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3D Least Squares Fit and some Linear Algebra

  1. Mar 4, 2010 #1
    Hello,

    I am trying to write an algorithm to calculate the Least Squares Fit Line of a 3D data set. After doing some research and using Google, I came across this document, http://www.udel.edu/HNES/HESC427/Sphere Fitting/LeastSquares.pdf (section 2 in page 8) that explains the algorithm for
    It uses something from Linear Algebra I have never seen called Singular Value Decomposition (SVD) to find the direction cosines of the line of best fit. What is SVD? What is a direction cosine? The literal angle between the x,y,z axes and the line?

    For simplicity's sake, I'm starting with the points (0.5, 1, 2) ; (1, 2, 6) ; (2, 4, 7). So the A matrix, as denoted by the document is (skipping the mean and subtractions)
    [tex]A = \left \begin{array} {ccc}
    [-1.6667 & -1.1667 & -2.8333 \\
    -2.0000 & -1.0000 & 3.0000 \\
    -2.3333 & -0.3333 & 2.6667 \end{array} \right][/tex]

    and the SVD of A is
    [tex]SVD(A) = \left \begin{array} {ccc}
    [6.1816 \\
    0.7884 \\
    0.0000 \end{array} \right][/tex]
    but the document says "This matrix A is solved by singular value decomposition. The smallest singular value
    of A is selected from the matrix and the corresponding singular vector is chosen which
    the direction cosines (a, b, c)" What does that mean?

    Any help will greatly be appreciated. Note: I am working in MATLAB R2009a

    Thank you in advance!

    *NOTE* I POSTED THIS IN THE WRONG MATH FORUM AND CANNOT DELETE THE FIRST POST.
     
  2. jcsd
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