lecture2 - VisualSimulation CAP6938 Dr.HassanForoosh...

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    Visual Simulation CAP 6938 Dr. Hassan Foroosh  Dept. of Computer Science UCF © Copyright Hassan Foroosh 2002
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    Today Single View Modeling Projective Geometry for Vision/Graphics on Monday (1/22) Multi-view modeling
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    3D Modeling from a  Photograph
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    Shape from X Shape from X Shape from shading Shape from texture Shape from focus Others… Make strong assumptions – limited applicability Lee & Kau 91 (from survey by Zhang et al., 1999)
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    Model-Based Techniques Blanz & Vetter, Siggraph 1999 Model-based reconstruction Acquire a prototype face, or database of prototypes Find best fit of prototype(s) to new image
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    User Input + Projective  Geometry  Horry et al., “Tour Into the Picture”, SIGGRAPH 96 Criminisi et al., “Single View Metrology”, ICCV 1999 Shum et al., CVPR 98
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    Projective Geometry for  Vision & Graphics Uses of projective  geometry Drawing Measurements Mathematics for projection Undistorting images Focus of expansion Camera pose estimation,  match move Object recognition via  invariants Today:  single-view projective  geometry Projective representation Point-line duality Vanishing points/lines Homographies The Cross-Ratio Later:  multi-view geometry
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    1 2 3 4 1 2 3 4 Measurements on Planes Approach: unwarp then measure
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    Planar Projective  Transformations Perspective projection of a plane lots of names for this: homography, texture-map, colineation, planar projective map Easily modeled using homogeneous coordinates = 1 * * * * * * * * * ' ' y x s sy sx H p p’ To apply a homography H compute p’ = Hp p’’ = p’ /s normalize by dividing by third component
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    Image Rectification To unwarp (rectify) an image solve for H given p’’ and p solve equations of the form: s p’’ = Hp linear in unknowns: s and coefficients of H – need at least 4 points
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    (0,0,0) The Projective Plane Why do we need homogeneous coordinates? represent points at infinity, homographies, perspective projection,  multi-view relationships What is the geometric intuition? a point in the image is a  ray  in projective space (sx,sy,s) Each point (x,y) on the plane is represented by a ray (sx,sy,s) all points on the ray are equivalent: (x, y, 1) 2245 (sx, sy, s) image plane (x,y,1) y x z
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    Projective Lines What is a line in projective space? A line is a plane of rays through origin all rays (x,y,z) satisfying: ax + by + cz = 0 [ ] = z y x c b a 0 : notation vector in A line is also represented as a homogeneous 3-vector l l p
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    l Point and Line Duality A line  l  is a homogeneous 3-vector (a ray)
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This note was uploaded on 06/13/2011 for the course CAP 6938 taught by Professor Staff during the Spring '08 term at University of Central Florida.

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lecture2 - VisualSimulation CAP6938 Dr.HassanForoosh...

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