# lec9 - Photometric stereo Illumination Cones and...

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1 CSE252A Illumination Cones and Uncalibrated Photometric Stereo CS252A, Fall 2010 Computer Vision I Photometric stereo Single viewpoint, multiple images under different lighting. 1. Arbitrary known BRDF, known lighting 2. Lambertian BRDF, known lighting 3. Lambertian BRDF, unknown lighting. CS252A, Fall 2010 Computer Vision I Three Source Photometric stereo: Step1 Offline: Using source directions & BRDF, construct reflectance map for each light source direction. R 1 (p,q), R 2 (p,q), R 3 (p,q) Online: 1. Acquire three images with known light source directions. E 1 (x,y), E 2 (x,y), E 3 (x,y) 2. For each pixel location (x,y), find (p,q) as the intersection of the three curves R 1 (p,q)=E 1 (x,y) R 2 (p,q)=E 2 (x,y) R 3 (p,q)=E 3 (x,y) 3. This is the surface normal at pixel (x,y). Over image, the normal field is estimated CS252A, Fall 2010 Computer Vision I Reflectance Map of Lambertian Surface What does the intensity (Irradiance) of one pixel in one image tell us? (e..g, let’s say the Then, the normal lies on 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 R(p,q) CS252A, Fall 2010 Computer Vision I One viewpoint, two images, two light sources Two super imposed reflectance maps E measured E measured 1 A third image would disambiguate between two possible n R 1 (p,q) R 2 (p,q) CS252A, Fall 2010 Computer Vision I Recovering the surface f(x,y) Many methods: Simplest approach 1. From estimate n =(n x ,n y ,n z ), p=n x /n z , q=n y /n z 2. Integrate p=df/dx along a row (x,0) to get f(x,0) 3. Then integrate q=df/dy along each column starting with value of the first row f(x,0)

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2 CS252A, Fall 2010 Computer Vision I Lambertian Surface At image location (u,v), the intensity of a pixel x(u,v) is: e(u,v) = [a(u,v) n (u,v)] · [s 0 s ] = b (u,v) · s where a(u,v) is the albedo of the surface projecting to (u,v). n (u,v) is the direction of the surface normal. s 0 is the light source intensity. s is the direction to the light source. ^ n ^ s ^ ^ a e(u,v) CS252A, Fall 2010 Computer Vision I Lambertian Photometric stereo If the light sources s 1 , s 2 , and s 3 are known , then we can recover b at each pixel from as few as three images. (Photometric Stereo: Silver 80, Woodham81). [e 1 e 2 e 3 ] = b T [ s 1 s 2 s 3 ] i.e., we measure e 1 , e 2 , and e 3 and we know s 1 , s 2 , and s 3 . We can then solve for b by solving a linear system. Surface normal is: n = b /| b |, albedo is: | b | CSE252A What is the set of images of an object under all possible lighting conditions? In answering this question, we’ll arrive at a photometric setereo method for reconstructing surface shape w/ unknown lighting. CSE252A The Space of Images Consider an n-pixel image to be a point in an n- dimensional space, x R n . Each pixel value is a coordinate of x . x 1 x 2 x n Many results will apply to linear transformations of image space (e.g. filtered images) Other image representations (e.g. Cayley-Klein spaces, See Koenderink’s “pixel f#@king paper”) x 1 x n x 2 CSE252A Assumptions For discussion, we assume: Lambertian reflectance functions.
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• Winter '08
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• Singular value decomposition, light source, Illumination Cone

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