.
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