MA/CS 375

1. MA/CS 375 Fall 2002 Lecture 23 MA/CS 375 Fall 2002

2. Class Exercise • Part 1: • Build a function called vandermonde.m which accepts a vector of x values and a polynomial order P • In the function find N=length of x • Function returns a matrix V which is Nx(P+1) and whose entries are: V(n,m) = (xn)(m-1) • Part 2: • Translate this pseudo-code to a Matlab script: • for N=1:5:20 • build x = set of N points in [-1,1] • build f = exp(x) • build xfine = set of 10N points in [-1,1] • build Vorig = vandermonde(N-1,x) • build Vfine = vandermonde(N-1,xfine) • build Finterp = Vfine*(Vorig\f); • plot x,f and xfine,Finterp on the same graph • end MA/CS 375 Fall 2002

3. Runge Phenomenon • Theorem 2 (van Loan page 90) • Suppose If(x) is an N-1’th order interpolating polynomial of the function f at N distinct points x1,x2,..,xN. If f is N times continuously differentiable on an interval I containing the {x1,x2,..,xN} then for any x in I=[a,b]: MA/CS 375 Fall 2002

4. Runge Phenomenon • If we base the interpolant on the equi-spaced points then after some basic math we can show:where MN bounds the N’th derivative MA/CS 375 Fall 2002

5. Runge Phenomenon • If we base the interpolant on the equi-spaced points then after some basic math we can show:where MN bounds the N’th derivative • Hence if the N’th derivative is badly behaved this bound suggests the quality of interpolation may decrease with increasing N!!. MA/CS 375 Fall 2002

6. Example: Runge Phenomenon • Use your Vandermonde routine to interpolate the function:for 10,20,30,40,50,60 points on the interval [-1,1] • Volunteer to describe what happens MA/CS 375 Fall 2002

7. Interpolating in Two Directions • A picture (e.g. jpeg, tif or gif) is actually an array of pixel values • There are a number of ways to represent color on the pixel. • gif uses a colormap, indexed by an integer • jpeg uses an 8 bit number for each component of red, green and blue MA/CS 375 Fall 2002

8. Loading an Image in Matlab • % read in the three dimensional array • pic = imread(‘foo.jpg’); • red = pic(:,:,1); • green = pic(:,:,2); • blue = pic(:,:,3); MA/CS 375 Fall 2002

9. Double The Resolution In Each Direction Using 4th order Interpolation MA/CS 375 Fall 2002

10. Results From Interpolation Notice 300 pixels Notice 600 pixels MA/CS 375 Fall 2002

11. Details of Interpolation MA/CS 375 Fall 2002

12. Details of Interpolation Notice reduced pixellation on new image MA/CS 375 Fall 2002

13. But Not So Fast MA/CS 375 Fall 2002

14. Now try 10th order interpolation MA/CS 375 Fall 2002

15. Not So Good! Note stripes appearing on new image MA/CS 375 Fall 2002

16. Close Up of 10th Order Interpolation Looks quite bad!!! MA/CS 375 Fall 2002

17. What’s Going On • We have divided each line of the picture into non-overlapping sections, containing 10 points each. • We have then interpolated the picture values at the 10 points to 20 points on the same section • The equispaced points lead to the Runge phenomenon i.e. highly oscillatory interpolants at the ends of each section • How can we make this better? MA/CS 375 Fall 2002

18. Team Exercise ( a little tricky – feel free to use loops) • Even for 10+ points the interpolant function is probably going to be ok at the center of the region. • build VDM for N points (N even) • build weights = VDM\(1;0;0;…;0) • load in your jpeg picture • now use these weights to build the interpolating values between each given points MA/CS 375 Fall 2002

19. Nth Order Mid-point Interpolation MA/CS 375 Fall 2002

20. Nth Order Mid-point Interpolation MA/CS 375 Fall 2002

21. Nth Order Mid-point Interpolation MA/CS 375 Fall 2002

22. Summary • So we have started to invert general matrix systems to obtain interpolating polyomial coefficients • We saw the Runge phenomenon in action when equi-spaced interpolation is used • We improved things somewhat by ignoring the interpolating polynomial away from the center of its range. MA/CS 375 Fall 2002

23. Next Time • Using Matlab’s built in interpolation routine • Morphing images class project MA/CS 375 Fall 2002