Differential Equations Solutions 49

Differential Equations Solutions 49 - Chapter 11 Solutions:...

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Unformatted text preview: Chapter 11 Solutions: Case Study: Classified Information: The Data Clustering Problem (coauthored by Nargess Memarsadeghi) CHALLENGE 11.1. The original image takes mpb bits, while the clustered image takes kb bits to store the cluster centers and mp log2 k bits to store the cluster indices for all mp pixels. For jpeg images with RGB (red, green, blue) values ranging between 0 and 255, we need 8 bits for each of the q = 3 values (red, green, and blue). Therefore, an RGB image with 250,000 pixels takes 24 ∗ 250, 000 = 6, 000, 000 bits, while the clustered image takes about 250, 000 log2 4 = 500, 000 bits if we have 3 or 4 clusters and 250,000 bits if we have 2 clusters. These numbers can be further reduced by compression techniques such as run-length encoding. CHALLENGE 11.2. (a) Neither D nor R is convex everywhere. Figure 11.2 plots these functions for a particular choice of points as one of the cluster centers is moved. We fix the data points at 0 and 1 and one of the centers at 1.2, and plot D and R as a function of the second center c. For c < −1.2 and c > 1.2, the function D is constant, since the second cluster is empty, while for −1.2 < c < 1.2, the function is quadratic. Since each function is above some of its secants (the line connecting two points on the graph), each function fails to be convex. (b) Neither D nor R is differentiable everywhere. Again see Figure 11.1. The function D fails to be differentiable at c = −1.2 and c = 1.2. Trouble occurs at the points where a data value moves from one cluster to another. 59 ...
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