（渲染）Veach博士论文3, 4章笔记

首先，老规矩：

未经允许禁止转载(防止某些人乱转，转着转着就到蛮牛之类的地方去了)

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Robust Monte Carlo Method For Light Transport Simulation

Eric Veach的博士论文笔记

Bilibili : Heskey0

space.bilibili.com/455965619

not to change the world, but to create one!

Chapter 3. Radiometry and Light Transport

3.1 Domains and measures

1. the whole scene surface denoted $ \Mu\subset\R^3 $, each surface filled with nonparticipating medium with a constant refractive index ( i.e. volume absorption, emission, and scattering are not allowed).

2. we define an area measure $ A $ on $ \Mu $ , so that $ A(D) $ denotes the area of a region $ D\subset\Mu $. The notation

   \[\int_{\Mu}f(x)dA(x) \]

   denotes the Lebesgue integral of function $ f:\Mu\rightarrow\R $ with respect to surface area.

3. Let $ \sigma $ be the usual surface area measure on $ S^2 $ .

   Given a set of directions $ D\subset S^2 $ ,

   the solid angle occupied by $ D $ is simply $ \sigma(D) $ .

4. the tangent space at the point x.

   \[T_{\Mu}(x)=\{y\in\R^3|y \cdot N(x)=0\} \]

5. the upward hemisphere $ H^2_+(x) $

   the downward hemisphere $ H^2_-(x) $

3.2 The phase space

1. the state of each photon can be represented by its position, direction of motion, and wavelength . for a system of N photons, the phase space would be 6*N-dimensional. ($ \Gamma $ phase space)

   \[\psi=\R^3\times S^2\times\R^+ \]

2. radiometric quantities can be defined by counting the number of photons in a given region of the phase space.

3.3 The trajectory space and photon events

1. if the phase space positions of all photons are graphed over time, we obtain a set of one-dimensional curves in the trajectory space

   \[\Psi=\R\times\psi \]

   where the first parameter represents time.

   radiometric measurements are defined by specifying a set of photon events along these curves, and then measuring the distribution of these events in various ways.

2. a photon event is a single point in the trajectory space . some events have natural definitions.

   For example :

   1. each emission, absorption, or scattering event corresponds to a single point along a photon trajectory.
   2. we could define the events to be the photon states at a particular time.
   3. given a plane P in $ \R^3 $, we could define a photon to be a crossing of P

   to define a radiometric quantity, we then measure the distribution of these events with respect to a suitable geometric measure .

3.4 Radiometric quantities

the discussion here is informal, a more detailed development is given in Appendix 3. B

3.4 : basic radiometry in PBRT

3.5 Incident and exitant radiance function

a radiance function is simply a function whose values correspond to radiance measurement . Most often, we will work with functions of the form

\[L:\Mu\times S^2\rightarrow\R \]

where $ \Mu $ is the set of scene surfaces . Occasionally, radiance functions of the form

\[L:\R^3\times S^2\rightarrow \R \]

will also be useful. $ L_i $ and $ L_o $ measure different sets of photon events, corresponding to the photon states just before their arrival at the surface, or just after their departure respectively.

To measure radiance, we define a photon event to be an intersection of one of these curves with the surface $ P=\R\times\Mu\times S^2\times\R+ $ in trajectory space.

3.6 The BSDF

1. The BSDF is not a standard concept in radiometry. The BRDF is obtained by simply restricting $ f_s $ to a smaller domain:

\[f_r:H^2_i\times H^2_r\rightarrow\R \]

where $ H_i^2 $ and $ H^2_r $ are often called the incident and reflected hemispheres respectively.

2. The BTDF is defined similarly to the BRDF, by restricting fs to a domain of the form
   \[f_t:H^2_i\times H^2_t\rightarrow\R \]

where the transmitted hemisphere $ H^2_t=-H2_i $ is the complement of $ H^2_i $ . As before $ H^2_i $ can represent either the upward hemisphere $ H^2_+ $ , or its complement $ H^2_- $ .

the BSDF is the union of two BRDF’s

Properties of BRDF: symmetric, energy conservation.

3.7 Introduction to light transport

3.7.1 The measurement equation

\[I=\int_{\Mu \times S^2}W_e(x,\omega)L_i(x,\omega)dA(x) d\sigma_x^\bot(\omega) \]

where:

$ I_j $ represents the value of a single pixel, and M is the number of pixels in the image, $ W_e $ the sensor responsivity, which specifies the “importance” of the light arriving along each ray to the corresponding measurement.

3.7.2 The light transport equation

3.7.3 importance transport equation

\[W(x,\omega)=W_e(x,\omega)+\int_{S^2}W((x_\Mu,\omega_i),-\omega_i)f_s(x,\omega_o\rightarrow\omega_i)cos\theta d\sigma_x(\omega_i) \]

where:

W the equilibrium importance function, measurements are computed by integrating the product $ W L_e $.

3.7.4 Bidirectional methods

1. finite element approaches [1992]
2. multi-pass methods [1991]
3. particle tracing algorithms [1990]
4. bidirectional path tracing [1993]

3.7.5 Sampling and evaluation of non-symmetric BSDF's

1. the bidirectional algorithms must take care when evaluating or sampling the BSDF, to ensure that $ \omega_i $ and $ \omega_o $ are ordered correctly

3.7.6 The adjoint BSDF

given an arbitrary BSDF $ f_s $ , the adjoint BSDF $ f_s^* $ is defined by

\[f_s^*(\omega_i\rightarrow\omega_o)=f_s(\omega_o\rightarrow\omega_i) \]

1. it lets the importance transport equation have the same form as the light transport equation .
2. provides a useful convention for sampling.

Appendix 3. A Field and surface radiance functions

defined by Arvo [1995]

field radiance $ L_f $ is similar to incident radiance $ L_i $,

surface radiance $ L_s $ is similar to exitant radiance $ L_o $

Field and surface radiance are defined only at surfaces, while incident and exitant radiance are defined in space as well.

Appendix 3. B Measure-theoretic radiometry

3. B. 1 Measure spaces

3. B. 2 The photon event space

the photon event space (P) contains all possible locations of the photon events we wish to count. depends on the definition of a photon event.

the photon event space is a subset of trajectory space  (\(\Psi\)).

For example:

1. volume emission : a photon be emitted from any point, any direction, any wavelength, at any time.
2. crossing a hypothetical surface : the photon event space would be a 6-dimensional manifold, within the 7-dimensional trajectory space $ \Psi $

\[P=\R\times S\times S^2 \times \R^+ \]

3. B. 3 The geometric measure

we define a geometric measure $ \rho $ :

1. defined on the photon event space

2. used to measure density of photon events.

3. defined as product of the natural Lebesgue measures on components of P.

For example:

volume emission is given by:

\[\rho=l\times v\times \sigma\times l^+ \]

where:

$ l $ and $ l^+ $ are the length measure on $ \R $ and $ \R^+ $

$ v $ the volume measure on $ \R^3 $

3. B. 4 The energy measure

we define an energy content function $ Q $:

\[Q:P\rightarrow [0,\infin] \]

1. to count photon events in various regions of $ P $

2. measures the total energy of the photon events in each measurable set D of the photon event space

【Theorem 3. 1】 Existence of Energy Measure

given a photon event space $ P $

with geometric measure $ \rho $

and energy content function $ Q $

\[Q:P\rightarrow [0,\infin]\\ Q(\bigcup_{i=1}^{\infin}D_i)=\sum_{i=1}^{\infin}Q(D_i)\\ \rho(D)<\infin \rightarrow Q(D)<\infin\\ \rho(D)=0 \rightarrow Q(D)=0 \]

3. B. 5 Defining radiometric quantities as a ratio of measures

radiometric quantities can be defined by measuring the density of $ Q $ with respect to $ \rho $, i.e. the ratio $ dQ/d\rho $ for a region D that becomes arbitrarily small.

【Theorem 3. 2】 Randon-Nikodym

if $ (P,P^\prime,\rho) $ is a σ-finite measure space, and if a σ-finite measure $ Q $ on $ P^\prime $ is continuous with respect to $ \rho $, then there exists a non-negative, real-valued, $ \rho $-measurable function f on $ P $ such that

\[Q(D)=\int_Dfd\rho \]

for every measurable set $ D\in P^\prime $. the function f is unique up to a set of \(\rho\)-measure zero. The function f is called the Randon-Nikodym derivative of $ Q $ with respect to $ \rho $, denoted

\[f=\frac{dQ}{d\rho} \]

【Theorem 3. 3】 Existence of energy density

given a photon event space $ P $

with geometric measure $ \rho $

and energy content function $ Q $

then there exists a $ \rho $-measurable function $ f:P\rightarrow(0,\infin) $, which is unique to within a set of $ \rho $-measure zero, satisfying

\[Q(D)=\int_Dfd\rho \]

where $ D\in P^\prime $ a measurable subset of $ P $

3. B. 6 Examples of measure-theoretic definitions

1. spectral radiant sterisent

while the geometric measure $ \rho $ is the whole trajectory space, we obtain a quantity

\[L_\lambda^*=\frac{dQ}{d\rho} \]

this quantity is called spectral radiant sterisent . it is used for the measurement of emission, scattering, and absorption within volumes.

2. spectral phase space density

3. spectral radiance

given the photon event space

\[P=\R\times S\times S^2\times \R^+ \]

with the geometric measure defined by

\[\rho=l\times A\times\sigma_x^\bot\times l^+ \]

thus the density

\[L_\lambda=\frac{dQ}{d\rho}=\frac{dQ}{dldAd\sigma_s^\bot dl^+} \]

corresponds to spectral radiance . we can use this equation to define $ L_\lambda $ anywhere in the trajectory space $ \Psi $

Chapter 4. A General Operator Formulation of Light Transport

In this chapter, we develop a light transport framework

4.1 Ray space

1. ray space

   the ray space consists of all rays that start at points on the scene surfaces

\[R=\Mu\times S^2 \]

​ where $ \Mu $ is the set of surfaces in the scene,

​ $ S^2 $ the set of all unit direction vectors.

​ ( The ray $ r=(x,\omega) $ has origin x and direction $ \omega $ )

2. The throughput measure

   we define a measure $ \mu $ on $ R $ , called the throughput measure , that is used to integrate functions on ray space.

   Consider a small bundle of rays around a central ray $ r = (x,\omega) $ , such that the origins of these rays occupy an area $ dA $ , and their directions lie within a solid angle of $ d\sigma $. Then the throughput of this small bundle is defined as :

\[d\mu(r)=d\mu(x,\omega)=dA(x)d\sigma_s^\bot(\omega) \]

​ that is, $ \mu $ is simply the product of the area and projected solid angle measures. This is known as the differential form of the throughput measure ,

we define $ \mu(D) $ for a set of rays $ D\subset R $

\[\mu(D)=\int_DdA(x)d\sigma_x^\bot(\omega) \]

​ the quantity $ \mu(D) $ measures the light-carrying capacity of a bundle of rays , and corresponds to the classic radiometric concept of throughput.

4. the throughput measure also allows us to define radiance in a simpler and more natural way, namely as power per unit throughput :

   \[L(r)=\frac{d\Phi(r)}{d\mu(r)} \]

5. Other representations of ray space

   a ray could be represented as a pair $ r=x\rightarrow x^\prime $

   Even when $ R $ is represented in different ways, the throughput measure $ \mu $ should be understood to have the same meaning.

4.2 Function on ray space

1. radiance functions

The distribution of radiance or importance in a given scene can be represented as a real-valued function on ray space, i.e. a function of the form

\[f:R\rightarrow \R \]

.

1. Norms

the $ L_p $ norms is defined by :

\[||f||_p=(\int_R|f(r)|^pd\mu(r))^{1/p} \]

where $ p $ is a positive integer. In the limit as $ p\rightarrow\infin $ , we obtain the $ L_\infin $ norm :

\[||f||_\infin=esssup|f(r)|, [r\in R] \]

where esssup denotes the essential supremum , i.e. the smallest real number m such that $ f(r)\le m $ almost everywhere.

​ the most commonly used norms are the $ L_1,L_2,L_\infin $ norms, which measure the average, root-mean-square, and maximum absolute value of a function respectively.

​ For the purposes of analysis, it is convenient to consider only the functions whose Lp norm is finite . The collection of all such functions (for a given value of p) is called an Lp space ( sometimes called Lebesgue space ), which we will denote by Lp(R) . These spaces have desirable analytic properties .

2. Inner products

the inner product of two functions on ray space is defined by

\[\left\langle f,g \right\rangle=\int_Rf(r)g(r)d\mu(r) \]

every inner product has an associated norm defined by

\[||f||=\left\langle f,f \right\rangle^{1/2} \]

which in this cast is identical to the $ L_2 $ norm.

4.3 The scattering and propagation operators

A linear operator is simply a linear function $ A:F\rightarrow F $ whose domain is a vector spaces F. F is a space of radiance function defined above. The notation $ Af $ denotes the an operator to a function, whose result is another function.

1. The local scattering operator

the local scattering operator defined by

\[(Kh)(x,\omega_o)=\int_{S^2}f_s(x,\omega_i\rightarrow\omega_o)h(x,\omega_i)d\sigma_x^\bot(\omega_i) \]

this operator is applied to an incident radiance function $ L_i $ , it returns the exitant radiance $ L_o = KL_i $ that results from a single scattering operation. $ K $ operates on entire radiance functions, rather than being restricted to a single point x. It maps one function $ L $ into another function $ KL $ , where each function is defined over the whole ray space $ R $ .

2. The propagation operator

let $ x_\Mu(x,\omega) $ the first point of $ \Mu $ that is visible from x in the direction $ \omega $ .

\[d_\Mu(x,\omega)=inf\{d>0|x+d\omega\in\Mu\} \]

which is called the boundary distance function . When the ray $ (x,\omega) $ does not intersect , we have distance $ d_\Mu(x,\omega)=\infin $ .

The propagation operator $ G $ (also called the geometric operator) defined by

\[(Gh)(x,\omega_i)=\Bigl\{{h(x_\Mu(x,\omega_i),-\omega_i) ,if d_\Mu(x,\omega_i)<\infin\\ 0,otherwise} \]

4.4 The light transport and solution operators

1. the light transport operator

The composition of the scattering and propagation operators is called the light transport operator

\[T=KG \]

the light transport equation is defined by :

\[L=L_e+TL \]

where L is the measured equilibrium radiance , $ L_e $ the emitted radiance function

in which case the solution is simply \(L=SL_e\)

2. the solution operator

the solution can be obtain by inverting the light transport equation

\[(I-T)L=L_e\\ L=(I-T)^{-1}L_e \]

where $ I $ is the identity operator. we get the solution operator :

\[S=(I-T)^{-1} \]

3. Conditions for invertibility

these formal manipulations are valid only if the operator $ I-T $ is invertible. a sufficient condition is that $ ||T||\lt1 $ , where $ ||T|| $ is the standard operator norm

\[||T||={\sup_{||f||\le1}}||Tf|| \]

where the norms on the right are function norms . Given that $ ||T||\lt1 $ , the inverse of $ I-T $ exists and is given by

\[S=(I-T)^{-1}=\sum_{i=0}^\infin T^i \]

This is called the Neumann series . this expansion has a physical interpretation when when applied to $ L=SL_e $ , since

\[L=L_e+TL_e+T^2L_e+... \]

For (one-sided) reflective surface :

$ ||G||_p\le1 $ for any $ 1\le p\le\infin $

$ ||K||_p\le1 $ as long as all BRDF's in the scene are energy-conserving and symmetric.

by making the additional assumption that no surface is perfectly reflective ,

$ ||K||_p\lt1 $ and thus

\[||T||_p=||KG||_p\le||K||_p||G||_p\lt1 \]

In the case of scattering (i.e. transmission + reflection)

\[||K||\lt\frac{\eta_{max}^2}{\eta_{min}^2} \]

where the $ \eta_{min},\eta_{max} $ denote the minimum and maximum refractive indices in the environment. This corresponds to the fact that radiance can increase during scattering, due to refraction.

4.5 Sensors and measurements

we can imagine that each pixel is a small piece of film within a virtual camera, and that the pixel value is proportional to the radiant power that it receives .

We will only deal with linear sensors , in which case the response is characterized by a function

\[W_e(x,\omega)=\frac{dS(x,\omega)}{d\Phi(x,\omega)} \]

where $ S $ is the unit of sensor response. that specifies the sensor response per unit of power arriving at $ x $ from direction $ \omega $ .

1. For real sensor $ W_e $ is called the flux responsivity of the sensor. Depending on the sensor, $ S $ could represent a voltage, current, change, in photographic film density, deflection of a meter needle, etc.

2. For hypothetical sensor used in graphics, $ W_e $ is called an exitant importance function. The corresponding sensor response is unitless, and thus importance has units of $ [W^{-1}] $ . We assume that $ W_e $ is defined over the entire ray space $ R $ , although it will be zero over most of this domain for typical sensors. In the case where measurements represent pixel values, note that $ W_e $ can model arbitrary lens systems used to form the image, as well as any linear filters used for anti-aliasing .

The measurement equation

To compute a measurement, we integrate the response

\[dS(r)=W_e(r)d\Phi(r)=W_e(r)L_i(r)d\mu(r) \]

for all the incident radiance falling on the sensor. this is summarized by Nicodemus' measurement equation , expressed in our notation as

\[I=\left\langle W_e,L_i\right\rangle=\int_RW_e(r)L_i(r)d\mu(r) \]

where $ I $ is a measurement, $ W_e $ is the emitted importance, and $ L_i $ is the incident radiance. It would not define $ L_e $ since the actual emission takes place somewhere else.

Notice that although we have defined the equilibrium solution $ L = SL_e $ as an exitant quantity, the measurement equation requires an incident function. This problem can be solved with the $ G $ operator, by using the relationship $ L_i = GL $ . Each measurement now has the form

\[I=\left\langle W_e,L_i \right\rangle=\left\langle W_e,GL \right\rangle=\left\langle W_e,GSL_e \right\rangle \]

it is the explicit inclusion of $ G $ in this equation that allows us to use the exitant forms of both $ L_e $ and $ W_e $ .

4.6 Importance transport via adjoint operators

The adjoint of an operator \(H\) is denoted \(H^*\) , and is defined by the property that

\[\left\langle H^*f,g\right\rangle=\left\langle f,Hg\right\rangle \]

An operator is self-adjoint if \(H=H^*\). This corresponds to familiar concept of a symmetric matrix in real linear algebra.

now we apply the identity to the measurement equation

\[I=\left\langle W_e,GSL_e\right\rangle=\left\langle(GS)^*W_e,L_e\right\rangle \]

the adjoint of \(K\) is given by

\[(K^*h)(x,\omega_o)=\int_{S^2}f_s^*(x,\omega_i\rightarrow\omega_o)h(x,\omega_i)d\sigma_x^\bot(\omega_i) \]

notice that \(K^*\) is the same as \(K\) , except that it use adjoint BSDF

\[f_s^*(x,\omega_i\rightarrow\omega_o)=f_s(x,\omega_o\rightarrow\omega_i) \]

let us suppose that \(f_s\) is symmetric at every point \(x\in\Mu\) , so that \(K=K^*\) . putting these facts together with standard identities (Appendix 4.B), it is easy to show that

\[(GS)^*=GS \]

i.e. the operator (GS) is self-adjoint as well

measurements can be evaluated

\[I=\left\langle W_e,GSL_e\right\rangle or I=\left\langle GSW_e,L_e\right\rangle \]

The only difference between these two expressions is that \(W_e\) and \(L_e\) have been exchanged.

Importance transport

the equilibrium importance function is given by \(W=SW_e\) , and satisfies the importance transport equation

\[W=W_e+TW \]

the relationship \(L_i = GL_o\) for incident radiance becomes \(W_i = GW_o\) for incident importance . we get the symmetric measurement equations

\[I=\left\langle W_e,L_i\right\rangle or I=\left\langle W_i,L_e\right\rangle \]

However, if the scene model contains any surface with a non-symmetric BSDF, then \(K \ne K^*\) . This does not affect the light transport operator \(T = KG\), which we will rename \(T_L\), but the importance transport operator becomes

\[T_w=K*G \]

This means that in general, light and importance obey different transport equations .