Announcement. Lighting and Photometric Stereo. Computer Vision I. Surface Reflectance Models. Lambertian (Diffuse) Surface.

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1 Lighting and Photometric Stereo CSE252A Lecture 7 Announcement Read Chapter 2 of Forsyth & Ponce Might find section of Forsyth & Ponce useful. HW Problem Emitted radiance in direction f r for incident radiance L i. (θ in,φ in ) ^ n (θ out,φ out ) ( ) = ρ x;θ in,φ in ;θ out,φ out ( ) L o x;θ out,φ out L i ( x;θ in,φ in )cosθ in dω L o ( x;θ out,φ out ) = ρ ( x;θ in,φ in ;θ out,φ out )L i ( x;θ in,φ in )cosθ in dω Where ρ is sometimes denoted f r. CSE 252A, Winter 2010 where ω i =(θ i, ϕ i ) Surface Reflectance Models Common Models Lambertian Phong Physics-based Specular [Blinn 1977], [Cook-Torrance 1982], [Ward 1992] Diffuse [Hanrahan, Kreuger 1993] Generalized Lambertian [Oren, Nayar 1995] Thoroughly Pitted Surfaces [Koenderink et al 1999] Phenomenological [Koenderink, Van Doorn 1996] Arbitrary Reflectance Non-parametric model Anisotropic Non-uniform over surface BRDF Measurement [Dana et al, 1999], [Marschner ] Specialized Hair, skin, threads, paper [Jensen et al] Lambertian (Diffuse) Surface BRDF is a constant called the albedo. ρ ( x;θ in,φ in ;θ out,φ out ) = ρ 0 Emitted radiance is NOT a function of the outgoing direction i.e. constant in all directions. For lighting coming in single direction ω ι, we can write L i (x,ω i ) = δ(ω-ω i ), and consequently the emitted radiance is proportional to cosine of the angle between normal and light direction L r = ρ 0 ˆ N ω i 1

2 Lambertian reflection Specular Reflection: Smooth Surface Lambertian Matte Diffuse N Light is radiated equally in all directions Light is emitted equally in all directions Rough Specular Surface Phong Lobe Non-Lambertian Reflectance General BRDF: e.g. Velvet [ After Koenderink et Computer al, 1998 Vision ] I 2

3 Three degrees of freedom spread among light source, detector, and/or sample Three degrees of freedom spread among light source, detector, and/or sample Can add fourth degree of freedom to measure anisotropic BRDFs Collect reflected light with hemispherical (should be ellipsoidal) mirror [SIGGRAPH 92] Result: each image captures light at all exitant angles For uniform BRDF, capture 2-D slice corresponding to variations in normals 3

4 Light Sources BRDF BSSRDF (Jensen, Marschner, Levoy, Hanrahan) CSE 252A Direction is a three vector s, with s = 1. Described as function on a sphere: radiance as a function of direction r(s) Single point source is a delta function at some direction Multiple point sources: sum of delta functions The sun illuminating the earth A source that is far enough away so that all light rays coming from the source can be treated as being parallel and of the same radiance. For a Lambertian surface, illuminated by a distant light source in direction S/ S and with strength S, image brightnes is: works because a ρo ( x ) N ( x ) S dot-product is basically a cosine ( N ) S CSE 252A CSE 252A N ( x ) S ( x ) ρ 0 ( x ) 2 r( x ) Order l=0 l=1 l=2... Green: Positive Blue: Negative CSE 252A m=-2 m=-1 m=0 m=1 N m=2 (Borrowed from: Ramamoorthi, Hanrahan, SIGGRAPH 01) S N is the surface normal ρo is the diffuse (Lambertian) albedo S is source vector - a vector from x to the source, whose length is the intensity term works because a dotproduct is basically a cosine CSE 252A 4

5 Examples: diffuser boxes, white walls. The radiosity at a point due to an area source is obtained by adding up the contribution over the section of view hemisphere subtended by the source change variables and add up over the source radiosity due to line source varies with inverse distance, if the source is long enough See Forsyth & Ponce or a graphics text for details. Conversely, the light emitted at a given point also is a function on a 2-D space (sphere) Conversely, the set of light rays emitted from all points From Leonard McMillan s, SIGGRAPH 99 course notes From Leonard McMillan s, SIGGRAPH 99 course notes A point that can t see the source is in shadow For point sources, the geometry is simple Cast Shadow Attached Shadow 1. Fully illuminated 2. Penumbra 3. Umbra (shadow) 5

6 Local shading model Surface has incident radiance due only to sources visible at each point Advantages: often easy to manipulate, expressions easy supports quite simple theories of how shape information can be extracted from shading Used in vision & real time graphics Global shading model surface radiosity is due to radiance reflected from other surfaces as well as from surfaces Advantages: usually very accurate Disadvantage: extremely difficult to infer anything from shading values Rarely used in vision, often in photorealistic graphics A view of a black room, under bright light. Below, we see a crosssection of the image intensity corresponding to the line drawn on the image. Figure from Mutual Illumination, by D.A. Forsyth and A.P. Zisserman, Proc. CVPR, 1989, copyright 1989 IEEE local shading model is a poor description of physical processes that give rise to images because surfaces reflect light onto one another This is a major nuisance; the distribution of light (in principle) depends on the configuration of every radiator; big distant ones are as important as small nearby ones (solid angle) The effects are easy to model It appears to be hard to extract information from these models At the top, geometry of a gutter with triangular cross-section; below, predicted radiosity solutions, scaled to lie on top of each other, for different albedos of the geometry. When albedo is close to zero, shading follows a local model; when it is close to one, there are substantial reflexes. Figure from Mutual Illumination, by D.A. Forsyth and A.P. Zisserman, Proc. CVPR, 1989, copyright 1989 IEEE Irradiance observed in an image of this geometry for a real white gutter. Photometric Stereo Figure from Mutual Illumination, by D.A. Forsyth and A.P. Zisserman, Proc. CVPR, 1989, copyright 1989 IEEE 6

7 Shading reveals 3-D surface geometry Two shape-from-x methods that use shading Shape-from-shading: Use just one image to recover shape. Requires knowledge of light source direction and BRDF everywhere. Too restrictive to be useful. Photometric stereo: Single viewpoint, multiple images under different lighting. 1. Arbitrary known BRDF 2. Lambertian BRDF, known lighting 3. Lambertian BRDF, unknown lighting. Photometric Stereo Rigs: One viewpoint, changing lighting Multi-view stereo vs. Photometric Stereo: Assumptions Multi-view (binocular) Stereo Multiple images Dynamic scene Multiple viewpoints Fixed lighting Photometric Stereo Multiple images Static scene Fixed viewpoint Multiple lighting conditions An example of photometric stereo Photometric Stereo: General BRDF and Reflectance Map 7

8 BRDF Coordinate system Bi-directional Reflectance Distribution Function ρ(θ in, φ in ; θ out, φ out ) f(x,y) Function of Incoming light direction: θ in, φ in Outgoing light direction: θ out, φ out Ratio of incident irradiance to emitted radiance (θ in,φ in ) ^ n (θ out,φ out ) y x Surface: s(x,y) =(x,y, f(x,y)) Normal vector Tangent vectors: n = s x s y = f x, f y, 1 Computer Vision I Gradient Space (p,q) n f(x,y) y x Gradient Space : (p,q) Normal vector 8

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