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Set Solution 8: More Inner Product Spaces First Question. For any matrix A, det A = det AT . But for an orthogonal matrix, AT = A 1 . So det A = det AT = det A 1 = (det A) 1 . Since det A is real, the only choices are det A = 1. p.346, #3b. With respect to the standard (orthonormal) basis, the matrix for T is so the matrix for T is 2 1+i i 0 . So if x = (3 i, 1 + 2i), we have 2 i 1 i 0 , T x = (2(3 i) + (1 + i)(1 + 2i), i(3 i)) = (5 + i, 1 3i). p.347, #13. a) Clearly ker T T ker T , because if T (v) = 0, then T T (v) = T ( 0) = 0. To prove that ker T T ker T , let v ker T T . Then T T (v) = 0, so T T v, v = 0. Using the property of the adjoint, we have T (v), T (v) = 0, and using the positivity property of the inner product yields T (v) = 0 So v ker T , and thus ker T T ker T . Since ker T T ker T and ker T T ker T , we have ker T T = ker T . If V is n-dimensional, then rank T = n dim ker T = n dim ker T T = rank T T . b) Choose an orthonormal basis for V . Then rank T = rank [T ] and rank T = rank [T ] . So we are reduced to proving that for any matrix A, rank A = rank A . If A is real, then this is true because A = AT , and the rank of A is the same as the rank of AT . If A has complex entries that are not real, we rst note that rank A = rank AT , and then we note that linear relations among the columns of a matrix are the same as linear relations among the conjugates of the columns of a matrix, provided we conjugate all of the coe cients. So the number of linearly independent columns in A is the same as the number of linearly independent columns of AT , that is, rank AT = rank A . Thus rank A = rank A . We conclude that rank T = rank T . From part (a), we have rank T T = rank (T ) T = rank T . From the preceeding paragraph, we have rank T = rank T . So rank T T = rank T . c) Given an n n matrix A, rank A = rank LA = rank L LA = rank LA A = rank A A, A where the second equality is from part (a) and the third equality follows from the fact that L = LA and the fact that composition of linear transformations corresponds to multiplication A of matrices. Likewise, rank A = rank LA = rank LA L = rank LAA = rank AA . A p.354, #2. a) The matrix of T with respect to the standard basis is 2 2 . This 2 5 is symmetric and real, so it is self-adjoint. So it is certainly normal. Since the matrix (with respect to an orthonormal basis) is normal and self-adjoint, The same holds for the operator. b) The matrix of T with respect to the standard basis is A = A = A, so T is not self-adjoint. However, AA = A A = T , is normal. 1 2i 21 . Since A = , 12 i 2 5 2 + 2i . So A, and thus 2 2i 5 c) We need to nd an orthonormal basis to write down the matrix for T . From the previous homework, we know that 1 and 12(x 1/2) are orthonormal. So to complete the orthonormal basis, we choose the vector x2 and then apply Gram-Schmidt, that is, we apply Gram-Schmidt to the basis 1, 12(x 1/2), x2 . We know that it will not a ect the frist two elements, because they are orthonormal. The third vector becomes x2 x2 , 1 x2 , 12(x 1/2) ( 12(x 1/2)) , x2 x2 , 1 x2 , 12(x 1/2) ( 12(x 1/2)) We which comes to 180(x2 x + 1/6). now have our orthonormal basis . Then, since out T (1) = 0, T ( 12(x 1/2)) = 12, and T ( 180(x2 x + 1/6)) = 2 180(x 1/2), we have that the matrix [T ] is equal to 0 12 0 0 0 2 15 . 00 0 12 0 0 Let A = [T ] . Then A = A , so T is not self adjoint. Also AA = 0 60 0 but 000 000 0 12 0 . Since they are not equal, T is not normal. A A= 0 0 60 #3. p.354, We know that (T U ) = U T . Since T and U are self-adjoint, whis is equal to U T . So (T U ) = U T , and thus T U is self-adjoint i T U = U T . 2 3 3i is (t 8)(t + 1), so the 3 + 3i 5 i 1 1 eigenvalues are 1 and 8. A basis for E 1 is { } and a basis for E8 is { }. 1 1+i (i 1)/ 3 1/ 3 Normalizing gives and . So these form the columns of the uni1/ 3 (1 + i)/ 3 (i 1)/ 3 1 0 1/ 3 tary matrix P = . In this case, P AP = , which is 08 1/ 3 (1 + i)/ 3 diagonal and has the eigenvalues along the diagonal. 1/3 2/3 2/3 1/ 1/ 2 . I found this by trial and 2 p.370, #11. One such matrix is 0 4/ 6 1/ 6 1/ 6 error, lling in each column at a time so that I would have orthonormal columns. But one can also take a basis for R3 where the rst element is (1/3, 2/3, 2/3), apply Gram-Schmidt, and use the resulting orthonormal basis to be the rows of the matrix. p.369, #2c. The characteristic polynomial of 31 . Since it is real sym13 metric, we can diagonalize it using an orthonormal basis of eigenvectors. The characteristic polynomial is (t 2)(t 4), so the eigenvalues are 2 and 4. The corresponding (normalized) p.372, #21d. The matrix of this quadratic form is A = 2 1/ 2 1/ 2 1/ 2 and . So if we let Q = , then Q is eigenvectors are 1/ 2 1/ 2 1/ 2 20 orthogonal, and Q 1 AQ = QT AQ = . As we showed in class on Wednesday, if we let 04 x x = QT , then the quadratic form becomes 2(x )2 + 4(y )2 . y y 1/ 2 1/ 2 p.406, #4. a) Yes, this is a bilinear form, as we have that both (cf1 + f2 )g = c and f (cg1 + g2 ) = c f g1 + f g2 . c) Not bilinear: For instance H(1, 2) = 5, but H(1, 1) + H(1, 1) = 6. d) Yes, this is bilinear, as one of the characterizing properties of the determinant is that it is linear in each column while the others are held xed. Second Problem. We see that H(cf1 + f2 , g) = (cf1 (3) + f2 (3))g(4) = cf1 (3)g(4) + f2 (3)g(4) = cH(f1 , g)+H(f2 , g). Likewise, H(f, cg1 +g2 ) = f (3)(cg1 (4)+g2 (4)) = cf (3)(g1 (4))+f (3)(g2 (4)) = cH(f, g1 ) + H(f, g2 ). So H is bilinear. Plugging in the pairs of basis vectors, we nd that the matrix (H) is given by: 1 4 16 3 12 48 . 9 36 144 f1 g + f2 g Extra Credit: pp.356-7, #16. Assume that two matrices M and N are similar, say M = Q 1 N Q. Then their characteristic polynomials are the same (call them f (t)), and f (M ) = Q 1 f (N )Q. So we see that f (M ) = 0 if and only if f (N ) = 0. So the CayleyHamilton theorem holds for M if and only if it holds for N . Thus, using Schur s theorem, we can replace A by a similar matrix which is upper triangular (such a matrix exists because the characteristic polynomial of A must split over the complex numbers). So assume that A is upper triangular. Then the characteristic polynomial of A is just the product of the diagonal entries of A tI, or n f (t) = i=1 n i=1 (Aii I (Aii t). n i=1 (Aii I A) = 0. To do this, it will su ce to show that We wish to show that A)(v) = 0 for all vectors v. Write this out as (A11 I A)(A22 I A) (Ann I A)(v) = 0. ? Let (e1 , . . . , en ) be the standard basis for Cn . Notice that (Ajj I A)(ei ) lies in the span of (e1 , . . . , ej 1 ) for any i j (this can be veri ed by writing down the matrix and performing the multiplication). This means that applying (Ann I A) to v yields something that lies in the span of (e1 , . . . , en 1 ). Then, applying An 1,n 1 I A to the result must yield something that lies in the span of (e1 , . . . , en 2 ). This process repeats itself until we see that nally applying A11 I A yields something that lies in the span of the empty set. Thus it must yield 0. So n i=1 (Aii I A) = 0, and we have the Cayley-Hamilton theorem. 3
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Path: UPenn >> SG0 >> 22 Fall, 2008
Description: Structural Studies of Model Metalloprotein Maquettes VectoriallyX22B Oriented at the Air/Water Interface J. Strzalka, X. Chen, C. C. Moser, P. L. Dutton and J. K. Blasie (U. Pennsylvania) and B. M. Ocko (BNL) De novo synthetic models or maquettes of...
x22b_abs98.pdf
Path: UPenn >> SG0 >> 22 Fall, 2008
Description: ...
makadia07ijcv.pdf
Path: UPenn >> CIS >> 07 Fall, 2008
Description: International Journal of Computer Vision 75(3), 311327, 2007 c 2007 Springer Science + Business Media, LLC. Manufactured in the United States. DOI: 10.1007/s11263-007-0035-2 Correspondence-free Structure from Motion AMEESH MAKADIA, University of Pen...
hicks06ao.pdf
Path: UPenn >> CIS >> 06 Fall, 2008
Description: Realizing any central projection with a mirror pair R. Andrew Hicks, Marc Millstone, and Kostas Daniilidiis We show that, for any rotationally symmetric projection with a single virtual viewpoint, it is possible to design a two-mirror rotationally s...
makadia06pami.pdf
Path: UPenn >> CIS >> 06 Fall, 2008
Description: 1170 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 28, NO. 7, JULY 2006 Rotation Recovery from Spherical Images without Correspondences Ameesh Makadia, Student Member, IEEE, and Kostas Daniilidis, Senior Member, IEEE Abstra...
yu05ivc.pdf
Path: UPenn >> CIS >> 05 Fall, 2008
Description: Image and Vision Computing 23 (2005) 377392 www.elsevier.com/locate/imavis Using skew Gabor lter in source signal separation and local spectral orientation analysis Weichuan Yua,*, Gerald Sommerb, Kostas Daniilidisc, James S. Duncand Department of D...
isler04pami.pdf
Path: UPenn >> CIS >> 04 Fall, 2008
Description: IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 26, NO. 5, MAY 2004 1 VC-Dimension of Exterior Visibility Volkan Isler, Student Member, IEEE, Sampath Kannan, Kostas Daniilidis, Member, IEEE, and Pavel Valtr AbstractIn this pa...
ansar03pami.pdf
Path: UPenn >> CIS >> 03 Fall, 2008
Description: IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 25, NO. 4, APRIL 2003 1 Linear Pose Estimation from Points or Lines Adnan Ansar and Kostas Daniilidis, Member, IEEE AbstractEstimation of camera pose from an image of n points o...
mulligan03csvt.pdf
Path: UPenn >> CIS >> 03 Fall, 2008
Description: 304 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 14, NO. 3, MARCH 2004 Stereo-Based Environment Scanning for Immersive Telepresence Jane Mulligan, Xenophon Zabulis, Nikhil Kelshikar, and Kostas Daniilidis, Member, IEEE Abst...
yu03ivc.pdf
Path: UPenn >> CIS >> 03 Fall, 2008
Description: Image and Vision Computing 21 (2003) 447458 www.elsevier.com/locate/imavis Three dimensional orientation signatures with conic kernel ltering for multiple motion analysis Weichuan Yua,*, Gerald Sommerb, Kostas Daniilidisc a Department of Diagnostic...
yu03cviu.pdf
Path: UPenn >> CIS >> 03 Fall, 2008
Description: Computer Vision and Image Understanding 90 (2003) 129152 www.elsevier.com/locate/cviu Multiple motion analysis: in spatial or in spectral domain? Weichuan Yu,a,* Gerald Sommer,b and Kostas Daniilidisc b Department of Diagnostic Radiology, Yale Univ...
geyer02pami.pdf
Path: UPenn >> CIS >> 02 Fall, 2008
Description: IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 24, NO. 5, MAY 2002 687 Short Papers _ Paracatadioptric Camera Calibration Christopher Geyer, Member, IEEE, and Kostas Daniilidis, Member, IEEE AbstractCatadioptric sensors re...
mulligan01ijcv.pdf
Path: UPenn >> CIS >> 01 Fall, 2008
Description: Trinocular Stereo: a Real-Time Algorithm and its Evaluation Jane Mulligan Dept. of Computer Science University of Colorado at Boulder janem@cs.colorado.edu Volkan Isler and Kostas Daniilidis University of Pennsylvania GRASP Laboratory {isleri,kostas}...
yu02pami.pdf
Path: UPenn >> CIS >> 02 Fall, 2008
Description: 1286 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 24, NO. 9, SEPTEMBER 2002 Oriented Structure of the Occlusion Distortion: Is It Reliable? Weichuan Yu, Member, IEEE, Gerald Sommer, Steven Beauchemin, Member, IEEE, and Ko...
geyer01ijcv.pdf
Path: UPenn >> CIS >> 01 Fall, 2008
Description: International Journal of Computer Vision 45(3), 223243, 2001 c 2001 Kluwer Academic Publishers. Manufactured in The Netherlands. Catadioptric Projective Geometry CHRISTOPHER GEYER AND KOSTAS DANIILIDIS University of Pennsylvania, GRASP Laboratory, 3...
ansar01cg.pdf
Path: UPenn >> CIS >> 01 Fall, 2008
Description: Computers & Graphics 25 (2001) 789798 Visual and haptic collaborative tele-presence Adnan Ansara,*, Denilson Rodriguesb, Jaydev P. Desaic, Kostas Daniilidisa, Vijay Kumara, Mario F.M. Camposd a GRASP Laboratory, University of Pennsylvania, Suite 30...
jmiv00.pdf
Path: UPenn >> CIS >> 00 Fall, 2008
Description: Complex Analysis for Reconstruction from Controlled Motion? R. Andrew Hicks David Pettey Kostas Daniilidis Ruzena Bajcsy GRASP Laboratory, Department of Computer and Information Science University of Pennsylvania frah, djpettey, kostasg@grip.cis.upe...
ijrr99.pdf
Path: UPenn >> CIS >> 99 Fall, 2008
Description: Konstantinos Daniilidis GRASP Laboratory University of Pennsylvania Philadelphia, Pennsylvania 19104-6228, USA kostas@grip.cis.upenn.edu Hand-Eye Calibration Using Dual Quaternions Abstract To relate measurements made by a sensor mounted on a mecha...
cviu97.pdf
Path: UPenn >> CIS >> 97 Fall, 2008
Description: Fixation simpli es 3D motion estimation A similar version appeared in the journal COMPUTER VISION AND IMAGE UNDERSTANDING 68:158-169, 1997. Konstantinos Daniilidis Computer Science Institute University of Kiel Preusserstr. 1-9, 24105 Kiel, Germany FA...
computing96.pdf
Path: UPenn >> CIS >> 96 Fall, 2008
Description: Computing (Suppl.) 11 (1995) 1-20 Special Issue on Theoretical Foundations of Computer Vision. Attentive visual motion processing: computations in the log-polar plane Kostas Daniilidis Computer Science Institute Christian-Albrechts University Kiel P...
ijcv93.pdf
Path: UPenn >> CIS >> 93 Fall, 2008
Description: Model-Based Object Tracking in Monocular Image Sequences of Road Trafc Scenes SIMILAR VERSION PUBLISHED IN INTERNATIONAL JOURNAL OF COMPUTER VISION 10:3 (1993) 257281. D. Kollery , K. Daniilidisy and H.-H. Nagelyz y Institut fur Algorithmen und Kogn...
makadia06cvpr.pdf
Path: UPenn >> CIS >> 06 Fall, 2008
Description: Fully Automatic Registration of 3D Point Clouds Ameesh Makadia, Alexander Patterson IV, and Kostas Daniilidis Department of Computer and Information Science University of Pennsylvania Philadelphia, PA, 19104, USA {makadia, aiv, kostas}@cis.upenn.edu ...
carceroni06cvpr.pdf
Path: UPenn >> CIS >> 06 Fall, 2008
Description: Structure from Motion with Known Camera Positions Rodrigo Carceroni Ankita Kumar Kostas Daniilidis Dept. of Computer & Information Science University of Pennsylvania 3330 Walnut Street, Philadelphia, PA 19104 {carceron,ankitak,kostas}@grasp.cis.upe...
nima06icra.pdf
Path: UPenn >> CIS >> 06 Fall, 2008
Description: Vision-based Control Laws for Distributed Flocking of Nonholonomic Agents Nima Moshtagh, Ali Jadbabaie, Kostas Daniilidis GRASP Laboratory, University of Pennsylvania, Philadelphia, PA 19104 Email: {nima, jadbabai, kostas}@grasp.upenn.edu Abstract We...
nima05cdc.pdf
Path: UPenn >> CIS >> 05 Fall, 2008
Description: Distributed Geodesic Control Laws for Flocking of Nonholonomic Agents Nima Moshtagh, Ali Jadbabaie, Kostas Daniilidis Abstract We study the problem of ocking and coordination of a group of kinematic nonholonomic agents in 2 and 3 dimensions. By anal...
mariottini05cdc.pdf
Path: UPenn >> CIS >> 05 Fall, 2008
Description: Vision-based Localization of Leader-Follower Formations Gian Luca Mariottini, George Pappas, Domenico Prattichizzo, Kostas Daniilidis Abstract This paper focuses on the localization problem for a mobile camera network. In particular, we consider the...
barreto05iccv.pdf
Path: UPenn >> CIS >> 05 Fall, 2008
Description: Fundamental Matrix for Cameras with Radial Distortion Joao P. Barreto Dept. of Electrical and Computer Engineering University of Coimbra 3030 Coimbra, Portugal jpbar@deec.uc.pt Kostas Daniilidis Dept. of Computer and Information Science University of...
nima05rss.pdf
Path: UPenn >> CIS >> 05 Fall, 2008
Description: Vision-based Distributed Coordination and Flocking of Multi-agent Systems Nima Moshtagh, Ali Jadbabaie, Kostas Daniilidis GRASP Laboratory University of Pennsylvania Philadelphia, Pennsylvania 19104 Email: {nima, jadbabai, kostas}@grasp.upenn.edu Ab...
zabulis05-3dim.pdf
Path: UPenn >> CIS >> 05 Fall, 2008
Description: Submitted for review to 3DIM2003 Digitizing Archaeological Excavations from Multiple Views Xenophon Zabulis Alexander Patterson Informatics and Telematics Institute University of Pennsylvania Thessaloniki, Greece Philadelphia, USA xenophon@iti.gr ai...
makadia05icra.pdf
Path: UPenn >> CIS >> 05 Fall, 2008
Description: Correspondenceless Ego-Motion Estimation Using an IMU Ameesh Makadia and Kostas Daniilidis GRASP Laboratory University of Pennsylvania, Philadelphia, PA 19104 {makadia, kostas}@grasp.cis.upenn.edu Abstract Mobile robots can be easily equipped with nu...
makadia05motion.pdf
Path: UPenn >> CIS >> 05 Fall, 2008
Description: Planar Ego-Motion Without Correspondences Ameesh Makadia, Dinkar Gupta, and Kostas Daniilidis GRASP Laboratory Department of Computer and Information Science University of Pennsylvania, Philadelphia, PA 19104 {makadia, dinkar, kostas}@cis.upenn.edu ...
barreto04liftings.pdf
Path: UPenn >> CIS >> 04 Fall, 2008
Description: Unifying Image Plane Liftings for Central Catadioptric and Dioptric Cameras Jo o P. Barreto and Kostas Daniilidis a GRASP Laboratory, University of Pennsylvania, Philadelphia PA, 19104 ISR/DEEC, University of Coimbra, Coimbra, Portugal jpbar,kosta...
barreto04calibration.pdf
Path: UPenn >> CIS >> 04 Fall, 2008
Description: Wide Area Multiple Camera Calibration and Estimation of Radial Distortion Jo o P. Barreto and Kostas Daniilidis a GRASP Laboratory, University of Pennsylvania, Philadelphia PA, 19104 ISR/DEEC, University of Coimbra, Coimbra, Portugal jpbar,kostas ...
zabulis04-3dpvt.pdf
Path: UPenn >> CIS >> 04 Fall, 2008
Description: Multi-camera reconstruction based on surface normal estimation and best viewpoint selection Xenophon Zabulis and Kostas Daniilidis GRASP Laboratory, Levine Hall L402, University of Pennsylvania, 3330 Walnut Street, Philadelphia, PA 19104-6228, USA za...
geyer03iccv.pdf
Path: UPenn >> CIS >> 03 Fall, 2008
Description: Mirrors in motion: Epipolar geometry and motion estimation Christopher Geyer University of California, Berkeley Berkeley, CA 94720 cgeyer@eecs.berkeley.edu Kostas Daniilidis University of Pennsylvania Philadelphia, PA 19104 kostas@cis.upenn.edu regi...
makadia03cvpr.pdf
Path: UPenn >> CIS >> 03 Fall, 2008
Description: Direct 3D-Rotation Estimation from Spherical Images via a generalized shift theorem Ameesh Makadia and Kostas Daniilidis Department of Computer and Information Science University of Pennsylvania Abstract Omnidirectional images arising from 3D-motion...
kelshikar03iccs.pdf
Path: UPenn >> CIS >> 03 Fall, 2008
Description: Real-time Terascale Implementation of Tele-immersion Nikhil Kelshikar1 , Xenophon Zabulis1 , Jane Mulligan4, Kostas Daniilidis1 , Vivek Sawant2 , Sudipta Sinha2 , Travis Sparks2 , Scott Larsen2 , Herman Towles2 , Ketan Mayer-Patel2 , Henry Fuchs2 , J...
isler03icra.pdf
Path: UPenn >> CIS >> 03 Fall, 2008
Description: Local Exploration: Online Algorithms and a Probabilistic Framework Volkan Isler, Sampath Kannan, and Kostas Daniilidis Department of Computer and Information Science, University of Pennsylvania, Philadelphia, PA, 19104 {isleri,kannan,kostas}@cis.upen...
geyer02eccv.pdf
Path: UPenn >> CIS >> 02 Fall, 2008
Description: Properties of the Catadioptric Fundamental Matrix Christopher Geyer and Kostas Daniilidis GRASP Laboratory, University of Pennsylvania, Philadlelphia, PA 19104 {cgeyer,kostas}@seas.upenn.edu Abstract. The geometry of two uncalibrated views obtained ...
geyer99iccv.pdf
Path: UPenn >> CIS >> 99 Fall, 2008
Description: Catadioptric Camera Calibration Christopher Geyer and Kostas Daniilidis GRASP Laboratory, Computer and Information Science Department University of Pennsylvania, fcgeyer,kostasg@grip.cis.upenn.edu Proceedings International Conference on Computer Visi...
dani02omnivis.pdf
Path: UPenn >> CIS >> 02 Fall, 2008
Description: Image Processing in Catadioptric Planes: Spatiotemporal Derivatives and Optical Flow Computation Kostas Daniilidis and Ameesh Makadia GRASP Laboratory and Department of Computer and Information Science University of Pennsylvania Thomas B low u Depart...
makadia08eccv.pdf
Path: UPenn >> CIS >> 08 Fall, 2008
Description: A New Baseline for Image Annotation Ameesh Makadia1 , Vladimir Pavlovic2, and Sanjiv Kumar1 Google Research, New York, NY Rutgers University, Piscataway, NJ makadia@google.com,vladimir@cs.rutgers.edu,sanjivk@google.com 2 1 Abstract. Automatically as...
makadia073dtv.pdf
Path: UPenn >> CIS >> 073 Fall, 2008
Description: HARMONIC SILHOUETTE MATCHING FOR 3D MODELS Ameesh Makadia, Mirk Visontai, and Kostas Daniilidis o Department of Computer and Information Science University of Pennsylvania Philadelphia, PA, 19104, USA {makadia, mirko, kostas}@cis.upenn.edu sections. ...
makadia06ijcvsubmitted.pdf
Path: UPenn >> CIS >> 06 Fall, 2008
Description: Correspondenceless Structure from Motion Ameesh Makadia and Kostas Daniilidis University of Pennsylvania Philadelphia, PA 19104 {makadia, kostas}@cis.upenn.edu Abstract We present a novel approach for the estimation of 3D-motion directly from two im...
makadia06pami.pdf
Path: UPenn >> CIS >> 06 Fall, 2008
Description: Rotation Recovery from Spherical Images without Correspondences Ameesh Makadia and Kostas Daniilidis GRASP Laboratory, University of Pennsylvania, Philadelphia, PA 19104 {makadia, kostas}@cis.upenn.edu Abstract This paper addresses the problem of r...
makadia05cvpr.pdf
Path: UPenn >> CIS >> 05 Fall, 2008
Description: Radon-based Structure from Motion Without Correspondences Ameesh Makadia Christopher Geyer University of Pennsylvania Philadelphia, PA 19104 {makadia, kostas}@cis.upenn.edu Abstract We present a novel approach for the estimation of 3Dmotion directly...
makadia05icra.pdf
Path: UPenn >> CIS >> 05 Fall, 2008
Description: Correspondenceless Ego-Motion Estimation Using an IMU Ameesh Makadia and Kostas Daniilidis GRASP Laboratory University of Pennsylvania, Philadelphia, PA 19104 {makadia, kostas}@grasp.cis.upenn.edu Abstract Mobile robots can be easily equipped with nu...
makadia05motion.pdf
Path: UPenn >> CIS >> 05 Fall, 2008
Description: Planar Ego-Motion Without Correspondences Ameesh Makadia, Dinkar Gupta, and Kostas Daniilidis GRASP Laboratory Department of Computer and Information Science University of Pennsylvania, Philadelphia, PA 19104 {makadia, dinkar, kostas}@cis.upenn.edu ...
makadia04icpr.pdf
Path: UPenn >> CIS >> 04 Fall, 2008
Description: Rotation Estimation from Spherical Images Ameesh Makadia, Lorenzo Sorgi and Kostas Daniilidis Department of Computer and Information Science University of Pennsylvania Abstract Robotic navigation algorithms increasingly make use of the panoramic eld...
makadia03cvpr.pdf
Path: UPenn >> CIS >> 03 Fall, 2008
Description: Direct 3D-Rotation Estimation from Spherical Images via a Generalized Shift Theorem Ameesh Makadia and Kostas Daniilidis GRASP Laboratory, University of Pennsylvania, Philadelphia, PA 19104 {makadia, kostas}@cis.upenn.edu Abstract Omnidirectional i...
dani02omnivis.pdf
Path: UPenn >> CIS >> 02 Fall, 2008
Description: Image Processing in Catadioptric Planes: Spatiotemporal Derivatives and Optical Flow Computation Kostas Daniilidis and Ameesh Makadia GRASP Laboratory and Department of Computer and Information Science University of Pennsylvania Thomas B low u Depart...
Math371_2008HW1.pdf
Path: UPenn >> MATH >> 371 Fall, 2008
Description: Due: Friday, Sept 19, 2008 Math 371 / Problem Set 1 1) Let X be a non-empty set, and P(X) its power set. Dene on P(X) the following composition laws: The symmetric dierence AB := (A\\B) (B\\A), and the usual intersection A B := A B, A, B P(X). Pro...
Math371_2008HW2.pdf
Path: UPenn >> MATH >> 371 Fall, 2008
Description: Due: Friday, Sept 26, 2008 Math 371 / Problem Set 2 1) Let (G, ) be a group. Prove or disprove: a) (x1 )1 = x for all x G; and if G is abelian, then (x y)n = xn y n for all x, y G. b) (x1 . . . xn )1 = x1 . . . x1 for all x1 , . . . , xn G. ...
Math371_2008HW3.pdf
Path: UPenn >> MATH >> 371 Fall, 2008
Description: Due: Friday, Oct 3, 2008 Math 371 / Problem Set 3 Products of groups/rings/elds In the sequel (G1 , ), (G2 , ) are groups, and G = G1 G2 is their product endowed with the composition law as dened in class (x1 , x2 ) (y1 , y2 ) := (x1 y1 , x2 y2 ). ...
Math371_2008HW4.pdf
Path: UPenn >> MATH >> 371 Fall, 2008
Description: Due: Friday, Oct 10, 2008 Math 371 / Problem Set 4 (two pages) Polynomial functions Let R S be rings with identity, and R[X] be the ring of polynomials in the variable X over R. Further let F(S, S) be the ring of all the maps from f : S S as introd...
Math371_2008HW5.pdf
Path: UPenn >> MATH >> 371 Fall, 2008
Description: Due: Friday, Oct 17, 2008 Math 371 / Problem Set 5 Modules and vector spaces (continued) 1) Let M be an R-module, and N M be an R-submodule. We set M := M/N and denote its elements by x := x + N for all x M . Suppose that x1 , x2 N are generators ...
Math371_2008HW6.pdf
Path: UPenn >> MATH >> 371 Fall, 2008
Description: Due: Friday, Oct 24, 2008 Math 371 / Problem Set 6 1) Let M be an R-module, and recall that for a subset T M , we denote by < T >R the R-submodule of M generated by T . Let X, Y M be subsets. Prove or disprove the following: a) < < X >R >R = < X >R...
Math371_2008HW7.pdf
Path: UPenn >> MATH >> 371 Fall, 2008
Description: Due: Friday, Oct 31, 2008 Math 371 / Problem Set 7 1) Let V1 , . . . , Vn be F -vector spaces, and let fi : Vi Vi+1 be an F -linear maps such that Ker(fi+1 ) = Im(fi ) for i = 1, . . . , n 1, and suppose that fn1 is surjective. a) Prove that if n =...
Math371_2008HW8.pdf
Path: UPenn >> MATH >> 371 Fall, 2008
Description: Due: Friday, Nov 7, 2008 Math 371 / Problem Set 8 Division in polynomial rings In the sequel, R is a commutative ring with 1R , and R[X] is the R-algebra of polynomials over R. Recall that a polynomial p(X) = an X n + . . . a0 R[X] with an = 0R is c...
Math371_2008HW9.pdf
Path: UPenn >> MATH >> 371 Fall, 2008
Description: Due: Friday, Nov 14, 2008 Math 371 / Problem Set 9 R-multilinear forms and Determinants 1) Form m, n 0, let I := { | = (i1 , . . . , in ), 1 i1 , . . . , in m} be as in the class. For every Sn dene a map : I I, (i1 , . . . , in ) (i(1) , . ....
Math371_2008HW10.pdf
Path: UPenn >> MATH >> 371 Fall, 2008
Description: Due: Friday, Nov 21, 2008 Math 371 / Problem Set 10 On the transpose Let R be a commutative ring with 1R . Let M, N be free R-modules, with R-bases A = (1 , . . . , m ), respectively B = (1 , . . . , n ), and let M , N be the R-dual modules, endowe...
Math371_2008HW11.pdf
Path: UPenn >> MATH >> 371 Fall, 2008
Description: Due: Friday, Nov 28, 2008 Math 371 / Problem Set 11 Diagonalization/Rank of a matrix/Linear systems 1) For A Mnm (F ), nd elementary transformation T1 , . . . , Tp Mnn (F ), S1 , . . . , Sq Mmm (F ) D such that Tp . . . T1 AS1 . . . Sq has the can...
Math371_2008HW12.pdf
Path: UPenn >> MATH >> 371 Fall, 2008
Description: Due: Friday, Dec 5, 2008 Math 371 / Problem Set 12 Roots of polynomials 1) Prove or disprove / answer the following: a) Every eld extension of degree 2 of Q is of the form K = Q[ d], where d Z is a square free integer. b) If d1 , d2 Z are two sq...
ExmnfnL.pdf
Path: UPenn >> MATH >> 371 Fall, 2008
Description: Due: Mo, Dec 15, 2008, at 3:00 PM Math 371 / Final Exam IMPORTANT: Points for each problem: 4pt for part a) / 6pt for part b) / 7pt for part a) and b) Grading: 20 < C, C, C+ 30 < B, B, B+ 40 < A, A, A+ 211 121 112 Consider the matrix A = 1) Answer...
Math502_2008PS1.pdf
Path: UPenn >> MATH >> 502 Fall, 2008
Description: Due: Friday, Sept 19, 2008 Math 502 / Problem Set 1 All the homework is from Hungefords Algebra book. Problems 5, 9, 10, 11, at the end of Introduction. Prove that the addition + and the multiplication on the natural numbers N as dened in class is...
Math502_2008PS2.pdf
Path: UPenn >> MATH >> 502 Fall, 2008
Description: Due: Friday, Sept 26, 2008 Math 502 / Problem Set 2 1) Unswer/prove the following: a) Let M be endowed with a composition law which is associative and satises: exists e M such that x e = x for all x M , and for every x M there exists x M such t...
Math502_2008PS3.pdf
Path: UPenn >> MATH >> 502 Fall, 2008
Description: Due: Friday, Oct 3, 2008 Math 502 / Problem Set 3 Recall that for a group G and a subgroup H, we denote by G/H and H\\G the set of left, respectively right, cosets of G with respect to H, and we say that |H\\G| =: (G : H) := |G/H| is the index of H in ...
Math502_2008PS4.pdf
Path: UPenn >> MATH >> 502 Fall, 2008
Description: Due: Friday, Oct 10, 2008 Math 502 / Problem Set 4 Groups/algebras of functions Let (M, ) be a set endowed with a composition law, X a non-empty set, and F(X, M ) be the set of all the maps f : X M . Dene a composition law on F(X, M ) as follows: Fo...
Math502_2008PS5.pdf
Path: UPenn >> MATH >> 502 Fall, 2008
Description: Due: Friday, Oct 17, 2008 Math 502 / Problem Set 5 Universal property of the polynomial ring/polynomial functions Let R be a commutative ring with 1R , and R[X] be the ring of polynomials in the variable X over R, viewed as an R-algebra via the canon...
Math502_2008PS6.pdf
Path: UPenn >> MATH >> 502 Fall, 2008
Description: Due: Friday, Oct 24, 2008 Math 502 / Problem Set 6 Problems 1, 4, 8, 9, 15, at the end of 4, Ch.3, of Hungefords Algebra book. Problems 1, 7, 10, at the end of 5, Ch.3, of Hungefords Algebra book. Problems 1, 2, 3, at the end of 1, Ch.4, of Hungef...
