Cameras and Radiometry. Last lecture in a nutshell. Conversion Euclidean -> Homogenous -> Euclidean. Affine Camera Model. Simplified Camera Models

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1 Cameras and Radiometry Last lecture in a nutshell CSE 252A Lecture 5 Conversion Euclidean -> Homogenous -> Euclidean In 2-D Euclidean -> Homogenous: (x, y) -> k (x,y,1) Homogenous -> Euclidean: (x, y, z) -> (x/z, y/z) 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) Z 1 Y X (x,y,1) (x,y) Affine Camera Model Take perspective projection equation, and perform Taylor series expansion about some point (x 0,y 0,z 0 ). Drop terms that are higher order than linear. Resulting expression is affine camera model Appropriate in Neighborhoo About (x 0,y 0,z 0 Simplified Camera Models erspective rojection Coordinate Changes: Rigid Transformations Affine Camera Model Scaled rthographic rojection Euclidean rthographic rojection Homogeneous 1

2 Some points about S(n) S(n) = { R R nxn : R T R = I, det(r) = 1} S(2): rotation matrices in plane R 2 S(3): rotation matrices in 3-space R 3 orms a Group under matrix product operation: Identity Inverse Associative Closure Closed (finite intersection of closed sets) Bounded R i,j [-1, +1] Does not form a vector space. Manifold of dimension n(n-1)/2 Dim(S(2)) = 1 Dim(S(3)) = 3 S(3) arameterizations of S(3) 3-D manifold, so between 3 parameters and 2n +1 parameters (Whitney s Embedding Thm.) Roll-itch-Yaw Euler Angles Axis Angle (Rodrigues formula) Cayley s formula Matrix Exponential Quaternions (four parameters + one constraint) Camera parameters Issue World units (e.g., cm), camera units (pixels) camera may not be at the origin, looking down the z-axis extrinsic parameters 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. Camera Calibration X U Transformation Transformation V = representing representing Y Z W intrinsic parameters extrinsic parameters T 3 x 3 4 x 4: Rigid transformation, estimate intrinsic and extrinsic camera parameters See Text book for how to do it. Camera Calibration Toolbox for Matlab (Bouguet) Qualcomm Augmented Reality Demo v=hxtq1qbmliw&feature=player_embedded rojective Transformation (Homography) 3 x 3 linear transformation of homogenous coordinates oints map to points, Lines map to lines 2

3 Image of a plane The mapping between coordinates in a plane in 3-D to the image plane under perspective is a projective transformation Beyond the pinhole Camera Getting more light Bigger Aperture A square maps to an arbitrary quadrilateral Under affine or scaled orthographic camera model, square maps to a parallelogram inhole Camera Images with Variable Aperture Limits for pinhole cameras 2 mm 1mm.6 mm.35 mm.15 mm.07 mm The reason for lenses Thin Lens ptical axis Rotationally symmetric about optical axis. Spherical interfaces. 3

4 Thin Lens: Center Thin Lens: ocus All rays that enter lens along line pointing at emerge in same direction. arallel lines pass through the focus, Thin Lens: Image of oint Thin Lens: Image of oint Z f Z All rays passing through lens and starting at converge upon Thin Lens: Image lane Thin Lens: Aperture Q Q Image lane A price: Whereas the image of is in focus, the image of Q isn t. Image lane Smaller Aperture -> Less Blur inhole -> No Blur 4

5 ield of View Deviations from the lens model Deviations from this ideal are aberrations Two types Image lane f ield of View 1. geometrical 2. chromatic spherical aberration astigmatism distortion coma Aberrations are reduced by combining lenses Compound lenses Spherical aberration Rays parallel to the axis do not converge uter portions of the lens yield smaller focal lengths Astigmatism An optical system with astigmatism is one where rays that propagate in two perpendicular planes have different foci. If an optical system with astigmatism is used to form an image of a cross, the vertical and horizontal lines will be in sharp focus at two different distances. object Distortion magnification/focal length different for different angles of inclination Chromatic aberration (great for prisms, bad for lenses) pincushion (tele-photo) barrel (wide-angle) Can be corrected! (if parameters are know) 5

6 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 Appearance and surface reflectance nly part of the light reaches the sensor eriphery of the image is dimmer Read Chapter 4 of once & orsyth Solid Angle Irradiance Radiance BRD Lambertian/hong BRD Radiometry A local coordinate system on a surface Consider a point on the surface Light arrives at from a hemisphere of directions defined by the surface normal N We can defined a local coordinate system whose origin is and with one axis aligned with N Convenient to represent in spherical angles. 6

7 oreshortening Measuring Angle The solid angle subtended by an object from a point is the area of the projection of the object onto the unit sphere centered at. Measured in steradians, sr Definition is analogous to projected angle in 2D If I m at, and I look out, solid angle tells me how much of my view is filled with an object By analogy with angle (in radians), the solid angle subtended by a region at a point is the area projected on a unit sphere centered at that point The solid angle subtended by a patch area da is given by Solid Angle ower is energy per unit time Radiance: ower traveling at some point in a specified direction, per unit area perpendicular to the direction of travel, per unit solid angle Symbol: L(x,θ,φ) Units: watts per square meter per steradian : w/(m 2 sr 1 ) (θ, φ) dω x da CSE 252A ower is energy per unit time Radiance: ower traveling at some point in a specified direction, per unit area perpendicular to the direction of travel, per unit solid angle Symbol: L(x,θ,φ) Units: watts per square meter per steradian : w/(m 2 sr 1 ) L = α (θ, φ) x da dω (dacosα)dω In free space, radiance is constant as it propagates along a ray Derived from conservation of flux undamental in Light Transport. CSE 252A ower emitted from patch, but radiance in direction different from surface normal CSE 252A, Winter

8 Crucial property: How much light is arriving at a Total Irradiance arriving at the surface? surface is given by adding Units of irradiance: Watts/m 2 irradiance over all incoming angles This is a function of incoming angle. Total irradiance is A surface experiencing radiance L (x,θ,φ) coming in from solid angle dω experiences irradiance: L(x,θ,φ) φ θ N What is image irradiance E for a radiance L emitted from a point? x x CSE 252A CSE 252A, Winter 2007 δa δa δa δa Let δω be the solid angle subtended by δa (or δa ) from the center of the lens Let Ω be the solid angle subtended by the lens from. CSE 252A, Winter 2007 CSE 252A, all 2009 δa δa δa δa The power δ emitted from the patch δa with radiance L and falling on the lens is: CSE 252A, Winter 2007 E = π 2 d cos 4 α L 4 z' E: Image irradiance L: emitted radiance d : Lens diameter z : depth of image plane CSE 252A, Winter 2007 α: Image Angle of Irradiance patch from optical axis 8