CSE 190 Spring 2015 Lecture 11 Ravi Ramamoorthi http wwwcsucsdedu ravir To Do Assignment 2 due May 15 Should already be well on way Contact us for difficulties etc ID: 312158
Download Presentation The PPT/PDF document "Advanced Computer Graphics" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Slide1
Advanced Computer Graphics
CSE 190 [Spring 2015], Lecture 11Ravi Ramamoorthi
http://
www.cs.ucsd.edu
/~
ravirSlide2
To Do
Assignment 2 due May 15Should already be well on way. Contact us for difficulties etc.This lecture on rendering, rendering equation. Pretty advanced theoretical material. Don’t worry if a bit lost; not directly required on the
homeworks. Slide3
Course Outline
3D Graphics Pipeline
Rendering
(Creating, shading images from geometry, lighting, materials)
Modeling
(Creating 3D Geometry)Slide4
Course Outline
3D Graphics Pipeline
Rendering
(Creating, shading images from geometry, lighting, materials)
Modeling(Creating 3D Geometry)
Unit 3: Advanced RenderingWeeks 6 – 8.
(Final Project)Unit 4: Animation, Imaging
Weeks 9, 10. (Final Project
)
Unit 2: Meshes, ModelingWeeks 3
–
5.
Assignment
2
Unit 1: Foundations of Signal and Image Processing
Understanding the way 2D images are formed and displayed, the important concepts and algorithms, and to build an image processing utility like Photoshop
Weeks 1 – 3.
Assignment 1Slide5
Illumination Models
Local IlluminationLight directly from light sources to surfaceNo shadows (cast shadows are a global effect)
Global Illumination: multiple bounces (indirect light)Hard and soft shadowsReflections/refractions (already seen in ray tracing)Diffuse and glossy interreflections (radiosity, caustics)
Some images courtesy Henrik
Wann JensenSlide6
Diffuse Interreflection
Diffuse interreflection, color bleeding [Cornell Box]Slide7
RadiositySlide8
Caustics
Caustics: Focusing through specular surfaceMajor research effort in 80s, 90s till todaySlide9
Overview of lecture
Theory for all global illumination methods (ray tracing, path tracing, radiosity)We derive Rendering Equation [Kajiya 86]
Major theoretical development in fieldUnifying framework for all global illuminationDiscuss existing approaches as special cases Fairly theoretical lecture (but important). Not well covered in textbooks (though see Eric Veach’s thesis).
See reading if you are interested.Slide10
Outline
Reflectance Equation (review)Global IlluminationRendering EquationAs a general Integral Equation and OperatorApproximations (Ray Tracing, Radiosity)Surface Parameterization (Standard Form)Slide11
Reflection Equation
Reflected Light
(Output Image)
Emission
Incident
Light (from
light source)
BRDF
Cosine of
Incident angleSlide12
Reflection Equation
Reflected Light
(Output Image)
Emission
Incident
Light (from
light source)
BRDF
Cosine of
Incident angle
Sum over all light sourcesSlide13
Reflection Equation
Reflected Light
(Output Image)
Emission
Incident
Light (from
light source)
BRDF
Cosine of
Incident angle
Replace sum with integralSlide14
Environment Maps
Light as a function of direction, from entire environmentCaptured by photographing a chrome steel or mirror sphereAccurate only for one point, but distant lighting same at other scene locations (typically use only one env. map)
Blinn
and Newell 1976, Miller and Hoffman, 1984
Later, Greene 86, Cabral et al. 87Slide15
Environment Maps
Environment maps widely used as lighting representationMany modern methods deal with offline and real-time rendering with environment mapsImage-based complex lighting + complex BRDFsSlide16
The Challenge
Computing reflectance equation requires knowing the incoming radiance from surfacesBut determining incoming radiance requires knowing the reflected radiance from surfacesSlide17
Rendering Equation
Reflected Light
(Output Image)
Emission
Reflected
Light
BRDF
Cosine of
Incident angle
Surfaces (
interreflection
)
UNKNOWN
UNKNOWN
KNOWN
KNOWN
KNOWNSlide18
Rendering Equation (Kajiya 86)Slide19
Rendering Equation as Integral Equation
Reflected Light
(Output Image)
Emission
Reflected
Light
BRDF
Cosine of
Incident angle
UNKNOWN
UNKNOWN
KNOWN
KNOWN
KNOWN
Is a Fredholm Integral Equation of second kind
[extensively studied numerically] with canonical form
Kernel of equationSlide20
Linear Operator Equation
Kernel of equation
Light Transport Operator
Can be discretized to a simple matrix equation
[or system of simultaneous linear equations]
(L, E are vectors, K is the light transport matrix)Slide21
Ray Tracing and extensions
General class numerical Monte Carlo methodsApproximate set of all paths of light in scene
Binomial TheoremSlide22
Ray Tracing
Emission directly
From light sources
Direct Illumination
on surfaces
Global Illumination
(One bounce indirect)
[Mirrors, Refraction]
(Two bounce indirect)
[Caustics etc]Slide23
Ray Tracing
Emission directly
From light sources
Direct Illumination
on surfaces
Global Illumination
(One bounce indirect)
[Mirrors, Refraction]
(Two bounce indirect)
[Caustics etc]
OpenGL ShadingSlide24
Outline
Reflectance Equation (review)Global IlluminationRendering EquationAs a general Integral Equation and OperatorApproximations (Ray Tracing, Radiosity)Surface Parameterization (Standard Form)Slide25
Rendering Equation as Integral Equation
Reflected Light
(Output Image)
Emission
Reflected
Light
BRDFCosine of
Incident angle
UNKNOWN
UNKNOWN
KNOWN
KNOWN
KNOWN
Is a Fredholm Integral Equation of second kind
[extensively studied numerically] with canonical form
Kernel of equationSlide26
Linear Operator Theory
Linear operators act on functions like matrices act on vectors or discrete representations Basic linearity relations holdExamples include integration and differentiation
M is a linear operator.
f and h are functions of u
a and b are scalars
f and g are functions Slide27
Linear Operator Equation
Kernel of equation
Light Transport Operator
Can also be discretized to simple matrix equation
[or system of simultaneous linear equations]
(L, E are vectors, K is the light transport matrix)Slide28
Solving the Rendering Equation
Binomial Theorem
Term n corresponds to n bounces of lightSlide29
Solving the Rendering Equation
Too hard for analytic solution, numerical methodsApproximations, that compute different terms, accuracies of the rendering equationTwo basic approaches are ray tracing, radiosity. More formally, Monte Carlo and Finite ElementMonte Carlo techniques sample light paths, form statistical estimate (example, path tracing)Finite Element methods discretize to matrix equationSlide30
Ray Tracing
Emission directly
From light sources
Direct Illumination
on surfaces
Global Illumination
(One bounce indirect)
[Mirrors, Refraction]
(Two bounce indirect)
[Caustics etc]Slide31
Ray Tracing
Emission directly
From light sources
Direct Illumination
on surfaces
Global Illumination
(One bounce indirect)
[Mirrors, Refraction]
(Two bounce indirect)
[Caustics etc]
OpenGL ShadingSlide32Slide33
Outline
Reflectance Equation (review)Global IlluminationRendering EquationAs a general Integral Equation and OperatorApproximations (Ray Tracing, Radiosity)Surface Parameterization (Standard Form)Slide34
Rendering Equation
Reflected Light
(Output Image)
Emission
Reflected
Light
BRDF
Cosine of
Incident angle
Surfaces (interreflection)
UNKNOWN
UNKNOWN
KNOWN
KNOWN
KNOWNSlide35
Change of Variables
Integral over angles sometimes insufficient. Write integral in terms of surface radiance only (change of variables)Slide36
Change of Variables
Integral over angles sometimes insufficient. Write integral in terms of surface radiance only (change of variables) Slide37
Rendering Equation: Standard Form
Integral over angles sometimes insufficient. Write integral in terms of surface radiance only (change of variables)
Domain integral awkward. Introduce binary visibility fn V
Same as equation 2.52 Cohen Wallace. It swaps primed
And unprimed, omits angular args of BRDF, - sign.
Same as equation above 19.3 in Shirley, except he has
no emission, slightly diff. notationSlide38
Radiosity Equation
Drop angular dependence (diffuse Lambertian surfaces)
Change variables to radiosity (B) and albedo (
ρ
)
Same as equation 2.54 in Cohen Wallace handout (read sec 2.6.3)
Ignore factors of
π
which can be absorbed.
Expresses conservation of light energy at all points in spaceSlide39
Discretization and Form Factors
F is the
form factor.
It is dimensionless and is the fraction of energy leaving the entirety of patch j (
multiply by area of
j to get total energy) that arrives anywhere in the entirety of patch i (
divide by area of i to get energy per unit area or radiosity). Slide40
Form FactorsSlide41
Matrix EquationSlide42
Summary
Theory for all global illumination methods (ray tracing, path tracing, radiosity)We derive Rendering Equation [Kajiya 86]Major theoretical development in field
Unifying framework for all global illuminationDiscuss existing approaches as special casesNext: Practical solution using Monte Carlo methods