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1 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). 2 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. 3 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. 4 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). 5 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. 6 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 ]. 7 Notes: • 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 you. 3. Optics is linear (this implies scaling and superposition). 8 A Thought Experiment Li T .......... . e .......... Camera dωe.... 33 e @@ . i4 .... ... .. .......... 4.. e ............. 4 D l dωr . D e l e dd e θi θr e 0 t 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. 9 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 t t t θ ..i..... .. ..... .... . ¢ ¢ ¢ θi Figure 4: Cosine foreshortening. dΩi = cos θi sin θidθidφi 10 • 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 11 • 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]. 12 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. 13 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. 14 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. 15 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). 16 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. 17 L0 g £ t 33@ g@ ¢£ g £ ¢ 4 t 4 .4 t gg c ££ ¢ D. . ¢ l d D. . t . . . ¢ l & G .. w t & . d. . . &d & a d . . d . . .. . . . . .d . . . . T t & . .. . . . . t . & e . . . . . . . . a & ¨ . . . . . . . . . s e ¨¨ & b e . . . . . . . . & . q . .. . . . . & ¨ %¨ g . ¤. .. . . . . . . . . . . . . . . ... & g . ... . . . . . . . . . . . & $$$i . . . . . . . . . . . . . . . . . . . .. && ¤ $ i $ ... W z . . . . . . . .. . . . . . i i . . . & . ... dA 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 18 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. 19 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 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. 20 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. 21 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- 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 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 1 Brightness ∝ cos(e) where e is the angle between the beam and the surface normal 23 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. 24 Chronology of “Shape from Shading” (See Figure 10.) Figure 10: Chronology of “shape from shading”