CS535 Fall Department of Computer Science Purdue University

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1 Omnidirectional Camera Models CS535 Fall 2010 Daniel G Aliaga Daniel G. Aliaga Department of Computer Science Purdue University

2 A little bit of history Omnidirectional cameras are also called panoramic cameras Panorama comes from the Greek phrase all sight Originally used for artistic purposes Robert Barker obtained a patent for the idea of a panorama in 1794 A Painting without Equal In 1800s, panorama became a common European word

3 Taxonomy of Omnidirectional Camera Designs Single center of projection Like a traditional camera, light rays meet at a single focal point Multiple center of projection Camera adoes not have aeaa single gefocal point Sampled surfaces can be missing or duplicated in full image Mathematical (re)projections are more complicated OR Si l C /I Single Camera/Image One view is acquired per image Multiple Camera/Image A single view composed by compositing several images

4 Some Omnidirectional Cameras Rotating camera design Fish eye ih lens design Multiple camera planar mirror design Single COP curved mirror design

5 Rotating Camera Design Place a camera on a tripod and spin it around snapping pictures every so often Pros Simple Cons Multiple centers of projection Multiple (overlapping images) to composite Vertical jitter Slow acquisition process

6 Rotating Camera Design original warped stitched

7 Rotating Camera Design Tienamen

8 Panoramic Video Textures [Agarwala-SIG05]

9 Panoramic Video Textures pvviewer [Agarwala-SIG05]

10 Panoramic Video Textures [Agarwala-SIG05]

11 Fish Eye Lens Design Use a wide field of view lens (~180 degrees) placed in front of a traditional camera Pros: Also relatively simple for users (making the lens can be troublesome for designers) Cons: Very severe image distortion Low resolution around perimeter of field of view Almost a single center ofprojection

12 Multiple Planar Mirror Design Catadioptric = reflective (mirror) + refractive (lens) p// [Nalwa96]

13 Multiple Planar Mirror Design [Fullview]

14 Multiple Planar Mirror Design [Fullview]

15 Multiple Planar Mirror Design

16 Fixed Viewpoint Constraint image point pinhole (0,h) image plane focal plane h z n c b a b+c world point d mirror point (r, z) r Property 1: c=90 o -a Property 2: a+d+2b+2c=180 o mirror base

17 Fixed Viewpoint Constraint Property 1: c=90 o -a Property 2: a+d+2b+2c=180 o 2b=a-d 2tan(b) = tan(a)-tan(d) tan(d) 1-tan 2 (b) = 1+tan(a)tan(d) r(h-2z)(dz/dr) 2-2(r 2 +hz-z 2 )(dz/dr) + r(2z-h) = 0 Quadratic first-order ODE (constraint equation)

18 Fixed Viewpoint Constraint Property 1: c=90 o -a Property 2: a+d+2b+2c=180 o 2b=a-d 2tan(b) = tan(a)-tan(d) tan(d) 1-tan 2 (b) = 1+tan(a)tan(d) (z-h/2) 2 -r 2 (k/2-1)=h 2 (k-2)/4k (z-h/2) 2 +r 2 (1+h 2 /2k)=(2k+h 2 )/4 for k 2 for k 0 Generalized solution to constraint equation

19 Single Curved Mirror Design Theoretical solutions to a single center of projection panoramic camera use mirrors that are subsets of swept conic sections Cones Spheres Ellipsoids Hyperboloids Paraboloids [ Panoramic Vision, Benosman/Kang]

20 Examples Walking in the mirror Museum

21 Conical Mirror

22 Spherical Mirror

23 Ellipsoidal Mirror

24 Hyperboloidal Mirror

25 Hyperboloidal Mirror ACCOWLE Co., LTD, A Spin-off at Kyoto University Spherical Mirror Hyperbolic Mirror Image: High res. in the top

26 Paraboloidal Mirror

27 Catadioptric Paraboloidal Camera Design by [Nayar97] Motorized cart with camera, computer, battery, radio remote control [Aliaga01,02]

28 Catadioptric Paraboloidal Camera z focal point x reference plane m mirror p image plane i Theoretical camera model

29 Catadioptric Paraboloidal Camera reference plane mirror m n^ p H image plane i lens [Aliaga ICCV01]

30 Catadioptric Paraboloidal Camera Calibration Assuming incident equals reflected angle: i i - - m m nˆ nˆ = p p - - m m nˆ nˆ And given a 3D point p, mirror radius r, convergence distance H, we group and rewrite in terms of m r : m 5 r -p r m 4 r + 2r 2 m 3 r + (2p r rh - 2r 2 p r )m 2 r + (r 4-4r 2 p H)m - (r 4 + 2r 3 z r p r Hp r ) = 0 [Aliaga ICCV01]

31 Omnidirectional Vision Home Page edu/~kostas/omni.html

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