EE263
S. Lall
2011.05.16.01
Homework 6
Due Thursday 5/19
1.
Image reconstruction from line integrals.
46040
In this problem we explore a simple version of a tomography problem.
We consider a
square region, which we divide into an
n
×
n
array of square pixels, as shown below.
x
1
x
2
x
n
x
n
+1
x
2
n
x
n
2
The pixels are indexed column first, by a single index
i
ranging from 1 to
n
2
, as shown
above.
We are interested in some physical property such as density (say) which varies
over the region. To simplify things, we’ll assume that the density is constant inside each
pixel, and we denote by
x
i
the density in pixel
i
,
i
= 1
, . . . , n
2
.
Thus,
x
∈
R
n
2
is a
vector that describes the density across the rectangular array of pixels. The problem is
to estimate the vector of densities
x
, from a set of sensor measurements that we now
describe.
Each sensor measurement is a
line integral
of the density over a line
L
.
In
addition, each measurement is corrupted by a (small) noise term.
In other words, the
sensor measurement for line
L
is given by
n
2
summationdisplay
i
=1
l
i
x
i
+
v,
where
l
i
is the length of the intersection of line
L
with pixel
i
(or zero if they don’t
intersect), and
v
is a (small) measurement noise. This is illustrated below for a problem
with
n
= 3. In this example, we have
l
1
=
l
6
=
l
8
=
l
9
= 0.
x
1
x
2
x
3
x
4
x
5
x
6
x
8
x
9
l
2
l
3
l
4
l
5
l
7
line
L
Now suppose we have
N
line integral measurements, associated with lines
L
1
, . . . , L
N
.
From these measurements, we want to estimate the vector of densities
x
. The lines are
characterized by the intersection lengths
l
ij
,
i
= 1
, . . . , n
2
,
j
= 1
, . . . , N,
1
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EE263
S. Lall
2011.05.16.01
where
l
ij
gives the length of the intersection of line
L
j
with pixel
i
. Then, the whole set
of measurements forms a vector
y
∈
R
N
whose elements are given by
y
j
=
n
2
summationdisplay
i
=1
l
ij
x
i
+
v
j
,
j
= 1
, . . . , N.
And now the problem: you will reconstruct the pixel densities
x
from the line integral
measurements
y
. The class webpage contains the Mfile
tomodata.m
, which you should
download and run in Matlab. It creates the following variables:
•
N
, the number of measurements (
N
),
•
n_pixels
, the side length in pixels of the square region (
n
),
•
y
, a vector with the line integrals
y
j
, j
= 1
, . . . , N
,
•
lines_d
, a vector containing the displacement
d
j
, j
= 1
, . . . , N
, (distance from the
center of the region in pixels lengths) of each line, and
•
lines_theta
, a vector containing the angles
θ
j
, j
= 1
, . . . , N
, of each line.
The file
tmeasure.m
, on the same webpage, shows how the measurements were computed,
in case you’re curious. You should take a look, but you don’t need to understand it to
solve the problem. We also provide the function
line_pixel_length.m
on the webpage,
which you do need to use in order to solve the problem. This function computes the pixel
intersection lengths for a given line.
That is, given
d
j
and
θ
j
(and the side length
n
),
line_pixel_length.m
returns a
n
×
n
matrix, whose
i, j
th element corresponds to the
intersection length for pixel
i, j
on the image. Use this information to find
x
, and display
it as an image (of
n
by
n
pixels).
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 Summer '08
 BOYD,S
 Least Squares, Regression Analysis, Linear least squares, Lall, S. Lall

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