Complete curves on the unit sphere As shown in Fig. Completeness of curves on the unit sphere. Clearly, half a great circle is the simplest complete curve. Again, in this paper all the formulae are based on complete curves unless otherwise stated. The 3D version of the two-step formula for parallel-beam problem As shown in Fig. A complete curve and its J k function. R2 In the real space R 3 , the plane Ots is called a projection plane Fig. When S moves along a continuous curved segment, the frequency plane moves continuously in the frequency space.
The segment can be divided at M points and approximated as a series of great arcs, and the motion of the frequency plane can be approximated a series of small rotations. For simplicity, we further assume that the segment is differentiable.
See Fig. The reconstruction process based on Eq. To our best knowledge, it is the first time that this formula is explicitly given, which can be considered as an improved form of the inverse Fourier transform in the cylindrical coordinate system. When an object can be fully covered by a parallel-beam, the reconstruction problem is solved by Eq. For a long object, such as human body, it is impossible for an x-ray beam to cover the entire object.
Therefore, a more practical problem is to reconstruct a part of the object from truncated projections pS t, s known in a region of the projection plane Ots. This is called the long object problem of 3D CT [12,28].
To deal with various cases, let us generalize Eq. Hence, Eq. Reconstruction formulae for 3D parallel-beam problem Let us develop new reconstruction formulae suitable for truncated parallel-beam projections. Our method is to choose some special weight functions so that the filtered projection in Eq.
Therefore, we consider a piecewise constant weight function with the same symmetry, dartboard function. Since the weight function defined in Eq. This completes the proof. Note that the 1D convolution can be done line by line.
The m-th filter and its related convolution operation are shown in Fig. In the projection plane, to calculate the filtered projection at a given point t, s , we use the projection data on several lines including its neighborhood for the derivative operation through the point t, s rather than projection data on the entire projection plane.
Now, let us study the case of the simple complete curve. This weight function is normalized, i. A simple justification for that normality goes as follows. Clearly, when the source point S moves from A to B, the frequency plane is turned over and every frequency point is swept once and only once if two passes in the opposite directions are not counted. To explain the normality of this function in detail, an intuitive analogy is provided in Appendix A.
For computation of the filtered projection at the origin, only the projection data indicated in e and f are involved. In a general sense, this remains an open problem. According to Eq. Therefore, Eq. The weight function is 1 for every point on the frequency plane. Cone-beam reconstruction Similar to the 2D case, let us translate the reconstruction formulae for 3D parallel-beam case to the cone-beam CT with the well-known relations between the parallel- and divergent-beam projections in Fig.
A trajectory and its complete region A trajectory C consists of a finite number of curve segments in the 3D space R 3 , along which an x-ray source goes Fig. Imaging geometries for cone-beam reconstruction at points O a and O trajectory C. The set of the complete points is called the complete Region of the locus C , denoted as R C. In contrast to the 2D case, integrals along some lines through a complete point may be unknown. This fact causes some difference between fan- and cone-beam reconstruction.
Reconstruction formulae for cone-beam CT Now, we translate the formulae for the parallel-beam to cone-beam case with the three-step method we employed for 2D CT. Imaging geometries for cone-beam reconstruction at points O a and O 5. Step 2. In reference to Fig. Then, Eq. The derivative operation in Eq. Similarly, the derivative operation in Eq. Formula 36 is a general cone-beam FBP formula, which does not need the assumption that the object must be supported inside the trajectory.
In the following, we directly give the associated cone-beam reconstruction formulae without describing the corresponding three steps. This formula is consistent with the filtered backprojection formulae FBP developed by several groups [13—17]. This formula is consistent to the backprojection filtration formulae [13,15,18—20]. In [13,18,20], the BPF is introduced based on the odd extension of the projection data. However, according to our current understanding, Theorem 3 in [15] is compromised by a minor conceptual flaw.
In the proof of Theorem 3 in [15], Eq. In other words, there exists an essential difference between the even and odd extensions of projection. O P2 Evidently, the approximation in the Palamodov cone-beam Formula P2 comes from the approximation of the associated parallel-beam formula P1. After we [43] pointed out the approximate nature of the Palamodov cone-beam formula [36], he modified his proof [44,45]. However, based on our new general reconstruction scheme, it is Theorem 3 in his paper [36] that leads to the approximate nature of his cone-beam formula.
Discussions and conclusion Evidently, we can extend the above discussion into the higher dimensional space to form a reconstruc- tion theory based on truncated projections. This is a promising direction of integral geometry [37]. We are working along this line and will report our results later.
Since cone-beam CT is practically important, most researchers have paid much more attention on cone-beam CT in hope to solve the cone-beam problems directly. This makes 3D CT problems quite different from and much more difficult than its 2D counterparts.
On the other hand, in this paper 3D parallel-beam problems are first carefully studied, and then cone-beam solutions come out in an easy way through the simple relations between parallel- and divergent-beam projection, as illustrated in Fig. This new methodology is a primary part of the originality of this paper.
However, since the three formulae are related to the different forms of the inverse Fourier transform, these three schemes should be essentially equivalent. The reader needs to choose the convenient one for their problems.
Meanwhile it is acknowledged that our scheme can generate many formulae in CT, but not all of them. Steps towards the solution of the cone-beam problem in our scheme. Applicability of our scheme in the cases of discontinuous trajectories. The traditional assumption that an object be compactly supported inside the locus is no longer necessary. Hence the reader still needs to pay attention to new methods and results in the CT field.
In conclusion, we have presented an intuitive and complete scheme for CT in different imaging geometries including 2D and 3D parallel- and divergent-beams.
A key step is the development of a new fundamental formula starting from the inverse Fourier transform in cylindrical coordinate system. Our results have been demonstrated to be not only consistent with the most latest main formulae but also valid under more general conditions including a non-continuous scanning trajectory and an extended object support Fig. Meanwhile, some minor conceptual flaws in the CT literature have been identified and fixed.
Finally, some open questions have been suggested. Our understanding is that Fourier analysis should be viewed as the theoretical foundation of CT and that this complete scheme of CT is just another example among many applications of Fourier analysis in modern sciences and technologies [49].
The coverage of dual-energy CT has been significantly expanded to include its background, theoretical development, and clinical applications. Posting Komentar. Rabu, 16 Juli [P Diposting oleh Armando di Label: Ebooks. Tidak ada komentar:. The situation has significantly changed since then, as dual-energy CT is now utilized in routine clinical applications to aid in disease diagnosis.
A significant expansion to Chapter 12 has been written to provide the technology background, theoretical development, and clinical applications of dual-energy CT. Many problems are open-ended and may not have uniquely correct solutions.
At the time of the publication of the second edition, the world was experiencing an unprecedented financial crisis that some called a financial "tsunami. Recent advances in CT have shown that the entire industry remains healthy, and the demand for advanced CT technologies has expanded beyond the developed counties.
The future of CT remains bright. Sign In View Cart Help. Email or Username Forgot your username? Password Forgot your password? Keep me signed in. Please wait No SPIE account? Create an account Institutional Access:. Author s : Jiang Hsieh. Jiang Hsieh July Hide Excerpt -. Buy this book on SPIE. This will count as one of your downloads. You will have access to both the presentation and article if available. This content is available for download via your institution's subscription.
To access this item, please sign in to your personal account. Create an account. Front Matter. Access to the requested content is limited to institutions that have purchased or subscribe to SPIE eBooks. To obtain this item, you may purchase the complete book in print or electronic format on SPIE. Image Reconstruction. Image Presentation. Major Components of the CT Scanner.
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