Concepts

[jiangyangmu]

Models

• Entity - Object's space properties
  • Vertex is a set of properties.
    • Required: Coordinates
    • Optional: Color, Normal information, Transparency/Texture ...
  • Surface is an interpolation of vertices.
  • Volume is an enclosed space constructed from surfaces.
• Sampling - From virtual to pixel
  • Interpolation
  • Occlusion: sampling of depth.
  • Super-sampling: sampling density.
  • Texture mapping: sampling of UV image. Texture can be color, normal, etc.
  • Alpha compositing: sampling combination (Alpha + Occlusion).
• Radiometry & Material - Object's light properties
  • Light model: path-tracing => straight path (wave length << objects)
  • Camera model: Lenses, Depth of Field, Exposure
  • Energy Loss model: incident sensor plane, distance from source
  • Others
    • Reflection & Refraction
    • BRDF (bidirectional reflectance distribution function): 描述光在非透明表面的反射分布
    • Mirror vs Diffuse
    • Phong, Micro-facet, Self-shadowing, Torrance-Sparrow
    • Radiance
• World - Coordinate System, Transformation
  • Space Normalization: world -> camera -> normalized

Pipeline

1. Normalize coordinate - Simplify virtual world
   • world space -> camera space -> normalized space(handle perspective, near-plane, far-plane) -> clipping
   • Now everything is in a cube!
   • (X,Y) to screen position, Z to depth.
2. Collect properties and do sampling - From virtual to samples/pixels
   • collect triangle attributes (depth, UV, ...) -> calculate attributes on samples within triangle (depth, UV mapped color, ...)
3. Test and draw - Final step
   • do depth test and update color buffer on samples

OpenGL/Direct3D graphics pipeline

Tools

• Monte Carlo: 简单粗暴的解决高维空间积分。
  • Path Tracing: 使用Monte Carlo计算渲染方程
• Signal Processing and Sampling: 纹理渲染
• Numerical Integration
  • calculus: 普通积分
  • Monte Carlo Integration: 投硬币积分

Framework

• TODO: need graph

Terms

• Photon: energy
• Solid Angle(ω): 三维中的“弧度”。就像圆中的弧度一样，定义基坐标，一个弧度值可以定位球体上的一个点。
• Flux(Φ): time density of energy (对象总能量的时间分布，对象单位时间内，发射、接收多少个光子, Watts)
• Irradiance(E): area density of flux (能量在对象表面的分布，<单位时间内>单位表面上，发射、接收多少个光子)
• Radiance(L): solid angle density of irradiance (能量的方向分布，<单位时间内>某个方向上，发射、接收多少个光子)
• 本质上，Radiance 描述了一种模型，使我们可以评估 在一定时间里，一定空间中，光子在各个方向上的流入流出数量情况。
• Monte-Carlo Estimator: 用于估计等效平均能量密度。比如，从前方光源射来的A个光子，相当于从 N 个方同同时射来 A/N 个光子，对这 N 坨 A/N 个光子积分，就可得出实际 A 个光子的能量估计值（这个例子不太好，观察方向不对）。

Understanding Render

• OpenGL/Direct3D is? CUDA is? Their relation?
• 显示（display）其实是一个采样（sampling）的过程。
• 锯齿（aliasing）：对（相对）高频信号采样不足（under-sampling）。
  • 例子1：锯齿三角形。用 super-sampling 解决（增加采样）。
  • 例子2：纹理平面
  • 例子3：旋转的轮子
  • 例子4：纹理贴图。用平滑滤波（降低数据源频率），mipmap。
  • 原理

Reference

• Rasterization vs Ray Tracing: 光栅化与光线追踪的区别与联系
• Rendering Equation - University of Freiburg
  • 清晰的阐述了渲染方程的原理，以及其各个部分的功能与 motivation。
  • 完美情况的直接计算，复杂情况的 Monte-Carlo 及其优缺点。
  • 采样数目与错误率的关系，采样中随机变量的 PDF(probability density function), CDF(cumulative distribution function)。
  • Expected Value & Variance 的意义。
• OpenGL 2.0 Specification
• CG: illumination model
• WebGL online demo：不错的在线交互式 path-tracing 渲染算法 demo。
• smallpt: 99 行的光线追踪器
• The Brigade Renderer: A Path Tracer for Real-Time Games
• 如何开始用 C++ 写一个光栅化渲染器？ - 知乎
• Rasterization on Larrabee: 一个不错的实现讲解

[terhuhf]

Modeling

Geometry

Smooth Curves Representation

• Principles
  • User specifies by control points and completely controlled by control points
  • We will interpolate the control points by a smooth curve
• Splines
  • A type of smooth curve in 2D/3D which results in a smooth parametric P(t)
• Cubic Bezier Curve
  • with 4 control points, the curve passes through the first and the last control point
• Cubic B-Splines
  • More than 4 control points
  • Logically cubic - cubics chained together
  • Curve is not contained to pass through any points
  • Automatically C2 level of smooth

    Surfaces Representation

• Triangles Meshes
  • Triangles: represented by 3 vertices
• Tensor Product Besieger Patches
  • Defined by 4 * 4 control points
  • Interpolates 4 corners, approximate others

    Hierarchical Modeling

• Scene Graph
  • Shapes (geometry)
  • Tranformations
  • Properties (materials, colors)
  • Groups (multiple instances)

    Color

• Map spectra (physical color) to perceptual color
  • Physical color: multiplication of light and reflection spectrum
  • Perceptual color: 3-number
  • Standard color space

Materials

Reflection Models

• Ideal diffuse
• Ideal specular
• Glossy: Most of the reflected light to travel in the direction of the ideal mirror ray
  • The Phong Specular Model

    Texture

• Texture Mapping
  • Associating each surface point with a 2D coordinate in the texture map
  • UV coordinates
  • Texture UV optimization: k_d
    • To flate 3D object onto 2D UV coordinates
    • For each vertex, find coordinates U,V such as distortion is minimized
  • Mathematical mapping
  • Projective Mapping
    • Given the camera H of the slide projectors, for a give 3D point P, project onto 2D space of slide projector: HP
  • Normal mapping: n
    • Use normal mapped from the 2D image in BRDF
    • Generating normal map

Lights

• Point lights
  • I_in = I_lightcos(theta)/r²
• Spotlights
  • Point light with non-uniform directional emission
  • Central direction
  • Hotpot angle
  • Falloff angle

    Radiance

• Incoming radiance: The amount of light energy received by a surface depends on incoming angle

Quantifying Reflection － BRDF

Ideal Diffuse Reflection

• Surface reflects equally in all direction
• L₀ = k_dmax(0, n•l)l_i/r²

  Ideal Specular Reflection

• Only at a mirror angle
• View dependent
• BRDF: A Dirac delta multiplied by a specular coefficient k_s

  Non-ideal Reflection

  The Phong Specular Model

• L₀ = [k_a + k_d(n • l) + k_s(v • r)^q] l_i/r²

  Microfacet Theory

Rendering

Ray Casting

Producing color values for pixels

• for every pixel
  • construct a ray from the eye
  • for every object in the scene
    • find intersection with the ray
    • keep if closest

      Intersection

      equation

• Ray Representation
  • Origin - Point
  • Direction - Vector
• Plane Representation
  • A point on the plane
  • Normal
• Sphere Representation
  • Center - Point
  • Normal

    Anti-aliasing

Theoretical Foundations

Radiometric Quantities

• Radiant energy: basic unit of electromagnetic energy
• Radiant flux: the amount of radiant energy that passes through a surface per second
• Radiant flux density: radiant flux per unit surface
• Irradiance E: flux density arriving at a surface
• Radiant existence: flux density leaving a surface
• Radiant intensity: flux density per unit solid angle
• Radiance L flux per unit projected area unit solid angle
  • Projected radiant intensity

    BRDF

• To represen the reflective properties of surfaces with materials
• Properties *Reciprocity
  • Linearity
  • Conservation of energy

    Reflectance

• Reflected radiant flux / incident radiant flux

  Questions

• Why using normals?
• Tensor Bezier Patches vs. Tensor Product Splines