Advanced Ray Tracing, Radiometry, BRDF, Path Tracing
Radiometry
 Motivation: BlinnPhong is experienced model and light intensity corresponding to the amount of shading is not explained well, the output is not realistic but plastic. Whittedstyle ray tracing sometimes doesn’t give correct result, especially involving how much light energy is refracted and reflected, and how much the rest is absorbed by the material, etc.
 All the answers can be found in radiometry!
 Measurement system and units for illumination
 Accurately measure the spatial properties of light (not temperal dimension, we still suppose light travel in straight line with out wave property)
 New terms: Radiant flux, intensity, irradiance, radiance (辐射通量, 光强, 照度, 亮度)
 Perform lighting calculations in a physically correct manner
Radiant Energy and Flux
 Radiant energy: the energy of electromagnetic radiation. It is measured in units of joules, and denoted by the symbol:
 Radiant flux (radiant power, luminous flux): the energy emitted, reflected, transmitted or received, per unit time.
 Flux: photons flowing through a sensor in unit time
Radiant Intensity
 Radiant Intensity: The radiant (luminous) intensity is the power per unit solid angle emitted by a point light source.
 The candela is one of the seven SI base units
Angles and Solid Angles
 Angle: ratio of subtended arc length on circle to radius
 $\theta = l / r$
 Circle has $2\pi$ radians
 Solid angle: ratio of subtended area on sphere to radius squared
 $\omega = A / r^2$
 Sphere has $4\pi$ steradians
 Differential Solid Angles
 Direction vector $\omega$ to denote direction vector of unit length
 $\theta$ : elevation, zenith angle (仰角)
 $\phi$ : azimuth, azimuth angle (方位角)
 Sphere: $S^{2}$
 Isotropic Point Source
 Modern LED Light
 Output: 815 lumens (11W LED replacement paifor 60W incandescent)
 Radiant intensity (assume isotropic) = 815 lumens / 4π sr = 65 candelas
Irradiance
 Irradiance: the power per (perpendicular / projected) unit area incident on a surface point.
 The lux is SI derived unit of illuminance, measuring luminous flux per unit area.
 Irradiance at surface is proportional to cosine of angle between light direction and surface normal
 Recall Lambert’s Cosine Law
Tip: Analogous Irradiance to Electric field intensity, Radiant flux to Magnetic Flux.
 The reason why we feel cold and hot in different seasons
 Earth’s axis of rotation: ~23.5° off axis
 Give rise to the different Solar elevation angle
 Correction of Irradiance Falloff
 Assume light is emitting power $\Phi$ in a uniform angular distribution
 Compare irradiance at surface of two spheres
Radiance
 Radiance (luminance): the power emitted, reflected, transmitted or received by a surface, per unit solid angle, per projected unit area.
 Radiance is the fundamental field quantity that describes the distribution of light in an environment
 Radiance is the quantity associated with a ray
 Rendering is all about computing radiance
 $\cos\theta$ accounts for projected surface area
 The nit, the candela per square metre, is the derived SI unit of luminance.
 Recall
 Irradiance: power per projected unit area
 Intensity: power per solid angle
 Radiance: power per solid angle, per projected unit area
 Thus
 Radiance: Irradiance per solid angle
 Radiance: Intensity per projected unit area
Incident Radiance
 Incident Radiance: the irradiance per unit solid angle arriving at the surface
 It is the light arriving at the surface along a given ray (point on surface and incident direction).
Exiting Radiance
 Exiting Radiance: the intensity per unit projected area leaving the surface
 For an area light, it is the light emitted along a given ray (point on surface and exit direction).
Irradiance vs. Radiance
 Irradiance: total power received by area $dA$
 Radiance: power received by area $dA$ from direction $d\omega$
 Unit Hemisphere $H^2$
Bidirectional Reflectance Distribution Function
 Radiance from direction $\omega_i$ turns into the power $E$ that $dA$ receives
 The npower $E$ will become the radiance to any other direction $\omega_o$

Differential irradiance incoming: $d E\left(\omega_{i}\right)=L\left(\omega_{i}\right) \cos \theta_{i} d \omega_{i}$

Differential radiance exiting (due to $dE(\omega_i)$): $dL_r(\omega_r)$

BRDF can be understanded in the way of energy distribution after the point on the surface absorbs the energy from directions all around and then reflects.
 For the specular reflection, BRDF specifies the reflected energy is mostly on the reflection direction
 For the diffuse reflection, BRDF specifies the reflected energy will distributed amoung all directions around.
BRDF
 Bidirectional Reflectance Distribution Function (BRDF): represents how much light is reflected into each outgoing direction $\omega_r$ from each incoming direction
 Conclusion: The BRDF, the ratio of exitant radiance to incident irradiance, is to characterize the directional properties of how a surface reflects light, which is independent of the strength of the light.
 In fact, we use cosine irradiance collector to measure incident irradiance, conical baffler to measure received radiance, and calculate the BRDF corresponding to the surface/texture by dividing the two. More info, and later notes
 In a word, BRDF describes how the light interacts with the object. Accordingly, BRDF describes the property of the texture.
 The quantity is between zero and one for reasons of energy conservation.
Reflection Equation
 Consider BRDF only takes one direction of incident light, thus integral light (energy) from all directions of hemisphere will get the entire light source directing to the reflection point.
 Understand in discrete way: traverse solid angles on the hemisphere, each incident solid angle will make contribution to the radiance of one specified direction. Sum these up will get the entire contribution, that is received energy.
 Challenege: recursion
 Reflected radiance $L_{r}\left(\mathrm{p}, \omega_{r}\right)$ on the left hand depends on incoming radiance $L_{i}\left(\mathrm{p}, \omega_{i}\right)$ on the right hand.
 But incoming radiance depends on reflected radiance at another point in the scene.
 Incoming radiance is not limited to the light source, but reflected light etc.
Rendering Equation
 Rewrite the reflection equation by adding an Emission term $L_{e}\left(p, \omega_{o}\right)$ to make it general:
Caveat: we assume that all directions are pointing outwards, and field of integration is hemisphere, which is the reason for $\cos\theta$ replacing $max(\cos\theta, 0)$.

Volume rendering equation is a more general form of rendering equation.

The solution of renderings equation will be narrated in the following part.

Understanding of the rendering equation
 for the single light source
 for the multiple light sources
 for the planar light sources
 for the surfaces with reflected light (interreflection)
Tip: $\omega$ is because the light direction is pointing opposite of the incident direction with regards to $x^\prime$.
 which is the canonical form of Fredholm Integral Equation of second kind
Tip: Use substitution rule for integrals from the current object to the object of reflected light source will achieve $d\omega_i$ to $dv$, which will be detailed in solution part.
 which can be discretized to a simple matrix equation or system of simultaneous linear equations, where $L,\ E$ are vectors, $K$ is the light transport matrix (reflection operator)
 rearrange terms and use taylor expassion
 final form of ray tracing is
 understanding of the expression
 $K$ : Reflection operator
 $E$ : Emission directly from light sources
 $KE$ : Direct illumination on surfaces
 $K^2E$ : Indirect illumination (One bounce indirect, mirros, refraction)
 $K^3E$ : Two bounce indirect illumination
 $\color{orange}{E + KE}$: Direct illumination is all the shading in rasterization can do
 Global illumination: the union of all direct and indirect light
 The more bounce of illumination, the more realistic. When the bounce of illumination goes to infinity, the image finally becomes factual as in the real world.
Mathematics Foundation
Fredholm Integral Equation

Equation of the first kind
\[g(t)=\int_{a}^{b} K(t, s) f(s) \mathrm{d} s\] Given the continuous kernel function $K$ and the function $g$, to find the function $f$

Equation of the second kind
\[\varphi(t)=f(t)+\lambda \int_{a}^{b} K(t, s) \varphi(s) \mathrm{d} s\] Given the kernel $K(t,s)$, and the function $f(t)$, the problem is typically to find the function $\varphi(t)$
Linear Operator
 Operator: generally a mapping or function acting on elements of a space to produce elements of another space.
 Linear Operator (kernel): $f$ is a linear

operator if it has two properties
\[f(x + y) = f(x) + f(y)\] \[f(cx) = cf(x)\]  Thus, $f(ax+by) = af(x)+bf(y)$
 If $M$ is a linear operator, $ab$ is constant, $fg$ is function

The most common examples of linear operators $K$ of differentiation and integration
\[(K \bullet f)(u)=\frac{\partial f}{\partial u}(u)\] \[(K \bullet f)(u)=\int k(u, v) f(v) d v\]  Proof of Fredholm Integral Equation as linear operator
 Want to show
 Then need to verify the two properties of linear operator
 Addition
 Multiplication
 More info refer to functional analysis
Monte Carlo Intergration
 It is too difficult to solve an integral analytically.
 Thus estimate the integral of a function by averaging random samples of the function’s value.
 Different with Riemann integral which solve an integral analytically, Monte Carlo estimates the value of integration
 Define the Monte Carlo estimnator for the definite integral of given function $f(x)$
 Definite integral
 Random variable, where $p(x)$ is probability density function (pdf)
 MonteCarlo estimator

Monte Carlo Integration
\[\int f(x) dx = \frac{1}{N} \sum_{i=1}^{N} \frac{f\left(X_{i}\right)}{p\left(X_{i}\right)}\] The more samples, the less variance
 Sample on x, integrate on x
 Don’t care about the domain of integration, but use numerical method sampling to estimate the value of integration
 When the random variable distributes between $a$ and $b$ uniformly
 Then the Basic Monte Carlo estimator will be
 If we want to keep the sampling number but increase the estimation accurarcy, we should sample from a distribution of interest than the uniform distribution, that is Importance Sampling
Path Tracing
Whittedstyle problem
 Whittedstyle ray tracing
 Always perform specular reflections or refractions
 Stop bouncing at diffuse surfaces

But these simplifications are not reasonable
 Whittedstyle ray tracing can not deal with the glossy materials where the ray reflects not the same as the way on mirror material
 Whittedstyle ray tracing has no reflection between diffuse material
Tip: In the left image of the above figure, the region that are not reached by direct illumination are shadowed in black, whereas the diffuse ambient actually illuminates the region. In the right image, the left side of rectangular is shaded in red, and right side of cubic is shaded in green, This phenomenon in which objects or surfaces are colored by reflection of colored light from nearby surfaces is called color bleeding.
Solution of rendering equation
 The rendering equation to be solved
 It involves two problems
 Solve an integral over the hemisphere
 Deal with recursion of $L_{i}\left(p, \omega_{i}\right)$
A Sample Monte Carlo Solution
 Suppose we want to render one pixel (point) in the following scene for direct illumination only
 No indirect reflection light is considered at this time

Use Monte Carlo method to solve this integration numerically
\[\int_{a}^{b} f(x) \mathrm{d} x \approx \frac{1}{N} \sum_{k=1}^{N} \frac{f\left(X_{k}\right)}{p\left(X_{k}\right)} \quad X_{k} \sim p(x)\] $f(x)$: $L_{i}\left(p, \omega_{i}\right) f_{r}\left(p, \omega_{i}, \omega_{o}\right)\left(n \cdot \omega_{i}\right)$
 $pdf$ (assume uniformly sampling the hemisphere): $pdf\left(\omega_{i}\right)=1 / 2 \pi$

Shading algorithm for direct illumination
def shade(p, wo)
Randomly choose **N** directions wi~pdf
Lo = 0.0
for each wi:
Trace a ray r(p, wi)
if ray r hit the light:
Lo += (1 / N) * L_i * f_r * cosine / pdf(wi)
return Lo
Introducing Global Illumination
 One more step forward: what if a ray hits an object
 Point $Q$ also reflects light to $P$, and the amount of radiance is the amount of radiance of direct illumination at point $Q$
def shade(p, wo)
Randomly choose **N** directions wi~pdf
Lo = 0.0
for each wi:
Trace a ray r(p, wi)
if ray r hit the light:
Lo += (1 / N) * L_i * f_r * cosine / pdf(wi)
# Add the following branch
elif ray r hit an object at q:
Lo += (1 / N) * shade(q, wi) * f_r * cosine / pdf(wi)
# minus wi is due to the reversed direction of the light towards object p
return Lo
Problem 1: Number Explosion
 Explosion of
num_rays
asnum_bounce
goes up

num_ray
will not explode if an only if $\color{red}{N = 1}$ 
Path Tracing: only 1 ray is traced at each shading point
def shade(p, wo)
# modify N to One
Randomly choose **One** directions wi~pdf
Trace a ray r(p, wi)
if ray r hit the light:
return (1 / N) * L_i * f_r * cosine / pdf(wi)
elif ray r hit an object at q:
return (1 / N) * shade(q, wi) * f_r * cosine / pdf(wi)
info: Distributed Ray Tracing: $ N != 1$
Ray Generation
 If the ray is only sampled once, the shaded point will be noisy!
 BUT it doesn’t matter, because just trace more paths thorugh each pixel and average their radiance.
 The goal is to render pixels of the image, instead of each points of objects in the scene.
def ray_generation(p, wo)
Uniformly choose **N** sample positions within the pixel
pixel_radiance = 0.0
for each sample in the pixel:
shoot a ray r(cam_pos, camera_to_sample)
if ray r hit the scene at p:
pixel_radiance += 1 / N * shade(p,  camera_to_sample)
return pixel_radiance
Problem 2: Infinite Recursion
 Dilemma: the light does not stop bouncing indeed in the real world
 Further, cut
num_bounce
is to cutenergy
and reduce authenticity  Solution: Russian Roulette (RR)
 With probability $ 0 \lt P \lt 1$, condition 1
 With probablity $1  P$, otherwise
 Previously, we always shoot a ray at a shading point and get the shading result $L_o$
 Suppose we manually set a probability $P$
 With probability $P$, shoot a ray and return the shading result divided by $P$: $L_o / P$
 With probablity $1P$, don’t shoot a ray and you’ll get $0$
 In this way, you can still expect to get $L_o$
def shade(p, wo)
# Add the following branch
Manually specify a probability p_RR
Randomly select ksi in a uniform distribution in [0, 1]
if ksi > p_RR:
return 0.0
Randomly choose **One** directions wi~pdf
Trace a ray r(p, wi)
if ray r hit the light:
return (1 / N) * L_i * f_r * cosine / pdf(wi) / p_RR
elif ray r hit an object at q:
return (1 / N) * shade(q, wi) * f_r * cosine / pdf(wi) / p_RR
 Expected value of bounce number, given probability $p$
Sampling the Light
 Path tracing is correct but not efficient actually
 The reason of being inefficient is the sampled 1 ray may not hit the light, so that a lot of rays are wasted if we uniformly sample the hemisphere at the shading point
 Sampling on the shading point will lead to waste of rays which fail to hit the light source
 Monte Carlo methods allows any sampling methods
 Hence, change the sampling point to the light
 Assume uniformly sampling on the light
 The rendering equation integrates on the solid angle
 Recall Monte Carlo Integration: Sample on $x$, and integrate on $x$
 Integral variable substitution
 Make the rendering equation converting converting from an integral of $d\omega$ to $dA$.
 The relationship between $d\omega$ and $dA$
 The projected area towards $x^\primex$ at point $x^\prime$ is $d A \cos \theta^{\prime}$
 Recall that $dS = R^2d\theta$
Final Form of Path Tracing
 Rewrite the rendering equation as final form with an integral on $dA$
 Now integration is on the light, so for Monte Carlo integration:
 $f(x)$ : $L_{i}\left(p, \omega_{i}\right) f_{r}\left(p, \omega_{i}, \omega_{o}\right) \frac{\cos \theta \cos \theta^{\prime}}{\leftx^{\prime}x\right^{2}}$
 $pdf$ : $1 / A$
 Previously, we assume the light is accidentally shot by uniform hemisphere sampling
 Now we consider the radiance coming from two parts:
 Light source (direct, no need to have RR), if the ray is not blocked in the middle
 Other reflectors (indirect, need to have RR)
def shade(p, wo)
# Contribution from the light source
Uniformly sample the light at x' (pdf_light = 1 / A)
L_dir = 0.0
Shoot a ray from p to x'
if the ray is not blocked in the middle:
L_dir = Li * f_r * cosθ * cosθ' / x'  p^2 / pdf_light
# Contribution from other reflections
## Test RR
Manually specify a probability p_RR
Randomly select ksi in a uniform distribution in [0, 1]
if ksi > p_RR:
return L_dir
## Ray Travce
L_indir = 0.0
Uniformly sample the hemisphere toward wi (pdf_hemi = 1 / 2pi)
Trace a ray r(p, wi)
if ray r hit a nonemitting object at q:
L_indir = shade(q, wi) * f_r * cosθ / pdf_hemi / p_RR
return L_dir + L_indir
 Path tracing is almost 100% correct, a.k.a Photorealistic
Summary & Future work
Summary
 Ray tracing: Previous vs. Modern Concepts
 Previous: Ray tracing == Whittedstyle ray tracing
 Modern
 The general solution of light transport, including:
 (Unidirectional or Bidirectional) Path tracing
 Photon mapping
 Metropolis light transport
 VCM / UPBP …
Future work
 Uniformly sampling the hemisphere
 How? and in general, how to sample any function? (sampling theory)
 Monte Carlo integration allows arbitrary $pdf$
 Instead of uniform sampling, what’s the best choice of sampling with regards to different shapes of function? (importance sampling)
 Do random numbers matter?
 Yes! (low discrepancy sequences)
 I can sample the hemisphere and the light
 Can I combine them? Yes! (multiple importance sampling)
 The radiance of a pixel is the averange of radiance on all paths passing through it
 Why? If different weighted sampling at the center of pixel or the edge is needed? (pixel reconstruction filter)
 Is the radiance of a pixel the color of a pixel?
 No. (gamma correction, curves concerning high dynamic range (HDR) image , color space, photography)