# lecture6 - EECS 442 Computer vision Epipolar Geometry Why...

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EECS 442 – Computer vision Epipolar Geometry • Why is stereo useful? • Epipolar constraints • Essential and fundamental matrix • Estimating F • Examples Reading: [AZ] Chapters: 4, 9, 11 [FP] Chapters: 10

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Pinhole perspective projection Recovering structure from a single view C O w P p Calibration rig Scene Camera K From calibration rig From points and lines at infinity + orthogonal lines and planes structure of the scene, K location/pose of the rig, K Knowledge about scene ( point correspondences, geometry of lines & planes, etc…
Pinhole perspective projection Recovering structure from a single view C O w P p Calibration rig Scene Camera K Why is it so difficult? Intrinsic ambiguity of the mapping from 3D to image (2D)

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Recovering structure from a single view Intrinsic ambiguity of the mapping from 3D to image (2D) Courtesy slide S. Lazebnik
Two eyes help!

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O 2 O 1 x 2 x 1 ? K =known Two eyes help! This is called triangulation K =known R, T
• Find X that minimizes ) , ( ) , ( 2 2 2 1 1 2 X P x d X P x d + O 1 O 2 x 1 x 2 X Triangulation

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Stereo-view geometry Scene geometry: Find coordinates of 3D point from its projection into 2 or images. Correspondence: Given a point in one image, how can I find the corresponding point x’ in another one ? Camera geometry: Given corresponding points in two images, find camera matrices, position and pose.
• Epipolar Plane • Epipoles e 1 , e 2 • Epipolar Lines • Baseline Epipolar geometry O 1 O 2 x 2 X x 1 e 1 e 2 = intersections of baseline with image planes = projections of the other camera center = vanishing points of camera motion direction

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Example: Converging image planes
O 1 O 2 X e 2 x 1 x 2 e 2 Example: Parallel image planes • Baseline intersects the image plane at infinity • Epipoles are at infinity • Epipolar lines are parallel to x axis

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Example: Parallel image planes
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