Tomografik görüntü oluşturulmasında doğrudan fourier yöntemine yeni bir yaklaşım

Abstract

ÖZET Bu çalışma, tomografık görüntü oluşturmada kullanılan ve merkezi kesit teoreminin bir uygulaması olan doğrudan Fourier yönteminin geliştirilmesi amacı ile yapılmıştır. Doğrudan Fourier yönteminde projeksiyonların Fourier transformlarının alınması ile cisim fonksiyonunun Fourier uzayında radyal hatlar boyunca değerleri elde edilir. Bu değerlerin kartezyen ızgara noktalarına interpolasyon ile taşınması gerekmektedir. İnterpolasyonun Fourier uzayında gerçekleştirilmesindeki güçlükler nedeni ile bu çalışmada interpolasyonun Fourier uzayına girmeden önce yapılması önerilmiştir. Bu önerinin temeli, projeksiyonun kartezyen eksen takımında gösterilebilen sürekli ve peryodik bir fonksiyon olmasına dayanmaktadır. Çalışmada, periyodik bir projeksiyon uzayında tanımlanan ve projeksiyon fonksiyonlarından oluşan yüzeyin frekans uzayına göre interpolasyona daha yatkın olduğu önerisi getirilmiştir. Bu şekilde oluşturulan projeksiyon haritası üzerinden interpolasyonla amaca uygun projeksiyonlar üretilmiştir. Çalışmada önerilen bir diğer yenilik de interpolasyon açılarının ve bu açılara bağlı örnekleme adımlarının seçimi ile ilgilidir. Merkezi kesit teoremine göre, bazı özel projeksiyon açılarında cismin yerleştirildiği kartezyen gride ait en az bir veya daha fazla nokta projeksiyon ekseni üzerine düşmektedir. İnterpolasyon açılan bu özellik kullanılarak cismin yerleştirildiği kartezyen grid noktalarından geçen açılar olarak seçilmiştir. Bu açılara bağlı örnekleme adımlan ile üretilen projeksiyonlann bir boyutlu Fourier transformları alınarak iki boyutlu Fourier uzayında cisme ait bilgiler elde edilmiş ve böylece oluşturulan kompleks matrisin iki boyutlu ters Fourier transformu alınarak görüntüye ulaşılmıştır. İnterpolasyona konu olan projeksiyon haritası, bir cismin etrafından alınan projeksiyonlardan oluşmaktadır. Önerilen yöntemin uygulanması amacı ile İ.T.Ü. Araştırma Fonu tarafından kısmen desteklenen ve Enstitü olanaklarının da kullanıldığı bir bilgisayarlı tomografi tarayıcısı kurulmuştur. Endüstriyel amaçlı tarayıcı, paralel demet geometrisine sahiptir ve kaynak olarak da 7.4X1010 Bq aktiviteli Cs-137 kullanılmıştır. Tarayıcı sistemde 1. 1X10`3 m çaplı tek delikli kurşun kolimatörler mevcuttur. Kolimatörün uzaysal ayırma gücü 1.2X1 0`3 m ve sistemin yoğunluk ayırma gücü %8,3'dür. Yaklaşık 0.6-0.7 m çapında metal numunelerin incelenebildiği tarayıcı sistemde elde edilen projeksiyonlar vasıtasıyla oluşturulan tomografık görüntülerde 1.5X10`3 m ve üzerindeki malzeme boşluklan ayırt edilebilmektedir. xii

The very concept and the mathematical background of tomography, which refers to sectional viewing or imaging, had been known for a long period of time when it suddenly emerged in the medical arena more than two decades ago. It was a major breakthrough in medical imaging technology. Even it had an enigmatic appearance in that the image reconstruction technique used in the first successfully installed CAT scanner was kept as a trade secret for a while, the scientific community was quick to discover some practical reconstruction algorithms soon: Production of clear brain images in spite of the interference of the dense skull material had revolutionary effects in the medical practice. Since then the tomography related research activity has increased by leaps and bounds in many fields of science and technology with practical applications including but not limited to nondestructive testing, seismic testing, three dimensional electron microscopy, ultrasound, microwave and magnetic resonance imaging, radioisotope scanning, positron emission tomography. In order to construct a tomogram of an object, it must be illuminated by an external source of energy. Beam of energy emanating from the source travels along a straight path with an accompanying attenuation. If the wavelength of the energy is comparable to the dimension of inhomogeneities, the beam is scattered; this case lends itself to diffracted tomography method. As a third alternative, the source may be distributed in the object to be imaged, as in the case of positron emission tomography. In this work we are concerned with parallel beam transmission tomography only. This work is confined of transmission tomography for nondestructive and nonmedical applications. In transmission tomography, the object is placed in between a well collimated (and aligned) source and a detector pair, and it is moved in the. transverse direction to the beam, in order to be able to measure, the attenuated beam intensity at each displacement step. Logarithm of the attenuation ratio as a xiiifunction of the displacement is referred to as a projection, P@(s), where s is the displacement of the beam from the center of rotational, and © is the rotation angle measured from a reference direction. A set of projection data is collected by repeating the scanning procedure all over again at each equal angular step by which the object is rotated around an axis perpendicular to the plane formed by the beam itself and the transverse scanning direction, which defines the cross sectional imaging plane as well. Tomographic imaging deals with reconstructing an image from its projections. Mathematically a projection is formed by combining a set of line integrals. Having a set of projections at hand, we need a prescription for the reconstruction of the tomographic image. For this purpose, some iterative algebraic equation solution techniques were used initially in the literature. Later on, methods based on `Fourier central section or slice, or projection` theorem have become more popular for reasons of improved image quality and reconstruction speed. The theorem basically states that `the (1 -dimensional) Fourier transform of a projection is the section of the (2-dimensional) Fourier transform of the image, at the same projection angle`. This theorem when implemented in the frequency domain, results in the so called `filtered backprojection` algorithm; on the other hand its equivalent in the spatial domain is known as the convolution backprojection algorithm. In back projection the measurement obtained at each projection are projected back along the same line, assigning the measured value at each point in the line. Thus the measured values are smeared across the unknown density function. A convolution or an equivalent filtering is necessary to recover a clear image from the back projected projections. A third and less popular algorithm utilizes the theorem in question directly, with some shortcomings in medical applications, where the range of density variations is relatively small. These direct methods provide a linear reconstruction formulation between a two-dimensional distribution and its projections. Essentially the Fourier transform of the projections are laid on a polar raster in the frequency plane. A 2-dimensional inverse transform of the data on this polar raster is necessary to construct the tomographic image. In practice only a finite number of projections of an object can be taken. In that case it is clear that the two-dimensional Fourier transform of the object function is only known along a finite number of radial lines. Therefore, inverse Fourier transform algorithms are suitable for the inversion of data on a square cartesian grid. At this moment data available on a polar raster must be interpolated into a square grid. It is common to determine the values on the square grid by some kind of nearest neighbour or linear interpolation XIVfrom the radial points. Since the density of radial points becomes sparser as one gets farther away from the center, the interpolation error also becomes larger. In this work a discrete version of the central slice theorem is proved for a lattice of point weights. The theorem stipulates that projection angles be selected with rational tangents (= ratio of two integers) for an exact reconstruction, with angle dependent sampling intervals. (We remember that implementation of direct Fourier method calls for an interpolation on the frequency plane). There is a multitude of projection angles such defined for a grid representing NxN pixels, which makes it very difficult if not impossible to obtain projection data as N is increased, for instance for N=8, there are 21 angular steps, whereas when N=64,that number grows to 1282. From the practical point of view getting as many projections as the number of those selected angles is a very time consuming and tedious task. To surmount this problem, we observe that, the projection PQ(s), which is known as the line integral of the attenuation coefficient distribution does not qualify to be a regular polar function for obvious reasons; namely it is not a single valued function at s=0. This fact suggests that P6(s) be imagined on an s-9 cartesian coordinate axes, which gives way to interpolations P6(s) is a periodic function of ©.To implement these ideas in a full reconstruction scheme, a set of regular projection data is obtained, i.e. at equally spaced angular and displacement intervals, and plotted on s-® axes, which is called a projection map. Finally knowing what special angle needs what sampling interval, one can interpolate data directly from the s-@ map. The discrete Fourier transform of such sampled data is guaranted to fall on the grid points of the frequency plane, accordance with the discrete version of the central section theorem. In summary, the frequency plane polar to cartesian interpolation stage of the reconstruction is replaced with an interpolation on s-0 map prior to 1-dimensional Fourier transformation of the projection data. A 2-dimensional inverse discrete transform of the data produces the image. In the study, the operation of Fourier transform is performed by using well-known radix-2 Cooley- Tukey type Fast Fourier Transform (FFT) algorithm available in open literature. This algorithm mentioned above has been modified in order to get the reconstruction to consume less time. The proposed algorithm has been tested on synthetic models as well as on real projection data of some phantoms, and compared with the results of a former direct Fourier reconstruction algorithm. A tomographic scanner driven by two stepping motors was assembled in our laboratory. On this device the transverse stepping interval is 0.25X10`3 m. with an angular displacement step of 0.9 degree. Activity of the Cs-137 gamma xvsource used, was 7.4X1010 Bq. The collimator is 1.1X10`3 m in diameter and the distance between the source and the 3`-NaI(Tl activated) is approximately 0.31 m. To evaluate the performance of the collimator, a 1 -dimensional spread function is determined experimentally by using the projection profile of an object with sharp edges. The spatial resolution of collimator which is calculated from 1 -dimensional spread function's full width at half maximum (FWHM) is 1.20X1 0`3m. A single channel analyzer is used to confine the counts to a narrow energy interval around the photopeak of Cs-137 at 660 keV. A NIM bin mounted counter is driven by the PC to gather the projection data at every sampling run. In most cases 10 sec. counting cycles proved to produce reasonable results with tolerable noise levels. Sampling interval of 1.0 mm was used because the spatial resolution is limited by the diameter of the collimator. To ensure that two small features can be resolved, they must be separated by at least two pixel widths. Thus, in this study the pixel width is taken equal to sampling interval. In order to save time, the scanner initially determines the borders of the object approximately. For this operation 2 sec. counting cycles are used. Under these conditions complete scanning of a 40 mm. diameter object at 40 angles would require approximately 7 hours. The number of independent measurements must be at least equal the number of free parameters in the image, in order to suppress the streak artifacts emanating from the sharp edged high contrast objects the minimum number of projections is rc/2 times the number of rays in each projection. Therefore, in this study the number of projection angles is selected approximately it/2 times the number of rays in each projection. Unlike medical applications, there is no time limitation in the tomography for industrial applications. However, the counts are affected negatively because of the drift in the parameters of the electronic instruments running for a very long time. The PC is located a safe distance away from the scanner to prevent any damage to its ROM. The programs for driving the system are written in Basic. Reconstruction algorithms are implemented in Fortran. All of the tomographic images are displayed on a 256 x 256 matrix, and the interpolation in a special cartesian` s-@ ` plane is made with 20037 interpolation angles. Reconstruction time depends on the number of projection angles and the number of rays in each projection. The tomographic images which is presented in this study required between 10 and 25 minus of reconstruction time. The tomographic images are printed using a graphics program and specially coded download characters representing various grey tones. Tomographic images are evaluated with respect to their quality. These include, a 2-dimensional point spread function and the density resolution function for the tomographic system. Some of the tomographic images are compared with images obtained by the direct XVIFourier transform. The spatial resolution of the former clearly surpasses that of the latter. The point spread function for the tomographic imaging system has 2 mm FHWM. The density resolution is measured as 8.3 % for steel. The developed system can be used for neutron tomography with some retrofitting. XVll