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Study of Radon transformation and

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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. Emiris

                                        4 July, 2005


Abstract –        NMR is used in order to        field. The nuclei absorb this
locate the positions of atoms in space. 2D       electromagnetic radiation in the radio-
projections of the real 3D space are the         frequency region at frequencies
only available information. In this paper        governed      by     their       chemical
we study projections of images as                environment. This environment is
generated by Radon transformation. We            influenced by chemical bonds,
implemented an image reconstruction              molecular conformations and dynamic
algorithm which receives three projections       processes.     By     measuring        the
of the original image as input. We
performed a series of experiments using
                                                 frequencies at which these absorptions
artificially created images in order to test     occur and their intensities, it is usually
and verify the algorithm. Pearson’s              possible to deduce facts about the
correlation coefficient was measured             structure of the molecule being
between the original and the reconstructed       examined [3].
images.                                               Up to 2003, the number of 3D
                                                 structures of macromolecules that has
I. Introduction                                  been deposited in the Protein Data
                                                 Bank (PDB) [4] was greater than 3150.
     The determination of the 3D                 NMR also is very significant in
structure of macromolecules is an                structural genomics. Many efforts are
important field of interest for biology.         being made in this filed to supplement
Nowadays, two methods dominate this              the knowledge on the sequence of
determination: X-ray crystallography             proteins by structural information on a
and Nuclear Magnetic Resonance                   genome-wide scale, determined either
(NMR) spectroscopy. They have the                experimentally or by theoretical
ability to produce a detailed picture of         homology modelling [2]. Finally,
the 3D structure of biological                   NMR methods have lead to the
macromolecules at atomic resolution              development of Magnetic Resonance
[1,2]. We focus on the NMR approach.             Imaging (MRI), an important medical
     NMR is a spectroscopic technique            imaging technique [5].
that reveals information about the                    For many years, NMR has been
environment of magnetically active               dominated virtually exclusively by the
nuclei. An external magnetic field is            Fourier Transformation (FT) [6,7]. FT
used to align them and this alignment            gives a simple graphical picture of
is perturbed by an electromagnetic               correlations among different nuclear
sites within a molecule. But as the        of the algorithm are 1D projections.
spectra is getting more complex due to     They are acquired using the Radon
more      intense    magnetic    fields,   transformation. The algorithm was
extension to three or even four            implemented and verified on artificial
dimensions is needed to resolve            images. The Pearson’s correlation
ambiguities.      This results to an       coefficient was chosen as a
increase to the amount of data acquired    measurement for the resemblance
and the required processing time [8].      between the reconstructed image and
     Figure 1 illustrates a typical 3D     the original one.
NMR spectrum. We need to determine              The rest of this paper is organized
the number of spots and their              as follows: Section II presents a
positions. If we were able to look at      mathematical background of Radon
the spectrum from different angles we      transformation and the reconstruction
could get this information. Currently      algorithm. In section III, the acquired
the only available information are         results are illustrated. Finally, we
projections of the spectrum from           discuss about future work and
different angles. Using that 2D            conclude our paper in section IV.
information, we try to reconstruct the
correct 3D image [8]. This image           II. Methods
reconstruction      approach,     using
different projections and angles of        A. The Radon Transformation
views, is very popular in many fields
such as x-ray scanning, tomography              The 2D Radon transformation is
and determination of protein structure.    the projection of the image intensity
If only two projections are used some      along a radial line oriented at a specific
resonances might be cut off by others.     angle [9]. Radon expresses the fact that
Thus, more projections may be              reconstructing an image, using
required depending on the problem          projections obtained by rotational
under study [8].                           scanning is 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
                                           and 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.
                                           Meneses-Fabian et al. [12] describe a
                                           novel technique for obtaining border-
                                           enhanced tomographic images of a
 Fig. 1 Example of a typical 3D NMR        slice belonging to a phase object.
             spectrum.                     Vítezslav       [13]    examines      fast
                                           implementations of the inverse Radon
    In this paper we present an image      transform for filtered backprojection
reconstruction algorithm introduced by     on computer graphic cards. Sandberg
E. Kupce and R. Freeman [18]. Inputs       et al. [14] describe a novel algorithm
for tomographic reconstruction of 3-D         points on the line whose normal vector
biological data obtained by a                 is in  direction and passes the origin
transmission electron microscope.             of    ( x, y ) -coordinate satisfy the
Milanfar [15] exploits the shift              equation:
property of Radon transformation to
image processing. Barva et al. [16]                           cos
present a method for automatic
                                              y
                                                 tan(  )                            
electrode localization in soft tissue         x          2     sin 
from radio-frequency signal, by
exploiting a property of the Radon                  x cos  y sin   0
Transform. Challenor et al. [17]
generalize the two dimensional Radon
transform to three dimensions and use
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 importance of the selection of the
“correct” projections for image
reconstruction.


                                                     Fig. 3 The Radon Transform
                                                             computation.


                                              The integration along the line whose
                                              normal vector is in  direction and
                                              that passes the origin of ( x, y ) -
                                              coordinate means the integration of
                                               f ( x, y ) only at the points satisfying
                                              the previous equation. With the help of
                                              the Dirac “function”  , which is zero
                                              for every argument except to 0 and its
     Fig. 2 Different projections of          integral is one, g (0, ) is expressed as:
      a three-dot image example.
                                              g (0, )   f ( x, y)   ( x cos  y sin  ) dxdy
      Suppose a 2-D function f ( x, y )
(Fig. 3). Integrating along the line,
whose normal vector is in  direction,        Similarly, the line with normal vector
results in the g ( s,  ) function which is   in  direction and distance s from the
                                              origin is satisfying the following
the projection of the 2D function
                                              equation:
 f ( x, y ) on the axis s of  direction.
When s is zero, the g function has            ( x  s  cos )  cos  ( y  s  sin  )  sin   0
the value g (0, ) which is obtained by             x cos  y sin   s  0
the integration along the line passing
the origin of ( x, y ) -coordinate. The
So the general equation of the Radon                       projections may be required. Because
transformation is acquired: [10, 11, 15,                   of the discrete nature of the NMR
16, 18]                                                    resonances, the problem converges
g (s, )   f ( x, y)   ( x cos  y sin   s) dxdy   very rapidly.

The inverse of Radon transform is
calculated by the following equation
[14] :
               
                   2
f ( x, y)         R sx, y  d
                
              
                   2

where R is the Radon transformation,
 is a filter and

sx, y   x cos  y sin 


B. Image Reconstruction Algorithm
                                                            Fig. 4 Peaks Using two projections.
     Kupce       and     Freeman     [18]
presented an image reconstruction
algorithm from a limited set of
projections. They suggest a method of
implementing the inverse Radon
transformation. Firstly, 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.
The 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: Suppose the
existence of two perpendicular
projections of four absorption peaks in
each one (Fig. 4). From these two                              Fig. 5 Using three projections.
projections, the potential peaks are 16,
but not all of them are true cross peaks.                        The algorithm we implemented is
If we take into account another                            based on the previous described
projection at a different angle and                        algorithm of Kupce and Freeman and
reapply the lower-value algorithm, we                      its steps are:
eliminate some potential as being false
peaks and we get the image shown in                        Step 1: Acquisition of three different
Fig. 5.                                                    projections.
     Another projection would refine                       Step 2: Expansion of the 1D projection
the solution even further. Usually three                   in 2D projection maps.
projections are enough to have an                          Step 3: Padding (with black) of the 2D
accurate definition of the peaks, but if                   projections maps in order not to loose
the original spectrum is complex more                      information due to the next step.
Step 4: Rotation of the maps to the        exists between the original image and
correct angle.                             the reconstructed one.
Step 5: Normalization of each map to            Figure 7 illustrates four images
the range 0 to 1.                          that were used to test the implemented
Step 6: Reconstruction of the original     algorithm and the corresponding
image by multiplying the maps pixel        reconstructed images which have the
by pixel.                                  optimal correlation coefficient. Figure
Step 7: Post-processing of the             8 presents the correlation coefficient
reconstructed image by normalization       for the different values of the variable
and cropping to the required size.         projection angle. Table 1 summarizes
                                           the optimal results for these images.

III. Results
     We       studied     the     Radon
transformation using Matlab and the
Image       Processing    Toolbox     in
particular. Initially we created a
collection of artificial images and             Image 1
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 angles: 0, 15, 30, 45, 90 and
135 degrees. Even though the image
consists of only three spots, in some           Image 2
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 existing
spots and their positions. Figure 6B
illustrates a more complicated example
of an image that consists of several            Image 3
small spots and the corresponding
spectra.
     We      also    implemented     the
reconstruction algorithm described in
section II to reconstruct artificially
created images. Three projections were
used to reconstruct the original images.        Image 4
The two projections are 0 and 90              Fig. 7 Artificial images and their
degrees. The third one is variable in        reconstruction using 3 projections.
the range of 1 to 89 degrees with a step
of one degree. The quality of the              The four presented images range
reconstruction     is   measured      by   in complexity and archived results.
calculating the absolute value of the      The first one, which is comprised of
2D correlation coefficient between the     only three spots, is the easier to
original image and the reconstructed       reconstruct reaching 0.88 to the
one. This measure receives a value         correlation coefficient. The second one
between 0 and 1. As the value              is comprised by spots of different
increases, so does the resemblance that
radius. The last two pictures contain a    [4] H.M. Berman, J. Westbrook, Z.
greater number of small spots. The         Feng, G. Gilliland, T.N. Bhat, H.
optimal projection angle for all           Weissig, I.N. Shindyalov, P.E. Bourne,
pictures ranges from 47 to 74 degrees.     Nucleic Acids Res. 28 (2000) 235.
                                           [5]     K.      Howard,     “Improving
     Table 1. Optimal Angle and            NMR/MRI”,         Princeton     Weekly
corresponding Correlation Coefficient.     Bulletin,     November     23,    1998,
  Image        Angle      Correlation      http://www.princeton.edu/pr/pwb/98/1
     1           47          0.88          123/nmr.htm
     2           62          0.75          [6] J. Jeener, Ampere International
     3           74          0.55          Summer        School,   Basko     Polje,
     4           60          0.59          Yugoslavia, 1971.
                                           [7] W.P. Aue, E. Bartholdi, and R.R.
                                           Ernst, J.Chem. Phys. Vol 64, 2229
IV. Conclusions                            (1976).
                                           [8] E. Kupce, R. Freeman, “Fast
                                           Multidimensional NMR Spectroscopy
        To summarize, in this paper we
                                           by the Projection – Reconstruction
tried to reconstruct an image using
                                           Technique”, Spectroscopy Vol. 19, pp.
projections from different perspectives,
                                           16-20, 2004.
which we obtained with the use of the
                                           [9]Mathworks,http://www.mathworks.
Radon transform. In order to achieve
                                           com, accessed 15 June 2005.
this, we implemented an algorithm,
                                           [10] A. Asano, “Radon transformation
based on the one proposed by Kupce
                                           and projection theorem”, Topic 5,
and Freeman [18]. In the presented
                                           Lecture notes of subject Pattern
examples we used three projections of
                                           information processing, 2002 Autumn
the input image, reaching a correlation
                                           Semester, http://kuva.mis.hiroshima-
coefficient of 0.88. Future perspectives
                                           u.ac.jp/~asano/Kougi/02a/PIP/
of the proposed work include the
                                           [11] A. Averbuch, R.R. Coifman, D.L.
application of the implemented
                                           Donoho, M. Israeli, J. Wald΄en, Fast
algorithm to real NMR data, the
                                           Slant Stack: A notion of Radon
application of more projections for the
                                           Transform for Data in a Cartesian Grid
image      reconstruction    and     the
                                           which      is   Rapidly Computible,
development of heuristics for the
                                           Algebraically Exact, Geometrically
determination of optimal projection
                                           Faithful and Invertible., to appear in
angles.
                                           SIAM J. Scientific. Computing, 2001
                                           [12]      C.     Meneses-Fabian,     G.
                                           Rodr´ýguez-Zurita, and J.F. V´azquez-
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Ultrasonics,    Ferroelectrics    and                            2004.




                                                     Original picture




                        0 degrees                      15 degrees                       30 degrees
             100                           50                                 50


              50


               0                            0                                  0
                   0       200       400         0         200          400         0       200       400
                        45 degrees                     90 degrees                       135 degrees
              50                           100                                100


                                           50                                 50


               0                            0                                  0
                   0       200       400         0         200          400         0       200       400


                       Fig 6A. Artificially created image and its Radon
                            based projections for different angles.
                                                                 Original picture




                                0 degrees                            15 degrees                      30 degrees
                 600                                  600                                 300


                 400                                  400                                 200


                 200                                  200                                 100


                  0                                    0                                   0
                       0           200       400            0           200         400         0        200        400

                                45 degrees                           90 degrees                      135 degrees
                 400                                  600                                 400

                 300                                                                      300
                                                      400
                 200                                                                      200
                                                      200
                 100                                                                      100

                  0                                    0                                   0
                       0           200       400            0           200         400         0        200        400




                            Fig 6A. Artificially created image and its Radon
                                 based projections for different angles.



                                                                Correlation
        1


       0.9


       0.8


       0.7


       0.6


       0.5


       0.4
                                         Img.1
                                         Img.2
       0.3                               Img.3
                                         Img.4
       0.2
             0             10        20          30             40            50      60            70         80         90


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

				
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