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# Geometry of perspective imaging

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Geometry of perspective imaging Coordinate transformations Image formation Vanishing points Stereo imaging Image formation Images of the 3 D world What is the geometry of the image of a three dimensional object Given a point in space where will we see it in an image Given a line segment in space what does its image look like Why do the images of lines that are parallel in space appear to converge to a single point in an image How can we recover information about the 3 D world from a 2 D image Given a point in an image what can we say about the location of the 3 D point in space Are there advantages to having more than one image in recovering 3 D information If we know the geometry of a 3 D object can we locate it in space say for a robot to pick it up from a 2 D image Image formation Euclidean versus projective geometry Euclidean geometry describes shapes as they are properties of objects that are unchanged by rigid motions lengths angles parallelism Projective geometry describes objects as they appear lengths angles parallelism become distorted when we look at objects mathematical model for how images of the 3D world are formed Image formation Example 1 Consider a set of railroad tracks Their actual shape tracks are parallel ties are perpendicular to the tracks ties are evenly spaced along the tracks Their appearance tracks converge to a point on the horizon tracks don t meet ties at right angles ties become closer and closer towards the horizon Image formation Example 2 Corner of a room Actual shape three walls meeting at right angles Total of 270o of angle Appearance a point on which three lines segments are concurrent Total angle is 360o Image formation Example 3 q B appears between A and C from point p But from point 1 A appears between B and C Apparent displacement of objects due to change in viewing position is called parallax shift Greeks knew that is the earth revolved around the sun there would be parallax shift of the stars Tycho Brahe looked for this

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