BMED-4800

1. BMED-4800/ECSE-4800Introduction to Subsurface Imaging Systems Lecture 7: CT Scanning II Kai E. Thomenius1 & Badri Roysam2 1Chief Technologist, Imaging Technologies, General Electric Global Research Center 2Professor, Rensselaer Polytechnic Institute

2. Outline of Course Topics THE BIG PICTURE What is subsurface sensing & imaging? Why a course on this topic? EXAMPLE: THROUGH TRANSMISSION SENSING X-Ray Imaging Computer Tomography INTRO INTO OPTICAL IMAGING AND SENSING COMMON FUNDAMENTALS propagation of waves interaction of waves with targets of interest  PULSE ECHO METHODS Examples MRI A different sensing modality from the others Basics of MRI MOLECULAR IMAGING What is it? PET & Radionuclide Imaging IMAGE PROCESSING & CAD

3. Recap of Last Lecture Introduction to CT scanners Emphasis on reconstruction algorithms Basic problem of reconstruction from projections Simple reconstruction using linear equations. Fourier Slice Theorem Backprojection & filtered backprojection

4. Quick X-Ray & CT Review

5. Quick X-Ray & CT Review

6. Hounsfield Units

7. Inverse Problems CT algorithms fall into a class of problems referred to as inverse problems. Typical statement: Given measured data, find a suitable model of what generated that data. In the CT setting, data are the projections find a 2D description of tissue attenuation that generated that data. Inverse Problems: Often ill posed No unique solution exists unstable

8. Getting into CT Details Relation of: Radon Transform Fourier Slice Theorem Filtered Backprojection All of these deal with making an image of m(x,y) from projections given by:

9. Radon Transform In 1917, Johann Radon published an exact solution to this problem, given below. None of the CT pioneers (e.g. Hounsfield, Cormack) were aware of Radon’s work (nor is the direct expression being used in any commercial scanners). This expression includes a derivative of P(t,q); given noisy x-ray data, this would make reconstructions challenging. Other problems include the discontinuity at t’-t, (this actually goes away if you do the q-integration first).

11. Sinogram Definition

12. Fourier Slice Theorem

14. Filtered Backprojection The most common reconstruction algorithm is the filtered backprojection method. Overview: FBP Reconstruction steps: Collect one projection immediately start processing the data Take its 1D Fourier Transform Multiply by desired filter response Take inverse 1D Fourier Transform Backproject across image space Repeat for all projection angles.

15. Filtered Backprojection Last time we discussed the Fourier Slice Theorem 1D FT of each projection gives one radial line of the final images 2D FT. The relation to FBP can be understood from: Each projection is nearly independent of each other. Can be more easily visualized in the frequency space. Only common point is at “DC”. Thus we can add their contributions to each other.

16. Filter for FBP Before we can add the individual contributions of the Fourier transform, we have to account for the non-uniform sampling. The ideal filter for this is the pie shaped wedges shown in Figure (a). Unfiltered processing would simply use response (b). To emulate response (a), people have used the filter response (c ). If we assume there are K projections over 180 degrees, then the width of the wedge will be 2p|w| / K. The ramp filter shown in (c ) is intended to model this response.

17. Filtered Backprojection Final reconstruction step is addition of the 2D inverse FT of each weighted projection. This step is called backprojection smearing of each filtered projection over the image plane. Advantages of this method over the Radon or 2D FFT methods: Process can be started after first projection data is in. Any interpolation is done as part of the backprojection.

18. Filtered Backprojection Finally, the filtering process can be adjusted to bring about any modifications in the data. The attached figure shows several such filters that have been used for this purpose. Two examples are shown. One can reduce higher frequency components by introducing a lower cutoff, see Hann or Parzen filters.

19. Getting Perfect CT Data Most of the code in today’s scanners does not deal with reconstruction, rather, data optimization. There is a real need to improve the quality of the raw data: Corrections for detector sensitivity variation Remember g = mx = ln(Io/I)? Our measurements are critically dependent on uniform responses from all the detectors. Correction for beam propagation effects and detector failings. As it turns out, doubling the path length seldom doubles the attenuation – things are actually worse. Beam hardening: lower frequency x-rays preferentially attenuated causing the remaining beam to have higher average energy Scattered radiation Detector nonlinearities: each detector has its own nonlinearities

20. Matlab & CT Reconstruction Matlab offers a forward radon and inverse iradon m-files. In our CT context, R = radon(I,theta) I contains an intensity image Theta is a vector of projection angles R will give the resulting projections, when plotted as a 2D image, it will be a sinogram I = iradon(R,theta) R is the projection data (this is the point where we usually start. Theta is the same vector as above.

21. Matlab & CT Reconstruction In addition to the radon m-files, Matlab gives you a Shepp-Logan Phantom m-file P = phantom(512); imshow(P) With those two statements, we get this gray scale Shepp-Logan image We can now take the radon transform of this: theta = 0:179; % projection angles [R, xp] = radon(P, theta); We can now display the resulting sinogram: figure, imagesc(theta3,xp,R3); colormap(hot); colorbar xlabel('\theta'); ylabel('x\prime'); Let’s check how much worse things get with going back with the iradon m-file P2 = iradon(R, 2);

22. Some Web Resources http://www.onid.orst.edu/~faridana/preprints/fbp.txt - matlab code for filtered backprojection Designer Shepp-Logan phantom Filter design possibilities Modified code available from http://www.ecse.rpi.edu/censsis/SSI-Course

23. Some Web Resources ftp://esftp.it.dtu.dk/pub/kursus/31655/demos/ - several demos involving projections, reconstructions, etc. Very nice demos for ct imaging in general and projections in particular

24. Some Web Resources http://www.ctsim.org - open source CT simulator, a very nice tool to play games with various reconstruction parameters Has to be downloaded to run on your PC Completely menu driven app

25. CT Scanner Evolution First Generation - 1970 Parallel beam design One/two detectors Translation/rotation 2nd Generation - 1972 Small fan beam Translation/rotation Larger no. of detectors 3rd Generation - 1976 Multiple detectors Large fan beam 4th Generation - 1978 Detector ring Source rotation Large fan beam

26. Fan Beam

28. CT Scanner Evolution Multi-slice scanners Helical (GE) or spiral (Siemens) scanners Simultaneous: Source rotation Table Translation Data Acquisition Electron Beam Tomography

29. Multi-slice CT Going beyond cone beam Area of keenest competition today Manufacturers are adding slices – 256 and 320 slice scanners are available. Difficult marketing choice – improved performance but adding slices increases cost dramatically. Are the extra slices worth it?

30. Multi-slice CT - Dates

31. New Challenges w. Helical CT In conventional CT, 360 degrees of data are acquired for one image. To get another image, the gantry is moved to next location. Helical CT covers a non-planar geometry Patient Table moves anywhere from 1 – 10 mm/sec. Most algorithms are the same as with conventional CT. An added interpolation step, z-interpolation, is required.

32. Homework 7a You will be supplied with a sinogram of an unknown image (unknown_hwk7a). Using Matlab’s iradon function, find out what the image is using theta = 0:179. D-theta increments will be 1, 2, 5, 10, 30 Form sinograms of the image with the radon function and varying numbers of projections (i.e. different lengths of theta). What happens to image quality with shorter vectors? At what point would you say the image quality is acceptable?

33. Homework 7b, extra credit Using the Farinada filtered backprojection code, change filter parameters for a lower bandpass. Demonstrate loss of spatial resolution.

34. Summary Several details involving CT scanner operation were reviewed: Distinctions among the major methods for image reconstruction. In particular, the role of the convolution filter in FBP was considered. CT related resources on the web were identified. The various generations of CT scanners were defined.

35. Instructor Contact Information Badri Roysam Professor of Electrical, Computer, & Systems Engineering Office: JEC 7010 Rensselaer Polytechnic Institute 110, 8th Street, Troy, New York 12180 Phone: (518) 276-8067 Fax: (518) 276-6261/2433 Email: roysam@ecse.rpi.edu Website: http://www.ecse.rpi.edu/~roysabm Secretary: Laraine Michaelides, JEC 7012, (518) 276 –8525, michal@rpi.edu

36. Instructor Contact Information Kai E Thomenius Chief Technologist, Ultrasound & Biomedical Office: KW-C300A GE Global Research Imaging Technologies Niskayuna, New York 12309 Phone: (518) 387-7233 Fax: (518) 387-6170 Email: thomeniu@crd.ge.com, thomenius@ecse.rpi.edu Secretary: Laraine Michaelides, JEC 7012, (518) 276 –8525, michal@rpi.edu