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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 Cnm  Anm  Bnm Anm  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  Anm Bm p compatible dimensions m cij   aik bkj k 1 Ann Bnn  Bnn Ann 1 0  0 Identity Matrix:    0 1  0 I  IA  AI  A       0 0  1 Matrices Transpose: Cmn  A nm 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 Ann A nn A nn Ann  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 )  Pt 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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