Announcements HW1 has been posted See links on web page for reading Introduction to Computer Vision CSE 152 Lecture 5 Coordinate Changes: Rigid Transformations both translation and rotatoin Rotation About (kx, ky, kz), a unit vector on an arbitrary axis (Rodrigues Formula) z y Rotate(k, θ) θ k x x' y' z' 1 = kxkx(1-c)+c kykx(1-c)+kzs kzkx(1-c)-kys kxky(1-c)-kzs kyky(1-c)+c kzky(1-c)-kxs kxkz(1-c)+kys kykz(1-c)-kxs kzkz(1-c)+c 1 x y z 1 where c = cos θ & s = sin θ Camera parameters Issue camera may not be at the origin, looking down the z-axis extrinsic parameters (Rigid Transformation) one unit in camera coordinates may not be the same as one unit in world coordinates intrinsic parameters - focal length, principal point, aspect ratio, angle between axes, etc. X U Transformation 1 Transformation V = representing 1 representing Y Z W intrinsic parameters 1 extrinsic parameters T 3 x 3 4 x 4 Camera Calibration, estimate intrinsic and extrinsic camera parameters See Text book for how to do it. 1
Limits for pinhole cameras Thin Lens: Image of Point P F Z f O Z P Thin Lens: Image Plane Deviations from the lens model Q F O P Q Image Plane A price: Whereas the image of P is in focus, the image of Q isn t. P Deviations from this ideal are aberrations Two types 1. geometrical 2. chromatic spherical aberration astigmatism distortion- pin-cushion vs. barrel coma Aberrations are reduced by combining lenses Compound lenses Lighting Radiometry, Lighting, Intensity Special light sources Point sources Distant point sources Strip sources Area sources Common to think of lighting at infinity (a function on the sphere, a 2-D space) Completely arbitrary lighting can be represented as a function on the 4-D ray space (radiances) 2
Radiance Irradiance Power traveling at some point How much light is arriving at a in a specified direction, per unit surface? area perpendicular to the Irradiance -- power per unit area: direction of travel, per unit solid angle Units: watts per square meter per steradian : w/(m 2 sr 1 ) θ (θ, φ) dω W/cm 2 Total power arriving at the surface is given by adding irradiance over all incoming angles Camera s sensor Measured pixel intensity is a function of irradiance integrated over pixel s area over a range of wavelengths For some time x da x Light at surfaces BRDF Many effects when light strikes a surface -- could be: transmitted Skin, glass reflected mirror scattered milk travel along the surface and leave at some other point absorbed sweaty skin Assume that surfaces don t fluoresce e.g. scorpions, detergents surfaces don t emit light (i.e. are cool) all the light leaving a point is due to that arriving at that point Bi-directional Reflectance Distribution Function ρ(θ in, φ in ; θ out, φ out ) Function of Incoming light direction: θ in, φ in Outgoing light direction: θ out, φ out (θ in,φ in ) ^ n (θ out,φ out ) Ratio of incident irradiance to emitted radiance Surface Reflectance Models in Graphics & Vision 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 ] Without shadows Lambertian Surface E(u,v) [ Important: We ll use this a lot ] ^ n At image location (u,v), the intensity of a pixel x(u,v) is: E(u,v) = [a(u,v) ^ n(u,v)] [s ^ 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 is the light source intensity. ^ s is the direction to the light source. ^ s. a 3
Specular Reflection: Smooth Surface Rough Specular Surface N Phong Lobe Phong rough, specular Shadows cast by a point source A point that can t see the source is in shadow For point sources, the geometry is simple Cast Shadow Attached Shadow Inter-reflections 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 Color Cameras Eye: Three types of Cones Cameras: 1. Prism (with 3 sensors) 2. Filter mosaic 3. Filter wheel and X3 4
Prism color camera Separate light in 3 beams using dichroic prism Requires 3 sensors & precise alignment Good color separation Filter mosaic Coat filter directly on sensor Demosaicing (obtain full colour & full resolution image) Filter wheel Rotate multiple filters in front of lens Allows more than 3 color bands new color CMOS sensor Foveon s X3 better image quality smarter pixels Only suitable for static scenes The appearance of colors Color appearance is strongly affected by (at least): Spectrum of lighting striking the retina other nearby colors (space) adaptation to previous views (time) state of mind From Foundations of Vision, Brian Wandell, 1995, via B. Freeman slides 5
Color Afterimage: South African Flag opponent colors Blue -> yellow Red -> green 6
Light Spectrum Talking about colors Color Reflectance Measured color spectrum is a function of the spectrum of the illumination and reflectance 1. Spectrum A positive function over interval 4nm-7nm Infinite number of values needed. 2. Names red, harvest gold, cyan, aquamarine, auburn, chestnut A large, discrete set of color names 3. R,G,B values Just 3 numbers From Foundations of Vision, Brian Wandell, 1995, via B. Freeman slides Illumination Spectra Blue skylight Tungsten bulb Violet Indigo Blue From Foundations of Vision, Brian Wandell, 1995, via B. Freeman slides Green Yellow Orange Red Measurements of relative spectral power of sunlight, made by J. Parkkinen and P. Silfsten. Relative spectral power is plotted against wavelength in nm. The visible range is about 4nm to 7nm. The color names on the horizontal axis give the color names used for monochromatic light of the corresponding wavelength --- the colors of the rainbow. Mnemonic is Richard of York got blisters in Venice. 7
Spectral albedoes for several different leaves, with color names attached. Notice that different colours typically have different spectral albedo, but that different spectral albedoes may result in the same perceived color (compare the two whites). Spectral albedoes are typically quite smooth functions. Measurements by E.Koivisto. Fresnel Equation for Polished Copper 8