Final Project

The goal of this project is to use tree regression to reconstruct an image that has been compressed by retaining only a tiny fraction of its pixels. A computed example can be bound in the slides Week10 TreeRegression.pdf posted on Canvas. The outline and the computational tips are given below. The data that you are provided in the  le MysteryImage.mat has been obtained as follows. The original is an RGB image of size m   n pixels, that is it consists of three matrices, each one of size n m with entries in the range [0; 1], de ning the intensities of red (R), green (G), and blue (B), respectively. We denote these matrices by XR, XG and XB, respectively. In this project, we have m = 1 456 and n = 2 592. The data consists of the RGB values at N = 15 000 pixels randomly chosen out of the n   m. Thus the data represents only of the full image data. The information is saved in sparse matrix form, the two vectors rows and cols of length N containing the row-column information of which pixels were saved, and a matrix vals of size N   3, whose jth row contains the RGB-values of the jth data pixel. The  rst step in this project is to download the Matlab  le MysteryImage.mat, where you  nd the variables m,n and rows, cols, vals described above. The problem is then to  nd out what the mystery image represent by using coding and using the tree regression procedure outlined in Chapter 10.8. of the textbook. Observe that since the image consist of three values per pixel, the optimal splitting requires that the mean square error is de ned in a vectorial form. Thus, if s is a split value either in the vertical (x) or horizontal (y) direction, the vectorial mean square error (see page 148) is now de ned as