现代图形学入门

Rasterization

What’s After MVP?

Model transformation (placing objects)
View transformation (placing camera)
Projection transformation
- Orthographic projection (cuboid to “canonical” cube)
- Perspective projection (frustun to “canonical” cube)

Sampling Artifacts in Computer Graphics

Signal are changing too fast (high frequency), but sampled too slowly
Sampling = Repeat Frequency Contents
Aliasing = Mixed Frequency Contents
Antialiasing = Limit, then repeating
- MSAA (Multi-Sample AA)
- FXAA (Fast Approximate AA 图像的后处理方法）
- TAA (Temporal AA 利用上一帧的信息）

Visibility / Occlusion

Painter’s Algorithm

Paint from back to front, overwirte in the framebuffer

Z-Buffer

Store current min. z-value for each sample (pixel)
Need an additional buffer for depth values
- frame buffer stores color values
- depth buffer stores (z-buffer) depth
z is always positive value
- samller z -> closer
- larger z -> further

Z-Buffer Algorithm

Initialize depth buffer to
During rasterization:

for (each triangle T)
    for (each sample (x, y, z) in T)
        if (z < zbuffer[x, y])
            framebuffer[x, y] = rgb;
            zbuffer[x, y] = z
        else
            ...

Shading

Definition

In Merriam-Webster Dictionary

The darkening or coloring of an illustration or diagram with parallel lines or a block of color.
In Compute Graphics

The process of applying a material to an object.
不同物体应用不用的材质。

Lambertian (Diffuse) Shading

: diffusely reflected light
: diffuse coefficient (color) 如果是一个表示颜色的向量，可以表示在 shading point 上吸收 / 反射的颜色
: engry arrived at the shading point
: engry received by the shading point

漫反射方向和视角方向没有关系

Specular Term (Blinn-Phong)

观察方向和镜面反射的方向接近的时候，就能得到高光

close to mirror direction half vector near normal

给定入射方向和法线方向，也可以算出反射方向进而计算出高光项，但计算量上要比半程向量复杂
指数：点乘向量能体现两个向量是否足够接近，用来控制高光的大小

Ambient Term

Add constant color to account for disregarded illumination and fill in black shadows
This is approximate

Graphics Pipeline

Interpolation Across Triangles

Why do we want to interpolate?
- Specify values at vertices
- Obtain smoothly varying values across triangles –希望在三角形内部实现平滑过渡
What do we want to interpolate?
- Texture coordinates, colors, normal vectors, … –可以对三角形上任意属性进行插值
How do we interpolate?
- Barycentric coordinates

Barycentric Coordinates

A coordinate system for triangles

重心坐标：三角形所在平面上任意一点都可以表示三个顶点坐标的线性组合，条件是线性组合的系数之和为 1
Inside the triangle if all three coordinates are non-negative

Texture Mapping

Simple Texture Mapping

for each rasterized screen sample (x, y): // usually a pixel's center
    (u, v) = evaluate texture coordinate at (x, y); // using barycentric coordinate（重心坐标）
    texcolor = texture.sample(u, v);
    set sample's color to texcolor; // using the diffuse albedo Kd

一种解释方法
假设需要用的纹理（比如是）渲染左边的两个三角形（比如是），但是纹理平面非常小，那么最后导致的结果就是一个纹素要对应 4 个像素，这样在实际渲染时产生的效果就是纹理被拉伸，视觉上会有模糊的感觉

另一种解释方法
纹理映射的过程会根据目标点离相机的远近，占用屏幕上不同大小的范围的像素，例如一个三角面在距离相机 20m 时占用 100 个屏幕像素，当三角面离相机更远时会看起来更小，此时可能占用 20 个屏幕像素，但是在两种情况下这个三角面使用的纹理贴图的大小是不变的。换句话说，由于纹理化表面可以相对于观察者处于任意距离和朝向，因此一个像素通常不直接对应于一个纹理像素。必须应用某种形式的滤波来确定像素的最佳颜色

Texture Magnification 纹理放大

What if the texture is too small?

就是说相对于要渲染的物体，纹理的分辨率不太高

由于要渲染的物体大于纹理平面，所以物体中的一些部分对应到纹理平面坐标可能就不是整数了，而是小数

Nearest
Bilinear

Linear interpolation(1D)

Two helper lerps

双线性插值其实就是横向和纵向两个方向做插值
Bicubic

What if the texture is too large?

另一种情况就是纹理相比于要渲染的物体大，这样就会导致纹理缩小，即一个像素会覆盖多个纹素

除了纹理分辨率大于要渲染的物体，在如下情况中也会出现 纹理缩小 的问题（通过透视投影将物体映射到平面，会造成近处纹理大，远处纹理小的视觉效果）。换句话说就是近处的一个像素可能只覆盖一个纹素，或者一个纹素覆盖多个像素（这种情况可用双线性插值解决），但对于远处的像素而言，一个像素会覆盖多个纹素，如下图所示（蓝点表示一个像素点，框表示该像素点所能覆盖的纹素数量）

Mipmap

Allowing (fast, approx., square) range quries.

“Mip” comes from the Latin “multum in parvo”, meaning a multitude in a small space.

Computing Mipmap Level `D`

利用重心坐标映射到纹理坐标
计算出纹理上近似正方区域的边长

Trilinear Interpolation

Linear interpolation based on continuous D value

Mipmap limitations: overblur

Geometry

Implicit Geometry

Points satisfy some specified relationship
E.g. sphere: all points in 3D, where . More generally, .
Constructive Solid Geometry
Combine implict geometry via Boolean operations.
Distance Functions
Fractals

Explict Geometry

All points are given directly or via parameter mapping.

Point Cloud

Curve

de Castekjau Algorithm

Evaluating Curves Algebraic Formula

Example: quadratic curve from three points

t可以理解为某一时刻

Bernstein form of a curve of order n:

: control points
: Bernstein polynomial

Surfaces

Shadow Mapping

Render from light

depth image from light source （只需要深度图）

Render from eye

Standard image (with depth) from eye

Project to light

Project visible points in eye view back to light source
Compare the depths from light and eye

Ray Tracing

Recursive (Whitted-Style) Ray Tracing

Ray-Surface Intersection

Ray equation:
Ray intersection with sphere

Sphere:
Ray intersection with implicit surface

General implicit surface:
Solve for real, positive roots
Ray intersection with triangle mesh

Plane equation (if satisfies it, then is on the plane):

Solve for intersection

A faster approach, giving barycentric coordinate（重心坐标） directly

Ray Intersection with Axis-Aligned Bounding Box
- The ray enters the box only when it enters all pairs of slabs
- The ray exits the box as long as it exits any pair of slabs
- For each pair, calcute the and (negative is fine)
- For the 3D box, ,
- if , we know ray stays a while in the box.

Spatial Partitions（空间划分）

Oct-Tree
KD-Tree
BSP-Tree

Object Partitions（物体划分） & Bounding Volume Hierarchy (BVH)

Radiometry（辐射度量学）

Radiant Energy and Flux (Power)
- 强度
- 能量
Import Light Measurements of Interest
- Radiant Intensity: power per solid angle
- Solid Angle（立体角）
- Irradiance（辐射照度）: power per unit area
- Radiance（辐射亮度）: power per unit solid angle, per projected unit area.
Irradiance vs.Radiance
- Irradiance: total power received by area
- Radiance: power received by area from “direction”
radiance 是来自四面八方的 irradiance 的和（积分）

Bidirectional Reflectance Distribution Function (BRDF)

Refection at a point

如果有一束光线进入，不同的反射方向上的能量分布。

The BRDF represents how much light is reflected into each outgoing direction from each incoming direction

The Reflection Equation

The Rendering Equation

by adding an Emission term ot make it general

: Reflected Light
: Emission（自发光）
: Incident Light
: BRDF
: Cosine of Incident angle

Rendering Equation as Intergral Equation

: Emission directly from light sources Shading in Rasterization
: Direct Illumination on surfaces Shading in Rasterization
: Indirect Illumination(one bounce indirect: mirrors, refraction)
: Two bounce indirect illum.

Path Tracing

Materials and Appearances

Diffuse / Lambertian Material

漫反射系数

Diffuse’s BRDF:
为反射率 (albedo)

Glossy Material

Perfect Specular Reflection

两个向量和的和一定沿着法向量的方向，长度是 2 倍的红颜色的长度，即入射方向投影到法线方向的长度（）的 2 倍

Specular Refraction

Snell’s Law

Fresnel Reflection / Term

MicroFacet Material

Microfacet Theory

Macroscale: flat & rough
Microscale: bumpy（凹凸的） & specular

Microface BRDF

: Frenel term
: shadowing-masking term（微表面的自遮挡、自投影处理）
: distribution of normals

Properties of BRDFs

Non-negativity
Linearity
Reciprocity principle（可逆性）
Energy conservation（能量守恒）
Isotropic（各项异性） & Anisotropic（各项同性）

Measuring BRDFs

Cameras, Lenses（棱镜） and Light Fields（光场）

Field of View
- It it common to refer to angular field of view by focal length（焦距） of a lens used on a 35mm-format film(36 24mm)
Exposure
- Exposure = time irradiance
  - Exposure time:
    - Control by shutter
  - Irradiance:
    - Power of light falling on a unit area of sensor
    - Controlled by lens aperture（光圈） and focal length
- Exposure Controls in Photography
  - Aperture size: Change the f-stop by opening / closing the aperture
    - f-stop 数越大光圈越小
    - F-Number definiton: the focal length divided by the diameter of the aperture
  - Shutter speed
  - IOS gain（感光度）
Depth of Field（指成像清晰的一段范围）

Color and Perception

Addtive Color（加色系统）

Given a set of primary lights, each with its own spectral distribution
- , ,
Adjust the brightness of these lights and add them together

Color Spaces

Standardized RGB(sRGB)
Perceptually Organized Color Spaces
- HSV Color Space(Hue-Saturation-Value)
  - Hue: 色调
  - Saturation: 饱和度
  - Value: 亮度
- L*a*b*
  - L* is lightness
  - a* and b* are color-opponent（互补色） pairs
  - a* is red-green
  - b* is blue-yellow
CMYK: A Subtractive Color Space
- Cyan, Magenta, Yellow and Key Widely

图形学

#Graphics #GAMES

现代图形学入门

https://silhouettesforyou.github.io/2021/11/04/2890ce8fb84c/

Author

Xiaoming

Posted on

November 4, 2021

Licensed under

OpenGL 基础 Previous

现代图形学入门

Rasterization

What’s After MVP?

Sampling Artifacts in Computer Graphics

Visibility / Occlusion

Painter’s Algorithm

Z-Buffer

Z-Buffer Algorithm

Shading

Definition

Lambertian (Diffuse) Shading

Specular Term (Blinn-Phong)

Ambient Term

Graphics Pipeline

Interpolation Across Triangles

Barycentric Coordinates

Texture Mapping

Simple Texture Mapping

一种解释方法

另一种解释方法

Texture Magnification 纹理放大

What if the texture is too small?

What if the texture is too large?

Mipmap

Computing Mipmap Level D

Trilinear Interpolation

Geometry

Implicit Geometry

Explict Geometry

Curve

de Castekjau Algorithm

Evaluating Curves Algebraic Formula

Surfaces

Shadow Mapping

Render from light

Render from eye

Project to light

Ray Tracing

Recursive (Whitted-Style) Ray Tracing

Ray-Surface Intersection

Spatial Partitions（空间划分）

Object Partitions（物体划分） & Bounding Volume Hierarchy (BVH)

Radiometry（辐射度量学）

Bidirectional Reflectance Distribution Function (BRDF)

Refection at a point

The Reflection Equation

The Rendering Equation

Rendering Equation as Intergral Equation

Path Tracing

Materials and Appearances

Diffuse / Lambertian Material

Glossy Material

Perfect Specular Reflection

Specular Refraction

Fresnel Reflection / Term

MicroFacet Material

Microfacet Theory

Microface BRDF

Properties of BRDFs

Measuring BRDFs

Cameras, Lenses（棱镜） and Light Fields（光场）

Color and Perception

Addtive Color（加色系统）

Color Spaces

Computing Mipmap Level `D`