Radiometry _Introduction_ Perfect specular reflection is

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                      Figure 1: perfect specular reflection

Radiometry (Introduction)
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).

                  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

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.

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).

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.

In the subsequent discussion, the following terminology will be used:
         L → Radiance [W m−2sr−1].
         M → Exitance [W m−2].
         E → Irradiance [W m−2].
         ω → Solid angle [sr].
         Ω → Projected solid angle [sr].
         Φ → Flux [W ].

 • Radiance and irradiance are radiometric concepts whereas lu-
   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
   3. Optics is linear (this implies scaling and superposition).

A Thought Experiment
                           Li                       T

                       dωe.... 33
                            e          @@               ––€                    .
                             ..                                  ˜˜..........
                        4.. e                                     .............
                       D                                          l dωr   .
                      D           e                               l
                                  e                                dd
                                   e θi
                                     e                         0
                                      e .....
                                      e ..                              t
                                        .              € ...
                                                       €                e
                                       e                   ...            e
                                         e                                gg
           ¤                                e           
                                              e                               i
          ¤                                                                     i
                                               e                                i
                                                   e                             i
                                            Ei dAΦi
         Figure 3: Diagram to understand the concepts of radiometry.

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        t
                       „         „ θ
                        „         „ ..i.....
                          „        „ .....
                            „       „
                             „           „
                               „     ....
                                „ . „
                           Figure 4: Cosine foreshortening.

                           dΩi = cos θi sin θidθidφi

• 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

• 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].

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
 – 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
 – 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.

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-
 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.

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.

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).

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),
                        fr =           0≤ρ≤1
    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:
                                      Lr (θn, φn) =
          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

  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.
  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

                          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:
        fr =       (cos θi cos θr )k−1,                         0≤k≤1

                   Lr (θn, φn) α cosk−1(e) cosk (i)

Some special cases of these surfaces are shown in Figure 6.

Background material: Horn Chapter 12 and FP 4
Properties of the BRDF

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

        viewing directions
                             scene radiance ≤ surf aceirradiance

 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.

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

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.
Brightness tells us about surface orientation but not about depth.
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-
 2. Analytical models based on surface microstructure (e.g., Re-
    flectance Spectroscopy: infer properties from “ground” powder
 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
Phong shading model (see Figure 8): λ cosn(s)+(1−λ) cos(i)   0≤
2 parameters λ, n (e.g., λ = .75, n = 10 is a good model of Al
This model follows our intuition but doesn’t obey Helmholtz reci-

                       Figure 8: Phong shading model

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
Brightness ∝ cos(e) where e is the angle between the beam and
the surface normal

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
                                                           p2 +qs +1
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.

Chronology of “Shape from Shading”
(See Figure 10.)

                   Figure 10: Chronology of “shape from shading”

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