Announcements. Image Formation: Outline. Homogenous coordinates. Image Formation and Cameras (cont.)
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1 Announcements Image Formation and Cameras (cont.) HW1 due, InitialProject topics due today CSE 190 Lecture 6 Image Formation: Outline Factors in producing images Projection Perspective Vanishing points Orthographic Lenses Sensors Quantization/Resolution Illumination Reflectance Pinhole Camera: Perspective projection Abstract camera model - box with a small hole in it Forsyth&Ponce Homogenous coordinates Our usual coordinate system is called a Euclidean or affine coordinate system Rotations, translations and projection in Homogenous coordinates can be expressed linearly as matrix multiplies Euclidean World 3D Convert Homogenous World 3D Projection Homogenous Image 2D Convert Euclidean World 2D Euclidean -> Homogenous-> Euclidean In 2-D Euclidean -> Homogenous: (x, y) -> k (x,y,1) Homogenous -> Euclidean: (u,v,w) -> (u/w, v/w) In 3-D Euclidean -> Homogenous: (x, y, z) -> k (x,y,z,1) Homogenous -> Euclidean: (x, y, z, w) -> (x/w, y/w, z/w) 1
2 The equation of projection Simplified Camera Models Perspective Projection Affine Camera Model Cartesian coordinates: Homogenous Coordinates and Camera matrix U X V = Y W Z f 0 T Scaled Orthographic Projection Orthographic Projection Affine Camera Model Appropriate in Neighborhood About (x 0,y 0,z 0 ) Perspective Assume that f=1, and perform a Taylor series expansion about (x 0, y 0, z 0 ) Take Perspective projection equation, and perform Taylor Series Expansion about (some point (x 0,y 0,z 0 ). Drop terms of higher order than linear. Resulting expression is affine camera model Dropping higher order terms and regrouping. Scaled Orthographic projection Starting with Affine camera mode Take Taylor series about (0, 0, z 0 ) a point on optical axis Rewrite Affine camera model in terms of Homogenous Coordinates Euclidean Coordinates Homogenous Coordinates 2
3 What if camera coordinate system differs from object coordinate system Coordinate Changes: Rigid Transformations {c} {W} P Rotation Matrix Translation vector A rotation matrix R has the following properties: Block Matrix Multiplication Its inverse is equal to its transpose R -1 = R T Its determinant is equal to 1: Or equivalently: det(r)=1. What is AB? Rows (or columns) of R form a right-handed orthonormal coordinate system. Homogeneous Representation of Rigid Transformations Transformation represented by 4 by 4 Matrix 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 Transformation V = representing representing Y Z W intrinsic parameters 0 0 1/ f 0 extrinsic parameters T 3 x 3 3x4 (Perspective) 4 x 4 Camera Calibration, estimate intrinsic and extrinsic camera parameters See Camera Calibration Toolbox for Matlab (Bouguet) 3
4 Perspective Projection of Plane: Homography Special Cases: y=hx h 11 h 12 h 13 H = h 21 h 22 h 23 General homography h 31 h 32 h 33 The mapping under perspective projection of a plane in the scene to the image plane is given by a homography. y=hx Where Y and X are respectively the homogenous 2-D coordinates of points on the plane and the homogenous 2-D image coordinates. H is a 3x3 matrix. h 11 h 12 h 13 H = h 21 h 22 h λr 11 λr 12 h 13 H = λr 21 λr 22 h Affine transformation Similarity transform Upper 2x2 is a scaled rotation matrix. Camera Obscura Getting more light Bigger Aperture "When images of illuminated objects... penetrate through a small hole into a very dark room... you will see [on the opposite wall] these objects in their proper form and color, reduced in size... in a reversed position, owing to the intersection of the rays". Da Vinci (Russell Naughton) Pinhole Camera Images with Variable Aperture The reason for lenses 2 mm 1mm.6 mm.35 mm.15 mm.07 mm 4
5 Thin Lens: Image of Point Thin Lens: Image Plane P F Z f O Z P Q F O P Q Image Plane A price: Whereas the image of P is in focus, the image of Q isn t. P Thin Lens: Aperture Field of View P Image Plane O P Smaller Aperture -> Less Blur Pinhole -> No Blur Image Plane f O Field of View Deviations from the lens model Deviations from this ideal are aberrations Two types 1. geometrical 2. chromatic spherical aberration astigmatism distortion coma Aberrations are reduced by combining lenses Spherical aberration rays parallel to the axis do not converge outer portions of the lens yield smaller focal lenghts CS252A, Fall 2009 Compound lenses Computer Vision I 5
6 Distortion magnification/focal length different for different angles of inclination Chromatic aberration Index of refraction of lens depends on wavelength of light pincushion (tele-photo) barrel (wide-angle) Can be corrected! (if parameters are know) Chromatic aberration Vignetting: Spatial Non-Uniformity rays of different wavelengths focused in different planes cannot be removed completely sometimes achromatization is achieved for more than 2 wavelengths camera Iris Litvinov & Schechner, radiometric nonidealities Vignetting Image Brightness Only part of the light reaches the sensor Periphery of the image is dimmer CS252A, Fall 2009 Computer Vision I 6
7 How Cameras Produce Images Camera s sensor Basic process: photons hit a detector the detector becomes charged the charge is read out as brightness Sensor types: CCD (charge-coupled device) high sensitivity high power cannot be individually addressed blooming CMOS most common simple to fabricate (cheap) lower sensitivity, lower power can be individually addressed Measured pixel intensity is a function of irradiance integrated over pixel s area over a range of wavelengths For some time 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 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 ] Lambertian Surface x(u,v) ^ n At image location (u,v), the intensity of a pixel x(u,v) is: x(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. ^ s. a 7
8 Specular Reflection: Smooth Surface Rough Specular Surface N Phong Lobe Phong rough, specular SUV Color Space Gloss removal From Foundations of Vision, Brian Wandell, 1995, via B. Freeman slides 8
9 Color Reflectance Color receptors Measured color spectrum is a function of the spectrum of the illumination and reflectance Red cone Green cone Blue cone Response of k th cone = From Foundations of Vision, Brian Wandell, 1995, via B. Freeman slides Color Cameras Prism color camera Separate light in 3 beams using dichroic prism Requires 3 sensors & precise alignment Good color separation Eye: Three types of Cones Cameras: 1. Prism (with 3 sensors) 2. Filter mosaic 3. Filter wheel and X3 Filter mosaic Lighting Coat filter directly on sensor What is color (spectrum) of light source? What is spatial distribution of light sources? Point source at infinity (directional source) Nearby point light source Strip light source Area light source Demosaicing (obtain full colour & full resolution image) 9
10 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) 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 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. CSE 252A Figure from Mutual Illumination, by D.A. Forsyth and A.P. Zisserman, Proc. CVPR, 1989, copyright 1989 IEEE CSE 252A Irradiance observed in an image of this geometry for a real white gutter. Figure from Mutual Illumination, by D.A. Forsyth and A.P. Zisserman, Proc. CVPR, 1989, copyright 1989 IEEE CSE 252A 10
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