1. Medical Image Analysis Image Reconstruction Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

2. Mathematical Preliminaries and Basic Reconstruction Methods Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003. An original image

3. Mathematical Preliminaries and Basic Reconstruction Methods Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003. Apply the Radon transform

4. Mathematical Preliminaries and Basic Reconstruction Methods Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003. After the inverse Radon transform

5. Mathematical Preliminaries and Basic Reconstruction Methods Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003. An test image

6. Mathematical Preliminaries and Basic Reconstruction Methods Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003. Apply the Radon transform

7. Mathematical Preliminaries and Basic Reconstruction Methods Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003. After the inverse Radon transform

8. Mathematical Preliminaries and Basic Reconstruction Methods Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003. x y q  p  p f(x,y) P(p,  ) Figure 2.8. Line integral projection P(p,q) of the two-dimensional Radon transform.

9. Mathematical Preliminaries and Basic Reconstruction Methods The Radon transform of an object Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

10. Central Slice Theorem The central slice theorem ◦ Called the projection theorem ◦ A relationship between the Fourier transform of the object function and the Fourier transform of its Radon transform or projection Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

11. Central Slice Theorem Figure comes from the Wikipedia, www.wikipedia.org.www.wikipedia.org

12. Central Slice Theorem Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

13. u v F(u,v)    Sk()Sk() S2()S2() S1()S1() Figure 5.1. The frequency domain of the Fourier transform F(u,v) with the Fourier transforms, S q (w) of individual projections J q (p).

14. Central Slice Theorem Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

15. Central Slice Theorem ◦ Represents the Fourier transform of the projection that is taken at an angle in the space domain with a rotated coordinate system Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

16. Inverse Radon Transform Where

17. Backprojection Method Modified projections ◦ Convolution-backprojection ◦ Filtered-backprojection

18. Backprojection Method From

19. Backprojection Method Ramakrishnan and Lakshiminarayanan In general

20. Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003. H(  ) 1/2  -1/2  1/2   Figure 5.2. A bandlimited filter function.

21. Backprojection Method The filter kernel function ◦ If the projections are sampled with a time interval of, the projections can be represented as, where is an integer

22. Backprojection Method For the bandlimited projections with a sampling interval of Then

23. Backprojection Method The quality of the reconstructed image ◦ The number of projections ◦ The spatial interval of the acquired projection ◦ Limited by the detector size and the scanning procedure ◦ Suffer from poor signal-to-noise ratio if there is an insufficient number of photons collected by the detector due to its smaller size

24. Backprojection Method Ramakrishnan and Lakshiminarayanan filter ◦ Has sharp cutoffs in the frequency domain at and ◦ Cause modulated ringing artifacts in the reconstructed image Hamming window function

25. Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003. H(  ) 1/2  -1/2   Figure 5.3. A Hamming window based filter kernel function in the frequency domain.

26. Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003. h(   h R-L (p) H Hamming (p) Figure 5.4. A comparison of the and convolution functions in the spatial domain.

27. Iterative Algebraic Reconstruction Methods Algebraic Reconstruction Techniques (ART) ◦ The raw projection data from the scanner are distributed over a prespecified image reconstruction grid such that the error between the computed projections from the reconstructed image and the actual acquired projections is minimized Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

28. Ray with ray sum p i f1f1 f2f2 f3f3 fNfN Overlapping area for defining w i,j Figure 5.5. Reconstruction grid with a ray defining the ray sum for ART.

29. Iterative Algebraic Reconstruction Methods : the projection data : the pixels of the image : weights ◦ Determined by geometrical consideration as the ratio of the area overlapping with the scanning ray to the total area of the pixel Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

30. Iterative Algebraic Reconstruction Methods : the computed ray sum in the iteration Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

31. Iterative Algebraic Reconstruction Methods The iterative ART ◦ Deal with the noise and random fluctuations in the projection data caused by detector inefficiency and scattering Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

32. Estimation Methods Statistical estimation ◦ Assume a certain distribution of the measured photons ◦ Find the parameters for attenuation function (in the case of transmission scans such as X- ray CT) or emitter density (in the case of emission scans such as PET) Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

33. Estimation Methods : measurement vector : the random variable representing the number of photons collected by the detector for the ray : the blank scan factor : the attenuation coefficients Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

34. Estimation Methods A line integral or ray sum for ray The Poisson distribution model for the photon counts

35. Estimation Methods The Maximum Likelihood (ML) estimate ◦ The log likelihood function

36. Estimation Methods The Maximum Likelihood (ML) estimate ◦ The log likelihood function ◦ Find

37. Estimation Methods Penalty functions ◦ Additional constraints such as smoothness ◦ Find

38. Estimation Methods Optimization methods ◦ Expectation Maximization (EM) ◦ Complex conjugate gradient ◦ Gradient descent optimization ◦ Grouped coordinated ascent ◦ Fast gradient based Bayesian reconstruction ◦ Ordered-subsets algorithms

39. Fourier Reconstruction Methods Direct Fourier reconstruction ◦ Use the central slice theorem ◦ Resampling the frequency domain information from a polar to a Cartesian grid ◦ Developing sinc-based interpolation method for the bandlimited functions in the radial direction

40. Image Reconstruction in Medical Imaging Modalities Choice ◦ Filtered backprojection (X-ray CT) ◦ Statistical estimation Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

41. C    B A S F R T D E x y O G Figure 5.6. A 2-D divergent beam geometry.

42. X-ray Computed Tomography : angular step : radial distance between the source and the origin : the angle that the source makes with its central reference axis : a fan projection from the divergent beam Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

43. X-ray Computed Tomography Objective: convert fan projections into the parallel-beam projection Sorted

44. X-ray Computed Tomography Backprojected : the total number of source positions : the angle of the divergent beam ray passing through the point : the distance between the source and the point for the source position

45. Nuclear Emission Computed Tomography: SPECT and PET X-ray CT ◦ Estimate the attenuation coefficient map SPECT or PET ◦ Reconstruct the source emission map within the object from the statistical distribution of photons that have gone through attenuation within the object but detected outside the object ◦ Attenuation correction Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

46. Nuclear Emission Computed Tomography: SPECT and PET The transmission scans in SPECT ◦ Computing attenuation coefficient parameter ◦ The iterative ML estimation-based algorithms have provided better results Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

47. Multi-Grid EM Algorithm Image reconstruction in PET Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

48. Figure 5.7. A flowchart of the MGEM algorithm for PET image reconstruction.

49. Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003. Figure 5.8. Wavelet based interpolation method.

50. Figure 5.9. Shepp and Logan phantom (top left) and reconstructed phantom images using WMREM algorithm (top right), ML-EM algorithm (bottom left) and filtered backprojection method (bottom right).

51. Figure 5.10. Four reconstructed brain images of a patient with a tumor from a PET scan. Images in the top row are reconstructed using filtered backprojection method, images in the middle row are reconstructed using WMREM algorithm. Images in the bottom row are reconstructed using a generalized ML-EM algorithm.

52. Image Reconstruction MRI Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

53. Image Reconstruction Ultrasound Imaging Point measurements ◦ Line scan ◦ Reduction of speckle noise  Image averaging  Image filtering: weighted median, Wiener filters Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.