1. Department of Computer Science
Medical Imaging
Mohammad Dawood
Department of Computer Science
University of Münster
Germany

2. Recap

3. α decay β- decay β+ decay
Gamma scintigraphy PET Sinogram

4. Ultrasound / Sonography

5. The frequency range of sound above 20kHz is known as ultrasound
Sound spectrum
The frequency range of sound above 20kHz is known as ultrasound

6. Sound wave propagation

7. Ultrasound

8. Ultrasound

9. Ultrasound
- is produced through the conversion of electrical energy into mechanical energy
- is detected by the reverse process, by converting mechanical energy into electrical energy.
- The transducer is a device that is both a transmitter and receiver of the ultrasound signal and it serves a dual role in pulse echo imaging.

10. Inge Edler und Helmut Hertz (1954)

11. Ultrasound

12. Ultrasound
Reflexion
Scatter
Transmission

13. Ultrasound
Resolution
Depth
Frequency

14. Ultrasound

15. Ultrasound

16. Medium
Speed of sound m/s
Air
331
Water
1483
Tissue
Liver
1549
Blood
1570
Glycerin
1923
Bones (Compact)
3600

18. Image Reconstruction

19. Reconstruction
Tomography

20. Reconstruction
Law of Attenuation

21. Reconstruction
Parallel projections of a plane

22. y
Reconstruction
Radon Transformation s f r n θ x

23. Reconstruction
Radon Transformation (Line Integrals at different angles)

24. Reconstruction
Radon Transformation
Original Sinogram (Radon Transform)

25. Reconstruction
Inverse Radon Transformation
H: Hilbert transform

26. Reconstruction
Inverse Radon Transformation
Problems with Missing data and Noise!

27. Reconstruction
Filtered Back Projection

29. Reconstruction
Filtered Back Projection

30. Reconstruction
FBP: Commonly used filters
1=Ram-Lak (ramp), 2=Shepp-Logan,
3=Cosine, and 4=Hamming

31. Reconstruction
Filtered Back Projection
2D/3D filtering is costly
Projections
Backproject
Filter 1D
Filter 2D
Image

32. Reconstruction
Fourier slice theorem
Take a two-dimensional function f(r), project it onto a line, and do a Fourier transform of that projection
Take that same function, but do a two-dimensional Fourier transform first, and then slice it through its origin parallel to the projection line

33. Reconstruction
Fourier Slice Theorem

36. Reconstruction
Iterative Reconstruction
b: measured values
x: unknown attenuation coefficients
aij: weights f1 f2 … fn
LOR1
LOR2
LORn

37. Reconstruction
Iterative Reconstruction
Kaczmarz Method (=ART: Algebraic Reconstruction Technique)

38. Reconstruction
Iterative Reconstruction
Kaczmarz Method (=ART: Algebraic Reconstruction Technique)
1. Start by setting x(0) = 0
2. Compute the forward projection from the n-th estimate, i.e. b(n) = A x(n)
3. Choose i and correct the current estimate x(n)
4. Iterate steps 2,3 until the difference between new forward projection b(n), computed in 2, and the old one is below tolerance

40. Reconstruction
Iterative Reconstruction
EM (Expectation Maximization)