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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 runlength 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.
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 Fall '11
 Dr.Robin

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