Multigrid Reconstruction of Micro-CT Data

Transcription

1 DEGREE PROJECT IN MEDICAL ENGINEERING, SECOND CYCLE, 30 CREDITS STOCKHOLM, SWEDEN 2017 Multigrid Reconstruction of Micro-CT Data BO GAO KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF TECHNOLOGY AND HEALTH

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3 Abstract The group of Medical Imaging at School of Health and Technology is developing a micro-ct scanner for small animal imaging. Micro-CT refers to reconstructing CT image with resolution in scale of micrometer, and this goal is achieved by acquiring projection data with high resolution. Nevertheless, high resolution projection data and high resolution reconstruction image have introduced the problem of memory insufficiency, as more points need to be processed during micro-ct reconstruction. For this reason, this paper has investigated how to alleviate the burden on computers memory through applying multi-grid reconstruction algorithms, which means to reconstruct region of interest (ROI) with high resolution while reconstructing background with lower resolution. By doing that, pixels being considered in the reconstruction space has been decreased and normal computers will be capable to handle reconstruction of micro-ct image. Through testing on numerical data (Shepp Logan Phantom), it can be concluded that multi-grid reconstruction algorithm could reconstruct high fidelity ROI with much faster speed comparing to full resolution reconstruction. Moreover, this proposed technique can also give decent reconstruction to data acquired from micro-ct scanner. Keywords: Micro-CT, CBCT, multi-grid reconstruction, DICOM 3

4 Acknowledgement Firstly, I would like to express my gratitude toward my supervisor, Professor Massimiliano Colarieti-Tosti, for offering me the opportunity to join his group and letting me participate in the development of micro-ct scanner. I also appreciate the guidance he provided in this half year, which has led me to finish my master thesis work on time with a decent quality. Secondly, I would like to thank my co-supervisor, David Larsson, for all the help he provides on operating micro-ct scanner, using relevant software and preparing the final presentation. Thirdly, I would like to show my appreciation toward my friends, Victor, Jaime, Basen, Patrick, etc for enriching my life outside school and offering their unconditional help whenever I need. Finally, I would like to direct special thanks to my family for supporting my decision of studying abroad and making everything mentioned above possible. 4

5 Contents 1 Introduction Purpose of Study Method Software Development Implementation of multi-grid algorithm Modifying ODL library Projection truncating function Open CT reader and converter Testing Proposed algorithm Testing the correctness of ODL modification Numerical data test with Shepp-Logan Phantom Test with Micro-CT projection data Results Test Results of functions invented to ODL Test Results with simulated Shepp-Logan Phantom Testing of multi-grid FBP reconstruction Testing of multi-grid CG reconstruction Testing of multi-grid TV reconstruction Validation of multi-grid reconstruction methods Validation of multi-grid FBP reconstruction Validation of multi-grid CG reconstruction Validation of multi-grid TV reconstruction Multi-grid reconstruction of low contrast phantom FBP multi-grid reconstruction of low contrast phantom CG multi-grid reconstruction of low contrast phantom TV multi-grid reconstruction of low contrast phantom Discussion Functionality of functions introduced to ODL Multi-grid Reconstruction with different algorithms Accuracy Assessment Timing Issue Conclusion 46 Appendices 47 Appendix A State of the Art 47 A.1 Introduction A.2 Computed tomography (CT) and micro-ct A.3 Image Reconstruction Algorithm A.3.1 Filtered backprojection algorithm A.3.2 Iterative reconstruction methods A.4 Techniques facilitate CT image reconstruction A.4.1 Hardware support for image reconstruction A.4.2 Software support for image reconstruction

6 A.5 Conclusion

7 1 Introduction According to [1], experiments using small animals is indispensable in the development of medical treatment. However, it is also mentioned in [1] that debate has never stopped on if it is morally right to use or sacrifice animals for research in medicine. To ease this conflict, the idea of small animal in-vivo imaging has been proposed as it is less invasive than conventional diagnostic and monitoring methods, ref [2]. Following the claim made in [2], most small animals imaging modalities are invented from similar devices used for human and modified to fit the needs for animals. One of the most successful examples is the invention of micro- CT, which is based on Computed Tomography (CT) and has improved the resolution of CT to scale of micrometer in order to get images of small animal with resolution comparable to what can be achieved on human. Nevertheless, this increased resolution is obtained at the price of acquiring higher resolution projection image at each angle, which will eventually lead to a dataset much larger than what is usually used in normal CT scanner. Moreover, micro-ct also requires a reconstruction space with more and smaller pixels to represent the cross-sectional image. Both of these will make the computation quite challenging for normal computers to carry out. To get out of the dilemma mentioned above, the most straightforward approach is to decrease the size of data processed by the computer. This can be achieved by multi-grid reconstruction, which means reconstructing region of interest (ROI) with high resolution while reconstructing other region (background) with low resolution. By doing this, the number of elements being considered in reconstruction space will be decreased, hence the computational burden will be eased. 1.1 Purpose of Study This thesis project aims at decreasing the computational burden of micro- CT reconstruction when only part of the reconstruction space is of interest. To achieve this goal, multi-grid reconstruction approach proposed above will be implemented into different reconstruction algorithms. By comparing the reconstruction result of multi-grid reconstruction with full resolution reconstruction, whether the ROI has been reconstructed accurately or not can be determined. In the end, a software with user-friendly interface will be designed to present this multi-grid reconstruction technique and users who are not familiar with principle of CT image reconstruction could do multi-grid micro-ct image reconstruction with the help of this software. 2 Method 2.1 Software Development Implementation of multi-grid algorithm As the name suggests, here multi-grid reconstruction means to reconstruct image on different grids, voxels on which do not have to have the same sizes. 7

8 By varying the size of voxels on a reconstruction space with fixed size, number of voxels being considered during reconstruction can be changed. Recalling the goal mentioned in Introduction, this property has brought the possibility of easing computational burden as now less points need to be considered in the background. However, there is actually one way that could further decrease points being considered in reconstruction space, that is: only reconstructing the ROI while ignoring other region. In principle, this approach can be carried out for Filtered Back Projection (FBP) while for iterative algorithm, this is impossible. In order to verify this claim, Algebraic Reconstruction Technique (ART, detailed description can be found at Appendix A.3.2.1) will be served as an example to explain why it is impossible to only reconstruct ROI with iterative algorithms. Assuming the goal has been set to reconstruct the image with ART on grid f (shown in Figure 1) with projection data P, during each iteration, ART will update value assigned to pixels in space f by comparing radon transform R of pixels with original projection data P. Based on the difference D between R and P, each pixel in space f can be modified with Equation (1). Figure 1: Full resolution reconstruction space f where f ij refers to each pixel in this space f k+1 ij = f k ij + p θn N l=1 f k lj N In Equation (1), f k+1 ij refers to the updated value after each iteration and fij k refers to the value before updating. p θn is the original projection data while N l=1 f lj k is the radon transform when the projection angle is parallel to each row in space f. From Equation (1), it can be learned that updating of each pixel not only requires the value of that pixel, but also intensities of all pixels on one line that goes across the whole reconstruction space. Therefore, only reconstructing the ROI is infeasible when the goal is to reconstruct image with iterative algorithms. For the same reason, multi-grid reconstruction is the only choice if the aim is (1) 8

9 to decrease points being considered during iterative reconstruction. In next paragraph, how to implement the notion of multi-grid reconstruction in ART will be presented. Assuming now points f 33, f 34, f 43, f 44 in Figure 1 are the ROI and the resolution of other region is not of interest, then one possible setting of reconstruction grids for multi-grid reconstruction can be given as Figure 2, where pixels labeled with F ij refers to points on coarse grid while pixels labeled with f ij refers to ROI. Had the same object is displayed on grid shown in Figure 1 and multi-grid shown on Figure 2, according to linearity of radon transform, radon transform of image in Figure 1 is identical to the sum of radon transform of coarse and fine grid shown in Figure 2. Therefore, when applying multi-grid ART to reconstruct the image, the updating rule for each pixel is the same as conventional ART and the update equations for these two different grids can be given as Equation (2) and Equation (3). Figure 2: Multigrid reconstruction space where F ij refers to pixel in coarse grid F while f ij indicates pixel in fine grid f f k+1 ij = f k ij + w pθn N1 l=1 w f k lj N 2 m=1 W F k mj N 1 w + N 2 W (2) F k+1 ij = F k ij + W pθn N1 l=1 w f k lj N 2 m=1 W F k mj N 1 w + N 2 W Comparing to Equation (1), Equation (2) and Equation (3) also update the pixel value based on the difference between original projection data and calculated radon transform. However, two differences can be observed at the updating equations for multi-grid ART. Firstly, for multi-grid reconstruction, projection data is compared with the sum of radon transform of all grids used during image reconstruction. Secondly, comparing with Equation (1), these two equations has further introduced weighting parameters W and w, which served to compensate the difference impact pixels in different grids have on projection. For instance, in Figure 2, the length of pixel in space F is two times larger then pixels in fine grid, then ratio between weighting parameters W and w should equals 2 as the impact each pixel has on the projection data is linear to its size. (3) 9

10 Based on description mentioned above, multi-grid ART reconstruction can be carried out with work flow as follow: 1. Create and initialize subspaces f and F with desired pixel size. 2. Perform weighted radon transform R on each subspace and sum them up. 3. Compare the difference between summed projection and the ground truth projection. 4. Using the difference between two set of projection data to update each grid with Equation (2) and Equation (3). 5. Repeat steps 1-4 numerous times. Although procedure described above sounds straightforward, implementation multi-grid reconstruction in computer is challenging. Due to the limitation of ODL (a Python library being used to do image reconstruction, refer to next section for the description of it), grid with shape as F shown in Figure 2 cannot be generated in ODL and that is why a masking operator has been introduced. The logic behind masking operator is: keep elements inside one selected region of the whole reconstruction space to zero. Nevertheless, masking operator will not remove any element in the reconstruction space and element inside the selected region will also be used during radon transform. For instance, in order to generate a coarse grid F as shown in Figure 2, coarse grid F as shown in Figure 3 needs to be generated firstly, after which this coarse grid F as well as the coordinate of pixel F 22 needs to be sent to masking operator as input, with which it will create a space F that is as large as space F but with value inside subspace F 22 set to zero. When implementing multi-grid ART in computer, two reconstruction grids being used are F and f (reconstruction grid f is the same as what is given in Figure 2). However, as region corresponding to f inside F will not pose any effect in iterative optimization problem, in principle this is same as using coarse grid F and fine grid f. Figure 3: Coarse grid F where F ij refers to pixel in this grid Above the procedure for reconstructing multigrid image with ART has been described and the notion of multigrid reconstruction can be implemented into 10

11 other iterative algorithms with similar principle. One thing can be noticed from Equation (1) to (3) is that for iterative algorithms, value assigned to each pixel depends on the projection data as well as intensity of other pixels on the reconstruction space. However, when reconstructing with FBP, intensity of each element is determined only by the projection data and value assigned to each pixel is independent of each other. Therefore, when using FBP, multigrid reconstruction can be simplified as follow: firstly, the whole space will be reconstructed with coarse resolution; Then, ROI will be reconstructed with high resolution. By substituting ROI in coarse space with the high resolution reconstruction, multi-grid reconstruction with FBP can be achieved Modifying ODL library All reconstruction or computation presented in this paper are carried out in Python, and mainly depends on open-source library: Operator Discretization Library (ODL, [3]). As ODL has embedded different reconstruction algorithms, focus can be put directly on implementing multi-grid reconstruction, instead of spending days to figure out how to do FBP or iterative algorithms in Python. In this thesis project, multi-grid reconstruction technique will be combined with three reconstruction algorithms, namely FBP, Conjugate Gradient Least Square (CG) and Iterative method with Total Variation Regularization (TV). ODL has developed functions for the first two reconstruction algorithms, while TV method can be achieved with the Chambolle-Pock Solver embedded in ODL. Despite the power of ODL, this library doesnt stay flawless, in order to fit our needs, 3 modifications have been applied to the library: modifying masking operator, adding projection comparison function to iterative algorithms and measuring computation time used by each iteration. How to achieve these three goals will be described as follow. Firstly, the key procedure in proposed multi-grid reconstruction algorithm lies in masking out ROI at the coarse reconstruction space. As ROI and the coarse reconstruction space has different pixel size and their boundaries may not coincide with each other, it is important to ensure each pixel in the coarse reconstruction space has been covered either by the masked coarse space or the defined high resolution ROI space. However, masking operator embedded in ODL fails to take this into consideration and obvious artifacts can be observed when reconstructing with the original masking operator (please refers to Section 3.1 for corresponding images). Due to this, masking operator has been modified with the work flow shown in Figure 4, which will automatically expand the size of ROI and force edges of two reconstruction spaces to have the same location when they don t coincide with each other. Projection comparison function is invented to store sum of absolute difference between original projection and radon transform of reconstructed multi-grid image after each each iterations. This goal can be achieved by five steps: 1. Do forward projecting of the reconstructed image inside iteration i and store is as matrix A. 2. Subtract A from original projection data. 3. Calculate the absolute value of each element inside the subtracted data. 4. Sum up all element inside A and add it at the end of an array. 11

12 Figure 4: Workflow for the modified masking operator 5. Repeat procedure mentioned above. To measure computation time in each iteration, time before any computation in one iteration t 1 and time after all computation in one iteration t 2 will be stored. Subtracting t 1 from t 2, time needed for each iteration will be determined Projection truncating function The proposed multi-grid reconstruction has alleviated the computational burden by decreasing the number of elements being considered in reconstruction space. Another way to ease the procedure of micro-ct reconstruction is to decrease size of projection been loaded in the computer. Instead of inputting the whole projection data acquired on the detector, pixels that do not correspond to the targeted reconstruction space could firstly be removed by function below. Assuming the origin of reconstruction space locates at the intersection of rotational axis (Z direction as default) and perpendicular line across the center of detector; size of reconstruction space is L mm, W mm, H mm along X, Y and Z direction. Then at one single projection image, projection data along each row that has a distance larger than L 1 pixels to the center pixel will be removed; Another criteria for removing projection points is that for data in each column, their distance to the point locates at the center of this specific column is larger than L 2. The value of L 1 and L 2 can be determined by two equations 12

13 listed as follow: src + det L 1 = Max( src L 2 W 2, src + det src W 2 L 2 src + det, 2 src Max(W, L) ) 1 +δ (4) 2 l 1 src + det L 2 = H L 2 + W δ (5) l 1 src 4 In these two equations, symbol MAX refers to function that would select the largest value among given components; src refers to the distance from the source to the center of rotation; det is the distance between the center and the detector; l 1 is the length of one single pixel on the detector, which in our case equals 0.05 mm; δ at the end of each equation is a buffer value, it serves to ensure targeted data will not be missed in truncated dataset Open CT reader and converter One gap between the micro-ct scanner and the developed reconstruction software lies in that micro-ct scanner can only generate files in binary form while the developed software is coded in Python and cannot do matrix calculation with binary dataset. Moreover, information in micro-ct binary file like the size of image, the time of acquiring and the raw data of the projection image are stored in a specific sequence which may not be known by potential users. In order to present all relevant informations in a clear manner to users and input data in the correct type to the reconstruction software, converting the binary file into a more straightforward format is needed, which in our case is achieved with DICOM-CT-PD format (also called Open CT format) introduced in [4]. In short, DICOM-CT-PD means to store each projection image in an extended version of DICOM (Digital Imaging and Communications in Medicine) format. As one imaging format designed specifically for medical images, DI- COM divides information of the projection image into two components: the header and the image, as shown below in Figure 5. Figure 5: Structure of the DICOM-CT-PD format Comparing with other image format like JPG, PNG and GIF, DICOM offers a header file where over 3,000 tags are designed to store information relevant 13

14 to all kinds of medical images. As an extended version of DICOM, DICOM- CT-PD has added more private tags to the header file in order to meet the need of CT image reconstruction. Apart from properties mentioned in header of Figure 5, this specific format can also store information like source to detector distance; number of projection; the frame time of X-ray, etc. By including all these parameters to projection image, software developers that are not familiar with the geometry of micro-ct scanner can reconstruct the image from the output DICOM file as well. Under the guidance of [4], one program has been developed in Python which can not only convert the input file but also perform necessary calibration on the projection image based on the result of [5, 6, 7]. Corresponding code of this program can be found in corresponding Github page of this project. Most of the data stored in DICOM-CT-PD images can be loaded into Python by an open source library Pydicom [8]. However, as Pydicom is designed for reading in standard DICOM file, private tags added in Open CT format, for instance: Flat Field Correction Flag, cannot be detected by Pydicom. To handle this problem, all needed private tags need to be created manually in dictionary of Pydicom. 2.2 Testing Proposed algorithm Testing the correctness of ODL modification In order to ensure modified ODL function works in the expected way, experiment has been carried out to test the output of each function. Firstly, to ensure modified masking operator works in the desire way, reconstruction will be performed with modified masking operator and with the original masking operator. Through visual observation, functionality of the modified masking operator can be determined. To assess functions developed for iterative algorithm, the best way is to call them when running iterative algorithms, those two functions are correct if output of them match expectation Numerical data test with Shepp-Logan Phantom There are two purposes for carrying out numerical data test. Firstly, as the numerical data is smaller than real dataset, it is more efficient to gain empirical knowledge on how different parameters affect the outcome of multi-grid reconstruction with numerical data. Apart from that, ground truth of numerical data is available beforehand and quality of proposed multi-grid reconstruction algorithms can be assessed by comparing its outcome with ground truth. In this thesis, all tests for numerical data is carried out on projection data of a Shepp-Logan Phantom discretized on a grid, the highest intensity of the Shepp-Logan has been set to 10. To simulate a situation closer to reality, white noise with 10% of the intensity of Shepp-Logan Phantom has been added on the projection data. Then, this generated projection data will be used as input for multi-grid reconstruction. For multi-grid reconstruction of the Shepp-Logan Phantom, ROI is always selected as the centered square region that contains three small ellipses at the bottom of the Shepp-Logan Phantom (as shown in the red square of Figure 6) as they correspond to the finest detail of the phantom. Another noticeable 14

15 property of this ROI reconstruction space is that it will always have the same size as the reconstruction space used to generate the original Shepp-Logan Phantom, which indicates it will always have the high resolution. Figure 6: Center cross-sectional image of the simulated Shepp-Logan Phantom and the selected region of interest To obtain empirical knowledge on how to optimize reconstruction through modifying parameters of FBP multi-grid reconstruction algorithm, test has been carried on how relative cut-off frequency of FBP s filter will affect the outcome of reconstruction algorithm. Basically, cut-off frequency means to set signal with frequency higher than certain value to zero, while word relative refers to normalize the range of signal frequency. During test, the coarse reconstruction space has been discretized into grid with size As the highest frequency of Shepp-Logan Phantom cannot be obtained without doing Fourier Transform, In order to gain experience on how to select the best iteration number for CG multi-grid reconstruction, experiment has been carried out on reconstructing Shepp-Logan Phantom with CG multi-grid reconstruction algorithm by different iteration number equals 3, 5, 10, 15, 20 and 50. In this experiment, coarse reconstruction space has been divided into either 10 3 or elements. Just like testing of CG multi-grid reconstruction, experiment has been carried out to determine the most suitable iteration number for TV multi-grid reconstruction. Apart from iteration number, how the selection of penalty weight (which control the level of total variance on reconstructed image) will affect the output of this specific reconstruction algorithm has also been tested. For all tests aim at gaining empirical knowledge on how to select parameters for TV regularization, coarse reconstruction space is discretized as In order to assess the quality of image reconstructed with multi-grid reconstruction approach, multi-grid reconstruction will be carried out for with coarse grid set to 10 3, 20 3, 50 3, 100 3, After that, result of multi-grid CG and multi-grid FBP will be compared with their corresponding full-resolution reconstruction. While for multi-grid TV, comparison has been carried out between multi-grid reconstruction and the original Shepp-Logan Phantom. Apart from the quality of reconstructed image, whether decreasing the number of element of coarse grid will save reconstruction time is also of interest in this projection. Therefore, for multi-grid CG and multi-grid TV, average calculation time for each iteration has also been calculated. 15

16 2.2.3 Test with Micro-CT projection data To test the performance of proposed algorithms in reality, all three multigrid reconstruction algorithms have been used to reconstruct a low contrast phantom. The low contrast phantom, as shown in Figure 7, is a cylindrical phantom made of resins. It contains four smaller cylindrical phantom inside with specified contrast to the background, which could be used for estimation of the low-contrast resolution of a Micro-CT scanner [9]. In order to avoid confusion, four low contrast cylindrical phantoms inside the QRM low contrast phantom will be called as inserts in this report. According to [9], among all four inserts, two of them have a diameter equals 2.5mm while the diameter of the other two equal 1mm; for phantoms with each diameter, one of them has a contrast difference with the background equals 4% while the other one has a contrast difference of 8%. Figure 7: Low Contrast Phantom made by QRM In order to obtain high quality projection data of this low contrast phantom, micro-ct scanner has applied parameters as follow: To begin with, projections will be taken at 300 different angles that cover 360 degrees. The number of projection is set to 300 to ensure a low noise level on reconstructed image. By increasing the projection number, a higher quality image can be expected to be obtained at the price of requiring a larger storage space for projection data. The contrast of projection acquired on detector depends on the selection of X-ray energy. In this specific case, X-ray tube voltage will be set to 40KV p, with current equals 600µA. In each period (also known as Frame time), X-ray will be turned on for 1500ms then shut down for 500ms. Such settings will guarantee an optimized contrast on projection. A smaller current or shorter turn on time may give images with poor contrast while parameters larger than those values could make the detector saturated and eventually decrease contrast of reconstructed image as well. As total size of projection data output by the micro-ct scanner is extremely large (3.3GB in this case), it may take several hours to obtain the reconstruction outcome had the whole projection been used as input of multi-grid reconstruction algorithms. In order to decrease reconstruction time, when reconstructing with FBP multi-grid reconstruction algorithm or TV multi-grid reconstruction algorithm, projection data will firstly be truncated by the projection truncating function described above. For all tests carried out for micro-ct projection data, region covers one of these four inserts will be selected as ROI and reconstructed with high resolution, 16

17 this test will repeat for all four inserts and corresponding contrast difference of the insert locates at high resolution reconstruction region will be calculated with rule as explained in paragraph below. After acquiring the reconstructed image, one region inside the area corresponds to the high resolution reconstructed insert will be selected and the mean value of this area will be recorded as CT insert. To minimize the effect of noise, this region should be selected as large as possible, but it is also important to not include any background pixel in this region. Apart from mean intensity inside the ROI, a relative large area at the high resolution reconstruction region that doesn t correspond to the ROI is also selected, and mean value of this area will be saved as CT background. After this, a large area that corresponds to air at the coarse reconstruction space will be selected and the average intensity of this region will be recorded as CT air. With this three parameter, the contrast of the insert been reconstructed with high resolution can be calculated by Equation (6). Contrast insert = CT insert CT background CT background CT air 100 (6) For the same reason as mentioned in Section 2.2.2, average computation time for each iteration will be recorded for CG and TV multi-grid reconstruction. 3 Results 3.1 Test Results of functions invented to ODL Multi-grid reconstruction of the low contrast phantom with original masking operator is shown on the left side of Figure 8 and multi-grid reconstruction with modified masking operator is shown as Figure 8(b). Figure 8: Multi-grid reconstruction with original masking operator, shown with tag (a) and modified masking operator, shown with tag (b) By calling projection comparison function when running CG multi-grid reconstruction and TV multi-grid reconstruction, difference between original projection and projection of the reconstructed image after each iteration can be output in an array, which can be plotted in Python as shown in Figure 9. 17

18 Figure 9: Projection comparison curve for CG multi-grid reconstruction is shown as 9(a) and projection comparison curve for TV multi-grid reconstruction is shown as 9(b) Using the same method as mentioned above, calculation time for each iteration when using TV multi-grid reconstruction can be shown as Figure 10. Figure 10: Computation time for each iteration takes when reconstructing with TV multi-grid reconstruction 3.2 Test Results with simulated Shepp-Logan Phantom For all images present in this section, the scale of intensity is set as 0 to 10 (same as the generated Shepp-Logan Phantom) unless a specific scale has attached in image to indicate the range of intensity Testing of multi-grid FBP reconstruction Following the instruction given in Method part, two sets of images can be presented in this section, where figure 11 aims at providing empirical knowledge on how the selection of cutoff frequency for FBP s filter will affect the quality of reconstruction while figure 12 shows performance of multi-grid reconstruction with coarse space discretized into different numbers, image (f) in Figure 12 18

19 is a full resolution reconstruction while all others in Figure 12 are multi-grid reconstruction. Figure 11: Reconstruction of Shepp Logan Phantom by FBP with Hann filter H, relative cut off frequency is set to 0.05, 0.1, 0.2, 0.4, 0.6 and 0.8 from image (a) to image (f) 19

20 Figure 12: Multi-grid FBP reconstruction of Shepp Logan Phantom with different setting on coarse grid Testing of multi-grid CG reconstruction Following the instruction given in Method part, four sets of images have been attached in this section as the result of multi-grid CG reconstruction. Figure 13 and Figure 14 show the procedure of determining the most suitable iteration number for reconstruction while Figure 15 provides information on how CG multi-grid reconstruction behaves when different discretization has applied to coarse reconstruction space, images there are presented in the same order as Figure 12. The focus of Figure 16 is computation time for each iteration when coarse reconstruction space has different setting. 20

21 Figure 13: Multi-grid CG reconstruction of Shepp Logan Phantom with different iteration number Figure 14: Multi-grid reconstruction of Shepp Logan Phantom with Conjugate Gradient method, where image (a) - (c) are reconstructed with corresponding iteration number equals 5, 7, 10 21

22 Figure 15: Multi-gridg CG reconstruction of Shepp Logan Phantom with different setting on coarse grid Testing of multi-grid TV reconstruction For the same motivation as mentioned in section above, four sets of images are presented below, where Figure 17 and Figure 18 aim at optimizing penalty weight for TV regularization and the third one (Figure 19) can be used to determine the performance of multi-grid TV reconstruction when coarse grid has been discretized into different grids, which also shares the same setting as Figure 12. In the end, there will be a figure (Figure 20) gives information on how computation time for each iteration varies with the setting of coarse grids. 3.3 Validation of multi-grid reconstruction methods As mentioned in chapter: Method, numerical data test not only includes reconstructing multi-grid image with different setting on coarse grids, but also 22

23 Figure 16: Average computation time for each iteration when coarse reconstruction space has different discretization comparing ROI output by multi-grid reconstruction with ROI reconstructed by full-resolution reconstruction algorithms. In this section, difference on ROI when reconstructing with different multi-grid reconstruction algorithms are listed Validation of multi-grid FBP reconstruction As mentioned in the second chapter, reconstruction of ROI is independent of the coarse grid when using FBP. Therefore, reconstructed ROI will not be affected by the setting of coarse grid. For this reason, only one error comparison image as shown in Figure 21 has been given. For the same reason, the maximum difference and mean difference will also stay constant despite the setting of coarse grid, and both of them are given in table 1. FBP Maximum Difference Mean Difference Table 1: Error Comparison between full resolution reconstruction and multi-grid reconstruction in ROI obtained by FBP Validation of multi-grid CG reconstruction Error comparison images for multi-grid CG reconstruction are given below, their corresponding maximum and average difference can be found in table 2. 23

24 Figure 17: Multi-grid reconstruction of Shepp Logan Phantom by TV with different regularization parameters Validation of multi-grid TV reconstruction Error comparison images for multi-grid TV reconstruction are given below, their corresponding maximum and average difference can be found in table 3. 24

25 Figure 18: Multi-grid reconstruction of Shepp Logan Phantom by TV with different regularization parameters between two reconstruction spaces Figure 19: Multi-reconstruction of Shepp Logan Phantom by TV 25

26 Figure 20: Average computation time for each iteration when coarse reconstruction space has different discretization Figure 21: Error Comparison between full-resolution reconstruction and FBP multi-grid reconstruction CG Maximum Difference Mean Difference coarse grid: coarse grid: coarse grid: coarse grid: coarse grid: Table 2: Error Comparison between full resolution reconstruction and Multi-grid reconstruction in ROI obtained by CG 3.4 Multi-grid reconstruction of low contrast phantom All images presented below represent space with size 20mm x 20mm in reality and the intensity scale is given in Hounsfield Unit, ranging from -100 to All curves plotted in this section represent the intensity change along one line across the inserts reconstructed in high resolution. Another thing about those curves is that vertical direction represent intensity, the unit of which is given in Hounsfield Unit and the horizontal direction indicates the index of pixel along 26

27 Figure 22: Error Comparison between full resolution reconstruction and CG multi-grid reconstruction TV Maximum Difference Mean Difference coarse grid: coarse grid: coarse grid: coarse grid: coarse grid: coarse grid: Table 3: Error Comparison between Original Shepp-Logan Phantom and Multi-grid reconstruction in ROI obtained by TV the plotted line. In each set, multi-grid cross-sectional images with different setting on coarse grid are presented, where coarse grid is discretized as 50 3 in image (a); in image (c) ; in image (e) while in image (g) the background space is a

28 Figure 23: Error Comparison between Original Shepp-Logan Phantom and TV multi-grid reconstruction grid. After each set of image, contrast difference calculation table for insert reconstructed in high resolution is attached. For those two iterative algorithms, information on computation time is also given through images FBP multi-grid reconstruction of low contrast phantom For FBP multi-grid reconstruction, cut-off frequency for Hann filter H is set to 0.1 in order to achieve multi-grid reconstruction as below. 28

29 Figure 24: FBP multi-grid reconstruction of Low Contrast Phantom with high resolution reconstruction focuses on 2.5mm 8% difference insert FBP 2.5mm 8% difference insert coarse grid: coarse grid: coarse grid: coarse grid: Table 4: Contrast difference calculation of 2.5 8% insert in low contrast phantom from the reconstructed images presented above 29

30 Figure 25: FBP multi-grid reconstruction of Low Contrast Phantom with high resolution reconstruction focuses on 1.0mm 8% difference insert FBP 1.0mm 8% difference insert coarse grid: coarse grid: coarse grid: coarse grid: Table 5: Contrast difference calculation of 1.0mm 8% insert in low contrast phantom from the reconstructed images presented above 30

31 Figure 26: FBP multi-grid reconstruction of Low Contrast Phantom with high resolution reconstruction focuses on 2.5mm 4% difference insert FBP 2.5mm 4% difference insert coarse grid: coarse grid: coarse grid: coarse grid: Table 6: Contrast difference calculation of 2.5mm 4% insert in low contrast phantom from the reconstructed images presented above 31

32 Figure 27: FBP multi-grid reconstruction of Low Contrast Phantom with high resolution reconstruction focuses on 1.0mm 4% difference insert FBP 1.0mm 4% difference insert coarse grid: coarse grid: coarse grid: coarse grid: Table 7: Contrast difference calculation of 1.0mm 4% insert in low contrast phantom from the reconstructed images presented above CG multi-grid reconstruction of low contrast phantom For multi-grid reconstruction with CG, iteration number is set to 10 to achieve the result as follow. 32

33 Figure 28: CG multi-grid reconstruction of Low Contrast Phantom with high resolution reconstruction focuses on 2.5mm 8% difference insert CG 2.5mm 8% difference insert coarse grid: coarse grid: coarse grid: coarse grid: Table 8: Contrast difference calculation of 2.5mm 8% inserts in low contrast phantom from the reconstructed images presented above 33

34 Figure 29: CG multi-grid reconstruction of Low Contrast Phantom with high resolution reconstruction focuses on 1.0mm 8% difference insert CG 1.0mm 8% difference insert coarse grid: coarse grid: coarse grid: coarse grid: Table 9: Contrast difference calculation of 1.0mm 8% inserts in low contrast phantom from the reconstructed images presented above 34

35 Figure 30: CG multi-grid reconstruction of Low Contrast Phantom with high resolution reconstruction focuses on 2.5mm 4% difference insert CG 2.5mm 4% difference insert coarse grid: coarse grid: coarse grid: coarse grid: Table 10: Contrast difference calculation of 2.5mm 4% insert in low contrast phantom from the reconstructed images presented above 35

36 Figure 31: CG multi-grid reconstruction of Low Contrast Phantom with high resolution reconstruction focuses on 1.0mm 4% difference insert CG 1.0mm 4% difference insert coarse grid: coarse grid: coarse grid: coarse grid: Table 11: Contrast difference calculation of 1.0mm 4% insert in low contrast phantom from the reconstructed images presented above 36

37 Figure 32: Average computation time for each iteration when coarse reconstruction space has different discretization TV multi-grid reconstruction of low contrast phantom When reconstructing low contrast phantom with TV multi-grid reconstruction, iteration number is selected as 150 to ensure the optimal reconstruction output. Different from multi-grid reconstruction of Shepp-Logan Phantom, here total variance regularization has only applied to ROI. Motivation behind this decision can be explained in two aspects: firstly, reconstruction result of the background is not of interest, hence denoising on background means nothing but heavier computation burden to computer. Moreover, the difference noise level on different reconstruction regions can also be used to prove the reconstruction is carried out on multi-grid. Below the computation time will be presented first, after which the reconstruction results will be given. Figure 33: Average computation time for each iteration when coarse reconstruction space has different discretization 37

38 Figure 34: TV multi-grid reconstruction of Low Contrast Phantom with high resolution reconstruction focuses on 2.5mm 8% difference insert TV 2.5mm 8% difference insert coarse grid: coarse grid: coarse grid: coarse grid: Table 12: Contrast difference calculation of 2.5mm 8% inserts in low contrast phantom from the reconstructed images presented above 38

39 Figure 35: TV multi-grid reconstruction of Low Contrast Phantom with high resolution reconstruction focuses on 1.0mm 8% difference insert TV 1.0mm 8% difference insert coarse grid: coarse grid: coarse grid: coarse grid: Table 13: Contrast difference calculation of 1.0mm 8% inserts in low contrast phantom from the reconstructed images presented above 39

40 Figure 36: TV multi-grid reconstruction of Low Contrast Phantom with high resolution reconstruction focuses on 2.5mm 4% difference insert TV 2.5mm 4% difference insert coarse grid: coarse grid: coarse grid: coarse grid: Table 14: Contrast difference calculation of 2.5mm 4% inserts in low contrast phantom from the reconstructed images presented above 40

41 Figure 37: TV multi-grid reconstruction of Low Contrast Phantom with high resolution reconstruction focuses on 1.0mm 4% difference insert TV 1.0mm 4% difference insert coarse grid: coarse grid: coarse grid: coarse grid: Table 15: Contrast difference calculation of 1.0mm 4% inserts in low contrast phantom from the reconstructed images presented above 4 Discussion 4.1 Functionality of functions introduced to ODL In this thesis project, one function (masking operator) inside ODL have been modified to improve the quality of reconstructed image. In order to make 41

42 more objective assessment on proposed multi-grid reconstruction approach, two functions have been added to iterative algorithms. From Figure 8(a), it can be noticed that before the modification on masking operator, a gap can be observed clearly at the bottom of ROI. And this is introduced to the reconstruction as that gap has not been assigned to ROI or coarse reconstruction region. By applying the modified masking operator, this gap can be removed, as can be seen in Figure 8(b). Apart from this gap effect, boundary between coarse reconstruction region and ROI appears to be either slightly brighter or bit darker than other region in both images presented in Figure 6, even though this intensity variation in Figure 8(b) is less obvious than what it is in Figure 8(a). After discussion with mathematician who invented this multi-grid reconstruction algorithm, conclusion has been made on that this edge effect is normal for the proposed algorithm and in general the reconstruction edge looks better when edges of two reconstruction space have matched each other. In another word, this dark edge effect is introduced by issues other than the masking operator. In Figure 9, numbers stored in projection comparison function has been plotted as a curve, which can be used to judge if the iterative algorithm has converged. From Figure 9(a), it can be noticed that the difference between projection data has decreased first and then increase. This is because CG is one semi-converged iterative reconstruction algorithm, which means it will first optimize the reconstruction while after certain iterations, optimization will go in the wrong direction and degrade the reconstruction completely. As a result of this degradation, difference between original projection and projection generated from the reconstruction will increase as shown in Figure 9(a) after 8 iterations. To the contrary, TV is one reconstruction algorithm that could always guarantee the converging of image. Therefore, in Figure 9(b), the projection comparison function will first go up and down (which means the algorithm is trying to find the optimization result) and then stay stable (best result achieved). Ground truth on how long should each iteration in iterative reconstruction algorithm take can never be acquired, so there is no way to determine if the timing function has been applied correctly. However, from Figure 10, the maximum difference between time takes for each iteration is roughly 0.25 seconds while each iteration takes more than 5 seconds. As the time variation for each iteration is as small as 5%, the correctness of this method should not be doubted. 4.2 Multi-grid Reconstruction with different algorithms Through experiment, the most distinct problem for multi-grid FBP reconstruction is that this algorithm is extremely sensitive to noise. To suppress the high noise level, a so-called Hann filter H has been applied. Apart from the selection of filter, relative cut-off frequency used by the filter can also help denoise. However, improper selection of relative cut-off frequency may remove fine details in the reconstruction and for the sake of obtaining the optimization reconstruction outcome, the best strategy is plotting reconstruction result with different relative cut-off frequency and select the most suitable one based on observation. With this thought on mind, the most suitable relative cut-off frequency in reconstruction of Shepp-Logan has been selected as 0.4 while for the Low Contrast Phantom, this value should equal 0.1. From reconstruction result of FBP multi-grid reconstruction, it can be seen that the setting of coarse grid will 42

43 not bring extra artifact to the image and the difference between full-resolution reconstruction and image reconstructed with multi-grid reconstruction is negligible. When reconstructing with CG, the selection of iteration number is crucial as CG is a semi-converge iterative reconstruction algorithm. Through experiment, it can be noticed that CG will always find the optimized reconstruction after 5 to 10 iterations. To determine the most suitable iterations, the best strategy is to first set the iteration number to 20 and plot the difference between the original projection and projection generated by the reconstructed image after each iterations. The optimized iteration number will give minimal difference between those two set of projection. However, even when the most suitable iteration number has been selected for CG, the quality of CG multi-grid reconstruction cannot be as high as multi-grid reconstruction given by FBP. For one thing, CG doesn t offer any parameter to suppress the noise level on the image therefore the reconstructed image will always contain a medium level of noise. Another issue about the multi-grid CG reconstruction is that low contrast phantom reconstructed by CG multi-grid reconstruction doesn t look homogeneous, more precisely, it seems the outer boundary of the phantom appears to be bit darker then the center of the phantom. However, judging from those plotted cross lines, the whole phantom doesn t have a large variation. One explanation for this is that the image has been scaled to display from -400 Hounsfield Units to -100 Hounsfield Units, therefore a small change on the image has been magnified by the scale procedure. Last but not least, a dark edge will always be observed at the boundary of high resolution ROI. By increasing elements included in coarse reconstruction space, this edge effect as well as the difference between full-resolution reconstruction and multi-grid reconstruction can be minimized, but not eliminated. Comparing with CG, time required to reconstruct multi-grid image with TV is quite long, as TV usually needs more than 100 iterations to converge. On the other hand, TV is an iterative algorithm that will eventually converge, hence the selection of iteration number for TV is out of the consideration of saving time only. In order to find a suitable iteration number, a iteration large enough (for instance, this number is set to 2000 for the low contrast phantom) will be used and the projection difference curve will be plotted. The suitable iteration number can be selected as any number on the plain region (for Figure 9(b), plain region refers to 1600 to 2000) of the curve. Comparing with other two algorithms, TV is capable of removing noise without degrading the fine details on the image, while the price to pay for this advantage is long testing time for finding the best regularization parameters. This procedure is time-consuming not only because there is literally no theoretical guide on parameter selection, but also because regularization parameters selected for coarse reconstruction space and ROI will also affect the iteration number for TV multi-grid reconstruction to converge. Another thing that has been noticed during penalty regularization parameter tuning procedure is that optimized parameters will also change with the setting of coarse grid. For this reason, when validating the correctness of TV multigrid approach, reconstruction is compared with original Shepp-Logan phantom instead of the full-resolution reconstruction, as different regularization parameters, other the the multi-grid reconstruction approach, may introduce the difference at ROI. For TV multi-grid reconstruction of Shepp-Logan Phantom presented in result part, parameters that could give best reconstruction 43

44 when coarse grid is discretized into a grid is used, and for the low contrast phantom, parameters are selected with the hope of giving best outcome when coarse grid has elements. Just like CG reconstruction algorithm, a edge effect can be observed at the reconstructed image, and this effect will be fainted by increasing the elements included in coarse reconstruction space. In [10], how to eliminate this edge effect through introducing another penalty term has been explored, the outcome of which may have the potential to improve TV multi-grid reconstruction proposed in this study. 4.3 Accuracy Assessment In this paper, accuracy in Shepp-Logan Phantom is assessed by comparing multi-grid reconstruction with full resolution reconstruction or original image. This method is evaluated as accurate if the difference between multi-grid reconstruction and compared ground truth is small. Through analysis, it is save to say this proposed method can reconstruct ROI accurately and the corresponding discussion gone as follow: From result presented in Section and 3.3.2, average difference between multi-grid reconstruction and full resolution reconstruction will never exceed 3%. Had the coarse reconstruction space been discretized into a large number, this difference can be suppressed to only 0.2%. From error comparison images given in this section, it can be concluded that fine details (for test with Shepp- Logan Phantom, this refers to those ellipses) in ROI of multi-grid image can be reconstructed with quality as high as what they are in full resolution reconstruction if the coarse grid has been discretized into a relatively large number. Otherwise, shape of those fine details could be preserved while the intensity of them may vary with their values in full-resolution reconstruction. Another thing can be learned from error comparison images is that when the coarse reconstruction grid is extremely coarse, error at the boundary of ROI will be quite large, and this is caused by the edge effect mentioned in section above. In order to avoid this error, users could either create a high reconstruction space larger than the actual size of ROI or increase the elements included in coarse reconstruction space. For reason mentioned in Section 4.1, TV multi-grid reconstruction has been compared with original Shepp-Logan Phantom. From table presented in Section 3.3.3, the biggest difference between multi-grid reconstruction and full-resolution reconstruction is 4.4% and it can be minimized to a value smaller than 1% if a suitable discretization has been selected for coarse reconstruction space. From presented error comparison images, it can be seen that apart from several pixels at the boundary of ellipses, difference between ROI of multi-grid reconstruction and full-resolution reconstruction is trivial. Those bright pixels at the boundary of ellipses are caused by the noise removing procedure, which will avoidably bring some difference to fine details on the image. Due to this, it is irrational to say multi-grid reconstruction has degraded fine details of reconstruction. Just like CG multi-grid reconstruction, large error can also be observed at the boundary of ROI when the coarse reconstruction space is extremely coarse. To move this artifact, please refer to strategy presented in paragraph above. Apart from the Shepp-Logan Phantom, reconstruction result of low contrast phantom may also provide key information on accuracy of this proposed multi-grid algorithm. However, it is hard to say this method is accurate as the 44

45 calculated contrast difference for those four inserts is quite different from the reality. Nevertheless, this difference is not only caused by the proposed multi-grid approach and several factors that may induce the difference have been concluded as follow. Firstly, reconstructed multi-grid images doesn t have a homogeneous value for region corresponds to low contrast insert. For instance, when reconstructing the 2.5 mm insert with 8% difference by TV, maximum individual pixel difference inside region corresponds to this insert will be 23 Hounsfield Unit. Therefore selected region for low contrast insert will cause variation in contrast difference calculation to some extent. Apart from this, reconstruction outcome depends highly on the quality of projection data. For low contrast phantom reconstruction, there is no way to guarantee the projection data has been acquired with the optimized setting on micro-ct scanner, or different kinds of artifacts have been removed from projection data perfectly. As there is no way to measure how much had these two factors affected the reconstructed multi-grid low contrast phantom images, it is not safe to make any judgment by looking at tables for contrast difference calculation. 4.4 Timing Issue Motivation for iterative multi-grid reconstruction is that less elements need to be calculated in each iteration (as coarse reconstruction space has been discretized into a grid with less elements), which will both decrease the requirement of memory storage space and increase the computational speed. More precisely, the coarser the background part is, the faster the reconstruction speed should be. From image presented above, this trend can be observed for timing of the numerical data while for micro-ct reconstruction, this trend is not that obvious and sometimes faster computation time cannot even be observed. This unexpected computation time can be addressed from several different aspects. Firstly, modification of the masking operator could have changed the rule of coarser background space corresponds to less reconstruction points. That is, the modified masking operator will first expand the size of ROI when its boundary doesn t coincide with boundary of coarse reconstruction space, as a result, number of pixels include in ROI at four different cases is not the same. To make multi-grid reconstruction of the Low Contrast Phantom as an example, when reconstructing with TV, the total pixels being calculated with coarse grid set to 50 3 equals , while this number decreases to when coarse grid is By further increasing the coarse grid to 160 3, the number of elements being calculated changed to Had the coarse grid being discretized as 200 3, pixels need to be updated in each iteration will increase to Due to this, it can be expected that the reconstruction time should be the shortest when the coarse grid is discretized into However, even with this though on mind, the computation time still does not match the expectation, and this can be explained as there is not a distinct difference between number of points being considered during multi-grid reconstruction of Low Contrast Phantom. According to data given above, the maximal difference for reconstruction of Low Contrast Phantom is approximately 50%, while for reconstruction of Shepp-Logan Phantom, this difference can be up to 12500%. Therefore, it should be expected that there is not a huge difference among reconstruction time when coarse grid has different settings. Moreover, as the data being processed during reconstruction of micro-ct image is extremely 45

46 large, time spend by computer other than reconstruction computation (read in data or store data to disk) will pose a more obvious effect on total computation time. As a result, no clear time decreasing trend can be observed for multi-grid reconstruction of micro-ct data. 5 Conclusion The notion of multi-grid reconstruction has been applied to different reconstruction algorithms and has been tested with both numerical data and projection data collected from micro CT scanner. From the testing result of numerical data, it can be concluded that multigrid reconstruction can make large projection reconstruction faster to solve. In the same time, this reconstruction approach will not degrade the quality of image at region of interest too much. In order to get reconstruction image of ROI with a quality similar to what can be achieved with full resolution reconstruction, the coarse grid needs to be discretized into a relatively large number. However, increasing the pixel points included in coarse grid will unavoidable increase the reconstruction time. Therefore, there should always be a trade-off between shorter reconstruction time and high quality image at ROI. Apart from test on numerical data, multi-grid reconstruction algorithms have also been applied to projection data obtained on micro-ct scanner, based on the reconstruction result, it is safe to say this method is robust enough to overcome the problem of memory insufficient in micro-ct reconstruction. One problem that hasn t been solved for this multi-grid reconstruction is that when implementing this approach to iterative methods, edge effect at the boundary between high resolution region and coarse resolution region will always appear. Methods to remove them need to be determined in the future work. 46

47 Appendices Appendix A State of the Art A.1 Introduction Small-animal in-vivo imaging (imaging small animals alive) has long been one focus in biomedical field as experiment on small animals can provide vital information in mechanisms of disease and drug discovery, cf. [11]. Nevertheless, as the characteristics of small animals are quite different from human, requirement on temporal and spatial resolution for small animal in-vivo imaging is extremely demanding. For instance, to acquire cross-sectional image on mouse heart with resolution comparable to what can be achieved with human means to increase the spatial resolution by 3000 times, cf. [11]. To make things worse, heart rate of mice is in the range of times per minute, in order to avoid blurring in image, specific design should be added on scanning time. Due to these challenges, researchers have optimized one conventional imaging modality for acquiring cross-sectional image of human to fit the need of small animals, which is known as micro-ct. However, micro-ct increases the quality of cross-sectional images at the price of requiring more input data, which has grown to be a burden too large for computer to carry. Therefore, how to balance the high requirement on input data with the fine resolution output image has been one problem that needs to be solved in micro-ct field. In this background paper, an introduction of micro-ct will be given firstly. Then the key technology for generating image in micro-ct, image reconstruction algorithms, will be discussed in details. From where techniques that facilitate the production of cross-sectional image will be presented. In the end, how to apply those methods for the problem in micro-ct will be illustrated. A.2 Computed tomography (CT) and micro-ct Since its discovery in 1895, X-ray technology has been widely used in the medical imaging field. The first medical imaging modality using X-ray is called conventional X-ray image, which can be used to distinguish components inside the patient body as different materials have different attenuation rate when being illuminated by X-ray. However, conventional X-ray imaging does not stay flawless. As X-ray imaging is merely a projection of the 3D object, shape and position information of the body cannot be be determined. Moreover, substances inside the object may remain invisible on the image, as elements closer to the X-ray source will overlay and cover things behind. To solve these problems, an alternative way of using X-ray for imaging, computed tomography (CT) scan, had been invented. According to [12], CT scan is a non-invasive image modality that makes use of x-ray images taken at different angles to acquire the cross-sectional image of one specific area of the target with the help of reconstruction algorithms. Compare with conventional x-ray image, CT scan offers higher image contrast and resolution. In the same time, it can accurately display the position of different components inside the object. A comparison between normal x-ray image (labelled as A) and CT image (with label B on up left corner) is shown 47

48 in Figure 38. Figure 38: Comparison between conventional X-ray and CT image,cf. [13] Advantages mentioned above has promoted the usage of CT scan in healthcare area. However, usage of CT scanner is not limited to taking cross-sectional image of patient. As mentioned in [14], researchers have also applied it into small-animal in-vivo imaging as X-ray CT is capable of providing soft-tissue structures, skeletal abnormalities, and tumors in live animals in a cost-effective manner. Due to the difference between small animals and human, modifications have been made on X-ray CT and the modified version of X-ray CT is commonly known as micro-ct. According to [15], micro-ct is scanner with resolution below 100 micrometers. In another word, the only difference between micro-ct and normal CT lies in that micro-ct can provide reconstructed images with higher spatial resolution. In order to acquire high quality reconstructed image, micro-ct requires projection data with a higher resolution. Therefore, micro-ct has been designed in a specific manner as shown in Figure 39, the principle of which can be explained as follow: The specimen will first be mounted on a rotation stage which serves to ensure projection data could be obtained at different angles. Then X-ray emitted from the X-ray tube will go through the specimen and hit on the fluorescent crystal plate, where the attenuated X-ray will be converted to visible light. After that light generated by the crystal plate will go through microscope objective, which is used to increase the resolution of projection data. For instance, if the magnification of the lens is 10 fold, 1cm 2 image on the len will cover 100cm 2 on the detector plane. After acquiring the data, tomography image can be obtained with reconstruction algorithms, which will be discussed in details in next chapter. A.3 Image Reconstruction Algorithm Following the claim made in [17], the core part of micro-ct (as well as conventional CT) lies in the computer algorithm for reconstructing the crosssectional image. Depending on what kind of CT scanners is used to acquire projection data, either it is parallel beam CT scanner, fan beam CT scanner or cone beam CT scanner (the difference of these methods lies in the way x-ray is emitted, as shown in Figure 40), different reconstruction algorithms will be applied on the projection data. However, the principle behind those algorithms is rather similar. Nowadays, reconstruction algorithms for CT scanner mainly fall 48

49 Figure 39: Structure of micro CT, with permission from [16] into two categories: filtered backprojection (FBP) and iterative reconstruction (IR) methods. How each method works is addressed as follows: Figure 40: Comparison of different CT scanners, with permission from [18] A.3.1 Filtered backprojection algorithm As one of the oldest image reconstruction algorithm, filtered backprojection has been widely used in CT reconstruction. In this section, 2-dimensional filtered backprojection will be introduced first, from where the scope will extend into 3-dimension space. The foundation of FBP lies in Radon transform, which is an integral transform that calculate the sum of intensity along line passes through an object at certain angle. As mentioned in [19], Radon transform can be written as Equation (7). Rf(t, θ) = f(x)ds(x) = f(x)δ(< x, n θ > t)dx (7) L t,θ 49

50 where n θ = (cos θ, sin θ) is the unit vector with angle θ. Rf(t, θ) is the integral of f along the line L t,θ whose direction is perpendicular to n θ and whose distance from the origin is t. In Equation (7), < x, n θ > means dot product of x and n θ, which is the same as x n θ. Please refers to Figure 41 for a better understanding of these parameters. Recalling the definition given in Chapter 2, conventional X-ray image can also be described as one set of radon transform on object at certain angle θ. Based on this claim as well as [12], which defines image reconstruction in CT as getting cross-sectional image from X-ray images taken at different angles around the object, the reconstruction problem can be understood as the process of acquiring information from the output of radon transform, or mathematically, radon inverse transform. Through mathematical derivations given in [19], it can be proved that the formula of radon inverse transform in 2D is Equation (8). f(x) = 1 (2π) 2 ˆ π 0 (Rf(, θ) h)(< x, n θ >)dθ (8) what Equation (8) intends to illustrate is that information about the object can be acquired through two steps. Firstly, radon transform along each direction needs to be convolved with a convolutional kernel h. After this, image of the original object can be acquired by integrating convolution results along all projection angles. In image reconstruction field, convolution performed in Equation (8) can be also called as filtering. Together with the fact that integral calculation in Equation (8) is in principle same as projection, this approach has been widely known as filtered backprojection algorithm. Figure 41: Mathematical model for radon transform [19] Although Equation (8) is the exact formula for FBP, applying this equation in reality is not as simple as this equation suggests. To begin with, this equation gives the solution for reconstruction in continuous domain while projection data of the signal, projection angles as well as reconstructed images are presented in discrete domain. Therefore, how to modify Equation (8) into discrete domain needs to be considered. Secondly, relevant data for reconstructing one point inside the object may not be easy to identify. In parallel beam CT model, each projection angle corresponds to the same angle of radon transform for all points inside the object, so convolution in radon inverse transform can be acquired by convolving the whole dataset acquired at each angle with a filter H. However, in 50

51 fan beam CT scanner, signal detected at one projection angle is radon transform of points at different angle. Therefore, convolution cannot be performed on the data acquired by the detector directly. Research has been carried out on how to perform filtered backprojection on fan beam CT scanner. Currently, the most successful approach is called rebinning. In short, this method starts with one specific projection angle for radon transform, from where it searches the entire fan beam projection dataset for radon transform along this direction; then it moves to the next projection angle and repeat the searching procedure; by carrying out this procedure along all possible angles for radon transform, data acquired by a fan-beam model can be converted to a form similar to the one acquired from parallel beam model, henceforth the FBP algorithm can be applied. Above is the description of applying FBP in 2D, however, rules for implementing this algorithm in 3D model is not as straightforward as in 2D case. The complexity of this process can be understood through the radon transform in 3D space, where the radon forward transform has the integral formula as Equation (9), cf. [20]. ˆ Rf(p, ˆn Ω ) = f(x)δ(p x ˆn Ω )d 3 x (9) V In Equation (9), δ function serves to pick up points on a place parallel to vector ˆn Ω and has a distance p to the origin point. In another word, radon transform in 3D is the integration on a plane, instead of a line. As the only data can be measured with X-ray is line integral, mathematicians had been working on derive 3D radon transform from line integral. In [21], Grangeat has come up with a derivative based method to solve this problem, however, as this is not the main focus of this paper, detailed explanation will not be presented here. In one word, the projection data acquired from cone beam CT needs certain preprocessing before applying in reconstruction algorithm. After obtaining the radon transform result, the inverse procedure can be achieved through Equation (10), which is given in [20]. What this equations does can be concluded in three steps: 1. Take second derivative of Radon transform; 2. Back-project over planes contain x; 3. Integrate over all angles. f(x) = 1 ˆ (8π) 2 R f(x ˆn Ω, ˆn Ω )dω (10) 4π Above is the mathematical instruction for applying FBP in 3D, in reality, implementation of FBP in Cone-Beam CT scanner needs to be addressed from other aspects like the trajectory of X-ray source, the shape of detectors etc [22, 23, 24, 25]. However, most papers in this area have focused on reconstruction of Cone Beam CT on long object (like human being, which cant be covered by the x-ray beam in one single position). As the focus of this paper is micro- CT image on small animals, assumption can be made on that the object has a size small enough to be covered by the x-ray emitted by the cone beam x-ray source. With this assumption, the reconstruction of object can be achieved with Feldkamp-Davis-Kress (FDK) method given in [26]. In short, FDK algorithm is an extension from Fan beam FBP reconstruction method. It starts with the reconstruction of slice on the mid-plane (plane perpendicular to rotation axis and includes x-ray source point as shown in Figure 41), which is exactly the same as 51

52 fan beam FBP algorithm in 2D. Then paper [26] has shown that any other row on detector plane can be projected on mid-plane with a cosine weighting. Based on this, reconstruction of points that don t locate on mid-plane can be achieved by weighting relevant projection data to mid-plane and applying fan-beam FBP algorithm there. Due to the simplicity of this approach, FDK algorithm has been widely used in commercial cone beam CT scan in all scale. Figure 42: Schematic physical arrangement of 3D tomographic system in FDK, with permission from [26] According to [27], FDK is an approximate reconstruction algorithm that doesn t satisfy the necessary condition for reconstruction mentioned in [23]. Therefore, effect of streaking and shading artifact could be obvious as the distance between the reconstructed slice and the mid-plane increases. However, as the object studied in micro-ct is quite small, effect of those artifacts can be ignored here. A.3.2 Iterative reconstruction methods Despite the efficiency and acceptable performance, FBP will suffer from streak artifact, image noise and experience low-contrast in certain scene, cf. [28]. All of this has stimulate the growing usage of iterative algorithm, which is a more accurate mathematical model that gives less artefact in reconstructed image, cf. [29]. According to [30], iterative method starts with the principle that reconstructing image through measured projection is the same as solving linear equations with the form: g = Af, in which g corresponds to measured projection data, A is a weighted matrix determined beforehand, while f stores intensity of reconstructed image. Therefore, iterative algorithm has transferred the problem of reconstruction to find solution of one set of linear equations, through which the value of f is adjusted by comparing projection of predicted image with the measured one. In reality, it is hard to find the matrix f that matches perfectly with the measured data f. Therefore, iterative algorithms usually developed with the aim of finding the minimal difference between the measured projection data and the expected data. To make the idea of IR method more clear, the introduction will start with algebraic iterative reconstruction (ART), from where more advanced methods can be illustrated. 52

53 A Algebraic Iterative Reconstruction ART algorithm is based on the idea that the density map of one object can be obtained by throwing back each projection across the reconstruction space through iteratively solving linear equations, cf. [29]. By iterative calculation, the density of each pixel will be modified until the reconstructed image has a projection dataset close to the measured data. Please refer to Figure 42 for a better understanding of this procedure. Figure 43: Principle of ART algorithm, redraw from [30] Figure 43 has given the simplest case for ART, while in reality, reconstructed image will be assumed to be one N x N pixels array (for the case in Figure 43, N equals 2), in which each pixel is uniform in density (displayed as grayness on image) while different from other pixels, cf. [29]. Projection in this case corresponds to sum of weighted density along one set of pixels between two parallel lines. For ART, the reconstructed image can be obtained by performing Equation (11) iteratively. f (k+1) j = f (k) j + g i N j=1 f (k) ji N In Equation (11), f (k+1) j is the updated value for pixel, while f (k) j is the intensity of one pixel during last iteration; g i is the measured projection value while N is the number of pixels along one axis. Through observation, it can be noticed that this equation modifies the result by adding a correction term in the estimated value, cf. [30], which is determined by the difference between the measured projection data and the estimated image. The update procedure will stop when projection of estimated image has a difference small enough with measured projection. (11) A Gradient Based Algorithm One problem with ART is that the estimated value for each pixel is updated by directly adding the difference between the measured data and estimated data, which may not be the most efficiency way for modifying. Due this this, gradient based iterative method has 53

54 been developed for image reconstruction, the principle of which is to calculate how intensity of each pixel could affect the difference between the measured projection and the predicted projection. The idea of this algorithm can be illustrated more clearly with Figure 43, in which the image is assumed to consist of only two pixels. Figure 44: Principle of Gradient based algorithm [30] In Figure 43, two axes along the horizontal plane correspond to the grayness of two pixels while vertical axis measures the difference between projection. As iterative reconstruction algorithms are aimed at finding the matrix f (reconstructed image) that minimized the difference between measured projection and the predicted projection. Iterative algorithm is in principle same as minimization problem. Mathematically, function will decrease the most along the gradient direction, therefore, minimization of difference can be achieved through modifying pixel value along dark lines in Figure 44, which mathematically can be written as Equation (12). f (k+1) = f (k) + α (k) p (k) (12) It can be noticed from Equation (12) that the updated value f (k+1) is the previous estimation plus a production of p (k) and α (k). In this formula, vector p (k) is the gradient of the measured projection and predicted projection while α (k) is the parameter to control the step of convergence. Although Equation (12) can ensure the difference is minimized in an efficient manner, it cannot ensure the gradient is decreased in the most efficient approach as the gradient is calculated locally. Therefore, a more efficient gradient based method, namely conjugate gradient (CG) algorithm has been invented. The principle of CG algorithm is the same as the gradient approach mentioned above. However, the vector p (k) is not determined by the direction of the gradient in the current location, but opposite to a combination of gradients found at previous location, cf. [30]. As CG algorithm has taken consideration of all previous estimation, the speed of convergence is faster than iterative method based on local gradients. A Expectation Maximization Algorithm One similar feature about two iterative algorithms mentioned above is that the updated reconstructed image is acquired by adding a modification value on the current estimated image. Apart from this adding approach, CT image can also be updated through multiplication. This multiplication based iterative method is called statistical esti- 54