jacco bikker - february april 2016 - utrecht university 01 - intro and whitted.pdf · abstract in...

𝑰 𝒙, 𝒙′ = 𝒈(𝒙, 𝒙′) 𝝐 𝒙, 𝒙′ + 𝑺

𝝆 𝒙, 𝒙′, 𝒙′′ 𝑰 𝒙′, 𝒙′′ 𝒅𝒙′′

INFOMAGR – Advanced GraphicsJacco Bikker - February – April 2016

Welcome!

Today’s Agenda:

Advanced Graphics

Recap: Ray Tracing

Whitted-style

Assignment 1

Website

http://www.cs.uu.nl/docs/vakken/magr

Site information overrules official schedule Downloads, news, slides, deadlines, links

Please check regularly.

INFOMAGR

Advanced Graphics – Introduction 3

Abstract

In this course, we explore physically based rendering, with a focus on interactivity.

At the end of this course, you will have a solid theoretical understanding of efficient physically based light transport.

You will also have a good understanding of acceleration structures for fast ray/scene intersection for static and dynamic scenes.

You will have hands-on experience with algorithms for efficient realistic rendering of static and dynamic scenes using ray tracing on CPU and GPU.

INFOMAGR


Topics

We will cover the following topics:

Ray tracing fundamentals (recap);

Whitted-style ray tracing;

Acceleration structure construction;

Acceleration structure traversal;

Data structures and algorithms for animation;

Stochastic approaches to AA, DOF, soft shadows, … ;

Path tracing;

Variance reduction in path tracing algorithms;

Various forms of parallelism in ray tracing.

INFOMAGR


Lectures

~13 lectures:Wednesday / Friday afternoon

~7 working colleges:Friday afternoon (starting week 2)

Attendance is not mandatory, but of course highly recommended.We move fast; missing a key lecture may be a serious problem.

INFOMAGR


Literature

Papers and online resources will be supplied during the course.

Slides will be made available after each lecture.

Recommended literature:

Physically Based Rendering, Second Edition – From Theory to Implementation, Pharr & Humphreys. Morgan Kaufmann, 2010.ISBN-10: 0123750792.

INFOMAGR


Dependencies

It is assumed that you have basic knowledge of rendering (INFOGR) and associated mathematics.

You also should be a decent programmer; this is explicitly not a purely theoretical course. You are expected to verify the theory and experience the good and the bad.

A brief introduction to GPGPU and SIMD will be provided for those that did not take INFOMOV.

INFOMAGR


Resources

You will develop a ray tracing testbed for assignment 1. As a starting point, several ‘templates’ are available:

Tmpl85.00aA basic C++ framework for graphics programming, which opens a window, provides a 32-bit RGB framebuffer, and some basic helper classes.

gpu_lab (available halfway the course)A basic C++ framework for GPGPU via CUDA and OpenCL.

Template_C#A basic C# template which uses OpenTK to provide a 32-bit RGB framebuffer as well as OpenGL functionality, and Cloo for using OpenCL.

INFOMAGR


Assignments

1. (not graded):Ray tracing framework

For this assignment, you prepare a testbed for subsequent assignments.

2. (weight: 1):Acceleration structures

In this assignment, you expand your testbed with efficient acceleration structure construction and traversal. This enables you to run Whitted-style ray tracing in real-time.

3. (weight: 2):Final assignment

In this assignment, you either implement an interactive path tracer, or a rendering algorithm you chose, using CPU and/or GPU rendering.

INFOMAGR


Exam

One final exam at the end of the block.

Materials to study:

Slides Notes taken during the lectures Provided literature Assignments

INFOMAGR


Grading & Retake

Final grade for assignments 𝑃 = (𝑃2 + 2 ∗ 𝑃3) / 3Final grade for INFOMAGR 𝐺 = (𝑃 + 𝐸) / 2

Passing criteria:

𝑃 ≥ 5.0 𝐸 ≥ 5.0 𝐺 ≥ 6.0

Repairing your grade using the retake exam:

only if 𝑃 ≥ 4.0 and 𝐸 ≥ 4.0 and 𝐺 < 6.0. 𝐺𝑟 = max( 𝐺, (𝑅 + 𝐺) / 2 )

INFOMAGR


Today’s Agenda:

Advanced Graphics

Recap: Ray Tracing

Whitted-style

Assignment 1

Ray

A ray is an infinite line with a start point:

𝑝(𝑡) = 𝑂 + 𝑡𝐷, where 𝑡 > 0.

The ray direction 𝐷 is usually normalized.

Recap


Scene

The scene consists of a number of primitives:

Spheres Planes Triangles

..or anything for which we can calculate the intersection with a ray.

We also need:

A camera (position, direction, FOV, focal distance, aperture size) Light sources

Recap


Recap

Ray Tracing

World space

Geometry Eye Screen plane Screen pixels Primary rays Intersections Point light Shadow rays

Light transport

Extension rays

Light transport


Ray setup

A ray is initially shot through a pixel on the screen plane. The screen plane is defined in world space:

Camera position: E = (0,0,0)

View direction: 𝑉

Screen center: C = 𝐸 + 𝑑𝑉Screen corners: p0 = 𝐶 + −1,−1,0 , 𝑝1 = 𝐶 + 1,−1,0 , 𝑝2 = 𝐶 + (−1,1,0)

From here:

Change FOV by altering 𝑑; Transform camera by multiplying E, 𝑝0 , 𝑝1 , 𝑝2 with the camera matrix.

Recap


Ray setup

Point on the screen:

𝑝 𝑢, 𝑣 = 𝑝0 + 𝑢 𝑝1 − 𝑝0 + 𝑣(𝑝2 − 𝑝0)𝑢, 𝑣 ∈ [0,1]

Ray direction (normalized):

𝐷 =𝑝 𝑢, 𝑣 − 𝐸

∥ 𝑝 𝑢, 𝑣 − 𝐸 ∥

Ray origin:

𝑂 = 𝐸

Recap

𝑝0

𝑝1

𝑝2

𝐸

u

v


Ray Intersection

Given a ray 𝑝(𝑡) = 𝑂 + 𝑡𝐷, we determine the closest intersection distance 𝑡 by intersecting the ray with each of the primitives in the scene.

Ray / plane intersection:

Plane: 𝑝 ∙ 𝑁 + 𝑑 = 0

Ray: 𝑝(𝑡) = 𝑂 + 𝑡𝐷

Substituting for 𝑝(𝑡), we get

𝑂 + 𝑡𝐷 ∙ 𝑁 + 𝑑 = 0

𝑡 = −(𝑂 ∙ 𝑁 + 𝑑)/(𝐷 ∙ 𝑁)

𝑃 = 𝑂 + 𝑡𝐷

Recap

𝑝0

𝑝1

𝑝2

𝐸


Ray Intersection

Ray / sphere intersection:

Sphere: 𝑝 − 𝐶 ∙ 𝑝 − 𝐶 − 𝑟2 = 0

Substituting for 𝑝(𝑡), we get

𝑂 + 𝑡𝐷 − 𝐶 ∙ 𝑂 + 𝑡𝐷 − 𝐶 − 𝑟2 = 0

𝐷 ∙ 𝐷 𝑡2 + 2𝐷 ∙ 𝑂 − 𝐶 𝑡 + (𝑂 − 𝐶)2−𝑟2 = 0

𝑎𝑡2 + 𝑏𝑡 + 𝑐 = 0 → 𝑡 =−𝑏 ± 𝑏2 − 4𝑎𝑐

2𝑎

𝑎 = 𝐷 ∙ 𝐷

𝑏 = 2𝐷 ∙ (𝑂 − 𝐶)𝑐 = 𝑂 − 𝐶 ∙ 𝑂 − 𝐶 − 𝑟2

Recap

𝑝0

𝑝1

𝑝2

𝐸

Negative: no intersections


Ray Intersection

Efficient ray / sphere intersection:

void Sphere::IntersectSphere( Ray ray ){

vec3 c = this.pos - ray.O;float t = dot( c, ray.D );vec3 q = c - t * ray.D;float p2 = dot( q, q );if (p2 > sphere.r2) return; t -= sqrt( sphere.r2 – p2 );if ((t < ray.t) && (t > 0)) ray.t = t;// or: ray.t = min( ray.t, max( 0, t ) );

}

Note:

This only works for rays that start outside the sphere.

Recap


O

𝐷

𝑐

t

𝑞

𝑝2

Observations

Ray tracing is a point sampling process:

we may miss small details; aliasing will occur.

Ray tracing is a visibility algorithm:

For each pixel, we find the nearest object (which occludes objects farther away).

Recap


Observations

Note: Rasterization (Painter’s or z-buffer) is also a visibility algorithm.

Rasterization:

loop over objects / primitives; per primitive: loop over pixels.

Ray tracing:

loop over pixels; per pixel: loop over objects / primitives.

Recap


Today’s Agenda:

Advanced Graphics

Recap: Ray Tracing

Whitted-style

Assignment 1

An Improved Illumination Model for Shaded Display

In 1980, “State of the Art” consisted of: Rasterization Shading: either diffuse (N · L) or specular ((N · H)n), both

not taking into account fall-off (Phong) Reflection, using environment maps (Blinn & Newell *) Stencil shadows (Williams **)

Goal: Solve reflection and refraction

Improved model: Based on classical ray optics

Whitted

** : Williams, L. 1978. Casting curved shadows on curved surfaces. In Computer Graphics (Proceedings of SIGGRAPH 78), vol. 12, 270–274.

* : Blinn, J. and Newell, M. 1976. Texture and Reflection in Computer Generated Images. Communications of the ACM 19:10 (1976), 542—547.


An Improved Illumination Model for Shaded Display*

Physical basis of Whitted-style ray tracing:

Light paths are generated (backwards) from the camera to the light sources, using rays to simulate optics.

Color Trace( ray r )I, N, mat = NearestIntersection( scene, r )return mat.color * DirectIllumination( I, N )

Whitted

* : T. Whitted. An Improved Illumination Model for Shaded Display. Commun. ACM, 23(6):343–349, 1980.



Color Trace( ray r )

I, 𝑁, mat = NearestIntersection( scene, r )

return mat.color * DirectIllumination( I, 𝑁 )

Direct illumination:

Summed contribution of unoccluded point light sources, taking into account:

Distance to I

Angle between 𝐿 and 𝑁 Intensity of light source

Note that this requires a ray per light source.

Whitted




I, 𝑁, mat = NearestIntersection( scene, r )if (mat == DIFFUSE)

return mat.color * DirectIllumination( I, 𝑁 )if (mat == MIRROR)

return mat.color * Trace( I, reflect( r.𝐷, 𝑁 )

Indirect illumination:

For perfect specular object (mirrors) we extend the primary ray with an extension ray:

We still modulate transport with the material color

We do not apply 𝑁 ∙ 𝐿 We do not calculate direct illumination

Whitted


Reflection

Given a ray direction 𝐷 and a normalized surface normal 𝑁, the

reflected vector 𝑅 = 𝐷 − 2(𝐷 ∙ 𝑁)𝑁.

Derivation:

𝑣 = 𝑁(𝐷 ∙ 𝑁)

𝑢 = 𝐷 − 𝑣

𝑅 = 𝑢 + (− 𝑣)

𝑅 = 𝐷 −𝑁 𝐷 ∙ 𝑁 − 𝑁(𝐷 ∙ 𝑁)

𝑅 = 𝐷 − 2(𝐷 ∙ 𝑁)𝑁

Whitted

𝑅 𝐷

𝑵 𝑣

𝑢


Question 1: For direct illumination, we take into account:

Material color Distance to light source

𝑁 ∙ 𝐿

Why?

Question 2: We use the summed contribution of all light sources.

Is this correct?

Question 3: Why do we not sample the light sources for a pure specular surface? (can you cast a shadow on a bathroom mirror?)

Question 4: Why do we not apply 𝑁 ∙ 𝐿 to reflections?

Question 5: Prove geometrically that, for normalized vectors 𝐷 and 𝑁, 𝑅 =

𝐷 − 2 𝐷 ∙ 𝑁 𝑁 yields a normalized vector.

Whitted



Handling partially reflective materials:


I, 𝑁, mat = NearestIntersection( scene, r )s = mat.specularityd = 1 – mat.specularityreturn mat.color * (

s * Trace( I, reflect( r.𝐷, 𝑁 )

d * DirectIllumination( I, 𝑁 )

Note: this is not efficient. (why not?)

Whitted



Dielectrics


I, 𝑁, mat = NearestIntersection( scene, r )if (mat == DIFFUSE)

return mat.color * DirectIllumination( I, 𝑁 )if (mat == MIRROR)

return mat.color * Trace( I, reflect( r.𝐷, 𝑁 )if (mat == GLASS)

return mat.color * ?

Whitted


Dielectrics

The direction of the transmitted vector 𝑇 depends on the refraction indices 𝑛1, 𝑛2 of the media separated by the surface. According to Snell’s Law:

𝑛1𝑠𝑖𝑛𝜃1 = 𝑛2𝑠𝑖𝑛𝜃2

or

𝑛1𝑛2𝑠𝑖𝑛𝜃1 = 𝑠𝑖𝑛𝜃2

Note: left term may exceed 1, in which case 𝜃2 cannot be computed. Therefore:

𝑛1𝑛2𝑠𝑖𝑛𝜃1 = 𝑠𝑖𝑛𝜃2⟺ 𝑠𝑖𝑛𝜃1 ≤

𝑛2

𝑛1 𝜃𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 = arcsin

𝑛2𝑛1sin𝜃2

Whitted

𝑇

𝐷𝑵

𝑛1

𝑛2

𝜃2

𝜃1

O. Alexandrov, Wikipedia


Dielectrics

𝑛1𝑛2𝑠𝑖𝑛𝜃1 = 𝑠𝑖𝑛𝜃2⟺ 𝑠𝑖𝑛𝜃1 ≤

𝑛2

𝑛1

𝑘 = 1 −𝑛1𝑛2

2

1− 𝑐𝑜𝑠𝜃12

𝑇 =

𝑇𝐼𝑅, 𝑓𝑜𝑟 𝑘 < 0

𝑛1𝑛2𝐷 + 𝑁

𝑛1𝑛2𝑐𝑜𝑠𝜃1 − 𝑘 , 𝑓𝑜𝑟 𝑘 ≥ 0

Note: 𝑐𝑜𝑠𝜃1 = 𝑁 ∙ −𝐷, and 𝑛1𝑛2

should be calculated only once.

* For a full derivation, see http://www.flipcode.com/archives/reflection_transmission.pdf

Whitted

𝑇

𝐷𝑵

𝑛1

𝑛2

𝜃2

𝜃1


Dielectrics

A typical dielectric transmits and reflects light.

Whitted

𝑇

𝐷𝑵𝑅


Dielectrics

A typical dielectric transmits and reflects light.

Based on the Fresnel equations, the reflectivity of the surface for non-polarized light is formulated as*:

𝐹𝑟 =1

2

𝑛1𝑐𝑜𝑠𝜃𝑖 − 𝑛2 1 −𝑛1𝑛2𝑠𝑖𝑛𝜃𝑖

2

𝑛1𝑐𝑜𝑠𝜃𝑖 + 𝑛2 1 −𝑛1𝑛2𝑠𝑖𝑛𝜃𝑖

2

+

𝑛1𝑐𝑜𝑠𝜃𝑖 − 𝑛2 1 −𝑛1𝑛2𝑠𝑖𝑛𝜃𝑖

2

𝑛1𝑐𝑜𝑠𝜃𝑖 + 𝑛2 1 −𝑛1𝑛2𝑠𝑖𝑛𝜃𝑖

2

* See “Physically Based Lighting Calculations for Computer Graphics”, Shirley, 1985: page 12-18.

Whitted


Dielectrics

In practice, we use Schlick’s approximation:

𝐹𝑟 = 𝑅0 + (1 − 𝑅0)(1 − 𝑐𝑜𝑠𝜃)5, where 𝑅0 =

𝑛1−𝑛2𝑛1+𝑛2

2.

Based on the law of conservation of energy:

𝐹𝑡 = 1− 𝐹𝑟

* An Inexpensive BRDF model for Physically-based Rendering, Schlick, Computer Graphics Forum 13, 1994.

Whitted

𝑇

𝐷𝑵𝑅


Ray Tracing

World space

Geometry Eye Screen plane Screen pixels Primary rays Intersections Point light Shadow rays

Light transport

Extension rays

Light transport


Whitted

Ray Tree

Using Whitted-style ray tracing, hitting a surface point may spawn:

a shadow ray for each light source; a reflection ray; a ray transmitted into the material.

The reflected and transmitted rays may hit another object with the same material.

A single primary ray may lead to a very large number of ray queries.


Whitted

Question 6: imagine a scene with several point lights and dielectric materials. Considering the law of conservation of energy, what can you say about the energy transported by each individual ray?


Whitted

Beer’s Law


Whitted

Beer’s Law

Light travelling through a medium loses intensity due to absorption.

The intensity 𝐼(𝑠) that remains after travelling 𝑠 units through a substance with absorption 𝑎 is:

𝐼 𝑠 = 𝐼 0 𝑒− ln 𝑎 𝑠

In pseudocode:

I.r *= expf( -a.r * s );I.g *= expf( -a.g * s );I.b *= expf( -a.b * s );


Whitted

Whitted - Summary

A Whitted-style ray tracer implements the following optical phenomena:

Direct illumination of multiple light sources, taking into account

VisibilityDistance attenuationA shading model: N dot L for diffuse

Pure specular reflections, with recursion

Dielectrics, with Fresnel, with recursion

Beer’s Law

The ray tracer supports any primitive for which a ray/primitive intersection can be determined.


Whitted

Today’s Agenda:

Advanced Graphics

Recap: Ray Tracing

Whitted-style

Assignment 1

Ray Tracing Testbed

Implement an experimentation framework for ray tracing. Ingredients:

Scene

Primitives: spheres, planes, trianglesI/O (e.g., obj loader)IntersectionMaterials: diffuse color, diffuse / specular / dielectric, absorption

Camera

Position, target, FOVRay generation

Ray

Renderer

Whitted-style

User interface

Input handlingPresentation


Assignment 1

Ray Tracing Testbed

Regarding the OBJ file loading requirement:

You may want to start with handcrafted scenes Only roll your own OBJ loader if it is really necessary Use assimp or tinyobjloader (C++) or find a lib if you’re using C#

http://www.stefangordon.com/parsing-wavefront-obj-files-in-c/http://www.rexcardan.com/2014/10/read-obj-file-in-c-in-just-10-lines-of-code/https://github.com/ChrisJansson/ObjLoader

Start with small files; minimize your development cycle.


Assignment 1

http://www.stefangordon.com/parsing-wavefront-obj-files-in-c/

http://www.rexcardan.com/2014/10/read-obj-file-in-c-in-just-10-lines-of-code/

https://github.com/ChrisJansson/ObjLoader

Ray Tracing Testbed

Intersecting triangles:

An easy to implement and quite efficient algorithm is:

Fast, Minimum Storage Ray/Triangle Intersection, Möller & Trumbore, 1997.

…which is explained in elaborate detail by scratchapixel.com:

http://www.scratchapixel.com/lessons/3d-basic-rendering/ray-tracing-rendering-a-triangle/moller-trumbore-ray-triangle-intersection


Assignment 1

http://www.scratchapixel.com/lessons/3d-basic-rendering/ray-tracing-rendering-a-triangle/moller-trumbore-ray-triangle-intersection

INFOMAGR – Advanced GraphicsJacco Bikker - February – April 2016

END of “Introduction”next lecture: “Acceleration Structures”

jacco bikker - february april 2016 - utrecht university 01 - intro and whitted.pdf · abstract in...

Documents