Reflectance 1
Lights, Surfaces, and Cameras Light sources emit photons Surfaces reflect & absorb photons Cameras measure photons 2
Light at Surfaces Many effects when light strikes a surface -- could be: Absorbed, reflected, scattered, and travel along the surface and leave at some other points 3
Reflectance Reflectance is all about the way light interacts with surfaces It is an entire field of study on its own The most important quantity is the BRDF 4
The BRDF Assumption: No fluoresce, light emission All the light leaving a point depends ONLY on the light arriving at that point ω o n ω i ρ ( ω ) o, ω i Bi-directional Reflectance Distribution Function (BRDF) 5
6 Fluorescence
The BRDF Given the incoming ray, it describes how reflected lights are distributed This distribution changes when the incoming ray changes 7 plot a BRDF as a function of ω o for a fixed ω i ; the radius along each direction is set to the radiance of the reflected light at that direction.
Questions? 8
Rendering Equation ω i θ ω r ω o 9 L Reflected Radiance (Pixel Intensity) o ( ωo) = ρbd ( ωo, ωi, ) Li ( ωi) cosθi BRDF Incident Cosine of radiance Incident angle (from light source)
Rendering Equation θ ω r 10 L Reflected Radiance (Pixel Intensity) o Sum over all light sources ( ω ) ρ ( ω, ω,) L ( ω ) o = i bd BRDF o i i i cosθ Incident Cosine of radiance Incident angle (from light source) i
Rendering Equation dω i θ ω r 11 L Reflected Radiance (Pixel Intensity) o Replace sum with integral ( ω ) o ρbd ( ωo, ωi, ) Li ( ωi) = cosθidωi Ω BRDF Incident radiance (from light source) Cosine of Incident angle
Radiometric Image Analysis illumination reflectance shape the rendering equation: determine pixel values from shape, lighting, or reflectance the radiometric image analysis: recover shape, lighting, or reflectance from intensity values 12
Radiometric Image Analysis Typical simplification assumptions: Single point light source No inter-reflection Simplified BRDF models (more details later) ω i θ ω r ω o L o ( ωo) = ρbd ( ωo, ωi, ) Li ( ωi) cosθi 13
Questions? 14
Diffuse & Specular Reflection Diffuse reflection: The surface look the same from all directions (many vision algorithms depend on this!) Matte surfaces Specular reflection: The surface look different from different directions (causes troubles to many vision algorithms) Shiny surfaces ideal diffuse reflection (e.g. walls) 15 ideal specular reflection (e.g. mirror) specular reflection (e.g. plastic, metal, porcelain)
Diffuse & Specular Reflection Diffuse reflection: has the same color as the object surface is unpolarized Specular reflection: has the same color as the light source has the same polarization as the light source 16 the original image diffuse reflection specular reflection
Questions? 17
Common BRDF Models Torrance-Sparrow Lambert s Model 18
Lambert s Model Empirical mathematic model for diffuse reflection Assume the BRDF is a constant ρ ( ω, o ωi) = ρ Observed Pixel intensity should be L o = L ρ cosθ = L ρν L i Features of this model: Named after Johann H. Lambert A pixel s brightness does not depend on viewing direction Brightness does depend on direction of illumination This is the model most commonly used in computer vision i i 19 Ν, L, V are unit vectors L, i L o are intensity of incoming and outgoing light
Lambert s Model plot a BRDF as a function of ω o for a fixed ω i 20
Questions? 21
Phong Model Mathematic model for specular reflection Assume light is concentrated on the mirrored direction Intensity of light falls off by cosine law Observed Pixel intensity should be Features of this model: Named after Bui Tuong Phong A pixel s brightness depends on viewing direction This is an empirical model, not physically correct! (e.g. violate energy conservation) 22 L = ( V R) R r L i = 2 n ( N L) N L
Phong Model Shininess n controls the size of the highlight spot n cos σ n=1 n=2 n=3 n=10 23 90 0 90
Phong Model Linear combination of Lambert s model and Phong Model L = k Ν L + k V R k s 0.1 o d s ( ) n 0.25 0.5 24 n = 3.0 n = 5. 0 n =10. 0 n = 27. 0 n = 200. 0
Phong Model plot a BRDF as a function of ω o for a fixed ω i 25
Phong Model plot a BRDF as a function of ω o for a fixed ω i 26
Cook-Torrance Model Mathematic model for specular reflection Assume the surface consists of small facets Derive the reflectance as the overall behavior of facets Features of this model: Named after Robert Cook and Kenneth Torrance Also known as Torrance-Sparrow model Assume surface consists of microfacets Assume each facet is a perfect mirror 27
Cook-Torrance Model 28
Questions? 29
Measuring the BRDF traditional Capture the BRDF by moving a camera + light source Need careful control of illumination, environment 30
Acquisition A spherically homogeneous sample of the material can make the acquisition simpler A sphere contains all kinds of normals So a single image contains many BRDF samples lamp controlled turntable sample Digital camera 31
Represent Measured Data Measure-then-fit analytic models Fitting can reduce noise but also is limited by the model Non-obvious error metric for fitting often biased to specular which has large values Difficult optimization nonlinear; depends on initial guess Tabulated BRDF 4D table Not editable 32
Acquisition 130 materials were scanned; 100 of them shown here MERL BRDF database, freely available online 33
Tabulated BRDF nickel hematite gold paint pink fabric 34
Questions? 35
PAMI 2005 Example-based Photometric Stereo 36 Aaron Hertzmann University of Toronto Steven M. Seitz University of Washington
Shiny things 37 Orientation consistency : points of similar orientation have similar intensity
38 One to many mapping. Normal at this pixel cannot be determined by referring to the sphere
Let s get multiple images. 39
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Virtual views 44
Velvet 45
Virtual Views 46
Brushed Fur 47
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Questions? 50
51 ( ) ( ) i o i o i i o o ϕ ϕ θ θ ρ ϕ θ ϕ θ ρ =,,,,, ( ) ( ) ( ),,,,,, o i i o o i i o i o i o ϕ ϕ θ θ ρ ϕ ϕ θ θ ρ ϕ ϕ θ θ ρ = = BRDF Symmetries Most of real materials are isotropic
Examples ISOTROPIC ANISOTROPIC [Westin et al. 92] 52