lecture11 - EECS 442 Computer vision Multiple view geometry...

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EECS 442 – Computer vision Multiple view geometry Perspective Structure from Motion Reading: [HZ] Chapters: 10,18,19 [FP] Chapter: 13 Some slides of this lectures are courtesy of prof. S. Lazebnik - Perspective structure from motion problem - Ambiguities - Algebraic methods - Factorization methods - Bundle adjustment - Self-calibration

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From the m x n correspondences x ij , estimate: •m projection matrices M i •n 3D points X j x 1 j x 2 j x m j X j motion structure M 1 M 2 M m Structure from motion problem
Structure from motion ambiguity - Position ambiguity: it is impossible based on the images alone to estimate the absolute location and pose of the scene w.r.t. a 3D world coordinate frame

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Structure from motion ambiguity -Scale ambiguity: it is impossible based on the images alone to estimate the absolute scale of the scene (i.e. house height)
Structure from motion ambiguity -The scene is determined by the images only up a similarity transformation (rotation, translation and scaling) = s / 1 0 t R 1 0 t R 1 s s H s

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= s / 1 0 t R 1 0 t R 1 s s H s [ ] i i i i T R K M = j i j X M x = j j i j s 1 s i j x X M X H H M x ~ = = = j s X H 1 s i H M [ ] [ ] i 1 i i 1 s i i i 1 s i T R R K H T R K H M = = The calibration matrix has not changed!
• The ambiguity exists even for calibrated cameras Structure from motion ambiguity • For calibrated cameras, the similarity ambiguity is the only ambiguity [Longuet-Higgins ’81]

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Structure from motion ambiguity [ ] i i i i T R K M = j i j X M x = j X H 1 j H M • In the general case (nothing is known) the ambiguity is expressed by an arbitrary affine or projective transformation ( )( ) j i j i j X H H M X M x -1 = =
Projective ambiguity

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Affine ambiguity
Structure from motion problem Given m images of n fixed 3D points x ij = M i X j , i = 1 , … , m, j = 1 , … , n x 1 j x 2 j x m j X j M 1 M 2 M m

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Structure from motion problem mcameras M 1 …M m x 1 j x 2 j x m j X j M 1 M 2 M m = 1 a a a b a a a b a a a M 33 32 31 2 23 22 21 1 13 12 11 i
The Structure-from-Motion Problem Given m images of n fixed points X j we can write Problem: estimate the m 3 × 4 matrices M i and the n positions X j from the m × n correspondences x ij . • With no calibration info, cameras and points can only be recovered up to a 4x4 projective j i ij X M x = • Given two cameras, how many points are needed?

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This note was uploaded on 10/26/2010 for the course EECS 442 taught by Professor Savarese during the Fall '09 term at University of Michigan.

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lecture11 - EECS 442 Computer vision Multiple view geometry...

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