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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 sx, y d 2 where R is the Radon transformation, is a filter and sx, 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. 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Challenor , P. Cipollini and Effect of Image Motion in the Radon D. Cromwell, “Use of the 3D Radon Transform Domain”, IEEE Transform to Examine the Properties Transactions on Image processing, vol. of Oceanic Rossby Waves”, Journal of 8, no. 9, September 1999 Atmospheric and Oceanic Technology, [16] M. Barva and J. Kybic with J. Volume 18, 22 January 2001. Mari and C. Cachard, “Radial Radon [18] E. Kupce, R. Freeman, “The Transform dedicated to Micro-object Radon Transform: A New Scheme for Localization from Radio Frequency Fast Multidimensional NMR”, Ultrasound Signal”, In UFFC '04: Concepts in Magnetic Resonance, Proceedings of the IEEE International Wiley Periodicals, Vol. 22, pp. 4-11, 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.