Warping this will leave holes because every pixel

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Unformatted text preview: Affine Transformation •  Ax = B •  Cannot take inverse directly •  Also, 4 correspondences may not be exactly represented by an affine transformation [Why ?] ⎡ྎ x1 ⎢ྎ 0 ⎢ྎ ⎢ྎ x2 ⎢ྎ ⎢ྎ 0 ⎢ྎ x3 ⎢ྎ ⎢ྎ 0 ⎢ྎ x ⎢ྎ 4 ⎢ྎ 0 ⎣ྏ y1 0 1 0 0 x1 0 y1 y2 0 y3 10 0 x2 10 0 y2 0 0 y4 0 0 x3 10 0 x4 y3 0 y4 0⎤ྏ ⎡ྎ x1 ' ⎤ྏ 1⎥ྏ ⎡ྎ a1 ⎤ྏ ⎢ྎ y1 ' ⎥ྏ ⎥ྏ ⎢ྎ ⎥ྏ ⎢ྎa ⎥ྏ ⎢ྎ x '⎥ྏ 0⎥ྏ ⎢ྎ 2 ⎥ྏ 2 ⎥ྏ ⎢ྎ ⎥ྏ 1⎥ྏ ⎢ྎ a3 ⎥ྏ ⎢ྎ y2 '⎥ྏ ⎢ྎ ⎥ྏ = ⎢ྎ ⎥ྏ 0⎥ྏ ⎢ྎa4 ⎥ྏ x3 ' ⎥ྏ ⎢ྎ ⎥ྏ ⎢ྎ a5 ⎥ྏ ⎢ྎ y3 '⎥ྏ 1⎥ྏ ⎥ྏ ⎢ྎa ⎥ྏ ⎢ྎ x '⎥ྏ 0 ⎢ྎ 6 ⎥ྏ ⎥ྏ ⎣ྏ ⎦ྏ ⎢ྎ 4 ⎥ྏ 1⎥ྏ ⎢ྎ y4 '⎥ྏ ⎦ྏ ⎣ྏ ⎦ྏ Pseudo inverse For an over-constrained linear system Ax = B A has more rows than columns Multiply by AT on both sides ATAx = ATB ATA is a square matrix of as many rows as x We can take its inverse x = (ATA)-1ATB Pseudo-inverse gives the least squares error solution! [Proof?] Recovering Best Affine Transformation n༆  n༆  n༆  n༆  n༆  n༆  In general, we may be given n correspondences Co...
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This note was uploaded on 03/02/2014 for the course CS 436 taught by Professor Sohaibkhan during the Winter '13 term at Lahore University of Management Sciences.

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