Assignment #4

Assignment Problems (Click on the link to view PDF):

Sample Codes and Illustration :

  1. Download and run the Matlab codes in
    http://www.cse.ucsc.edu/~hongwang/Codes/Find_period
    Learn how to use the golden search method combined with the cubic spline interpolation to find the period of a function when the function is known only on a set of data points.

  2. Download and run the Matlab codes in
    http://www.cse.ucsc.edu/~hongwang/Codes/Dist_to_trajectory
    Learn how to use the golden search method combined with the cubic spline interpolation to find the distance from a point to a trajectory where the trajectory is only described by N data points.

  3. Download and run the Matlab codes in
    http://www.cse.ucsc.edu/~hongwang/Codes/Least_square
    Try to see how the codes work and learn how to use the least square method to fit a linear combination of a set of given functions to the data.
    "LSd5_sc1.m" fits q1(x) = b1+b2*sin(x)+b3*cos(x) to data5.txt.
    "LSd5_sc2.m" fits q2(x) = b1+b2*sin(x)+b3*cos(x)+b4*sin(2*x)+b5*cos(2*x) to data5.txt.
    "LSd6_p2.m" fits p2(x) = b1+b2*x+b3*x^2 to data6.txt.

  4. Download and run the Matlab codes in
    http://www.cse.ucsc.edu/~hongwang/Codes/FFT_applications
    Try to see how the codes work and learn how to use "fft" and "ifft" in Matlab.
    "data_compression.m" shows that a smooth function can be very well approximated by a small fraction of Fourier modes.
    "noise_removal.m" uses the Fourier transform and a filter to remove noise from the data.
    "heat_equation.m" uses the Fourier transform to solve the heat equation.

  5. Computer demonstration of implementing FFT
    http://www.cse.ucsc.edu/~hongwang/Codes/FFT_implement

  6. Computer demonstration of image compression
    http://www.cse.ucsc.edu/~hongwang/Codes/Image_compression
    "demo_grayscale.m" demonstrates the effect of image compression on a grayscale image. "demo_color.m" demonstrates the effect of that on a color image. Image compression is achieved by storing only those Fourier coefficients that are larger than a threshold. Fourier coefficients that are smaller than the threshold are set to zero.
    The compression method used here is very rudimentary. The JPEG compression uses the same principle but a more sophisticated implementation.