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Exploring the Mathematics of Stereo Imaging Varieties

Delve into the world of stereo imaging with a focus on the Geometry of Stereo Panoramas, Epipolar Geometry, and Mathematics for Stereo Views. Learn how to define epipolar geometry, projection, and triangulation in stereo imaging. Discover the Stereo Classification Theorem and how to generate stereo views using the Pushbroom Stereo technique. This comprehensive guide offers insights into different stereo imaging varieties, their properties, and applications. Explore the future possibilities of multiperspective camera design and image analysis in the realm of stereo imaging.

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Exploring the Mathematics of Stereo Imaging Varieties

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  1. The Space of All Stereo Images Steve Seitz University of Washington

  2. Stereo Imaging • Key property: horizontal parallax • Enables stereoscopic viewing and stereo algorithms • Gives rise to epipolar geometry • What other image varieties have this property?

  3. left right Stereo Panoramas • [Ishiguro, Yamamoto, Tsuji, 92] • [Peleg and Ben-Ezra, 99] • [Shum, Kalai, Seitz, 99] • [Nayar and Karmarkar, 00] I

  4. Problem Statement • Suppose you could capture any set of rays • Which rays result in a stereo pair? • satisfy horizontal parallax constraint • How to define epipolar geometry? • How to express projection, triangulation?

  5. Geometry of Stereo Panoramas

  6. Geometry of Stereo Panoramas

  7. Epipolar Geometry

  8. There are only 3 doubly-ruled surfaces • Extension of classical result [Hilbert] hyperboloid hyperbolic paraboloid plane Stereo Classification Theorem • Two images form a stereo pair iff • Each row lies on a doubly ruled surface • Corresponding rows are the same surface

  9. Generating Stereo Views • Recipe book for stereo views • All stereo pairs are generated by moving a camera along a conic curve • line, ellipse, parabola, hyperbola

  10. Pushbroom Stereo

  11. parabolic panorama Parabolic Panorama perspective image

  12. Stereo Cyclographs input cyclographs

  13. Stereo Cyclograph Reconstruction • Computed from two cyclograph images • Using unmodified stereo matcher [Zitnick & Kanade]

  14. Mathematics for Stereo Views • Defined by two families of quadric surfaces • Rays in row u lie on quadric Qu • Rays in column v lie on quadric Qv

  15. Mathematics for Stereo Views • Projection • Solve for u, v, given X • Triangulation • Solve for X, given u1, u2, v • Unified treatment for all stereo varieties • Reduces to standard eqns for the perspective case

  16. Conclusions • Main results • Stereo Classification Theorem • Recipe book for generating stereo pairs • Mathematics of stereo imaging • Future work • Exploring interesting new varieties • Multiperspective camera design • Multiperspective image analysis • Thanks to Jiwon Kim

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