Announcements Image ormation and Cameras CSE 252A Lecture 3 Assignment 0: Getting Started with Matlab is posted to web page, due Tuesday, ctober 4. Reading: Szeliski, Chapter 2 ptional Chapters 1 & 2 of orsyth & Ponce The course Part 1: The physics of imaging Part 2: Early vision Part 3: Reconstruction Part 4: Recognition Image ormation: utline actors in producing images Projection Perspective/rthographic Projection Vanishing points Projective Geometry Rigid Transformation and S(3) Lenses Sensors Quantization/Resolution Illumination Reflectance and Radiometry Earliest Surviving Photograph How Cameras Produce Images Basic process: photons hit a detector the detector becomes charged the charge is read out as brightness irst photograph on record, la table service by Nicephore Niepce in 1822. Note: irst photograph by Niepce was in 1816. Sensor types: CCD (charge-coupled device) high sensitivity high power cannot be individually addressed blooming CMS simple to fabricate (cheap) lower sensitivity, lower power can be individually addressed 1
Images are two-dimensional patterns of brightness values. Effect of Lighting: Monet igure from US Navy Manual of Basic ptics and ptical Instruments, prepared by Bureau of Naval Personnel. Reprinted by Dover Publications, Inc., 1969. They are formed by the projection of 3D objects. Change of Viewpoint: Monet Pinhole Camera: Perspective projection Abstract camera model - box with a small hole in it Haystack at Chailly at sunrise (1865) orsyth&ponce Camera bscura Camera bscura "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. --- Leonardo Da Vinci Used to observe eclipses (e.g., Bacon, 1214-1294) By artists (e.g., Vermeer). http://www.acmi.net.au/aic/camera_bscura.html (Russell Naughton) 2
Camera bscura Distant objects are smaller Jetty at Margate England, 1898. http://brightbytes.com/cosite/collection2.html (Jack and Beverly Wilgus) (orsyth & Ponce) Purely Geometric View of Perspective Geometric properties of projection 3-D points map to points 3-D lines map to lines Planes map to whole image or half-plane Polygons map to polygons The projection of the point P on the image plane Π is given by the point of intersection P of the ray defined by P with the plane Π. Important point to note: Angles & distances not preserved, nor are inequalities of angles & distances. Degenerate cases: line through focal point project to point plane through focal point projects to a line Equation of Perspective Projection Cartesian coordinates: We have, by similar triangles, that (x, y, z) -> (f x/z, f y/z, f ) Establishing an image plane coordinate system at C aligned with i and j, we get (x,y,z) ( f x z, f y z ) 3
Parallel lines meet in the image Vanishing point Vanishing points H VPL VPR Image plane ormed by line through Parallel to the given line(s) A single line can have a vanishing point Different directions correspond different vanishing points VP 1 VP 2 VP 3 Vanishing Points Simplified Camera Models Perspective Projection Affine Camera Model Scaled rthographic Projection rthographic Projection Affine Camera Model Appropriate in Neighborhood About (x 0,y 0,z 0 ) Perspective 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 Assume that f=1, and perform a Taylor series expansion about (x 0, y 0, z 0 ) u = 1 x 0 1 x 0 2 ( z z v z 0 z 0 ) + 1 1 ( x x 0 z 0 0 0 ) y 0 + 1 0 y y z 0 1 0 y 0 ( ) + 1 2 2 3 z 0 Dropping higher order terms and regrouping. u 1 x 0 v z 0 y 0 + 1/z 0 x /z x 2 0 0 0 2 y 0 1/z 0 y 0 /z = Ap + b 0 z x 0 y 0 z z 0 ( ) 2 + 4
rthographic projection Starting with Affine Camera Model, take Taylor series about (x o, y 0, z 0 ) = (0, 0, z 0 ) a point on the optical axis Two topics that we won t cover today 1. Homogenous coordinates and projective transforms 2. Rigid transforms what happens if coordinate system isn t as convenient. (0, 0, z 0 ) u = 1 x v z 0 y ther camera models Some Alternative Cameras Generalized camera maps points lying on rays and maps them to points on the image plane. mnicam (hemispherical) Light Probe (spherical) Lenses Beyond the pinhole Camera Getting more light Bigger Aperture 5
Pinhole 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. Thin Lens: Center Thin Lens: ocus All rays that enter lens along line pointing at emerge in same direction. Parallel lines pass through the focus, 6
Thin Lens: Image of Point Thin Lens: Image of Point P P P Z f Z P All rays passing through lens and starting at P converge upon P Thin Lens: Image Plane Thin Lens: Aperture Q P P P Q Image Plane A price: Whereas the image of P is in focus, the image of Q isn t. P Image Plane Smaller Aperture -> Less Blur Pinhole -> No Blur Image Plane ield of View f ield 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 Compound lenses 7
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) 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 8
Vignetting nly part of the light reaches the sensor Periphery of the image is dimmer 9