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Image Reconstruction Methods

Image Reconstruction Methods. 東海大學資管系助理教授 余心淳. Outlines. Basic concepts Image processing and its applications Image reconstruction from projections Algorithms of image reconstruction Image reconstruction from limited data Prior Discrete Fourier Transform (PDFT). 2014/9/13. 2.

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Image Reconstruction Methods

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  1. Image Reconstruction Methods 東海大學資管系助理教授 余心淳

  2. Outlines • Basic concepts • Image processing and its applications • Image reconstruction from projections • Algorithms of image reconstruction • Image reconstruction from limited data • Prior Discrete Fourier Transform (PDFT) 2014/9/13 2

  3. Basic Concepts of Image Reconstruction

  4. Image processing A kind of signal processing Analog, digital, optical, and acoustic Computerized operation Applications Nuclear medicine Computer vision Feature detection Microscope image Remote sensing Mine detection Lane departure warning system Non-photorealistic rendering Image Processing and Its Applications 2014/9/13 4

  5. Projection Angular projections Recording projections Reconstruction Image reconstruction from projections Reconstruction from Projections 2014/9/13 5

  6. Projection and Reconstruction Angular Projection Reconstruction 2014/9/13 6

  7. Projection and Reconstruction angle distance Projection Slice Sinogram 2014/9/13 7

  8. Most Popular Applications in Nuclear Medicine • X-ray • Positron emission tomography (PET) • Single photon emission computed tomography (SPECT/SPET) • Computed tomography (CT) • Electron beam CT(EBCT) • Magnetic resonance imaging (MRI) • Ultrasonic

  9. 3D Positron Emission Tomography Projection 2014/9/13 9

  10. 3D PET Image Reconstruction 2014/9/13 10

  11. 3D SPECT Image Reconstruction 2014/9/13 11

  12. Computed Tomography Image Reconstruction 2014/9/13 12

  13. Magnetic Resonance Imaging 2014/9/13 13

  14. Taxonomy of Reconstruction Methods • Analytical algorithms • Inverse matrix • Filtered backprojection (FBP) • Fourier transform (FT) • Iterative algorithms • Conventional algebraic methods • Algebraic reconstruction technique (ART) • Simultaneous iterative reconstruction technique (SIRT) • Iterative least-squares technique (ILST) • Statistical reconstruction methods • Gradient and conjugate gradient (CG) • Maximum likelihood expectation maximization (MLEM) • Ordered-subsets expectation maximization (OSEM) • Maximum a posteriori (MAP) 2014/9/13 14

  15. Analytical Reconstruction Algorithm- Inverse Matrix Detector Electromagnetic Beam 2014/9/13 15

  16. Analytical Reconstruction Algorithm- Filtered Backprojection Projections Backprojections 2014/9/13 16

  17. Analytical Reconstruction Algorithm- Filtered Backprojection Negative Wings (Kernel Function) Filtered Backprojections 2014/9/13 17

  18. Analytical Reconstruction Algorithm- Filtered Backprojection Backprojections Filtered Backprojections 2014/9/13 18

  19. Analytical Reconstruction Algorithm- Filtered Backprojection 1 2 3 4 Projection Phase 2014/9/13 19

  20. Analytical Reconstruction Algorithm- Filtered Backprojection Reconstruction Phase 2014/9/13 20

  21. Analytical Reconstruction Algorithm- Filtered Backprojection 1 2 Backprojection Phase 3 4 2014/9/13 21

  22. Analytical Reconstruction Algorithm- Filtered Backprojection 5 6 FilteredBackprojection 7 8 2014/9/13 22

  23. Analytical Reconstruction Algorithm- Filtered Backprojection 2014/9/13 23

  24. Analytical Reconstruction Algorithm- Filtered Backprojection Original 1st Backprojection 2014/9/13 24

  25. Analytical Reconstruction Algorithm- Filtered Backprojection Original 2nd Backprojection 2014/9/13 25

  26. Analytical Reconstruction Algorithm- Filtered Backprojection Original 4th Backprojection 2014/9/13 26

  27. Analytical Reconstruction Algorithm- Filtered Backprojection Original 8th Backprojection 2014/9/13 27

  28. Analytical Reconstruction Algorithm- Filtered Backprojection Original 10th Backprojection 2014/9/13 28

  29. Analytical Reconstruction Algorithm- Filtered Backprojection Original 30th Backprojection 2014/9/13 29

  30. Analytical Reconstruction Algorithm- Filtered Backprojection Original 60th Backprojection 2014/9/13 30

  31. Analytical Reconstruction Algorithm- Filtered Backprojection Original 90th Backprojection 2014/9/13 31

  32. Analytical Reconstruction Algorithm- Filtered Backprojection Original 180th Backprojection 2014/9/13 32

  33. Analytical Reconstruction Algorithm- Fourier Transform Original 2D Fourier Transform 2014/9/13 33

  34. Analytical Reconstruction Algorithm- Fourier Transform Y V one line FT X U Projections 2D Fourier Transform 2014/9/13 34

  35. Analytical Reconstruction Algorithm- Fourier Transform Projections Fourier Space 2014/9/13 35

  36. Analytical Reconstruction Algorithm- Fourier Transform IFT Reconstruction Fourier Space 2014/9/13 36

  37. Iterative Reconstruction Algorithm Flow chart of iterative image reconstruction scheme

  38. Iterative Reconstruction Algorithm Bit 0s Initial Image (guess) Original (unknown)

  39. Iterative Reconstruction Algorithm

  40. Comparison Between Analytic and Iterative Reconstruction Algorithms • FBP (Analytic technique) • Quick and efficiency • Accuracy ??? • Direct inversion of the projection formula • Become more popular and commercialized in nuclear medicine • Need a lot of filtering - trade-off between blurring and noise • Quantitative imaging difficult

  41. Comparison Between Analytic and Iterative Reconstruction Algorithms • ART (Iterative technique) • Inaccuracy • Allow for a rich description and modeling of blurring, attenuation, and scatter, etc. • Long calculation time • Amplification of noise

  42. Image Reconstruction From Limited Data

  43. Image Reconstruction From Limited Data • Digital and computerized data storage and processing • Being widely applied to the image reconstruction problems from limited Fourier data • Examples: X-ray diffraction, electron microscopy • Underdetermined problem • Inaccurately reconstructed image • Estimation solutions

  44. Prior Discrete Fourier Transform (PDFT) • A Fourier transform reconstruction method • An linear spectral estimator • Using prior information to enhance the performance of image reconstruction • Working well in case of the presence of noise and other objects around the target • Appling to many image reconstruction problems

  45. Why PDFT ? • Great potentiality to challenge the ill-posed inverse problem (underdetermined) described by Fourier Integral equation. • PDFT obtains a high resolution of the target by projecting a minimum norm solution of the estimate on the suitably designed Hilbert space that is created by prior knowledge.

  46. The PDFT Algorithm: Example FT Data in Fourier Space

  47. The PDFT Algorithm: Example The original object DFT estimate DFT estimate The original object Prior function PDFT estimate PDFT estimate

  48. Diffraction Tomography • The far field scattered by the incident plane wave based on the first-order Born approximationis represented as • With the geometric approximation • Then

  49. Single Scattering Approximation ky Scattering Wave ks2 ks2 ks3 ks1 ks3 ks1 kx -kso radius = k kso Object Diffraction Tomography The Scattered Field mapped on Fourier Space for one incident wave with a specific frequency Incident Plane Wave

  50. Single Scattering Approximation ky y kx x Reconstructed Image Inverse Fourier Transform

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