Photometric Image Formation CSE 559: Computer Vision Guest Lecturer: Austin Abrams Images/Demo from Steve Seitz, Wikipedia How are images made? • One half: geometric vision – “how the pixel projected onto the image” • Today: photometric vision (aka radiometric) – “how the pixel got its color” Vision and Graphics Computer Graphics Properties of a scene Image Vision Image Formation Approach • Come up with a model for how the scene was created • Given images, find the most likely properties that fit that model Diffuse Surfaces Brightness of a pixel depends on: • object color • lighting direction • surface normal But NOT view direction! Lambertian Cosine Law • The intensity of an observed diffuse object is proportional to the cosine of the angle between the normal and lighting direction L θ N I = ρ cos θ = ρ |L||N| cos θ =ρL N = LN = L N x = I = ρ L N Recovering Albedo and Normals • Can you decompose a single image into its albedo and normal images? x = x x Photometric Stereo • Given multiple images taken with varying illumination, recover albedo and normals. – take pictures in dark room with varying illumination. – estimate lighting directions L. – recover albedo and normals. Side note 1: How to get the lighting direction? • Put a shiny sphere in the scene • Sphere’s geometry (normals) are known • Find specular highlight Side-note 2: Why “Stereo”? Surface normals provide constraints on depth differences Photometric Stereo • If L is known, and albedo is grayscale this is a linear problem. I = ρ(L N) = ρ (Lx Nx + Ly Ny + Lz Nz ) = Lx Nxρ + Ly Nyρ + Lz Nzρ = Lx a + Ly b + Lz c I = ρ(L N) = L x a + Ly b + Lz c For each pixel: I1 I2 I3 … In = Lx1 Ly1 Lz1 Lx2 Ly2 Lz2 Lx3 Ly3 Lz3 … Lxn Lyn Lzn Then: ρ = sqrt(a2 + b2 + c2) N = (a,b,c) / ρ a b c Demo When does this model fail? I ≠ ρ (L N) Attached shadows L N=0 L L N<0 I = ρ max(L N, 0) N>0 Cast Shadows, Ambient Light I = ρ (S L N + a) S = 0 or 1 Radiometric Camera Calibration • Pixel intensities are usually not proportional to the energy that hit the CCD RAW image Published image Radiometric Camera Calibration Published f RAW Radiometric Camera Calibration Observed = f(RAW) (Grossberg and Nayar) f -1 (Observed) = RAW Radiometric Camera Calibration • How do you model f -1? f -1(x) = xγ f -1(x) = c0 + c1x + c2x2 + c3x3 + … f -1(x) = f0(x) + f1(x) c1 + f2(x)c2 + … mean camera curve basis camera curves Radiometric Camera Calibration I = f (ρ (S L N + a)) Adding exposure: I = f (e ρ (S L N + a)) Heliometric Stereo Given lots of images from a stable webcam, use lighting from the sun to recover: I = f (e ρ (S L N + a)) Heliometric Stereo Heliometric Stereo Heliometric Stereo