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# Computer Vision II CSE 252B

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This 5 page Class Notes was uploaded by Jacey Olson on Thursday October 22, 2015. The Class Notes belongs to CSE 252B at University of California - San Diego taught by Serge Belongie in Fall. Since its upload, it has received 32 views. For similar materials see /class/226789/cse-252b-university-of-california-san-diego in Computer Science and Engineering at University of California - San Diego.

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Date Created: 10/22/15

CSE 252B Computer Vision II Lecturer Serge Belongie Scribes Jia Mao Andrew Rabinovich LECTURE 9 Af ne and Euclidean Reconstruction 91 Strati ed reconstruction Recall that in 3D reconstruction from two uncalibrated views7 we can only obtain the structure of the scene up to a projective transformation denoted p The true structure X is related to Xp by Xp HpHageX where X5 95X differs from X by a Euclidean transformation gs RWTB7 Xa ageXe differs from X5 by a general af ne transforma tion7 and XI differs from Xa by a general projective transformation Figure 1 illustrates the different structures obtained at each stratum of the recon struction process 1Department of Computer Science and Engineering7 University of California7 San Diego April 26 2004 2 SERGE BELONGIE CSE 252B COMPUTER VISION II Figure 1 Projective structure Xp a ine structure X and Euclidean structure X 2 obtained in different stages of reconstruction MaSKS g610 In this lecture we will describe the strati ed reconstruction approach in 3D Namely rst recover the af ne structure Xa by identifying the matrix Hgl and then recover the Euclidean structure XE by identifying the matrix Hgl Moreover we will see that one can go directly from projective to Euclidean structure if we know the ground truth77 coordinates of 5 points in the scene Analogous to the strati ed reconstruction in 2D the tools for us to do the af ne and Euclidean upgrades in 2D and 3D are listed in Table 91 Af ne upgrade Euclidean upgrade I 2D line at in nity Zoo 001T circular points IJ Iii0T 3D plane at in nity woo 0001T absolute conic Zoo Table 1 Useful entities for 2D vs 3D strati ed reconstruction 92 Af ne upgrade The line at in nity Zoo is used to do af ne upgrade in the 2D reconstruction because it stays xed under an af ne transformation In the 3D case the plane at in nity 00 is the corresponding entity that allows us to achieve the same goal 921 Plane at in nity woo Parallelism is preserved in the af ne structure The plane at in nity woo enables us to identify parallelism In particular 0 Two planes are parallel iff the line of intersection is on 00 0 Two lines are parallel iff the point of intersection is on 00 LEXJTUEEB Armquot mo mqu moumvmou 3 pm on we ee me me e we m x new New me We 0amp0an o u we m menuty the mega dim mmd w new 011m we m an me We upyeda we I u H n e meme XeHquotXend rte W J A me W Free we nee a mm m We We we we en w m We meum emguey ee meme 9 2 2 meme We peepwe mm mm Meme e page he e me mm on me we pee me me Wampum e mm m e 2 he emmemmmem M meeeeumemme 3D We hgnr 2 Yardklnmauduumnkdwm mgym s mm meeememmmeuemBmmemm amenth peeve mmsmwtm mu In many mlmsea at me seme venuhmg Pam Hewevee mmemmg hues m an s mememem m we meme dnmse A en exmneme we can nd me an mdmsles d e vemshmg pm we mm m dslam me Wm meg meme leZ smt the fallowmg elgunhm a and venishmgpmnl 2quot m 1mg 1 mm epeu 0 mega pexenex hues 2 Campum epipnlsx hue m use 2 22 F2 me punt m the 5mm use correspondinch 4 must he an 011 hue a 1mm 2e wnh emsmmng pexella hm m xmea z een n 239 me that step 2 ensures lhsl the hackpmpc39ed Keys Lhmugq 4 end 239 m an wl mmsect 4 SERGE BELONGIE7 CSE 252B COMPUTER VISION H 923 71390 for a ine upgrade MaSKS Example 65 Now that we have three vanishing points Xi j 17 237 we can then solve for 39UT 17027037U4X0 j17273 which then fully speci es H51 IT 0 J as desired 4 114T from the linear system 1 1 Other alternative methods also exist for af ne upgrade7 with more of them coming out each year MaSKS 643 gives two examples7 one of which is to exploit the case of pure translation ie R I7 the other is to exploit the equal modulus constraint 93 Direct upgrade from X to Xe using groundtruth points An example of direct upgrade from projective to Euclidean structure in the 2D case is the four point algorithm aka DLT It utilizes four ground truth correspondences of which no three in each image are collinear to recover the planar homography matrix Similarly7 in 3D7 H has size 4 gtlt 47 which has 16 71for scale15 degrees of freedom Therefore7 5 ground truth points suf ce as long as they are in general position ie7 no 4 points are coplanar An example of this direct upgrade is shown in Figure 3 The analogous ve point algorithm is left as an exercise Figure 3 Direct upgrade from projective to Euclidean using 5 ground truth points7 Which are in general position Le7 no 4 points are coplanar HampZ Figure 96 94 The Absolute Conic In the absence of suf cient ground truth points7 a versatile alternative is to use the absolute conic 900 or its dual7 the absolute dual quadm39c Q What is the absolute conic The absolute conic lives on the plane at in nity 00 It cannot be seen 7 it can only be inferred It is a point conic LECTURE 9i AFFINE AND EUCLIDEAN RECONSTRUCTION 5 as opposed to a line conic points of the form X XYZ7 lVT on 900 satisfy 91 X2Y2ZZ 0 92 W 0 Thus for ideal points ie points with W 0 on 00 the de ning equation is 100 X XYZ010 Y0 001 Z From this we see 900 I This is an imaginary conic 941 Some Properties of the Absolute Conic 1 All spheres intersect the plane at in nity in the absolute conic 2 All circles intersect the absolute conic in two points Suppose a circle lives in a plane 7T Then 7T intersects 71390 in a line7 and that line intersects 900 in two points These two points are the circular points of the plane 7T 942 Absolute Conic and the Calibration Matrix Hartley has shown how to nd the image of the absolute conic lAC 9 S and factor it to get K7 the calibration matrix In particular7 9 KKT 1 This is the metric of the uncalibrated space Consider the relationship between points on the plane at in nity and the image plane We can write these points as X00 dT7 0T7 and when imaged by a general carnera7 we get d Aw HXOQ KRIT 0 J KRd which does not depend on T It only depends on K the calibration rnatrix7 and R the orientation wrt the world frarne7 which will drop out You can observe this phenomenon by looking out the window of a moving car at a distant point7 eg on a mountain or on the moon While objects near the car appear to move rapidly7 these distant points appear xed on your retina Recall that the absolute conic lives on 71390 and we know how to compute its image under the hornography H Under the mapping 2 Hana the conic C gt gt H TCH l7 so 900 I H KR TIKR 1 K TRRTK l KKT 1

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