LIMITED ANGLE IMAGE RECONSTRUCTION USING FOUR

HIGH RESOLUTION PROJECTION AXES AT CO-PRIME

RATIO VIEW ANGLES

Anastasios L. Kesidis

Computational Intelligence Laboratory, Institute of Informatics and Telecomunications,

National Center for Scientific Research “Demokritos”, GR-153 10 Agia Paraskevi, Athens, Greece

Keywords: Image reconstruction, limited angle tomography, coprime view angles, inverse problems.

Abstract: This paper proposes a sequential image reconstruction algorithm for the exact reconstruction of an image

from a limited number of projection angles. Specifically, four projection axes oriented at coprime ratio view

angles are used. The set of proper values for the view angles as well as the overall number of samples on the

projection axis are explicitly defined and are related only to the dimensions of the image. The slopes of the

four projection axes are calculated according to the chosen view angle and are symmetrically oriented with

respect to the horizontal and the vertical axis. The reconstruction is a non-iterative, one pass process based

on a decomposition sequence which defines the order in which the image pixels are restored. Several

simulation results are provided that demonstrate the feasibility of the proposed method.

1 INTRODUCTION

Computerized tomography (CT) is an important and

effective tool for a large number of imaging

applications allowing the observation of the internal

structure of objects. The majority of applications

refer to the medical sciences including x-ray

tomography, magnetic resonance imaging, electron

microscopy tomography, and diagnostic radiology.

(CT) is also extensively used in the industry for non-

destructive evaluation allowing the determination of

defects and abnormalities in industrial objects. In

general, the object is reconstructed from the

projection data acquired by projecting it into several

fairly equidistant view angles. However, there are

several cases in which projection data can be

acquired only in a limited range of projection view

angles. The limited angle tomography problem

concerns a wide range of applications in surgical

imaging, dental radiology and positron emission

mammography as well as in electron microscopy

and in astronomy. It is well known that applying the

conventional filtered backprojection reconstruction

method and assuming zero values for the missing

view angles produces approximations of the original

image containing significant artefacts (Natterer,

2001). Several approaches have been proposed that

address this problem including sinogram techniques,

Bayesian methods, projections onto convex sets,

maximum entropy techniques and many others.

Recently, Rantala et al (Rantala, 2006) addressed the

reconstruction from limited angle data by using a

wavelet expansion approximation and Besov space a

priori information in order to compute a maximum a

posteriori estimation for the original image. A pre-

thresholding method is also proposed in which

thresholding is applied to the wavelet coefficients

prior to the computation of the reconstruction.

Delaney and Bresler (Delaney, 1998) formulate the

reconstruction problem as a regularized weighted

least-squares optimization problem, and propose a

family of regularization functionals that are meant to

apply a constraint of piecewise smoothness on the

image. Clackdoyle et al (Clackdoyle, 2004) focus on

2-D reconstruction processes where data from entire

projection directions are unmeasured or unavailable

and state that region-of-interest reconstruction from

these truncated projections is possible under certain

conditions. Both direct and statistically based

(iterative) reconstruction algorithm can be used for

the image reconstruction. Schule et al (Schule, 2005)

describe how multi-valued objects can be

reconstructed by combining binary decisions. They

use convex-concave regularization to improve the

reconstruction quality as well as the EM-algorithm

L. Kesidis A. (2008).

LIMITED ANGLE IMAGE RECONSTRUCTION USING FOUR HIGH RESOLUTION PROJECTION AXES AT CO-PRIME RATIO VIEW ANGLES.

In Proceedings of the Third International Conference on Computer Vision Theory and Applications, pages 50-57

DOI: 10.5220/0001086300500057

 SciTePress

to motivate the adaption of the absorption

coefficients as hidden data estimation.

In this paper we propose a method for exact

image reconstruction when a limited number of

projections axes (i.e. four) is available that consists

of projection samples with higher resolution than the

pixels of the reconstructed image. Reconstruction

under such conditions may occur in cases where it is

very hard to obtain projections; however, it is

possible in these projections to use high resolution

detectors, or in cases that is not necessary to

reconstruct high resolution images. Examples of

such cases are in surgical imaging, in imaging

nuclear waste cites or in non destructive testing.

More specifically, we propose a method that allows

the exact reconstruction of an image assuming four

projection axes oriented at coprime ratio view

angles. The original grayscale image is projected

into the four projection axes and the collected

projection data are stored in an accumulator array.

The set of proper values for the view angle as well

as the overall number of samples on the projection

axis are related only to the dimensions of the image.

The slopes of the four projection axes are calculated

according to the chosen view angle and are

symmetrically oriented with respect to the horizontal

and vertical axes. The reconstruction is a non-

iterative, one pass process that uses a decomposition

sequence which defines the order in which the image

pixels are restored. The decomposition sequence is

determined so that, a unique correspondence

between a pixel in the exterior of the image and a

projection ray that intersects only this pixel, is

preserved during the reconstruction process.

Initially, the determination of the view angle and the

decomposition sequence is based on the pixels in the

octant of the image. However, during the

reconstruction process, the symmetrical geometric

properties of the pixels in the other octants are used

in order to restore all the image’s pixels.

The rest of this paper is organized as follows:

Section 2 presents an initial approach to determine a

proper view angle that allows the exact

reconstruction of the image. Besides the view angle,

certain projection parameters are also defined that

determine the geometry of the reconstruction

utilization. In section 3 we extend the methodology

in order to determine a set of several other coprime

ratio view angles that can also be used in the

proposed reconstruction scheme. We provide

generalized versions of the projection parameters

that are related only to the image dimensions and the

slope of the projection axis. Time and memory

complexity issues are also considered and the

applicability of the proposed method using various

projection view angles is also demonstrated. Finally,

section 4 draws the conclusion.

2 AN INITIAL APPROACH FOR

EXACT IMAGE

RECONSTRUCTION

2.1 Accumulator Array

Without loss of generality let us suppose an input

image I of size N×N pixels where N is assumed to be

an even positive integer. A pixel in the input image

at position (i,j) has grayscale (intensity) value I(i,j),

with i, j = −N/2…N/2−1, and is assumed to be a

square area of unit size with constant intensity value.

The projection data obtained by projecting the image

into K

projection axes is stored in an accumulator

array C. Each projection axis consists of a finite

number of projection rays s

where l=1,2,…,K

Clearly, each sample in array C(θ,s) corresponds to a

projection ray identified by the combination of the

slope θ

with the displacement value s

. If I

θ,s

denotes

the set of image pixels that a projection ray r

θ,s

intersects, then for each pixel (i,j)∈I

θ,s

we define the

weighting factor w

θ,s

(i,j) to be the area of the portion

of r

θ,s

inside pixel (i,j). Thus, each sample of C is

calculated as

C(θ,s)=

∑

∈

Iji

jiwjiI

),(

),(),(

(1)

2.2 Projection Parameters

Determination

The image is divided into 8 octants as shown in Fig.

1 where the gray shaded pixels denote the 1

octant.

A pixel (i,j) in the 1

octant has coordinates

0≤i≤N/2−1, 0≤j≤i. The black pixels in Fig. 1 denote

pixels in all octants that have symmetrical geometric

properties. There are also K

=4 projection axes

shown at slopes

=( uuuu −+−

, ) for k=1…K

(2)

where u is the projection view angle. The following

discussion refers to pixels in the 1

octant and will

be later generalized for the pixels in all octants based

on the symmetrical positioning of the pixels

relatively to the image’s center.

LIMITED ANGLE IMAGE RECONSTRUCTION USING FOUR HIGH RESOLUTION PROJECTION AXES AT

CO-PRIME RATIO VIEW ANGLES

Let us consider the four instances shown in Fig.

2 as the first steps of a decomposition sequence

regarding the pixels in the 1

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

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

and r

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

octant are ignored. Hence, in

Fig. 2b pixel (i,i) is ignored, and pixel (i,i−1) is

considered. Clearly, any line r

further away from

the origin does not affect the intersection area. On

the other hand, if line r

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

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

is determined by points p

and p

and

is the closest to the origin line for which the

projection ray intersects only pixel (i,i−n). Line r

a line parallel to r

that intersects the upper right

vertex of pixel (i,i−n), namely point p

. It can be

noticed that for n>1 point p

corresponds to the

upper right vertex of pixel (i−1,i−1) while points p

and p

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/

+− NN ,

w=1/(N-2), K

/2 and K

= 2/

(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

and

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

θ,s

(i,j) in (1) equals

{

}

⎩

⎨

⎧

otherwise2

orif

),(

ssssw

jiw

(4)

where s

and s

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

samples. The overall projection data, i.e. K

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

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

and

displacement value s

that corresponds to this

projection ray. This sample is used in order to

determine the corresponding pixel’s grayscale value

(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

{t} initially

defined for the pixels in the 1

octant. The sequence

{t} contains the pixels of the 1

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

{t}, pixel (i,j)=(N/2−1, N/2−1).

Let

),( jis

denote the furthest from the origin

projection ray of projection axis θ

that intersects

pixel (i,j). For

=θ

and s

),( jis

projection ray

intersects only pixel (i,j). The accumulator array

sample C(

, s

), that holds the value of projection

VISAPP 2008 - International Conference on Computer Vision Theory and Applications

ray

, equals the weighted portion ),(

jiw

pixel’s (i,j) grayscale value. Hence, pixel I

(i,j) in

the reconstructed image can be calculated as

(i,j)=

),(

)

(

jiw

for

=θ

and

),( jis

(5)

After the determination of I

(i,j) the contribution of

pixel (i,j) is removed from the other samples of C.

Specifically, if P

(i,j) denotes the set of projection

rays r

θ,s

that intersect pixel (i,j) at any angle θ

then

the accumulator array is updated as follows

C(θ,s) ← C(θ,s)

−

(i,j)w

(i,j)

(6)

for {(θ,s) : r

θ,s

∈ P

(i,j)}.

So far, we considered pixels in the 1

octant.

Applying proper re-indexing allows the

manipulation of pixels in all other octants. Thus,

before continuing with the next member of T

{t} the

above process is applied to pixels in the other

octants that are symmetrical to pixel (i,j). There are

four or eight of them depending on whether pixel

(i,j) lies on the diagonal axis or not. Thus, if M

(i,j)

denotes the set of pixels that are symmetrical to

pixel (i,j) then

{}

{

}

⎪

⎩

⎪

⎨

⎧

≠−−−−−−−−

−−−−−−−−

=−−−−−−−−

jiji,ij,i,j

j,ijiijijji

jiiiiiiiii

jiM

if)1( ),1( ),11(

),11( ),,1( ),,1( ),,( ),,(

if)1,(),1,1(),,1(),,(

),(

(7)

The process continues with the next element of the

decomposition sequence until all the pixels in all

octants are examined. Finally, the accumulator array

becomes empty and the reconstructed image equals

the original one, that is, corresponding pixels in I

and I

have identical intensity values.

3 GENERALIZATION FOR

RATIONALLY SLOPED

PROJECTION AXES

In this section we show that the determination of the

decomposition sequence T

{t} by sorting the pixels

column-wise from the periphery to the inner of the

image is not unique. In fact, it corresponds to a

special case of a general set of parameter settings

that allow the reconstruction of the image from

several other projection axes that are rationally

sloped.

3.1 Projection Slope Determination

In the following by projection line we mean a line

that defines the border between adjacent projection

ray, e.g. lines r

and r

in Fig.2. Also, the term

lattice points refers to points on the image plane that

correspond to the upper right vertex of the pixels in

the 1

octant. Let p

(a)=(N/2−a, N/2−a) be a lattice

point corresponding to the upper right vertex of a

diagonal pixel with offset a relatively to lattice point

(N/2,N/2). Points p

(a) and p

=(N/2,0) form a

projection line with slope

)(

,1,3

app

−

=1−

(8)

The view angle u of a projection axis perpendicular

to m is given by

=arctan(−

)=arctan

⎟

⎠

⎞

⎜

⎝

⎛

− aN

(9)

Since the range of view angles in the 1

octant is

≤u≤π/4 it follows from (9) that valid integer values

for offset a are in the range

∈

{1…

⎣

⎦

4/N }

(10)

where symbol

⎣

⎦

x stands for the largest integer less

than or equal to x.

Our intention in the selection of a proper slope m

(which in turn affects the value of the perpendicular

view angle u) is to form a utilization of adjacent

projection rays in such a way that the upper right

area of each pixel in the 1

octant is intersected by

only one projection ray. Reconsidering the example

in Fig. 2d it is clear that the slope of the projection

ray is m

=−3 which follows from (8) for N=8 and

=1. It can be noticed that for a=1, projection line

intersects only one lattice point in the 1

octant,

that is p

(a)=(N/2−1, N/2−1). This is not the case for

all values of a. Our main requirement that only one

projection ray intersects each pixel’s upper right area

can be interpreted as a requirement for the projection

line l connecting points p

(a) and p

to intersect only

one lattice point. Indeed, if line l intersects more

than one lattice points then the projection ray that

follows l intersects the upper right area of more than

one pixel in this octant. This can be clearly seen in

the examples of Fig. 3b and 3d. Therefore, from all

⎣

⎦

4/N possible values of offset a we concern only

those for which point p

(a) is mutual visible from

point p

as in the case of Fig. 3a and 3c.

LIMITED ANGLE IMAGE RECONSTRUCTION USING FOUR HIGH RESOLUTION PROJECTION AXES AT

CO-PRIME RATIO VIEW ANGLES

Typically, two lattice points (x

) and (x

) are

mutually visible if the line segment joining them

contains no further lattice points (Boca, 2000).

Moreover, they satisfy the relation gcd(x

−x

−y

)=1, where gcd(a,b) denotes the greatest

common divisor of a and b. In our case this

corresponds to

gcd(p

1,x

−p

3,x

, p

1,y

−p

3,y

)

gcd(a,a

−

N/2)=

=gcd(a,N/2)=1

(11)

This is an important relation in our method since it

states that p

(a) and p

are mutually visible if offset

a is coprime to N/2. Let F

denote the set of all

values of a for which the above relation holds, that is

⎣⎦

{}

1)2/,gcd(and4/: =≤ NaNaa

(12)

The cardinality K

of F

is given by the Euler

function φ(n). Function φ(n), also known as totient

function (Finch, 2003), is defined as the number of

positive integers less than or equal to n that are

relatively prime to n, that is

φ(n)

= #{a∈N: a≤n and gcd(a,n)=1}

(13)

where 1 is counted as being relatively prime to all

numbers. For a prime p it is φ(p)

=p−1 since all

numbers less than p are relatively prime to p. Since

≤

⎣⎦

4/N the overall number of proper offset values

()

2/N

(14)

Indeed, for N

=16, it is F

={1,3}, K

=2 and (8) gives

=−7/1 and m=−5/3 as shown in Fig. 3a and 3c.

Summarizing, if the image is projected on a

projection axis at angle u given by (9) for a

∈F

then

each projection ray does not intersect the upper right

area of more than one pixel in the 1

octant.

3.2 Determination of the other

Projection Parameters

The width d of the projection ray is given by the

width of the broadest projection ray that does not

contain any lattice points of the 1

octant in its

interior. Indeed, if one or more lattice points are

contained in the projection ray then the upper right

area of more that one pixels are intersected by this

specific projection ray, which contradicts our main

hypothesis. It can be shown that, if the projection

lines have a direction tanθ

=m=a/b, with gcd(a,b)=1

then there are projection ray paths of width d>0

containing no lattice points (Olds, 2001). The width

of the broadest such ray is

bqap

),gcd(

ba +

(15)

We have already defined point p

(a) as a lattice

point in the diagonal with offset a from point (N/2,

N/2). Hence b

= N/2−a and the above equation results

aaNN +−

(16)

This equation provides the width of each projection

ray (and consequently the width of the samples on

the projection axis) as a function of offset a.

According to Fig. 4 the area of the triangle that

forms the intersection between a projection ray and

the upper right area of a pixel equals

))(( KLML

cossin2

2sin

(17)

Using equations (9) and (16) this can be written as a

function if a as

⎟

⎠

⎞

⎜

⎝

⎛

−

+− )

arctan(2sin)84(

aaNN

aaN )2(

−

(18)

The intersection area w

θ,s

(i,j) between any pixel (i,j)

and a projection ray perpendicular to angle θ

and

displacement s can be calculated as a multiple of w.

Indeed, as shown in Fig. 4, any intersection area can

be constructed as a sum of one or more right

triangles of area w. For a projection axis at angle

tan(θ)

=a/b, where a, b are coprime numbers with

a<b and b≠0, the displacement k inside the pixel’s

area is

⎩

⎨

⎧

≤≤−

otherwise0

fcc

sssss

(19)

with s

=aj+bi and s

−1. The number of

triangles inside the projection area is given by

m(k)

⎪

⎩

⎪

⎨

⎧

−≤≤−−

<≤

<≤+

1if1)(2

if2

0if12

KkbkK

bkaa

akk

(20)

Hence, the intersection area in the k-th displacement

(i,j)

m(k)w

(21)

VISAPP 2008 - International Conference on Computer Vision Theory and Applications

Equation (18) implies also that for given image

dimension N, the area w is maximum for a

=1 and

decreases as a reaches

⎣⎦

4/N . Regarding the

parameters K

and K

it follows from Fig. 4 that

CF)(

uu )sin()cos( +

(22)

and

(23)

These relations are in accordance with (3) and show

that the number of projection ray intersecting each

pixel as well as the overall number of samples in the

projection axis do not dependent on offset a.

3.3 Image Reconstruction

The reconstruction methodology described in section

2.3 can be applied for any value of a∈F

. Indeed,

equations (9), (16) and (18) provide the view angle

u, the width of the projection ray d and the

intersection area w as a relation of image dimension

N and offset a. Thus, if the original image is

projected onto four projection axes whose

parameters are defined by these relations then the

original image pixels can be recovered from

accumulator array C. The process described in

section 2.3 can be considered as a special utilization

using parameter settings calculated for a=1.

However there is a significant difference concerning

the decomposition sequence. In section 2.3 sequence

{t} is defined by the pixels of the 1

octant sorted

column-wise from the periphery to the inner of the

image. In its general form, the decomposition

sequence T

{t} holds the pixels of the 1

octant

sorted decreasingly according to their s

value, that is

the furthest from the origin projection ray that

intersects each pixel. Fig. 5 depicts two

decomposition sequences for N=16. On the left

example, it is a=1 while on the right example, the

offset is a=3 resulting to a different re-ordering of

the pixels in the decomposition sequence.

Fig. 6 depicts the reconstruction of the well

known phantom image (Shepp, 1974) of size

N×N=256×256 pixels using three different view

angles. The image is projected into four projection

axes given by (2) which are symmetrically oriented

with respect to the horizontal and vertical axis, as

shown in Fig. 1. In all the cases the pixels in the

periphery of the image are reconstructed first

followed by the pixels in the center of the image. For

each pixel in the decomposition sequence its

symmetrical pixels in the other octants are also

reconstructed leading to a symmetrical outer-to-

inner reconstruction of the image. However, the

order in which the pixels are considered depends on

the view angle u which, in turn, is directly related to

the applied offset value. In the left column of Fig. 6

the image is reconstructed using an offset value a=1

which corresponds to a view angle u=0.45

. It can be

clearly seen that pixels are reconstructed column-

wise or row-wise, depending on the octant in which

the process is applied. In the middle and the right

column the offset values are a=23 and a=63,

corresponding to view angles u=12.36

and

u=44.10

, respectively. The later is the highest value

of a that can be used for the given image dimensions

according to (12). Clearly, there is a different

decomposition sequence for any of the K

available

values of a, but all of them lead finally to the exact

reconstruction of the original image. It should be

also noticed, that if N=2p where p a prime number

then according to (14) there is a maximum of

−

1)/2

(N−2)/4

(24)

available offset values and consequently (N−2)/4

different view angles u according to which the four

projection axes can be oriented.

3.4 Complexity and Applicability

Let n=N

denote the overall number of pixels in the

image. The memory requirements of the proposed

methods is O(n). Indeed, the accumulator array

requires K

=4(n/2) =2n memory units and there

are n/8 entries in decomposition sequence

}{tT

Considering time complexity, the most consuming

processes are the decomposition sequence

determination and the reconstruction process.

Although sorting the pixels during the determination

}{tT

is O(n

) in the worst case, the average time

is O(nlogn) (Havil, 2003). The complexity for the

reconstruction process is related to the

decomposition of the accumulator array C which is

O(n

). It should be noticed that }{tT

is not

related to the image’s contents. Therefore, it is not

necessary to determine the decomposition sequence

each time an N×N image is considered. Instead, it

can be determined once for a given set of parameter

settings N and a

and then retrieved from a lookup

table anytime an image with the same parameters is

considered.

LIMITED ANGLE IMAGE RECONSTRUCTION USING FOUR HIGH RESOLUTION PROJECTION AXES AT

CO-PRIME RATIO VIEW ANGLES

4 CONCLUSIONS

In this paper we presented a sequential

reconstruction method that allows the exact

reconstruction of an image when it is projected into

four projection axes which are symmetrically

oriented with respect to the horizontal and the

vertical axis at coprime ratio view angles. Analytical

relations are provided that determine the parameter

settings, namely the set of proper view angles, the

density of samples in each projection axis and the

intersection area between a pixel and a projection

ray. The chosen view angle affects the

decomposition sequence which determines the order

in which the pixels are restored. The image is

reconstructed by a one pass decomposition process

where the external pixels are restored first followed

by the pixels in the image’s center. It should be

noticed that we addressed the proposed method as a

quantitative reconstruction process problem and did

not considered optimization of noise propagation.

Future work includes a detailed analysis of the

algorithm’s behavior when noisy data are present as

well as the formulation of the proposed method in an

increasingly detailed hierarchical reconstruction

approach.

REFERENCES

Boca, F. P., Cobeli, C, Zaharescu, A. , 2000. Distribution

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6 7

Figure 1: The image is divided into eight octants. The gray

shaded pixels denote the 1st octant. The black pixels

denote pixels with symmetrical geometric properties.

N/2-1

(a) (b)

N/2-1

Figure 2: Intersection of a projection ray and the pixel (i,

i−n) in the 1

octant of an N×N=8×8 pixels image. Four

cases are shown where i=N/2−1 and n=0, 1, 2 and 3,

respectively. In each case the light shaded strip depicts the

projection ray and the dark shaded area denotes the

intersection between the pixel and the projection ray.

VISAPP 2008 - International Conference on Computer Vision Theory and Applications

(a) (b)

Figure 3: The number of pixels whose upper right area is

intersected by a projection line equals the number of

lattice points of the 1

octant joining the line segment l

between p

(a) and p

(denoted by a circle). There is a

unique such pixel if offset a is coprime to N/2 (subfigures

(a) and (c)), and more than one, otherwise (subfigures (b)

and (d)). In any case the dark shaded area denotes the

intersection between the upper right area of a pixel and the

projection ray.

{

Figure 4: Each pixel is intersected by K

projection rays.

The dark shaded area denotes the intersection between

pixel (i,j) and the projection ray s

that is the furthest from

the origin ray that intersects the pixel.

3536

=16,

(a) (b)

Figure 5: Decomposition sequence T

{t} of an

N×N=16×16 sized image for (a) a=1 and (b) a=3. In each

case the pixels are sorted according to the furthest from

the origin projection ray that intersects them.

(a)

(b)

(c)

Figure 6: Reconstruction of a 256×256 phantom image

using three different projection angles. The offset values

are (a) a=1, (b) a=23 and (c) a=63. In each case, the

symmetrical orientation of the four projection axes around

the horizontal and the vertical axis result to an outer-to-

inner reconstruction of the image.

LIMITED ANGLE IMAGE RECONSTRUCTION USING FOUR HIGH RESOLUTION PROJECTION AXES AT

CO-PRIME RATIO VIEW ANGLES