![](bg3.png)
Let us consider the four instances shown in Fig.
2 as the first steps of a decomposition sequence
regarding the pixels in the 1
st
octant. By convention,
the diagonal pixels are attributed to the odd octants.
The decomposition sequence refers to pixels in the
last column of the 1
st
octant, namely pixels (i,i−n),
where i=N/2−1 and n=0…i denotes an positive offset
from the diagonal. Our intention is to identify the
width of the projection ray as well as its
displacement on the projection axis that provide the
largest possible intersection area without intersecting
any other pixel of the current or the adjacent octant
(octants 1 and 2 in this case). Lines r
1
and r
2
are
perpendicular to the projection axis and define the
boundaries of the projection ray. For each n, the
intersection between projection ray r
θ,s
and pixel
(N/2−1, N/2−1−n) is considered. For all forthcoming
values of n, the pixel just examined and its
counterpart in the 2
nd
octant are ignored. Hence, in
Fig. 2b pixel (i,i) is ignored, and pixel (i,i−1) is
considered. Clearly, any line r
2
further away from
the origin does not affect the intersection area. On
the other hand, if line r
1
is shifted in parallel towards
the origin then pixels (i−1,i) and (i,i−2) are also
intersected by projection ray r
θ,s
which is
undesirable. Again, in the following steps pixel
(i,i−1) as well as its counterpart in 2
nd
octant, pixel
(i−1,i), are ignored.
The process is repeated in Figs. 2c and 2d for
pixels (i,i−2) and (i,i−3), respectively. As shown in
Fig. 2d line r
1
is determined by points p
1
and p
3
and
is the closest to the origin line for which the
projection ray intersects only pixel (i,i−n). Line r
2
is
a line parallel to r
1
that intersects the upper right
vertex of pixel (i,i−n), namely point p
2
. It can be
noticed that for n>1 point p
3
corresponds to the
upper right vertex of pixel (i−1,i−1) while points p
1
and p
2
are related to n. Setting i=N/2−1 and n=i, and
based on the geometry of the utilization as shown in
Fig. 2d allows the determination of the projection
parameters
u=arctan(2/(N-2)), d=2/
84
2
+− NN ,
w=1/(N-2), K
p
=
Ν
/2 and K
= 2/
2
N
(3)
where u is the view angle of the projection axis, d
denotes the width of each projection ray, w is the
area between any pixel and the furthest from the
origin projection ray that intersects it, while K
p
and
K
l
denote the number of rays that intersect a pixel
and the overall number of samples on the projection
axis, respectively. These parameters are related only
to the image dimension N. The weighting factor
w
θ,s
(i,j) in (1) equals
⎩
⎨
⎧
==
=
otherwise2
orif
),(
,
w
ssssw
jiw
fc
s
θ
(4)
where s
c
and s
f
denote the nearest and furthest from
the origin projection ray that intersect pixel (i,j),
respectively.
2.3 Image Reconstruction
Let us suppose that the original image is projected
onto the four projection axes each one consisting of
K
l
samples. The overall projection data, i.e. K
θ
K
l
samples, are stored in accumulator array C. In the
following we present how the original image can be
exactly reconstructed from the samples in array C.
Let
I
R
denote the reconstructed image. In section 2.2
we stated that the main criterion for the
determination of the projection parameters is that the
furthest from the origin projection ray that intersects
a pixel does not intersect the upper right area w of
any other pixel in the same octant. Thus for each
pixel (i,j) there is a specific sample at slope θ
k
and
displacement value s
l
that corresponds to this
projection ray. This sample is used in order to
determine the corresponding pixel’s grayscale value
I
R
(i,j). Having obtained its value, the pixel’s
contribution is removed from the accumulator array
i.e., all the samples of C affected by this pixel
decrease their value by an amount proportional to
the weighting factor w
θ,s
(i,j).
The order in which the pixels are examined is
given by a decomposition sequence T
1
{t} initially
defined for the pixels in the 1
st
octant. The sequence
T
1
{t} contains the pixels of the 1
st
octant sorted
column wise from the periphery to the inner of the
image, i.e. pixels (N/2−1, N/2−1), (N/2−1, N/2−2),
…, (N/2−1, 0), (N/2−2, N/2−2), (N/2−2, N/2−3),…
etc, down to (0,0). For each pixel of the
decomposition sequence its symmetric counterparts
in the other octants are also examined before the
decomposition continues with the next sequence
element. The reconstruction process starts with the
first member of T
1
{t}, pixel (i,j)=(N/2−1, N/2−1).
Let
),( jis
f
denote the furthest from the origin
projection ray of projection axis θ
1
that intersects
pixel (i,j). For
θ
ˆ
=θ
1
and s
ˆ
=
),( jis
f
projection ray
s
r
ˆ
,
ˆ
θ
intersects only pixel (i,j). The accumulator array
sample C(
θ
ˆ
, s
ˆ
), that holds the value of projection
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