Study of Radon Transformation and Application of its Inverse to NMR

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Study of Radon Transformation and

Application of its Inverse to NMR


S. Venturas,
I. Flaounas


Dept. of Informatics & Telecommunications,

National and Kapodistrian University of Athens


Paper for “
Algorithms in Molecular Biology
” Course

Assoc. Prof. I. Emi
ris


4 July
,

2005



Abstract


NMR is used in order to
locate the positions of atoms in space.
2D
p
rojections of the real 3D space are the
only available
information
.

In

this paper
we study

projections of images as
generated by Radon transformation. We
im
plemented an image reconstruction
algorithm which receives three projections
of
the

original image as input
.

We
performed a series of experiments using
artificially created images in order to test
and verify the algorithm.
Pearson’s

correlation coefficient

was measured
between the origina
l and the reconstructed
images.


I. Introduction


The determination of the 3D
structure of macromolecules is an
important field of interest for biology.
Nowadays, two methods dominate this
determination: X
-
ray crystallograp
hy
and Nuclear Magnetic Resonance
(NMR) spectroscopy. They have the
ability to
produce a detailed picture of
the 3D structure of biological
macromolecules at atomic resolution

[
1
,2
]. We focus on the NMR approach.

NMR is a spectroscopic technique
that reve
als information about the
environment of magnetically active
nuclei. An external magnetic field is
used to align them and this alignment
is perturbed by an electromagnetic
field. The nuclei absorb this
electromagnetic radiation in the radio
-
frequency regio
n at frequencies
governed by their chemical
environment. This environment is
influenced by chemical bonds,
molecular conformations and dynamic
processes. By measuring the
frequencies at which these absorptions
occur and their intensities, it is usually
pos
sible to deduce facts about the
structure of the molecule being
examined [
3
].

Up to

2003, the number of
3D

structures of macromolecules that has
been deposited in the Protein Data
Bank (PDB)
[
4
]

was greater than 3150.
NMR also is very significant in
struc
tural genomics. Many efforts are
being made in this filed to supplement
the knowledge on the sequence of
proteins by structural information on a
genome
-
wide scale, determined either
experimentally or by theoretical
homology
modelling

[
2
]. Finally,
NMR meth
ods have lead to the
development of Magnetic Resonance
Imaging (MRI), an important me
dical
imaging technique
[
5
].

For many years, NMR has been
dominated virtually exclusively by the
Fourier Transformation (FT)

[6,7]
. FT
gives a simple graphical picture of
correlations among different nuclear
sites within a molecule. But as the
spectra is getting more complex due to
more intense magnetic fields,
extension to three or even four
dimensions is needed to resolve
ambiguities.
This results to

an
increase to the
amount of data acquired
and the required processing time

[
8
].

Figure 1 illustrates a typical 3D
NMR spectrum.
We

need to
determine

the number of spots and their
positions. If we were able to look at
the spectrum from different angles we
could get this info
rmation. Currently
the only available information are
projections of the spectrum
from

different angles. Using that 2D
information, we try to reconstruct the
correct 3D image [
8
]. This image
reconstruction approach
,

using
different projections and angles o
f
views
,

is very popular in many fields
such as x
-
ray scanning, tomography
and determination of protein structure.
If only two projections are used some
resonances might be cut off by others.
Thus, more projections may be
required depend
ing on the problem
under study
[
8
].



Fig. 1 Example of a typical 3D NMR
spectrum.


In this paper we
present an image
reconstruction algorithm introduced by
E. Kupce and R. Freeman [18]
. Inputs
of the algorithm are

1D projections
.
They

are

acquired using the Radon
transfor
mation. The algorithm

was

implemented and verified

on
artificial

images
.

The Pearson’s correlation
coefficient was chosen as a
measurement for the resemblance
between

the reconstructed image
and

the original one.


The rest of this paper is organized
as fol
lows: Section II presents a
mathematical background of Radon
transformation and the reconstruction
algorithm. In section III, the acquired
results are illustrated. Finally, we
discuss about future work and
conclude our paper in section IV.


II. Methods


A.

The
Radon Transformation


The 2D Radon transformation is
the projection of the image intensity
along a radial line oriented at a specific
angle [
9
]. Radon expresses the fact that
reconstructing an image, using
projections obtained by rotational
scanning i
s feasible. His theorem is the
following:
The value of a 2
-
D function
at an arbitrary point is uniquely
obtained by the integrals along the
lines of all directions passing the point.

The
Radon transformation
shows the
relationship between the

2
-
D object
an
d
its

projections
[
10
]
.

The Radon Transformation is a
fundamental tool which is used in
various applications such as radar

imaging, geophysical imaging,
nondestructive testing and medical
imaging [
11
]. Many publication
exploit the Radon Transformation.
Men
eses
-
Fabian et al. [12]

describe a
novel technique for obtaining border
-
enhanced tomographic images of a
slice belonging to a phase object.
Vítezslav
[
13
]

examines fast
implementations of
the inverse Radon
transform for filtered backprojection
on
computer
graphic cards.

Sandberg
et al.

[
14
] describe a novel algorithm
for tomographic reconstruction of 3
-
D
biological data obtained by a
transmission electron microscope.
Milanfar

[
15
] exploits the shift
property of Radon transformation to
image processing.
Barv
a

et al.

[
16
]

present a method for automatic
electrode localization in soft tissue
from radio
-
frequency signal, by
exploiting a property of the Radon
Transform
. Challenor et al.

[
17
]
generalize the two dimensional Radon
transform to three dimensions and us
e
it to study atmospheric and ocean
dynamics phenomena.

Figure 2 illustrates several 1D
projections from different angles of an
image consisting of three spots in the
2D domain. In some of the projections,
only two spots are shown. This reveals
the importa
nce of the selection of the
“correct” projections for image
reconstruction.



Fig. 2 Different projections of

a three
-
dot image example.


Suppose a 2
-
D function
)
,
(
y
x
f

(
Fig. 3).

Integrating along the line,
whose normal vector is in


direction,
results in the
)
,
(

s
g

function which is
the
projection of the 2D function
)
,
(
y
x
f

on the axis
s

of


direction.

When
s

is zero
, the
g

function has
the value
)
,
0
(

g

which is obtained by
the integration along the line passing
the origin of
)
,
(
y
x
-
coordinate.

The
points on the line whose normal vector
is in


direction and passes the origin
of
)
,
(
y
x
-
coordinate satisfy the
equation:






sin
cos
)
2
tan(




x
y






0
sin
cos




y
x



Fig.
3

The Radon Transform
computation.



The integration a
long the line whose
normal vector is in


direction and
that passes the origin of
)
,
(
y
x
-
coordinate means the integration of
)
,
(
y
x
f

only at the points satisfying
the previous equation.

With the help o
f
the Dirac

function



,
which
is
zero
for every argument except

to
0

and
it
s

integral is

one
,
)
,
0
(

g
is expressed as:






dxdy
y
x
y
x
f
g
)
sin
cos
(
)
,
(
)
,
0
(






Similarly
,

the line
with

normal vector
in


direct
ion and distance

s

from the
origin

is

satisfying

the following
equation:


0
sin
)
sin
(
cos
)
cos
(












s
y
s
x



0
sin
cos



s
y
x





So

the general equation of the Radon
transformation
is acquired
: [
10, 11, 15,
16, 18]






dxdy
s
y
x
y
x
f
s
g
)
sin
cos
(
)
,
(
)
,
(





The inverse of Radon transform is
calculated by the following equation
[14]

:










d
y
x
s
R
y
x
f




2
2
,
)
,
(

where

R

is the Radon transformation,


is a filter and






sin
cos
,
y
x
y
x
s





B.
Image
Reconstruction A
lgorithm


Kupce and Freeman [1
8
]
presented an image
reconstructi
on

algorithm

from a limited set of
projections
. They
suggest a method of
implementing the inverse Radon
transform
ation. First
ly
,
they

get the
projections from different perspectives.
Then
they

expand every 1D projection
at right angles, so as to create a 2D
map

that consists of parallel

ridges.
T
he superposition and the comparison
of the created 2D projection maps

result in the final reconstructed image
.

Their technique can be explained
by the
following example
:

S
uppos
e
the
existence of

two

perpendicular
projections
of

four absorption peaks

in
each
one

(Fig.
4
).

From these two
projections, the potential peaks are 16,
but not all of them are true cross peaks.
If we take into account another
proje
ction at a different angle and
reapply the lower
-
value algorithm, we
eliminate some potential as being false
peaks and we get the image shown in
Fig. 5.

Another projection would refine
the solution even further. Usually three
projections are enough to have

an
accurate definition of the peaks, but if
the original spectr
um

is complex more
projections

may be required
.

B
ecause
of the discrete nature of the NMR
resonances, the problem converges
very rapidly.



Fig. 4 Peaks Using two projections.




Fig. 5 Usi
ng three projections.


The algorithm we implemented is
based on the previous described
algorithm of Kupce and Freeman

and

its

steps are:


Step

1:

Acqui
sition of

three different

projections
.

Step

2:

Expan
sion

of
the 1D projection
in
2D projection maps
.

Ste
p

3:

Padding (with black) of

the 2D
projections maps

in order not to loose
information due to the next step.

Step

4:

Rotat
ion

of
the maps

to the
correct angle.

Step

5:

Normaliz
ation of

each map

to
the range 0 to 1
.

Step

6
:

Reconstruct
ion of

the

original

image by multiplying the maps pixel
by

pixel.

Step

7
:

Post
-
processing
of
the
reconstructed
image
by normaliz
ation
and

c
rop
ping

to the required size.



III. Results


We studied the Radon
transformation using Matlab and the
Image Processing Toolbox in
partic
ular. Initially we created a
collection of artificial images and
applied the Radon transformation in
order to construct the corresponding
projections
. Figure
6A

presents an
example of these images along with
the corresponding spectra for six
different angl
es: 0, 15, 30, 45, 90 and
135 degrees. Even though the image
consists of only three spots, in some
projections (0, 90, 135 degrees), there
seem to be only 2 spots. This proves
the need of several projections in order
to verify the correct number of existin
g
spots and their positions. Figure
6B

illustrates a more complicated example
of an image that consists of several
small spots and the corresponding
spectra.


We also implemented the
reconstruction algorithm described in
section
II

to reconstruct artificia
lly
created images. Three projections were
used to reconstruct the original ima
ges.
The two projections are

0 and 90
degrees. The third one is variable

in
the range

of

1 to 89 degrees with a step
of one degree. The quality of the
reconstruction is measured

by
calculating the absolute value of the
2D correlation
coefficient

between the
original image and the reconstructed
one. This measure receives a value
between 0 and 1.
As
the value

increases
,

so does the resemblance

that

exists between the original image

and
the reconstructed one.

Figure
7

illustrates four images
that were used to test the implemented
algorithm and the corresponding
reconstructed images

which have the
optimal correlation coefficient. Figure
8

presents the correlation coefficient
for the
different values of the variable
projection angle. Table 1 summarizes
the optimal results for these images.



Image 1



Image 2



Image 3



Image 4


Fig.
7

Artificial images and their
reconstruction using 3 projections.


The four
presented

im
ages range
in complexity and archived results
.
The first one, which is comprised of
only three spots, is the easier to
reconstruct reaching 0.88 to

the

correlation

coefficient
.

The second one
is comprised by spots of different
radius. The last two pictures

contain a
greater number of small spots.

The
optimal

projection

angle for all
pictures range
s

from 47 to 74 degrees.


Table 1. Optimal Angle and
corresponding Correlation Coefficient.

Image

Angle

Correlation

1

47

0.88

2

62

0.75

3

74

0.55

4

60

0.59



IV. Conclusions


To summarize, in this paper we
tried to reconstruct an image
using

projections from different perspectives,
which we obtained with the use
of
the
Radon transform.
In order to achieve
this, w
e
implemented

an algorithm,
based on the one pro
posed
by

Kupce
and Freeman [
18
]
.

In

the presented

examples we used three projections of
the input image
,

reaching a correlation
coefficient of 0.88.

F
u
ture perspectives
of the proposed work
include

the

application of the implemented
algorithm to real NMR d
ata, the
application of more projections for the
image reconstruction and the
development of heuristics for the
determination of optimal projection
angles.




References

[1] Flemming M. Poulsen, “A brief
introduction to NMR spectroscopy of
proteins”, 20
02.

[2] P. Guntert, “Automated NMR
protein structure calculation”, RIKEN
Genomic Sciences Center, 1
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7
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22
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0045, Japan, Accepted 23 June 2003.

[3] U. Oehler, “NMR
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A short
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,
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[
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mprov
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[
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[
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64
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[
8
] E. Kupce, R. Freeman, “Fast
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by the Projection


Reconstruction
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16
-
20, 2004.

[
9]
http://www.mathwork
s.com
, accessed 15 June 2005
.

[
10
]
A. Asano,

“Radon transformation
and projection theorem”
,

Topic 5,
Lecture notes of subject Pattern
information processing
, 2002 Autumn
S
emester,
http://kuva.mis.hiroshima
-
u.ac.jp/~asano/Kougi/02a/PIP/

[
11
]
A. Averbuch, R.R. Coifman, D.L.
Donoho, M. Israeli, J. Wald
΄
en,
Fast
Slant Stack:

A notion of Radon
Transform for

Data in a Cartesian Grid
which is Rapidly Computible,
Algebraically Exact, Geometrically
Faithful and Invertible.,
to appear in
SIAM J. Scientific. Computing, 2001

[
12
]
C. Meneses
-
Fabian, G.
Rodr´ýguez
-
Zurita, and J.F. V´azquez
-
Castillo

“Optical tomograph
y of phase
objects with parallel projection
differences and ESPI”, Investigacion
revista mexicana de fisica


49

(3) 251

257 Junio 2003
.

[
13
]

V
.
V. Vlcek, “
Computation of
Inverse Radon Transform on Graphics
Cards”,

International Journal of Signal
Processing
1(1) 2004 1
-
12
.

[
14
]

K. Sandberg
,
D. N. Mastronarde,
G. Beylkin
a,


A fast reconstruction
algorithm for electron microscope
tomography”,

Journal of Structural
Biology 144 (2003) 61

72,

3
September 2003.

[
15
]

P
.

Milanfar, “
A Model of the
Effect of Image Moti
on in the Radon
Transform Domain”,

IEEE
Transactions on Image processing, vol.
8, no. 9, September 1999

[
16
]

M. Barva

and
J. Kybic with J.
Mari and

C. Cachard,
“Radial Radon
Transform dedicated to Micro
-
object
Localization from Radio Frequency
Ultrasound S
ignal”,

In
UFFC '04:
Proceedings of the IEEE International
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Conference
. Piscataway: IEEE, 2004,
p.

1836
-
1839. ISBN 0
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7803
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8412
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1.

[
17
]

P.G. Challenor , P. Cipollini and
D. Cromwell, “
Use of the 3D Radon
Transform to Examine t
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Journal of
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18
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] E. Kupce, R. Freeman, “The
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New Scheme for
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Original picture
0
200
400
0
50
100
0 degrees
0
200
400
0
50
15 degrees
0
200
400
0
50
30 degrees
0
200
400
0
50
45 degrees
0
200
400
0
50
100
90 degrees
0
200
400
0
50
100
135 degrees

Fig
6
A. Artificially created image and its Radon

based projections for different angles.


Original picture
0
200
400
0
200
400
600
0 degrees
0
200
400
0
200
400
600
15 degrees
0
200
400
0
100
200
300
30 degrees
0
200
400
0
100
200
300
400
45 degrees
0
200
400
0
200
400
600
90 degrees
0
200
400
0
100
200
300
400
135 degrees
Fig
6
A
.
Artificially

created image

and
its

Radon

based projections for different angles.





0
10
20
30
40
50
60
70
80
90
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Correlation
Img.1
Img.2
Img.3
Img.4

Fig 7. Correlation Coefficient for four different images and variable projection angle.