Capturing Light: Geometry of Image Formation
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3 Capturing Light: Geometry of Image Formation Computer Vision James Hays Slides from Derek Hoiem, Alexei Efros, Steve Seitz, and David Forsyth
4 Administrative Stuff My Office hours, CoC building 35 Monday and Wednesday -2 TA Office hours To be announced Project goes out today Piazza should be your first stop for help Matlab is available for students from OIT software.oit.gatech.edu
5 Previous class: Introduction Machine Learning Computer Vision Scope of CS 4476 Robotics Human Computer Interaction Graphics Computational Photography Optics Image Processing Geometric Reasoning Recognition Deep Learning Neuroscience Medical Imaging
6 The Geometry of Image Formation Mapping between image and world coordinates Pinhole camera model Projective geometry Vanishing points and lines Projection matrix
7 What do you need to make a camera from scratch?
8 Image formation Slide source: Seitz Let s design a camera Idea : put a piece of film in front of an object Do we get a reasonable image?
9 Pinhole camera Idea 2: add a barrier to block off most of the rays This reduces blurring The opening known as the aperture Slide source: Seitz
10 Pinhole camera f c f = focal length c = center of the camera Figure from Forsyth
11 Camera obscura: the pre-camera Known during classical period in China and Greece (e.g. Mo-Ti, China, 47BC to 39BC) Illustration of Camera Obscura Freestanding camera obscura at UNC Chapel Hill Photo by Seth Ilys
12 Camera Obscura used for Tracing Lens Based Camera Obscura, 568
13 Accidental Cameras Accidental Pinhole and Pinspeck Cameras Revealing the scene outside the picture. Antonio Torralba, William T. Freeman
14 Accidental Cameras
15 First Photograph Oldest surviving photograph Took 8 hours on pewter plate Photograph of the first photograph Joseph Niepce, 826 Stored at UT Austin Niepce later teamed up with Daguerre, who eventually created Daguerrotypes
16 Today s class: Camera and World Geometry How tall is this woman? How high is the camera? What is the camera rotation? What is the focal length of the camera? Which ball is closer?
17 Dimensionality Reduction Machine (3D to 2D) 3D world 2D image Point of observation Figures Stephen E. Palmer, 22
18 Projection can be tricky Slide source: Seitz
19 Projection can be tricky Slide source: Seitz
20 Projective Geometry What is lost? Length Who is taller? Which is closer?
21 Length and area are not preserved A C B Figure by David Forsyth
22 Earth as an example
23 ISS timelapse. 4 kilometers from Earth
24 The Blue Marble, taken on December 7, 972, by the crew of the Apollo 7 spacecraft, at a distance of about 45, kilometers
25
26 Earth from Curiosity Rover, 24, 6 million kilometers from Earth
27 Earth from Curiosity Rover, 24, 6 million kilometers from Earth
28 Consider again that dot. That's here. That's home. That's us. On it everyone you love, everyone you know, everyone you ever heard of, every human being who ever was, lived out their lives. The aggregate of our joy and suffering, thousands of confident religions, ideologies, and economic doctrines, every hunter and forager, every hero and coward, every creator and destroyer of civilization, every king and peasant, every young couple in love, every mother and father, hopeful child, inventor and explorer, every teacher of morals, every corrupt politician, every "superstar," every "supreme leader," every saint and sinner in the history of our species lived there on a mote of dust suspended in a sunbeam. Carl Sagan Pale Blue Dot from Voyager, February 4, 99, 6 billion kilometers from Earth
29 Projective Geometry What is lost? Length Angles Parallel? Perpendicular?
30 Projective Geometry What is preserved? Straight lines are still straight
31 Vanishing points and lines Parallel lines in the world intersect in the image at a vanishing point
32 Vanishing points and lines Vanishing Line Vanishing Point o Vanishing Point o
33 Slide from Efros, Photo from Criminisi Vanishing points and lines Vertical vanishing point (at infinity) Vanishing point Vanishing point
34 Projection: world coordinatesimage coordinates.. f Z Y. P X Y Z. U V Camera Center (,, ) U p V If X = 2, Y = 3, Z = 5, and f = 2 What are U and V?
35 Projection: world coordinatesimage coordinates.. f Z Y. P X Y Z. U V Camera Center (,, ) U p V U X * V Y * Sanity check, what if f and Z are equal? f Z f Z U 2* V 3*
36 Projection: world coordinatesimage coordinates Optical Center (u., v ) f Z Y.. P X Y Z. u v u p v Camera Center (t x, t y, t z )
37 Interlude: why does this matter?
38 Relating multiple views
39
40 Homogeneous coordinates Conversion Converting to homogeneous coordinates homogeneous image coordinates homogeneous scene coordinates Converting from homogeneous coordinates
41 Homogeneous coordinates Invariant to scaling x k y w kx ky kw Homogeneous Coordinates kx kw ky kw x w y w Cartesian Coordinates Point in Cartesian is ray in Homogeneous
42 Projection matrix Slide Credit: Savarese R,t j w X k w O w x i w x K R X t x: Image Coordinates: (u,v,) K: Intrinsic Matrix (3x3) R: Rotation (3x3) t: Translation (3x) X: World Coordinates: (X,Y,Z,)
43 X x K I z y x f f v u w K Slide Credit: Savarese Projection matrix Intrinsic Assumptions Unit aspect ratio Optical center at (,) No skew Extrinsic Assumptions No rotation Camera at (,,) X x
44 Remove assumption: known optical center X x K I z y x v f u f v u w Intrinsic Assumptions Unit aspect ratio No skew Extrinsic Assumptions No rotation Camera at (,,)
45 Remove assumption: square pixels X x K I z y x v u v u w Intrinsic Assumptions No skew Extrinsic Assumptions No rotation Camera at (,,)
46 Remove assumption: non-skewed pixels X x K I z y x v u s v u w Intrinsic Assumptions Extrinsic Assumptions No rotation Camera at (,,) Note: different books use different notation for parameters
47 Oriented and Translated Camera R j w t X k w O w x i w
48 Allow camera translation X t x K I z y x t t t v u v u w z y x Intrinsic Assumptions Extrinsic Assumptions No rotation
49 3D Rotation of Points Rotation around the coordinate axes, counter-clockwise: cos sin sin cos ) ( cos sin sin cos ) ( cos sin sin cos ) ( z y x R R R p p y z Slide Credit: Saverese
50 Allow camera rotation X t x K R z y x t r r r t r r r t r r r v u s v u w z y x
51 Degrees of freedom X t x K R z y x t r r r t r r r t r r r v u s v u w z y x 5 6
52 Orthographic Projection Special case of perspective projection Distance from the COP to the image plane is infinite Also called parallel projection What s the projection matrix? Image World Slide by Steve Seitz z y x v u w
53 Field of View (Zoom, focal length)
54 Beyond Pinholes: Radial Distortion Corrected Barrel Distortion Image from Martin Habbecke
55 Things to remember Vanishing points and vanishing lines Vanishing line Vanishing point Vertical vanishing point (at infinity) Vanishing point Pinhole camera model and camera projection matrix Homogeneous coordinates x K R tx
56 Next class Light, color, and sensors
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