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Reflection Models and Radiometry Remote Sensing, Etc 12/02/2017 Applications To render realistic looking materials Applications also in computer vision, optical engineering, Reflection models and radiometry remote sensing, etc. To understand Advanced Graphics how surfaces reflect light Rafał Mantiuk Computer Laboratory, University of Cambridge Last year you learned about Phong Applications reflection model Many applications require faithful reproduction of material appearance Source: http://ikea.com/ Good for plastic objects, not very accurate for other materials Source: http://www.mercedes-benz.co.uk/ Pictures courtasy of MIT, Lecture Notes 6.837 Rendering equation Radiance Most rendering methods require solving an (approximation) of Power of light per unit projected area per unit solid angle the rendering equation: Integral over the Symbol: 퐿(풙, 흎 ) Incident light hemisphere of 푊 incident light Units: Steradian 푚2푠푟 퐿푟 흎풓 = 휌 흎풊, 흎풓 퐿 휔 푐표푠휃푑흎풊 Ω BRDF Reflected light The solution is trivial for point light sources Much harder to estimate 흎풓 the contribution of other surfaces 흎풊 = 휙푖, 휃푖 1 12/02/2017 Solid angle Radiance Power of light Angle in 2D Equivalent in 3D Position 푑휙 퐿 풙, 흎 = 풊 푑휔 푑퐴 푐표푠휃 휃 Incoming direction Projected area 퐿 퐴 Power per solid angle per projected surface area 휃 = [푟푎푑] 휔 = [푠푟] 푅 푅2 Invariant along the direction of propagation (in vacuum) Response of a camera sensor or a human eye is related to radiance Pixel values in image are related to radiance (projected Full circle = 2π radiance Full sphere = 4 π steradians along the view direction) Relation between Irradiance and Irradiance and Exitance Radiance Power per unit area Irradiance is an integral over all incoming rays Irradiance: H(x) – incident power per unit area Integration over a hemisphere Ω: Exitance / radiosity: E(x) – exitant power per unit area 퐻 = 퐿 풙, 흎풊 푐표푠휃 푑휔 Units: 푊 Ω In the spherical coordinate system, the differential solid angle 푚2 is: Irradiance Exitance / Radiosity 푑휔 = sin 휃푑휃 푑휙 Therefore: 휋 2휋 2 퐻 = 퐿 풙, 흎풊 푐표푠휃 푠푛휃 푑휃 푑휙 휙=0 휃=0 For constant radiance: 퐻 = 휋퐿 BRDF: Bidirectional Reflectance Distribution Function BRDF of various materials Reflected light Differential radiance of reflected light Incident light 푑퐿푟(흎풓) 푑퐿푟(흎풓) The diagrams show 휌 흎풊, 흎풓 = = the distribution of 푑퐻(흎풊) 퐿 흎풊 푐표푠휃푑흎풊 reflected light for the Aluminium; λ=2.0μm Source: Wikipedia given incoming Differential irradiance of incoming light direction The material samples Aluminium; λ=0.5μm are close but not BRDF is measured as a ratio of reflected radiance to irradiance accurate matches for the diagrams Because it is difficult to measure 퐿 흎풊 , BRDF is not defined as the ratio 퐿푟 흎풓 /퐿 흎풊 Magnesium alloy; λ=0.5μm 2 12/02/2017 Other material models Sub-surface scattering Source: Guarnera et al. 2016 Light enters material and is scattered several times before it exits Examples - human skin: hold a flashlight next to your hand and see the color of the light The effect is expensive to compute Bidirectional Scattering Surface Reflectance Distribution F. But approximate methods exist Bidirectional Reflectance Distribution Function Bidirectional Transfer Distribution Function But also: BTF, SVBRDF, BSDF In this lecture we will focus on BRDF only Subsurface scattering - examples BRDF Properties Helmholtz reciprocity principle BRDF remains unchanged if incident and reflected directions are interchanged 휌 흎풓, 흎풊 = 휌(흎풊, 흎풓) Smooth surface: isotropic BRDF reflectivity independent of rotation around surface normal BRDF has only 3 instead of 4 directional degrees of freedom 휌(휃푖, 휃푟, 휙푟 − 휙푖) BRDF Properties BRDF Measurement Characteristics Gonio-Reflectometer BRDF units [1/sr] BRDF measurement Not intuitive point light source position (,) Range of values: light detector position (o ,o) From 0 (absorption) to (reflection, δ -function) 4 directional degrees of freedom Energy conservation law BRDF representation 휌 휔푟, 휔푖 푐표푠휃푖푑휔푖 ≤ 1 – m incident direction samples (,) Ω – n outgoing direction samples (o ,o) No self-emission – m*n reflectance values (large!!!) Possible absorption Reflection only at the point of entry (xi = xr) No subsurface scattering Stanford light gantry 3 12/02/2017 BRDF Modeling Diffuse Reflection It is common to split BRDF into diffuse, mirror and glossy components Light equally likely to be reflected in any outgoing Ideal diffuse reflection Lambert’s law direction (independent of incident direction) Matte surfaces Constant BRDF 휌 휔푟, 휔푖 = 푘푑 = 푐표푛푠푡 Ideal specular reflection Reflection law Mirror 퐿푟 휔푟 = 푘푑퐿푖 휔푖 푐표푠휃푖푑휔푖 = 푘푑 퐿푖 휔푖 cos 휃푖푑휔푖 = 푘푑퐻 Glossy reflection Ω Ω Directional diffuse k : diffuse coefficient, material property [1/sr] Shiny surfaces d I N Lr= const Diffuse reflection Reflection Geometry N Cosine term Direction vectors (all normalized): -(I•N)N R(I) The surface receives N: surface normal less light per unit area I: vector to the light source I -(I•N)N at steep angles of V: viewpoint direction vector incidence H: halfway vector H= (I + V) / |I + V| I Rough, irregular R(I): reflection vector R(I)= I - 2(I•N)N Top view surface results in R(V) diffuse reflection Tangential surface: local plane H N Light is reflected in N R(I) I R(I) random direction I R(V) (this is just a model, V V light interaction is H more complicated) Glossy Reflection Glossy Reflection Due to surface roughness Empirical models Phong Blinn-Phong Physical models Blinn Cook & Torrance 4 12/02/2017 Phong Reflection Model Phong Exponent ke 푘푒 Cosine power lobe 휌 휔푟, 휔푖 = 푘푠 푅 ∙ 푉 푘푒 휌 휔푟, 휔푖 = 푘푠 푅 ∙ 푉 Determines the size of highlight Dot product & power Not energy conserving/reciprocal Plastic-like appearance N R(I) I V Blinn-Phong Reflection Model Phong Illumination Model Blinn-Phong reflection model Extended light sources: l point light sources 푘푒 휌 휔푟, 휔푖 = 푘푠 퐻 ∙ 푁 ke LightLr source,kaLi, aviewerkd farL awayl(IlN)ksLl(R(Il)V)(Phong) l l I, R constant: H constant ke Colour of specular reflection equal to light source H lessLr expensivekaLi,a tok computedLl(IlN)ksLl(HlN)(Blinn) l l Heuristic model Contradicts physics H N Purely local illumination I Only direct light from the light sources V No further reflection on other surfaces Constant ambient term Often: light sources & viewer assumed to be far away Micro-facet model Ward Reflection Model We can assume that at small scale materials BRDF ∠(퐻, 푁) are made up of small facets exp − tan2 푘푑 1 휎2 휌 = + 푘푠 Facets can be described by the distribution of 휋 (퐼 ∙ 푁)(푉 ∙ 푁) 2휋휎2 their sizes and directions D Rendering of glittery materials Some facets are occluded by other, hence there is [Jakob et al. 2014] also a geometrical attenuation term G standard deviation (RMS) of surface slope And we need to account for Fresnel reflection (see next slides) Simple expansion to anisotropic model (x, y) 퐷 ∙ 퐺 ∙ 퐹 Empirical, not physics-based 휌 휔 , 휔 = 푖 푟 4 cos 휃 휃 푖 푟 Inspired by notion of reflecting microfacets Convincing results Good match to measured data 5 12/02/2017 Cook-Torrance model Fresnel term Can model metals and dielectrics Sum of diffuse and specular components 휌 휔푖, 휔푟 = 휌푑 휔푖 + 휌푠(휔푖, 휔푟) Specular component: 퐷 ℎ 퐺 휔푖, 휔푟 퐹(휔푖) 휌 휔푖, 휔푟 = 휋 cos 휃푖 cos 휃푟 훼 2 − Distribution of microfacet orientations: 퐷 ℎ = cos 휃푟 푒 푚 The light is more likely to be Example from: https://www.scratchapixel.com/lessons/3d- basic-rendering/introduction-to-shading/reflection-refraction- Roughness reflected rather than fresnel Geometrical attenuation factor parameter transmitted near grazing angles To account to self-masking and shadowning The effect is modelled by Fresnel equation: it gives the 2(푛 ∙ ℎ)(푛 ∙ 휔푟) 2(푛 ∙ ℎ)(푛 ∙ 휔푖) probability that a photon is reflected rather than 퐺 휔푖, 휔푟 = 푚푛 1, , 휔푟 ∙ ℎ 휔푟 ∙ ℎ transmitted (or absorbed) Fresnel equations Fresnel term Reflectance for s-polarized light: In Computer Graphics the Fresnel equation is approximated by Schlick’s formula [Schlick, 94]: 5 푅 휃, 휆 = 푅0 휆 + 1 − 푅0 휆 1 − 푐표푠휃 Reflectance for p-polarized light: where 푅0 휆 is reflectance at normal incidence and 휆 is the wavelength of light For dielectrics (such as glass): 2 푛 휆 − 1 푅 휆 = 0 푛 휆 + 1 Which one is Phong / Cook-Torrance ? Image based lighting (IBL) 1. Capture an HDR image of a light probe 2. Create an 3. Use the illumination illumination (cube) map map as a The scene is source of light surrounded by a in the scene cube map 36 Source: Image-based lighting, Paul Debevec, HDR Symposium 2009 6 12/02/2017 37 38 Blender monkeys + IBL (path tracing) 39 Further reading A. Watt, 3D Computer Graphics Chapter 7: Simulating light-object interaction: local reflection models Eurographics 2016 tutorial D. Guarnera, G. C. Guarnera, A. Ghosh, C. Denk, and M. Glencross BRDF Representation and Acquisition DOI: 10.1111/cgf.12867 Some slides have been borrowed from Computer Graphics lecture by Hendrik Lensch http://resources.mpi- inf.mpg.de/departments/d4/teaching/ws200708/cg/slides/CG07-Brdf+Texture.pdf 7.
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