BRDFs. Steve Rotenberg CSE168: Rendering Algorithms UCSD, Spring 2017

Transcription

1 BRDFs Steve Rotenberg CSE168: Rendering Algorithms UCSD, Spring 2017

2 The Rendering Equation

3 Radiance Radiance is a measure of the quantity of light radiation reflected (and/or emitted) from a surface within a given solid angle in a specified direction It is typically measured in units of W/(sr m 2 ) or Watts per steradian meter squared (Note there are 4π steradians in a sphere, just like there are 2π radians in a circle) It is the property we usually associate as light intensity and is what we measure when we are computing the color to display Technically, radiance refers to a single wavelength, and spectral radiance refers to the radiance across a spectrum Spectral radiance is what we associate as the perceived color (including the intensity), and is what we use to determine the pixel color

4 Area of a Hemisphere We can compute the area of a hemisphere by integrating rings ranging over a second angle θ, which ranges from 0 (at the north pole) to π/2 (at the equator) The area of a ring is the circumference of the ring times the width rdθ The circumference is going to be scaled by sinθ as the rings are smaller towards the top of the hemisphere area = 0 π 2 2π sin θ r 2 dφ 0 dθ = 2πr 2

5 Irradiance Now, let s assume that we are interested in computing the total amount of light arriving at some point on a surface We are essentially integrating over all possible directions in a unit hemisphere above the surface (assuming it s opaque and we re not receiving translucent light from below the surface) We are integrating the light intensity (or radiance) coming in from every direction The total incoming radiance over all directions in the hemisphere is called the irradiance We will say that the incoming radiance from any one direction is a function of the direction: L i θ, φ The incoming radiance from each direction is scaled by cosθ to account for the fact that light coming in from a horizontal direction is more spread out than light coming from directly above

6 Irradiance The area of a unit hemisphere is area = 0 π 2 2π sin θ dφ 0 dθ We can compute the irradiance as an integral of the incoming radiance L i over the hemisphere of incoming light directions: irrad = 0 π 2 2π L i θ, φ cos θ sin θ dφ 0 dθ

7 Irradiance irrad = 0 π 2 2π L i θ, φ cos θ sin θ dφ 0 dθ We can simplify the notation a bit. Instead of writing it as a double integral over two angles, we can write it as a single integral over the domain Ω of all possible directions ω irrad = Ω L i ω cos θ dω

8 Irradiance irrad = Ω L i ω cos θ dω ω is a unit vector for some particular direction. To be consistent with convention, these vectors will always point away from the surface regardless of whether they represent incoming of outgoing light Ω is the hemisphere domain, which can be thought of as a (infinite) set of all possible directions in a hemisphere dω is an infinitesimal area element (around direction ω) on the surface of the hemisphere. We don t really care about it s exact shape, just that it represents an infinitesimal portion of the total surface area We know that the surface area of a unit hemisphere is 2π, so by definition: area = Ω dω = 2π

9 BRDFs A bidirectional reflectance distribution function is a four dimensional function that describes how light is reflected off a surface It relates light coming from the incident direction ω i to light reflected in some outward direction ω r f r ω i, ω r Note that because the two vectors are unit length direction vectors, they really only have two independent variables each (the two polar angles describing the direction). Sometimes, BRDFs are written like this: f r θ i, φ i, θ r, φ r

10 Reflected Radiance Let s say that L i ω i is a function that measures the incident radiance from direction ω i When this hits a surface with normal n, this gets scaled by cos θ i to account for the light getting spread out at shallower angles (Note that cos θ i = ω i n) If a beam of light is shining on the surface from one particular direction ω i, then the reflected radiance we see coming off in direction ω r is f r ω i, ω r L i ω i cos θ i

11 Radiance Equation The reflected radiance in direction ω r from a single direction of incoming light ω i is: f r ω i, ω r L i ω i cos θ i We can compute the total reflected radiance L r in direction ω r from the integral of the reflected radiance from every direction of incoming light L r ω r = f r ω i, ω r L i ω i cos θ i dω i Ω This is the radiance equation

12 Rendering Equation The radiance equation calculates the total light reflected off a surface in direction ω r, from all incoming light across the hemisphere L r ω r = f r ω i, ω r L i ω i cos θ i dω i Ω If we include the possibility that the surface is also emitting light, then the total radiance out from the surface in direction ω r is: L out = L emitted + L reflected L ω r = L e ω r + L r ω r This is the rendering equation L ω r = L e ω r + f r ω i, ω r L i ω i cos θ i dω i Ω

13 Rendering Equation L ω r = L e ω r + f r ω i, ω r L i ω i cos θ i dω i Ω When we render an image, we are attempting to evaluate the rendering equation for every pixel in the image We trace a ray from the camera and hit a surface. We then want to calculate the radiance L coming off of that surface going back towards the camera in direction ω r (the opposite of the ray direction)

14 BRDFs

15 BRDF Properties There are two important properties that are required for BRDFs to be physically valid: Conservation of Energy Reciprocity

16 Conservation of Energy Conservation of energy is a fundamental principle in physics It says that energy is never created or destroyed- only changed from one form to another With respect to light scattering from materials, it tells us that the total light reflected off of the material must be less than the light incident on the material ω i, Ω f r ω i, ω r cos θ r d ω r 1

17 Reciprocity Hermann von Helmholtz ( ) was a German physicist who made many contributions to optics, theory of vision, color, thermodynamics, electrodynamics and more In a famous treatise on physiological optics, published in 1856, he formulated the Helmholtz reciprocity principle The principle states that if we follow a beam of light on any path through an optical system, the loss of intensity is the same as for a beam traveling in the reverse direction f r ω i, ω r = f r ω r, ω i

18 Isotropic & Anisotropic A BRDF can be either isotropic or anisotropic An isotropic BRDF doesn t change with respect to surface rotation about the normal Brushed metals and animal fur are good examples of anisotropic materials that do change with respect to orientation For an isotropic BRDF, there is no fixed frame of reference for the φ angles. The absolute angles are irrelevant, but the difference between the two is still important. This effectively reduces the isotropic BRDF to a 3 dimensional function: f r θ i, θ r, φ i φ r

19 Fundamental BRDFs

20 Idealized Surfaces The most fundamental of the physically based BRDFs are the idealized Fresnel specular and Lambert diffuse models Fresnel metal Fresnel dielectric Lambert diffuse

21 Fresnel Metal The Fresnel metal BRDF is 0 everywhere except in the direction of perfect specular reflection, which happens when ω r = reflect(ω i,n)= (ω i n)n-2ω i In that case, the reflectivity is based on the Fresnel equation for metals The reflectivity varies based on the wavelength of the light leading to coloring of the reflection (especially for copper and gold) The coloring can be modeled based on actual optical properties of the metals or can be approximated as just constant colors multiplied times the Fresnel equation

22 Fresnel Dielectric Fresnel dielectrics are clear and split an incident ray into a reflected ray and a refracted ray The Fresnel dielectric is actually a BSDF, since it contains reflecting and transmitting components It is 0 everywhere except where ω r is in the direction of perfect reflection or refraction

23 Lambert Diffuse The Lambert diffuse BRDF can be written as: f r ω i, ω r = c diff π where c diff is the diffuse color Note that the BRDF itself is constant. The cosine term usually associated with Lambert reflection comes from the cosine term in the radiance equation itself

24 Microfacet Models The Fresnel metal, dielectric, and Lambert BRDFs are all based on the assumption of a perfectly smooth surface These have been extended to roughened surfaces made up of microfacets, where each microfacet is an ideal Fresnel or Lambert reflector The roughened models typically account for shadowing and masking effects between microfacets, as well as interreflections between them

25 Cook-Torrance BRDF The Cook-Torrance (1981) model extends the Fresnel metal model to handle rough metal surfaces The reflectivity is modeled as the product of three terms: Fresnel F(), Geometry G(), and Distribution D() f r ω i, ω r = F ω i h G ω i, ω r D h n 4 ω i n ω r n where h is the halfway vector between ω i and ω r

26 Oren-Nayar BRDF The Oren-Nayar (1994) BRDF takes the ideas from Cook-Torrance and applies them to perfectly diffuse microfacets The resulting model is a more realistic diffuse model that accounts for backscattering due to the opposition effect

27 Oren-Nayar Reflectance Model f r ω i, ω r = ρ π A + B max 0, cos φ i φ r sin α tan β where A = B = 0.45 σ 2 σ σ 2 σ α = max θ i, θ r β = min θ i, θ r σ = roughness (ranges from 0 to around 0.5) φ i φ r =angle between incident and reflected rays projected onto the plane

28 Walter BSDF The BSDF of Walter, Marschner, Li, and Torrance (2007) applies the same concepts to microfacets made of perfect Fresnel dielectrics It is a BSDF (scattering) because it accounts for both reflection and transmission They also explore the use of various different expressions for microfacet distributions D() and geometric attenuation G()

29 Fundamental BRDFs Diffuse Metal Dielectric Ideal (smooth) Lambert Fresnel metal Fresnel dielectric Microfacet (rough) Oren-Nayar Cook-Torrance Walter

30 Empirical BRDFs

31 Empirical Models The fundamental BRDFs that we looked at so far are all based on real physics to varying degrees of approximation There are also several popular BRDFs that are based on empirical formulations that attempt to model the resulting visual phenomena rather than the underlying causes Some examples include: Phong Ward Schlick Ashikhman

32 Phong The Phong model (1971) is a classic empirical model in computer graphics It is still widely in use due to its simplicity and intuitive control When the research focus shifted heavily towards physically based rendering, it was observed that the classic Phong model was not physically valid A modern Phong BRDF was developed that is a minor modification of the original in order to enforce reciprocity and conservation of energy

33 Schlick Christophe Schlick created An Inexpensive BRDF Model for Physically-based Rendering in 1994 It makes some approximations to the Fresnel equation and uses some other fast empirical functions to model diffuse, specular, and a simple layered surface mode It was also designed with sampling in mind and has appropriate functions for generating sample ray directions

34 Ashikhmin In 2000, Michael Ashikhmin and Peter Shirley published a BRDF that remains popular today It is an empirical BRDF intended to model general purpose surfaces from diffuse to plastics to metals They listed 6 main goals for the BRDF: Obey conservation of energy and reciprocity Allow anisotropic reflection Controlled by intuitive parameters Accounts for Fresnel effect Non-constant diffuse term (diffuse decreases as specularity increases) Well suited to Monte Carlo methods

35 Ashikhmin BRDF ρ k 1, k 2 = ρ s k 1, k 2 + ρ d k 1, k 2 ρ s k 1, k 2 = n u + 1 n v + 1 8π n h n ucos 2 φ+n v sin 2 φ h k max n k 1, n k 2 F k h F k h = R s + 1 R s 1 k h 5 ρ d k 1, k 2 = 28R d 23π 1 R s 1 1 n k n k 2 2 5

36 Ashikhmin

37 Ashikhmin One of the nicest features of the Ashikhmin BRDF is the effectiveness of the sampling functions

38 Ashikhmin BRDF

39 Direct Light Only

40 Advanced BRDF Concepts

41 Layered Models Many real surfaces are built up of multiple layers of primers, pigments, scattering particles, and clear coatings There have been various attempts to develop BRDFs that account for the interaction of light between these different layers One early model was Reflection from Layered Surfaces due to Subsurface Scattering by Pat Hanrahan and Wolfgang Krueger

42 Multiscale BRDFs

43 Measured BRDFs It is possible to measure a BRDF from a real world material by using a special tool called a gonioreflectometer It takes a sample of the material and illuminates it with a light source and measures the reflection with a light sensor Both the light source and sensor can move to various angles around the sample to measure the BRDF across the whole range of incident and reflected angles

44 Virtual Gonioreflectometer It is also possible to simulate a virtual gonioreflectometer and build BRDF models from actual modeled microgeometry A sample of the surface is modeled with actual geometry Basic material properties are assigned to the microsurfaces such as Fresnel reflection/refraction, particle scattering, absorption, and diffuse reflection Millions of random rays are shot at the material and then bounced around within the material The rays that escape off the surface are measured and contribute to the final BRDF

45 Virtual Gonioreflectometer

46 Westin, Arvo, Torrance A classic paper on multiscale BRDFs and virtual gonioreflectometers is Predicting Reflectance Functions from Complex Surfaces by Stephen Westin, James Arvo, and Kenneth Torrance from 1992

47 Wavelength Dependence Jay Gondek, Gary Meyer, and Jonathan Newman wrote a paper called Wavelength Dependent Reflectance Functions that used a virtual gonioreflectometer to model complex materials They account for wavelength dependent properties such as interference that lead to iridescent effects

48 BRDF Representations Imagine if you were measuring a real material BRDF or simulating one with a virtual setup If you want reasonable resolution, you probably want to sample everything at 1 degree increments or so For an isotropic BRDF, this means sampling two variables over 90 degrees and one over 180 degrees, leading to 90x90x180=1,458,000 data points An anisotropic BRDF would add another dimension of 180, raising it to 262 million data points Various hierarchical, multidimensional data structures have been devised to store and compress BRDF data, and various mathematical tools have been developed to represent and manipulate the data

49 Midterm

50 Midterm The midterm will be in class on Wednesday There will be 10 questions, and it is worth 10% of your grade Topics include everything we ve discussed so far Guidelines: Not very math heavy, but be familiar with core vector & matrix math Pay attention to any terms in italics that are defined in the lecture notes The test will focus more on concepts and understanding In situations where multiple options are discussed (such as BRDF types, data structure types, number sequence types, etc.), be prepared to compare and contrast the different options

51 Camera & Scene Cameras Camera ray generation Look-at function Objects & Scene Objects & ray intersections Instancing logic Spatial data structures

52 Intersections Ray equation: r(t)=p+td Ray-sphere concepts Ray-plane math: d = q n = (p+td) n = p n + td n t = (d-p n) / (d n) Ray-box concepts Ray-triangle concepts Barycentric coordinates: q = a + α(b-a) + β(c-a)

53 Spatial Data Structures Render performance Hierarchical data structures Tree construction Ray traversal BVH vs. spatial partitions AABB, sphere tree, K-Tree, BSP, nested grid, octree Scene graph & animation issues

54 Fresnel Dielectric behavior & concepts Snells law Total internal reflection Metal behavior Concept and trends in Fresnel equation Fresnel effect Beer-Lambert law concepts Recursive ray tracing

55 Materials Diffuse material behavior Microgeometry Shadowing/masking Distribution Opposition effect Oren-Nayar model Cook-Torrance model Isotropic vs. anisotropic

56 Shadows & Area Lights Umbra & penumbra Area light sampling Solid angles

57 Antialiasing Aliasing problems & their causes: shimmering, Moiré, stairstepping, strobing Signals, sampling, & reconstruction concepts Low/high frequency signals Pixel sampling (uniform, jittered, random, weighted )

58 Texture Wrap modes Sampling modes Minification / magnification EWA sampling Normal & displacement mapping concepts

59 Random Sampling Random & jittered numbers Quasi-random numbers Concepts behind mappings

60 BRDFs Bidirectional reflectance distribution function Physical validity: ω Conservation of energy i, f r ω i, ω r cos θ r d Ω Reciprocity f r ω i, ω r = f r ω r, ω i Isotropic vs. anisotropic Qualities of different BRDF models BRDF sampling Radiance equation ω r 1 L r ω r = f r ω i, ω r L i ω i cos θ i dω i Ω

61 Path Tracing Path tracing Monte Carlo integration Rendering equation Local/global illumination