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Geometry 2: A taste of projective geometry Introduction to Computer Vision Ronen Basri Weizmann Institute of Science

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Summery of last lecture

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Material covered Pinhole camera model, perspective projection Two view geometry, general case: Epipolar geometry, the essential matrix Camera calibration, the fundamental matrix Two view geometry, degenerate cases Homography (planes, camera rotation) A taste of projective geometry Stereo vision: 3D reconstruction from two views Multi-view geometry, reconstruction through factorization

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Camera matrix

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The uncalibrated case: the Fundamental matrix

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The Fundamental matrix

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Geometry Geometry – Greek: earth measurement Geometry concerns with shape, size, relative positions, and properties of spaces Euclidean geometry: Point, line, plane Incidence Continuity Order, “between” Parallelism Congruence = invariance: angles, lengths, areas are preserved under rigid transformations

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Projective geometry How does a plane looks after projection? How does perspective distorts geometry?

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Plane perspective Pencil of rays

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Plane perspective Pencil of rays

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Projective transformation

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How these change from Eucleadian geometry? Point, line, plane Incidence Continuity Order, “between” Parallelism Congruence Under projective transformation A (straight) line transforms to a line and a conic to a conic But order and parallelism are not preserved Likewise, angles, lengths and areas are not preserved

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Projective coordinates

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Projective line

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Intersection and incidence

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Ideal points

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Line at infinity

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Homography

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Hierarchy of transformations Rigid Preserves angles, lengths, area, parallelism SimilarityPreserves angles, parallelism AffinePreserves parallelism Homography Preserves cross ratio

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Camera rotation

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Planar scene

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Summary HomographyPerspective (calibrated) Perspective (uncalibrated) Orthographic Form PropertiesOne-to-one (group) Concentric epipolar lines Parallel epipolar lines DOFs 8(5) 8(7)4 Eqs/pnt 2111 Minimal configuration 45+ (8,linear)7+ (8,linear)4 DepthNoYes, up to scale Yes, projective structure Affine structure (third view required for Euclidean structure)

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Recovering epipolar constraints

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Interest points (Harris)

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Descriptor: SIFT ( Scale invariant feature transform)

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SIFT matches

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RANSAC

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Epipolar lines

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