estimate pru s result by Kermack and McKendrick in 19 3. A famous s that kill on

Question

estimate pru s result by Kermack and McKendrick in 19 3. A famous s that kill only a small fraction of a susceptibl 7.3. e population the death rate as a function of time is well modeled by o(t) = A sech2 [B (t-C), for constant values of the parameters A, B C. Since the ch(0-1, A is the maximum death rate and C is the time of is s peak deaths. You will use this model to fit the deaths per week from plague recorded in Bombay (now Mumbai) in a period during 1906: 5, 10, 17, 22, 30, 50, 51, 90, 120, 180, 292, 395, 445, 775, 780, 780, 698, 880, 925, 800, 578, 400, 350, 202, 105, 65, 55, 40, 30, 20

Explanation / Answer

Let

f

be a given image represented as a

m

r

by

m

c

matrix. By applying the singular value

decomposition (SVD) to

f

, we can write

f

=

U

V

T

, where

U

is an

m

r

by

m

r

orthogonal

matrix (

U

T

=

U

1

), is an

m

r

by

m

c

diagonal matrix (0 except on its main diagonal) and

V

is a

m

c

by

m

c

orthogonal matrix (

V

T

=

V

1

). The diagonal entries of are the singular

values of

f

, and by convention they can be ordered by decreasing magnitu

de,

i

=

i,i

i

+1

,i

+1

=

i

+1

.

We can use the SVD to compress

f

by defining approximations

f

s

to

f

by

f

s

=

U

s

V

T

,

where

s

keeps only the

s

largest singular values,

1

, ...,

s

, replacing the rest with zeros.

This is not necessarily the best way to compress images, but i

t is an interesting illustration

of the SVD and that the largest singular values tend to corres

pond to the most important

information.

By the Eckart-Young Theorem,

f

s

is the best rank

s

approximation to

f

in the sense of

minimizing

i,j

(

f

i,j

g

i,j

)

2

over all matrices

g

having exactly

s

nonzero singular values.

Note that with only

s

nonzero singular values of

f

s

, it is only necessary to store the first

s

columns of

U

and

V

in order to represent

f

s

. Thus the total number of elements we need to

store is

s

(1 +

m

r

+

m

c

), which is less than

m

r

m

c

for

s <

m

r

m

c

1+

m

r

m

c

.

MATLAB:

To run the accompanying code, SVD

compress.m, on the elvis test image, type

g = SVD_compress(’elvis.bmp’,r);

where

r

is the ratio of the retained singular values to the total numb

er. For example, the

result with

r

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