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The Cone beam System on Princess Margaret Hospital, University health network.

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The C-arm Cone beam System on Princess Margaret Hospital, University health network

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Summarizing the work in Local Tomograhy Dr. Shuangren Zhao Research Associate Radiation Physics Department Princess Margaret Hospital

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What is local Tomography? Ordinary tomography is global since reconstruction at a point x requires integrals over lines far from the point x. Local tomography uses only lines close to the point x

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Compare the local and non- Local Tomography Ordinary Tomography Uses the projection data far away from the local region Local Tomography Only uses the projection data closing to the local region

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Why use Local Tomography? Only small region of the object is interested. Large dose of radiation is avoided. Increase the Speed of the reconstruction algorithm. Only the discontinuity of the object function is interested. If projection data are truncated, there are truncated artifacts on non-local Tomography.

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Simulation with Truncated data using non Local Tomography truncated data, truncated object, truncated effect

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What difficulties does Local Tomography have ? If f(x) is a solution of equation A f = p, some different functions for example g(x) satisfies A g= 0, f(x)+ g(x) is still the solution of this problem. This kind of functions g(x) are called null functions. If null function is not zero, the problem is not unique. Non Local tomography is with uniqueness. Local Tomogaphy entails the loss of the uniqueness.

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1. Known Theory for Local Tomography (λ f) tomography (λ f+ μ/λ f) tomography Pseudolocal Tomography Wavelet-based local Tomography Extrapolation for the missing data

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Definition of operator λ F is Fourier Transform operator

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Important formula about λ

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Non local reconstruction

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(λ f) tomography E.I. Vainberg 1981 K.T. Smith 1985 Cupped reconstruction Pure local reconstruction

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(λ f + μ/λ f) tomography A Faridani (1992) With Cup correction

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Pseudolocal Tomography A. I. Katsevich 1996 A. G. Ramm 1996 Using Radon reconstruction Cut the kernel length of Hibert transform short What errors are introduced from this method?

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Wavelet-based local Tomography D. Walnut (1992) F. Rashi-Frarokhi (1997) Filter is implemented with wavelets. Wavelets are back projected.

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The principle of the wavelet Reconstruction: with different basis functions For filter backprojection method, we use δ function as basis For Wavelet reconstruction, they use wavelet basis functions A) δ function, B) Harr wavelets, C) Daubchies wavelets At the backprojection process, δ function spread from a point, however wavelet will be more localized to a point. The wavelet method will have exactly same results as filter back projection method if global projections are available, however if projection is truncated we could not expected wavelets methods have the same results as filter backprojection methods. I will make a simulation with one truncated object to check these theory.

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Wavelet method could be checked with one cylinder simulation truncated data, look at the truncated effect

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Simple Extrapolation The projection data are just extrapolated according to the value in the boundary of truncated projection data.

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λ reconstruction

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Comparing local filter to non-local filter

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(λ f + μ/λ f) filter k=k(l, μ)

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2 With Simulation Data Truncated data (f ) (λ f) (λ f + μ/λ f) l=4 μ=0.01,0.03,0.05,0.2,1.0

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Simulation with Truncated data using non Local Tomography

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Simulation with Truncated data using Local (λ f + μ/λ f) Tomography μ=0.0μ=0.01

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Simulation with Truncated data using Local (λ f + μ/λ f) Tomography μ=0.03μ=0.05

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Simulation with Truncated data using Local (λ f + μ/λ f) Tomography μ=0.2μ=1.0

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Profile for μ=0.0, 0.01, 0.03, 0.05, 0.2, 1.0

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The Influence of Truncated projection data

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Balance the truncated effect The Influence of Truncated data Using non local tomography (The detector size is just half the size of the view of object), for Cylinder_1 with f= 1 and Cylinder_2 with f= - 1..

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Important simulation results: (1) See the above example we could obtain the results that if the truncated object is outside of ROI with high frequency, it will has very small influence to the inside of ROI. (2) If we filter out only the low frequency, the out side object should has very small influence to the inside. λ Local tomography utilize this mechanism. (3) We could use an anti object, which has negative value and looks like the original object to balance the influence of the original object. The anti object does not have to be the same as the original object. This method will offer a correction of the truncation effect.

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3. With measured data not truncated data (f ) (λ f) (λ f + μ/λ f) Truncated data (f ) (λ f) (λ f + μ/λ f)

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Rabbit White, Mode2_Rump

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See shell reconstruction f, λf, λf +0.02 1/λ f

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4. The work I have done (Between the theory and the implementation) Proved with understanding that (λ f + μ/λ f) = (λ + μ/λ) f. The normalization of the kernel of local tomography filter Parallel-beam => fan-beam => cone beam Match the out put of simulation to the input of our Iview3D software

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Prove with understanding that (λ f + μ/λ f) =? (λ + μ/λ) f. Using the left side, the back projection process need to be done two times. The advantage is that μ could be easily adjust. If the value of μ is known, we could use the right side. In this case the back projection process need only to be made once. Hence the speed of the calculation is increased in practical use. In general these two sides are not equal. However if the operators λ and 1/λ are linear they will be equal. The problem is whether or not they are linear. I found that the two operator will be linear if and only if the interpolation of the back projection is linear. As we know, the interpolation of our software Iview3D is linear. So that for our λ tomography reconstruction the right side of the above formula could be used.

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The normalization of the kernel of local tomography filter λ f, 1/λ f and f have different units. It is not required that the reconstruction of those 3 functions have similar values. However our software Iview3D could only show the reconstruction value between 0 to 1. This suggest that any reconstruction values should be at this range. A Faridani use μ=47 to balance two operator λ and 1/λ. It is clear he did not normalize the three functions suitable. If λ and 1/λ are normalized in the same order then μ should be in the range 0 ~ 1. I made this normalization, which is shown in the following.

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Nomalization

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for μ=0.03, the reconstructed value is close to the value of phantom

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Parallel-beam=> fan-beam=>cone beam The theory of λ local tomography is written for parallel beam case. Parallel beam and cone beam cases still have some differences. Implementation for his theory to the cone beam case is required to find out whether or not some correction necessary. After checking the theory, I find that the reconstruction formula for the cone beam case is just the same as the parallel beam case. We are very lucky. However these does not mean cone beam λ tomography are easy. The λ tomography began in 1981. The first cone-beam implementation was reported at 2000[3]

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Match the output of the simulation to the input of our Iview3D The simulation program we have is not designed for our IView3D software. Hence the output has a different format. We need to match the simulation output results to our IView3D software and produce correct header files. The format transformation is done with Matlab. Now it could be used but still it still has problem. The size of variables U and V of simulated program could not be greater than 512 (the program will crash). The size of variable V and U could not take different number than 128, otherwise the object leave the the center of the view.

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5. The work I am doing now Speeding up the Local Tomography Optimization of the kernel of the local filter Iterated extrapolation to the projection data Correction of the influence of the truncation effect Reconstruction with general Hankel Transform

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Optimization of the kernel of local filter Optimize the kernel in Fourier frequency domain or in wavelet frequency domain. For a given length of the kernel we could modify the kernel so that it is the closest kernel in frequency domains to the Ram-Lak filter. Optimize the kernel with the goal function in the spatial domain. For a given length of the kernel we could modify the kernel so that the reconstructed object is as close to the phantom as possible. Optimize the kernel so it is not sensitive to noises.

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Iterated extroplation The truncated objects are obtained by first reconstruction The extrapolation data are calculated from the truncated objects. The second reconstruction is done through the truncated data together with the extrapolated projection data.

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Iterated correction Given a negative value to the truncated object that is calculated from the first reconstruction. The truncated projection is calculated from the negative object. Correction to the reconstruction is calculated from the above calculated projections by some reconstruction algorithm. The final reconstruction is obtained by adding up the first reconstruction and the correction. Since the corrections are only with low frequency, low frequency reconstruction algorithm could be used.

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Balance the truncated effect The Influence of Truncated data Using non local tomography (The detector size is just half the size of the view of object), for Cylinder_1 with f= 1 and Cylinder_2 with f= - 1..

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General Hankel Transform Hankel transform could be used for parallel beam reconstruction. General Hankel transform is developed for fan beam reconstruction, (which is one of my work in Julich research center of Germany). General Hankel transform could be utilized If the projection and back projection process are in a iterated way. Using General Hankel transform, the calculations of the algorithms could be faster, especially if the object has only low angle-frequency.

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Conclusions Different technologies could be utilize to implement the local tomograpy (1) Filter out low frequency: λ tomography (2) Choose different basis function: wavelet methods (3) simple extrapolation, iterated extrapolation (4) iterated correction. Some of the above technology could be implemented together.

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Acknowledgments Douglas J. Moseley for offering the input projection data simulation program and Matlab program to see the results of IView3D Steve M. Ansell for developing the software interface Sami Siddique for the data acquisition Graham A. Wilson for the working environment Jeffrey H. Siewerdsen for introducing me the basic knowledge x- ray equipment David A. Jaffray for introducing the technology--reconstruction with large size. Because of this technology, I could see immediately the influence of the truncation, the shadow. The shadow is produced by the truncated object outside VOI.

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References [1]A Faridani, Local Tomography, SIAM J. APPL. MATH Vol 32, No 2, pp459-484, April 1992 [2]F. Rashid-Farrokhi, Reconstruction singularities……, Math. And Comut. Modelling, 18(1993), pp. 109-138 [3] P. Huabsomboon, 3D Filtered Backprojection Algorithm for local Tomography, M.S. Paper, Dept. of Mathematics, Oregon State University, Corvallis, Or 97331, U.S.A., (2000). [4] S.R. Zhao, H. Halling: Image Reconstruction for fan beam tomography using a new interal trasform pair. International Symposium on Computerized Tomography in Novosibirsk, Russia August 10-14, 1993. Abstracts ed. M.M. Lavrentev, p125.

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