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Announcements ã Class mailing list: send email to Hyoungjune Yi: aster@cs.umd.edu ã Homework at the end of class. ã Text is on reserve in the CS library.…

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Announcements ã Class mailing list: send email to Hyoungjune Yi: aster@cs.umd.edu ã Homework at the end of class. ã Text is on reserve in the CS library. ã Powerpoint should be available by 10am class day. Matlab tutorial and Linear Algebra Review ã Today’s goals: ã Learn enough matlab to get started. ã Review some basics of Linear Algebra ã Essential for geometry of points and lines. ã But also, all math is linear algebra. ã (ok slight exaggeration). ã Many slides today adapted from Octavia Camps, Penn State. Starting Matlab ã For PCs, Matlab should be a program. ã For Sun’s: Numerical Analysis and Visualization Matlab 6.1 Help ã help ã help command Eg., help plus ã Help on toolbar ã demo ã Tutorial: http://amath.colorado.edu/scico/tutorials /matlab/ Matlab interpreter ã Many common functions: see help ops Vectors ã Ordered set of v ( x1 , x2 , , xn ) numbers: (1,2,3,4) n 2 v x ã Example: (x,y,z) i 1 i coordinates of pt in space. If v 1, v is a unit vecto r Indexing into vectors Vector Addition v w ( x1 , x2 ) ( y1 , y2 ) ( x1 y1 , x2 y2 ) V+w v w Scalar Product av a( x1 , x2 ) (ax1 , ax2 ) av v Operations on vectors ã sum ã max, min, mean, sort, … ã Pointwise: .^ Inner (dot) Product v w v.w ( x1 , x2 ).( y1 , y2 ) x1 y1 x2 . y2 The inner product is a SCALAR! v.w ( x1 , x2 ).( y1 , y2 ) || v || || w || cos v.w 0 v w Matrices a11 a12 a1m a a2 m Sum: 21 a22 Cnm Anm Bnm Anm a31 a32 a3m cij aij bij an1 an 2 anm A and B must have the same dimensions Matrices Product: A and B must have Cn p Anm Bm p compatible dimensions m cij aik bkj k 1 Ann Bnn Bnn Ann 1 0 0 Identity Matrix: 0 1 0 I IA AI A 0 0 1 Matrices Transpose: Cmn A nm T ( A B) A B T T T cij a ji ( AB) B A T T T If AT A A is symmetric Matrices Determinant: A must be square a11 a12 a11 a12 det a11a22 a21a12 a21 a22 a21 a22 a11 a12 a13 det a21 a22 a22 a23 a21 a23 a21 a22 a23 a11 a12 a13 a32 a33 a31 a33 a31 a32 a31 a32 a33 Matrices Inverse: A must be square 1 1 Ann A nn A nn Ann I 1 a11 a12 1 a22 a12 a a 21 a 22 a11a22 a21a12 21 a11 Indexing into matrices Euclidean transformations 2D Translation P’ t P 2D Translation Equation P’ t ty P P ( x, y ) y t (t x , t y ) x tx P' ( x t x , y t y ) Pt 2D Translation using Matrices P’ P ( x, y ) t ty P t (t x , t y ) y t P x tx x x t x 1 0 t x P' y y t y 0 1 t y 1 Scaling P’ P Scaling Equation P’ s.y P ( x, y ) P P' ( sx, sy ) y P' s P x s.x sx s 0 x P' sy 0 s y P' S P S Rotation P P’ Rotation Equations Counter-clockwise rotation by an angle x' cos sin x P’ y ' sin cos y Y’ P y P' R.P x X’ Degrees of Freedom x' cos sin x y ' sin cos y R is 2x2 4 elements BUT! There is only 1 degree of freedom: The 4 elements must satisfy the following constraints: R RT RT R I det( R ) 1 Stretching Equation P ( x, y ) P’ P ' ( s x x, s y y ) Sy.y P sx x sx 0 x P' y s y 0 sy y y x Sx.x S P' S P Stretching = tilting and projecting (with weak perspective) sx sx x sx 0 x s 0 x P' s y y y y s s y 0 y y 0 1 Linear Transformation a b x P' c d y SVD sin cos s x 0 sin cos x 0 cos sin s y cos sin y sx sin cos 0 sin cos x sy sin y sy cos sin cos 0 1 Affine Transformation x a b tx P' y c d ty 1 Files Functions ã Format: function o = test(x,y) ã Name function and file the same. ã Only first function in file is visible outside the file. Images Debugging ã Add print statements to function by leaving off ; ã keyboard ã debug and breakpoint Conclusions ã Quick tour of matlab, you should teach yourself the rest. We’ll give hints in problem sets. ã Linear algebra allows geometric manipulation of points. ã Learn to love SVD.

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