Today’s Class

1. Illumination
2. Spherical and Hemispherical Functions
3. Environment Mapping

Directional/Punctual Lights (Review)

• Consider each light to have a colour $\bm{c}_\text{light}$ representing reflected radiance from a white Lambertian surface facing towards the light ($\bm{n}=\bm{l}_c$)

   $$L_o(\bm{v}) = \int_{\bm{l}\in\omega_l} \, f(\bm{l},\bm{v}) \, L_l \, (\bm{n}\cdot\bm{l})^+ d\bm{l} \approx
   \pi \, f(\bm{l}_c,\bm{v}) \, \bm{c}_\text{light} \, (\bm{n}\cdot\bm{l}_c)^+$$ 

• Directional light:

   $$L_o(\bm{v}) = \pi \, f(\bm{l}_c, \bm{v}) \, \bm{c}_\text{light} \, (\bm{n}\cdot\bm{l}_c)^+$$ 

• Punctual lights

   $$L_o(\bm{v}) = \pi \sum_{i=1}^n f(\bm{l}_{c_i}, \bm{v}) \, \bm{c}_\text{light} \, (\bm{n}\cdot\bm{l}_{c_i})^+$$ 

Area Light Sources I

• For Lambertian surface, using a point light for an area light can be exact. Outgoing radiance is proportional to irradiance:

   $$L_o(\bm{v}) = \frac{\rho_\text{ss}}{\pi} E$$ 

• We can then compute irradiance as:

   $$E = \int_{\bm{l}\in\omega_l} L_l(\bm{n}\cdot\bm{l})^+ \, d\bm{l} \approx \pi \bm{c}_\text{light} (\bm{n}\cdot\bm{l}_c)^+$$ 

  • Vector irradiance allows converting an area light source of arbitrary size and shape into a point or directional light source

Area Light sources II

• Assuming $L_i$ is wavelength-independent, we can compute the irradiance at a point $\bm{p}$ as

   $$\bm{e}(\bm{p}) = \int_{\bm{l}\in\Theta} L_i(\bm{p},\bm{l})\,\bm{l}\,d\bm{l}$$ 

  • This time done over unitary sphere $\Theta$

Area Light Sources III

• We can use the vector irradiance to compute the net irradiance at $\bm{p}$ through a plane with normal $\bm{n}$

   $$\underbrace{E(\bm{p},\bm{n})}_\text{"positive side"} - \underbrace{E(\bm{p},-\bm{n})}_\text{"negative side"} = \bm{n}\cdot\bm{e}(\bm{p})$$ 

• When light distribution has no parts where the angle between $\bm{l}$ and $\bm{n}$ exceeds 90° $\implies E(\bm{p},-\bm{n})=0$

   $$E(\bm{p},\bm{n}) = \bm{n}\cdot\bm{e}(\bm{p})$$ 

  • This can be used to light lambertian surfaces
• Can then be used to create an exact directional light source

   $$\bm{l}_c = \frac{\bm{e}(\bm{p})}{\|\bm{e}(\bm{p})\|}, \qquad
   \bm{c}_\text{light} = \bm{c}' \frac{\|\bm{e}(\bm{p})\|}{\pi}$$ 

Area Light Source Example

• For simple cases we can solve $\bm{e}(\bm{p}) = \int_{\bm{l}\in\Theta} L_i(\bm{p},\bm{l})\,\bm{l}\,d\bm{l}$
• For a sphere with center at $\bm{p}_l$ with radius $r_l$ that emits $L_l$ from all points in all directions:

   $$\bm{l}_c = \frac{\bm{p}_l - \bm{p}}{\|\bm{p}_l - \bm{p}\|}, \qquad
   \bm{c}_\text{light} = \frac{r_l^2}{\|\bm{p}_l-\bm{p}\|^2} L_l$$ 

• Note that this is only correct with no “negative” irradiance, e.g., a surface with no light “under the horizon”

Glossy Materials

• Non-Lambertian surfaces are much more complicated
• Important is the specular highlight
  • Size and shape are similar to area light
  • Edges are blurred based on roughness of the surface
• Almost all approaches are approximations based on finding equivalent punctual lighting per shaded point
  • Doesn’t require rewriting shaders

Mittring’s Roughness Modification

• Core idea: find a cone that contains most of the light source irradiance onto the hemisphere of directions incident to the surface and fit a similar cone around the specular lobe containing “most” of the BRDF
• Approximated by only changing the roughness of the BRDF
• Can be applied to Trowbridge-Reitz BRDF with spherical area light to obtain:

   $$\alpha_g' = \left( \alpha_g + \frac{r_l}{2\|\bm{p}_l - \bm{p}\|} \right)^\mp$$ 

Most Representative Point Solution

• Alternate view that modifies the light vector based on shading point
• Mean value theorem

   $$\int_D f(x) \, dx = f(c) \int_D 1, \qquad c \in D$$ 

• Consider product of BRDF and irradiance over hemisphere as integral
• Problem is determining $c$ and $\int_D 1$
• Generally many approximations are used

Karis Representative Point Approximation

• For homogeneous sphere lights only
• Compute point on reflection ray closest to sphere center $\bm{p}_\text{cr} = (\bm{l}\cdot\bm{r})\,\bm{r}-\bm{l}$
• Point on sphere surface is then $\bm{p}_\text{cs} = \bm{l} + \bm{p}_\text{cr} \cdot \min(1, \frac{\text{radius}}{\|\bm{p}_\text{cr}\|})$

Sphere Light Example

vec3 lighting( vec3 p, vec3 v, vec3 n ) {
   const vec3 albedo = vec3( 0.5, 0.0, 0.0 );
   const vec3 light_pos = vec3(1.0);
   const float radius = 0.2;
   /* Karis most representative point approximation. */
   vec3 l = normalize( light_pos - p );
   vec3 r = reflect( v, n );
   vec3 pcr = dot(l,r) * r - l;
   vec3 pcs = l + pcr * min( 1.0, radius / length(pcr) );
   vec3 fdiff = pow(radius,2.0) / pow(distance(light_pos,p),2.0) * vec3(10.0) * albedo;
   return M_PI * (fdiff + fspec( normalize(pcs), v, n )) * max(0.0, dot(n,l));
}

Comparison

vec3 lighting( vec3 p, vec3 v, vec3 n ) {
   const vec3 albedo = vec3( 0.5, 0.0, 0.0 );
   vec3 light_pos = vec3(1.0);
   vec3 l = normalize( light_pos - p );
   vec3 f =  albedo / M_PI + fspec( l, v, n );
   return M_PI * f * vec3(1.0) * max( 0.0, dot( n, l ) );
}

General Light Shapes: Tube Lights

• Useful for representing fluorescent tube lights
• For Lambertian BRDFs there is a closed-form solution

   $$\int_{\bm{p}_0}^{\bm{p}_1} \left( \bm{n}\cdot\frac{\bm{x}}{\|\bm{x}\|} \right) \frac{1}{\|\bm{x}\|^2} dx =
   \frac{ \frac{\bm{n}\cdot\bm{p}_0}{\|\bm{p}_0\|^2} + \frac{\bm{n}\cdot\bm{p}_1}{\|\bm{p}_1\|^2} }
      { \|\bm{p}_0\|\,\|\bm{p}_1\|+(\bm{p}_0\cdot\bm{p}_1) }$$ 

  • Equivalent to evaluating the lighting from two point lights at the extremes with falloff function
• Solution for Phong specular BRDF also exists

General Light Shapes: Card Lights

• Card lights are rectangular planar area lights
• Lambertian BRDF lighting integral:

   $$L_i \int_{\bm{l}\in\omega_l} (\bm{n}\cdot\bm{l})^+ \frac{1}{r_l^2} d\bm{l}$$ 

  • $r_l$ is the length of the ray from surface to light plane in direction $\bm{l}$
• Maximum of $(\bm{n}\cdot\bm{l})^+$ is $\bm{p}_c$ on the boundary of the light region closest to $\bm{p}'$ and found by intersecting the normal with the light plane
• Maximum of $1/r_l^2$ is point $\bm{p}_r$ on the boundary closest to the point $\bm{p}''$ on the light plane closest to point being shaded
• Global maximum can be found as $\bm{p}_\text{max} = t_m \bm{c}_c + (1-t_m)\bm{p}_r$ and can be found by numerical integration

Linearly Transformed Cosines (LTC)

• Siggraph 2016 Paper
• Practical, accurate, and general technique based on highly expressive and easily integrated spherical functions
• LTC are just a cosine lobe transformed by a $3 \times 3$ matrix
• Integral is straight-forward to compute
• Complexity is in approximating the BRDF function as one or more LTC and tabulating in lookup arrays
• Can find solutions for general textured polygonal area light sources

Environmental Lighting

• Reflectance equation does not distinguish between direct and indirect light
• In practice, direct light has small solid angles with high radiance, while indirect tends to be diffuse with low radiance
  • Best to treat both separately
• Global illumination can consider bouncing light
• For Lambertian surfaces with fixed radiance $L_A$:

   $$L_o(\bm{v}) = \frac{\rho_\text{ss}}{\pi} L_A \int_{\bm{l}\in\Omega} (\bm{n}\cdot\bm{l})\,d\bm{l} = \underbrace{\rho_\text{ss} L_A}_\text{ambient light}$$ 

• For arbitrary BRDF:

   $$L_o(\bm{v}) = L_A \int_{\bm{l}\in\Omega} f(\bm{l},\bm{v}) \, (\bm{n}\cdot\bm{l})\,d\bm{l}$$ 

Representation Trade-offs

• Efficient encoding (projection) and decoding (lookup)
• Representation power vs representation complexity (few coefficients)
• Rotational invariance
  • Rotating the projection of a function = rotating function then projection
• Computation of sums and products of encoded functions
• Computation of spherical convolutions and integrals

Simple Tabulated Forms

• Simplest approach is to store values for several directions
• Evaluation becomes finding samples and reconstruction with interpolation
• Simple and expressive
• Distributing points is also not simple
  • Commonly unwrapped to rectangle for GPU acceleration
• Drawbacks:
  • Combining spherical functions is non-trivial
  • Resolution is critical
  • Computational complexity depends on #samples
  • Not invariant under rotation
• Commonly used for high frequency complex functions

Spherical Bases

• Infinite number of representations with fixed #coefficients
• Most simple representation is $F_c(\theta,\phi) = c\cdot 1$
  • Projection becomes $c = \frac{1}{4\pi} \int_\Omega f(\theta,\phi)$
  • Can’t represent most functions well
• More complex approximation with coefficients $a$ and $b$:

   $$F_\text{hemi}(\theta,\phi) = a + \frac{\cos(\theta)+1}{2}(b-a)$$ 

  • Not invariant to rotations
  • Can be seen as a tabular form with two samples on poles

Spherical Radial Basis Functions (SRBF)

• Radially symmetric and function of only one argument: the angle between defined axis and evaluated direction
• Basis consists of a set of functions known as lobes
• Consists of a parameter for each lobe
• Due to the add complexity, lobe directions are usually assumed to be fixed or spread uniformly
• Representations:
  • Spherical Gaussians (SG)
  • Spherical Harmonics (SH)
  • Linearized Transformed Cosines (LTC)
  • …

Spherical Gaussians

• Spherical Gaussian (SG) or von Mises-Fisher distribution:

   $$G(\bm{v},\bm{d},\lambda) = e^{\lambda(\bm{v}\cdot\bm{d}-1)}$$ 

  • $\bm{v}$ is evaluation direction (unit vector)
  • $\bm{d}$ is lobe direction axis (unit vector)
  • $\lambda \geq 0$ is lobe sharpness
• Construction becomes

   $$F_G(\bm{v}) = \sum_k w_k G(\bm{v}, \bm{d}_k, \lambda_k)$$ 

• Projection is finding parameters $\{ w_k, \bm{d}_k, \lambda_k \}$ that minimize reconstruction error by optimization

Combining SG

• Product of two SG is a SG:

   $$G_1G_2 = G\left( \bm{v},\, \frac{\bm{d}'}{\|\bm{d}'\|},\, \lambda' \right) \\
  \bm{d}'=\frac{\lambda_1\bm{d}_1 + \lambda_2\bm{d}|2}{\lambda_1+\lambda_2},\quad \lambda' = (\lambda_1 + \lambda_2)\|\bm{d}'\|$$ 

• Integral of an SG over a sphere is:

   $$\int_\Omega G(\bm{v})\,d\bm{v} = 2\pi\frac{1-e^{2\lambda}}{\lambda}$$ 

• Strength: can be used to represent both light radiance and BRDF in a single framework
• Main drawback: each lobe is nonzero over all the sphere and this increases computational complexity

Spherical Harmonics

• Orthogonal set of basis functions over sphere
  • Inner product of any two from the set is 0
• Inner product of functions

   $$\langle f_i(x),\, f_j(x) \rangle \equiv \int f_i(x) f_j(x) \, dx$$ 

• For spherical functions

   $$\langle f_i(\bm{n}),\, f_j(\bm{n}) \rangle \equiv \int_{\bm{n}\in\Theta} f_i(\bm{n}) f_j(\bm{n}) \, d\bm{n}$$ 

• Orthonormal set $\implies$ inner product is 0 when $i\neq j$ and 1 otherwise

Spherical Harmonics Projections

• Projection

   $$k_j = \langle f_\text{target} (),\, f_j() \rangle, \qquad f_\text{target}() \approx \sum_{j=1}^n k_j f_j()$$ 

• Strengths:
  • Rotationally invariant
  • Computationally efficient
• Weaknesses:
  • Global support (cover entire sphere)
  • Can have ringing artifacts

Hemispherical Bases

• Necessary for BRDF, incoming radiance, irradiance, etc…
• Ambient/Highlight/Direction (AHD) basis is the simplest and used commonly for irradiance
  • Consists of constant ambient light, directional light that approximates “highlight” irradiance, and direction where most light is concentrated
  • Consists of 8 parameters
  • Expensive to compute, usually heuristics are used

     $$E(\bm{n}) = \bm{c}_\text{ambient} + (\bm{n}\cdot\bm{d})^+ \bm{c}_\text{diffuse}$$ 

Radiosity Normal Mapping

• Also known as Half-Life 2 Basis
• Designed to store precomputed diffuse lighting while normal mapping
• Uses three directional vectors:

   $$\bm{m}_0 = \left( \frac{-1}{\sqrt{6}},\, \frac{1}{\sqrt{2}},\, \frac{1}{\sqrt{3}} \right), \quad
   \bm{m}_1 = \left( \frac{-1}{\sqrt{6}},\, \frac{-1}{\sqrt{2}},\, \frac{1}{\sqrt{3}} \right), \\
   \bm{m}_2 = \left( \frac{\sqrt{2}}{\sqrt{3}},\, 0,\, \frac{1}{\sqrt{3}} \right), \quad$$ 

• Reconstruction is done by interpolation:

   $$E(\bm{n}) = \frac{ \sum_{k=0}^2 \max(\bm{m}_k\cdot\bm{n},\,0)^2\,E_k }{ \sum_{k=0}^2 \max(\bm{m}_k\cdot\bm{n},\,0)^2 }$$ 

Reflection Mapping

• Simplest approach for environment mapping
• Assumes the BRDF is a perfect mirror
• Reflects incoming ray of light to light’s reflection direction $\bm{r}_i$
• Reflected view vector $\bm{r}$:

   $$\bm{r} = 2(\bm{n}\cdot\bm{v})\bm{n} - \bm{v}$$ 

• Reflectance equation for mirrors:

   $$L_o(\bm{v}) = F(\bm{n},\bm{r})\, L_i(\bm{r})$$ 

  • Note Fresnel term $F$ uses angle between $\bm{n}$ and $\bm{r}$ (not $\bm{h}$)
• Incoming radiance $L_i$ depends on direction and is stored in 2D table
• Compute $\bm{r}$ for each point and look up radiance

Reflection Mapping Algorithm

1. Load/generate texture of environment
2. For each reflective pixel, compute normal
3. Compute reflected view vector $\bm{r}$ from $\bm{v}$ and $\bm{n}$
4. Compute index in map from $\bm{r}$
5. Use the texel data from the map as incoming radiance in $L_o(\bm{v}) = F(\bm{n},\bm{r})\, L_i(\bm{r})$

Notes

• In general, flat surfaces do not work well with reflection mapping due to vectors $\bm{r}$ clustering in small parts of the map
• Illuminating scenes with textures is known as Image-Based Lighting (IBL)

Cube Mapping

• Most popular and fast approach today
• Project environment onto a cube and use faces as map
  • Created by rendering with 6 viewpoints or panorama stitching
• Is view-independent
• Supported directly by hardware with gl.TEXTURE_CUBE_MAP without edge problems

Specular Image-Based Lighting

• Extends environment mapping to glossy reflections
• Environment maps become specular light probes
  • Capture radiance from all directions at a point
  • Used to evaluate BRDF
  • Modified and called specular cube maps
• Surface roughness can be simulated by non-linear blurring (prefiltering)
• More correct approach is to consider BRDF function when filtering

Prefiltered Environment Mapping

• Approximates BRDF to be independent of $\bm{v}$ and $\bm{n}$
• Usually restrict BRDF to glossiness with radial symmetry

Convolving the Environment Map

• Compute for every texel corresponding to a direction $\bm{v}$ the specular lobe $D$:

   $$\sum_\Omega D(\bm{l},\bm{v}) \, L_i(\bm{l}) \, d\bm{l}$$ 

• No analytical form usually adopting Monte Carlo methods

   $$\sum_\Omega D(\bm{l},\bm{v}) \, L_i(\bm{l}) \, d\bm{l} \approx \lim_{N\to\infty} \frac{1}{N} \sum_{k=1}^N \frac{ D(\bm{l}_k,\bm{v}) \, L_i(\bm{l}_k)}{p(\bm{l}_k,\bm{v})}$$ 

  • $\bm{l}_k$ with $k=1,2,\dots,N$ are discrete samples over the unit sphere
  • $p(\bm{l}_k,\bm{v})$ is probability associated with direction $\bm{l}_k$
• Importance sampling and other tricks are used to improve results

Split-Integral Approximations

• Up until now assuming Phong lobe and post-multiplying by perfect-mirror Fresnel term:

   $$\int_{\bm{l}\in\Omega} f(\bm{l},\bm{v}) \, L_i(\bm{l}) \, (\bm{n}\cdot\bm{l}) \, d\bm{l} \approx F(\bm{n},\bm{v}) \int_{\bm{l}\in\Omega} D_\text{Phong}(\bm{r}) \, L_i(\bm{l}) \, (\bm{n}\cdot\bm{l}) \, d\bm{l}$$ 

• For specular macrofacet BRDF (reminder):

   $$f_\text{smf}(\bm{l},\bm{v}) = \frac{F(\bm{h},\bm{l}) \, G_2(\bm{l},\bm{v},\bm{h}) \, D(\bm{h})}{4|\bm{n}\cdot\bm{l}|\,|\bm{n}\cdot\bm{v}|}$$ 

• Even assuming $D(\bm{h}) \approx D_\text{Phong}(\bm{r})$ is valid, we are significantly simplifying!
• Split-integral approach

   $$\int_{\bm{l}\in\Omega} f_\text{smf}(\bm{l},\bm{v}) \, L_i(\bm{l}) \, (\bm{n}\cdot\bm{l}) \, d\bm{l} \approx
   \int_{\bm{l}\in\Omega} D(\bm{r}) \, L_i(\bm{l}) \, (\bm{n}\cdot\bm{l}) \, d\bm{l}
   \int_{\bm{l}\in\Omega} f_\text{smf}(\bm{l},\bm{v}) \, (\bm{n}\cdot\bm{l}) \, d\bm{l}$$ 

Split-Integral Notes

• Assume all terms are symmetric around reflection vector
• Both terms can be computed efficiently
• First only depends on roughness and reflection vector, commonly imposing $\bm{n}=\bm{v}=\bm{r}$ on the lobe
  • Usually precomputed with modified roughness
• Second depends on elevation angle $\theta$, roughness $\alpha$, and Fresnel term $F$
  • Fresnel is typically approximated with Schlick
  • Precomputed tables or curve fitting is used
• Exact solution for constant environment map

Irradiance Mapping

• Further extends environmental mapping to diffuse
• Only depend on surface normal $\bm{n}$ and contain irradiance values
• Allows for exact computation of BRDF lobe
• Usually stored independently from specular/reflection map with a cube map
• Created with extremely wide filters
  • Can be sped up by computation at low resolutions or in frequency domain
  • Commonly handled with Spherical Harmonics