Interpolation of 3D magnetic resonance data J. Mikulka 1, E. Gescheidtova 2 and K. Bartusek 3 1, 2 Department of Theoretical and Experimental Electrical Engineering, Brno University of Technology, Kolejni 2906/4, 612 00 Brno, Czech Republic 3 Institute of Scientific Instruments, Academy of Sciences of the Czech Republic, Kralovopolska 147, 612 64 Brno, Czech Republic Abstract- This article deals with three-dimensional reconstruction methods of nuclear magnetic resonance images. The testing images were observed by tomography with basic magnetic field of 4.7 T at the Institute of Scientific Instruments (Academy of Sciences of the Czech Republic). The 20 slices of the test phantom were acquired. Two methods were found with the aim of getting utmost information about the shape of the testing phantom. One possible way is to increase the count of the sensed slices, but it implies decreasing of the signal to noise ratio. The second approach is finding the compromise between the effective count of slices and the following interpolation of other slices between the sensed ones. The both approaches were compared. Only 10 slices were used to compute the in-between others. It is better to use this approach because the computed slices can be compared with the real slices obtained by MR tomography. The results are described in the article. The images were interpolated in order to improve the following three-dimensional model creation. 1. INTRODUCTION The advantages of the nuclear magnetic resonance (NMR) were described in many publications. It is approach to acquisition of spatial data of soft tissues. The main advantage is absolutely the fact of unproved negative effects of the electromagnetic radiation to human organism subject to prescribed hygienic regulations. Against to other tomography approaches the magnetic resonance images are with the higher resolution and quality. The observed images of sensed object can be used for three-dimensional model creation after the application of suitable preprocessing methods. The reconstructed object can be useful for example to the better diagnosis in medical sciences, for quantitative or qualitative description of tissues, tumors etc. The requirement is to have utmost slices for the three-dimensional model of sensed tissue. The signal-to-noise ratio (SNR) decreases with the increasing number of slices [1]. We have to specify compromises between the number of observed slices and satisfying SNR in images individually. However, if we actually need more slices for the 3D model creation, we can calculate the other images between observed ones. Two basic approaches are described for images interpolation of MR spatial data in this paper. The first method is based on ordinary averaging of neighboring pixels intensities with the same position in the frequency domain. We can get the images describing the real scene by Fourier transformation of each slices in the k-space, which we observed by tomography. We can get (2n-1) images from original n images. This algorithm is repeatable. By the second approach we process directly the data in k-space. A vector of complex numbers is created by data observing with the same positions in the k-space. Then the frequency domain of this vector is obtained by Fourier transform [2], [3]. The frequency domain is extended by zero values for higher frequency parts. This extended spectrum is transformed back to the time domain by the inverse Fourier transformation. The length of the obtained vector depends on number of zeros included to frequency spectrum of the original vector. The flask with water was inserted to work space of
tomography with the magnetic field 4.7 T (200 MHz for 1 H nuclei) for the testing. This article follows the previous research in this area, which was described in [4]. 2. ASSEMBLY OF EXPERIMENT The images obtained by MR tomography in the time domain (k-space) are then transformed by Fourier transform to the frequency domain. These images in the frequency domain represent the real scenes. The first method interpolates the three-dimensional data in the transformed frequency domain. The second method shows the possibility of data processing and interpolating the three-dimensional data in the original time domain k-space. The example of experiment assembly is shown in figure 1. In this case only one flask filled by water was used. In figure 2 the procedure of scanned spatio-temporal slices processing and the transformation of the slices to the frequency domain is shown. Figure 1: The example of phantom for testing process of proposed interpolation methods of spatial MR data. Figure 2: The procedure of obtaining the images by MR tomography. Three selected slices in the k-space on the left, three selected slices of phantom in the frequency domain on the right. 3. AVERAGING The ordinary averaging method is used for nearby pixel intensities in the frequency domain for comparison of the proposed method of signal interpolation in the k-space, which are generated between two existing images obtained by MR tomography. The intensity of a new pixel is then given by the equation: (, ) (, ) + (, ) Inew x y. (1) 2 new-1 new-1 = I x y I x y 4. INTERPOLATION IN k-space This approach is a little bit complicated but it gives better results. The basic principle is to extend the spectrum of the obtained signal in the k-space, which arises by values of the nearby pixel intensities with the same positions. In figure 4 we can see the original complex signal generated by nearby pixel intensities. The same resampled signal transformed to frequency domain with including the zero values for higher frequencies and transformed back to the k-space is shown on the right. It is clear that by this approach the signal was supplemented by new samples, witch interpolates the k-space in the space where no slices were obtained by the tomography. This interpolated k-space is then transformed by FT to frequency domain with (2n-1) images.
Figure 3: Interpolation of pixel intensity in the image between original slices by averaging. The pixel intensities vector (module, phase) is on the left, the pixel intensities resampled vector (module, phase) is on the right. Figure 4: Interpolation of images between obtained slices by the spectrum adjustment in k-space. The original vector of k-space values (module, phase) is on the left, the resampled vector (module, phase) is on the right. 5. EXPERIMENTAL RESULTS The both methods (averaging in the frequency domain, extending the spectrum of k-space by zero values) were tested on 21 original slices of proposed phantom obtained by MR tomography at the Institute of Scientific Instruments of the ASCR in Brno. The algorithms were implemented in the Matlab software [3]. The set of 11 original slices was interpolated to 21 slices. The set of 10 new images were computed between the 11 original odd slices and compared with the original even ones. Several slices from the middle of the measured object are shown in figure 5. Calculated slices with use of the method of averaging in the frequency domain are shown in figure 6. Calculated slices with use of the method of the spectrum extending in k-space are shown in figure 7. All the calculated slices are red marked.
Figure 5: Several original slices in the middle of the measured object. Blue color signs images to be calculated. Figure 6: Calculation of images by averaging. Red color signs calculated images.
Figure 7: Calculation of images by extending the spectrum of signals in the k-space. Red color signs calculated images. 6. CONCLUSIONS We can compare results obtained by both tested methods. The first one the averaging method of the neighbor pixel intensities with the same position in the frequency domain gives the good results. The intensity of a new pixel in the interpolated image always reflects the values of obtained pixel intensities. There is a problem, after the segmentation of this image, the size of the segmented object will be always the same as in the neighbor image. But there is no other error brought into the image. It is clear that this method doesn t give any new spatial information, only the average number of hydrogen nucleus in the area of interpolation we can get. By the second method interpolation in k-space an error due to resampling of the signal by extending its spectrum by zero values is brought. This signal is devaluated by harmonic spatial signal. There are spatial artifacts in more images. The processed images will be segmented, whereas the suitable level of intensity for the segmentation will be found. The resultant contours will be used for accurate three-dimensional model creation and it will be made the comparison of the both methods. ACKNOWLEDGEMENT This work was supported within the framework of the research plan MSM 0021630513 and projects ME10123 and GACR 102/11/0318. REFERENCES 1. Gescheidtova, E., Bartusek, K. Kriteria pro vyber vlnek pri zpracovani MR obrazu. Elektrorevue [online - (http://www.elektrorevue.cz], 2009, n Czech. 2. Vich, R., Smekal, Z. Cislicove filtry. ACADEMIA Praha, 2000, in Czech. 3. Gonzales, R., et col. Digital Image Processing using Matlab. 2009. 4. Mikulka, J., Bartusek, K. 3D reconstruction in magnetic resonance imaging. In Proceedings of PIERS 2010 in Cambridge. 2010. pp. 1043-1046.