Multi-slice CT Image Reconstruction Jiang Hsieh, Ph.D.

Transcription

1 Multi-slice CT Image Reconstruction Jiang Hsieh, Ph.D. Applied Science Laboratory, GE Healthcare Technologies 1

2 Image Generation Reconstruction of images from projections. textbook reconstruction advanced acquisition (helical, cone beam) advanced application (cardiac) Formulation of 2D images to 3D volume. Reconstruction Presentation 2

3 Textbook Reconstruction The mathematical foundation of CT can be traced back to 1917 to Radon. The algorithms can be classified into two classes: analytical and iterative. Some of the commonly used reconstruction formulae were developed in the late 70s and early 80s. With the introduction of multi-slice helical CT, new cone beam reconstruction algorithms become the focus of research area. 3

4 CT Data Measurement -Under Ideal Conditions x-ray attenuation follows Beer s law. I o e µ x µ x µ 1 e 2 e n x I o µ I µ x o I = I e x o µ 1 µ 2 µ 3 µ 4 µ n ( µ + µ + + µ ) x x = I e 1 2 o x 0, n x-ray tube detector P I = ln = µ ( x ) dx I o 4

5 Measured Projections The measured data are not line integrals of attenuation coefficients of the object. beam hardening scattered radiation detector and data acquisition non-linearity patient motion others The data need to be calibrated prior to the tomographic reconstruction to obtain artifact-free free images. 5

6 Sampling Geometries The sampling geometry of CT scanners can be described three configurations. Due to time constraints, we will not present in-depth discussions on each geometry. detector detector detector source source source parallel beam fan beam cone beam 6

7 Fourier Slice Theorem (Central Slice Theorem) 4000 i 0( x) = f ( x, y dy P u = f x y e 2 πux 0 ( ) (, ) dxdy P( u,0) p ) i2πux = f ( x, y) e dxdy 2000 FT 500 = PROJECTION v=0 y 2D FT v x u f ( x, y) F( u, v) = f ( x, y) e i2π ( ux+ vy) dxdy 7

8 Fourier Slice Theorem (central slice theorem) Fourier transform of projections at different angles fill up the Fourier space. Inverse Fourier transform recovers the original object. p θ (x) FT P θ (ω) 2D FT 8

9 Filtered Backprojection The filtered backprojection formula can be derived from the Fourier transform pair, coordinate transformation, and the Fourier slice theory: f backprojection filtering = π j 2 πω t Pθ ( ω ) ω e 0 ( x, y) dωdθ K(ω) pre-processed processed data filter the data -w w ω backprojection 9

10 Filtering Consider an example of reconstructing a phantom object of two rods. Original Sinogram Filtered Sinogram views Object single projection detector sample 10

11 Backprojection 0 o -30 o 0 o -60 o 0 o -90 o 0 o -120 o 0 o -150 o 0 o -180 o 11

12 Fan Beam Reconstruction Each ray in a fan beam can be specified by β and γ. Reconstruction process is similar to parallel reconstruction except additional apodization step and weighting in the backprojection. y β γ x pre-processed processed data Apodizaton filter the data backprojection fan beam geometry fan beam reconstruction 12

13 Equiangular Fan Beam Reconstruction f ( x, y) = 2π 0 L 2 dβ γ m γ m D cos γ p( γ, β ) h( γ ' γ ) dγ backprojection filtering The projection is first multiplied by the cosine of the detector angle. Filtering is applied across detector channels in a similar fashion as the parallel beam reconstruction. In the backprojection process, the filtered sample is scaled by the distance to the source. Because of the distance-dependent dependent weighting in the backprojection, impact on computation and noise result. 13

14 Fan-Para Reconstruction Alternatively, the fan beam data can be converted to a set of parallel samples. Parallel reconstruction algorithms can be used for image formation. projection angle, β β=β 0 γ parallel samples detector angle, γ 14

15 MIP & MPR Image Comparison MIP fan beam backprojection parallel beam backprojection MPR fan beam backprojection parallel beam backprojection 15

16 Helical Scanning In helical scanning, the patient is translated at a constant speed while the gantry rotates. Helical pitch: h = q d distance gantry travel in one rotation collimator aperture q 16

17 Helical Scanning Advantages of helical scanning nearly 100% duty cycle (no inter- scan delay) improved contrast on small object (reconstruction at any z location) improved 3D images (overlapped reconstruction) z 17

18 Helical Scanning The helical data collection is inherently inconsistent. If proper correction is not rendered, image artifact will result. z reconstructed helical scan without correction 18

19 Helical Reconstruction The plane of reconstruction is typically at the mid-point between the start and end planes. Interpolation is performed to estimate a set of projections at the plane of reconstruction. start of data set plane data sampling helix end of data set plane plane of reconstruction 19

20 Helical Reconstruction -360 o interpolation Samples at the plane-of of-reconstruction is estimated using two projections that are 360 o apart. p' ( γ, β ) = wp( γ, β ) + (1 w) p( γ, β + 2π ) where w = q q x x q data sampling helix p(γ,β) p (γ,β γ,β) p(γ,β+2π) 20

21 Helical Reconstruction -180 o interpolation In fan beam, each ray path is sampled by two conjugate samples that are related by: γ ' = γ β ' = β + π + 2γ For helical scan, these two samples are taken at different z location because of the table motion. 21

22 Helical Reconstruction -180 o interpolation Linear interpolation is used to estimate the projection samples at the plane of reconstruction. Because samples are taken at different view angles, the weights are γ and β dependent. wp( γ, β) + (1 w) p( γ, β + π 2γ ) plane of reconstruction p n (-γ,β+π 2γ) p k (γ,β) z-axis 22

23 Artifact Suppression Helical reconstruction algorithm effectively suppresses helical artifacts. without helical correction with helical correction 23

24 Multi-slice CT Multi-slice CT contains multiple detector rows. For each gantry rotation, multiple slices of projections are acquired. Similar to the single slice configuration, the scan can be taken in either the step-and and-shoot mode or helical mode. x-ray source detector 24

25 Advantages of Multi-slice Large coverage and faster scan speed Better contrast utilization Less patient motion artifacts Isotropic spatial resolution 25

26 Cone Beam Artifact center slice z edge slice multi-slice 26

27 Multi-slice Helical When acquiring data in a helical mode, the N detector rows form N interweaving helixes. Because multiple detector rows are used in the data acquisition, the acquisition speed is typically higher. h = q d distance gantry travel in one rotation collimator aperture plane-of-reconstruction d multi-slice 27

28 Cone Beam Helical Reconstruction Exact algorithms produce mathematically exact solutions when input projections are perfect. Katsevich Grangeat Rebin PHI FBP PHI Approximate algorithms, although non-exact, generate clinically accurate images. FDK-type N-PI CB-virtual circle Tilted Plane ZB Generally speaking, approximate algorithms are computationally less expensive and more flexible. 28

29 Cone Beam Algorithm small cone angle From a computational point of view, 3D backprojection is more expensive than 2D backprojection. To reduce computation, reconstruction planes are defined as planes that best fit the helix so that 2D reconstruction algorithm can still be used. interpolated sample plane of reconstruction z source helix tilted plane conventional POR 29

30 Tilted Plane Reconstruction For small cone angles, the flat plane and source helix match quite well. When the same weighting function is used, reconstructions with the tilted plane produces better image quality than the conventional reconstruction plane with 2D backprojection. conventional plane tilted plane 30

31 Cone Beam Reconstruction moderate cone angle For larger cone angles, tilted plane reconstruction is no longer sufficient, due to the larger difference between the flat plane and the curved helix. FDK-type algorithm with appropriate weighting is often used. z z helical path tilted plane multi-slice conventional POR 31

32 Cone Beam Reconstruction FDK Algorithm Each ray in a cone beam can be specified by β, γ, and α. FDK algorithm was derived from fan-beam algorithm by studying the impact of cone angle to the rotation angle. Unlike parallel or fan beam algorithms, FDK algorithm is an approximation. z pre-processed processed data y weighting γ α β x x filter the data along row 3D backprojection 32

33 FDK-type Algorithm FDK-type algorithm can be combined with different weighting functions to optimize its performance in different performance parameters. Cone beam artifacts are suppressed but not completely eliminated. original FDK-based 33

34 Tangential Filtering Conventional filtering process is carried out along detector rows. Tangential filtering is carried out along the tangential direction of the source trajectory. z tangential filtering x O γ α conventional filtering β y S 34

35 Tangential Filtering conventional filtering tangential filtering 35

36 3D Helical Weighting The helical weighting function changes with projection angle β, detector angle γ, and cone angle α. z Experiments show that 3D weighting function provides significant improvement in image quality. β γ α 36

37 3D Helical Weighting For moderate cone angle, FDK with 3D weighting can provide equivalent image quality as the exact algorithms. off the shelf recon 3D weighting more expensive exact recon 37

38 Reconstruction Half-scan (180 o + fan angle) for reconstruction Volume covered non-uniformly due to cone beam Extrapolation to estimate the missing data Adjacent scans to provide complimentary information desired ROI detector z scan n+1 scan n source trajectory scan n-1 38

39 39 Cone Beam Reconstruction For each image For each image voxel voxel, a quality factor is generated based on, a quality factor is generated based on the amount of extrapolation used in the reconstruction. the amount of extrapolation used in the reconstruction. The final image is the weighted sum of adjacent scans. + = m d v z y x γ π β β ϕ η 0 ), ( ),, ( The final image is the weighted sum of adjacent scans. ),, ( ),, ( ),, ( ),, ( ),, ( ),, ( ),, ( ),, ( ),, ( z y x f z y x z y x z y x z y x f z y x z y x z y x z y x f B B A A A B A B η η η η η η = y ROI scan n-1 scan n+1 scan n β A β B x

40 Phantom Results Heart phantom scanned on LightSpeed VCT 64 (64x0.625mm) Image for the outer most slice conventional reconstruction proposed reconstruction 40

41 Slice Thickness Change With Algorithm Slice thickness can be selected by modifying the reconstruction process. By low-pass filtering in the z-z direction, the slice sensitivity profile can be broadened to any desired shape and thickness. From an image artifact point of view, images generated with the thinner slice aperture is better. Filtering z 41

42 Example Z filtering can be applied in either the projection domain or the image domain. In general, z-smoothing z provides artifact suppression capability. 16x0.625mm detector aperture at 1.75:1 helical pitch FWHM=0.625mm FWHM=2.5mm 42

43 Cardiac Scans The most challenging problem in cardiac scanning is motion. Unlike respiratory motion, cardiac motion cannot be voluntarily controlled. For motion suppression, we could either reduce the acquisition time and/or acquire the data during the minimum cardiac motion. In cardiac motion, there are relative quiescent period: diastolic phase of the heart motion. 43

44 Halfscan In fan beam, each ray path is sampled by two conjugate samples. We need only fan angle data for complete reconstruction. 360 o 180 o +fan angle 0 o detector channels 44

45 Single-cycle Reconstruction Cardiac imaging takes advantage of the quiescent periods in cardiac motion with EKG-gating. The duration of quiescent periods change with heart rates. image set 1 image set magnitude diastasis (msec) time (sec) heart rate (bpm) 45

46 Multi-cycle Cardiac Reconstruction A complete set of projections is acquired over multiple cardiac cycles to improve temporal resolution. The method relies on the regularity of heart motion. image set magnitude time (sec) 46

47 Cardiac Imaging curved reformation volume rendering 47

48 Summary Advanced CT Image reconstruction techniques have been continuously developed over the years to keep up with advancements in acquisition hardware and acquisition protocols. With the large number of images generated by new CT scanners, advanced visualization tools are required to improve the productivity of radiologists. Image reconstruction algorithm not only complement scanner hardware, but also extends the hardware capability. 48

49 References J. Hsieh, Computed Tomography: principles, design, artifacts, and recent advances,, SPIE Press, J. Hsieh, CT Image Reconstruction, in RSNA Categorical Course in Diagnostic Radiology Physics: CT and US Cross-sectional sectional Imaging 2000,, ed. L. W. Goldman and J. B. Fowlkes, RSNA, 2000; pp A. Kak and M. Slaney, Principles of Computed Tomographic Imaging, IEEE Press,

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