1. A New Modality for Microwave Tomographic Imaging: Transit Time Tomography
Matt Trumbo
3/30/06

2. Overview History of Computerized Tomography Radon Transform
Transit Time Tomography: The Concept
Experimental Setup
Results and Conclusion

3. History of Computerized Tomography
Initially conceived by Johann Radon in 1917
Developed by Godfrey Hounsfield in 1972
Initially used X-rays
Nobel Prize in 1979

4. History of Computerized Tomography cont.
Since the initial development of the CT scanner in 1972, many improvements to the concept have been added
Fan-beam projections
Multiple detectors

5. Math of Computerized Tomography

6. Parallel Projections
Using these line integrals, projections for all angles are taken
These parallel projections make reconstruction easier, but image acquisition takes longer

7. Fourier Slice Theorem What do we do with these projections?
Start by defining….

8. Fourier Slice Theorem For the projection at v = 0,
Notice the center integral is merely a projection

9. Fourier Slice Theorem cont.
Wait a second…that’s the Fourier transform of a projection!
Since this result is independent of the orientation between the object and the coordinate system, this result holds for any PΘ(t).

10. Fourier Slice Theorem cont.
More formally stated:
The Fourier Transform of parallel projection of an image f(x,y) taken at an angle Θ gives a slice of the 2 dimensional transform F(u,v), subtending an angle Θ with the u-axis.
This means we could reconstruct the two-dimensional Fourier transform of our object from just the projections given by the Radon transform!

11. Image Reconstruction …but it’s not that simple.
Density of sampling causes high-frequency components to suffer in reconstruction

12. Filtered Backprojection
An algorithm to better interpolate the high frequency components is necessary
Once we’ve interpolated the transform, we can simply take an inverse Fourier transform and sums across our image domain to achieve our image reconstruction

13. Sinograms
Stacked Projections

14. Transit Time Tomography: The Concept
New way of calculating the line integral for the Radon transform
Assume an object with a relative permittivity distribution εr(x, y) , and velocity of propagation v(x,y) x y
v(x,y).

15. The Concept cont.
Consider movement in 1-D at a positional dependent velocity u(x).
And this velocity is given by the equation for speed of electromagnetic propagation

16. The Concept cont. Combining these two equations, we obtainx y
. a b
Combining these two equations, we obtain
To solve for the time required to traverse the ray trace shown, we see that

17. The Concept cont.
Using this as our approximation of the line integral for the Radon transform, we could reconstruct an image of the refractive index distribution using a standard inverse Radon transform with filtered backprojection.

18. What could possibly go wrong?
Transit Time tomography does require several assumptions
Thin Lens assumption
Ignoring of reflection and scattering
Vertically polarized antennas
Isolation of the object being imaged

19. Experimental Setup
A transmitter and receiver pair of antennas are mounted on a single bracket, which moves back and forth on a Velmex Bislide 3MN10, controlled by a 2 phase stepper motor.
A Vexta CFK II 5-phase stepper motor rotates the platform upon which the object to be imaged is actually placed.

20. Experimental Setup cont.
Motor and Scope control are handled by a Visual Basic program, shown at right
Direct motor control -- pulses to the motor and switch detection -- is handled by a Basic Stamp microcontroller

21. Experimental Setup cont.
Microwave data is taken and recorded by the Hewlett-Packard VNA 8720ET.
It generates chirp microwave pulses ranging from 1GHZ to 20GHZ in frequency, and records the response.
The VNA interfaces with the Visual Basic control program via a GPIB interface.

22. Experimental Setup cont.
The antennas used for the transmitter and receiver are simple bi-conic circular horn antennas.
They were selected for their uniform frequency transmission pattern, vertical polarization, and ease of construction.

23. Experimental Setup cont.
Typical operation of the Transit Time Tomography system follows this simple algorithm:
User initiates data acquisition.
Visual Basic program requests to the BS2E that the antennas be reset to their starting positions.
The BS2E sends a constant stream of pulses to the linear slide motor, until a limit switch is activated.
Response of a chirp pulse is recorded and saved in a Microsoft Excel file on the laptop.
The Visual Basic program requests an antenna position increment.
The BS2E sends the appropriate number of pulses to increment the antennas 1/500th of a full sweep.
Measurements are repeated 500 times for the entire parallel projection.
The Visual Basic program requests a rotational increment of the object.
The BS2E sends the appropriate number of pulses to increment the rotational position 18/25th of a degree.
Go back to the start, and repeat for 180 degrees of projections.

24. Delay Acquisition
How do we obtain our single measurement of delay to approximate the line integral?

25. Delay Acquisition cont.
It was initially thought that peak tracking would provide a simple way of measuring delay
After repeated attempts at peak tracking, the idea was abandoned

26. Delay Acquisition cont.
The solution was to use a center of mass calculation on the waveform, within a given time region.

27. Results
Using the center of mass delay acquisition technique, excellent projections were obtained
For our first proof of concept testing, small circular jars of cooking oil (dielectric constant of approximately 2.5) were placed at varying distances from each other
This allowed us to test resolution and the class of circular objects simultaneously

28. 4 inches of separation

29. 6 inches of separation

30. 2.7 inches of separation

31. Rectangle Another class of objects was desired for comparison
Many more issues were experienced with rectangles
Reflection
Attenuation

32. Rectangle

33. Conclusions Results match physical reality
Circle imagery is almost perfect
Rectangle imagery has some issues, but still provides a reasonably good image

34. The future
The primary advantage of this system is the simplicity of every aspect
Medical Applications
Industrial Applications

35. Acknowledgements Chris Anderson Eric Green Dr. Randall Jean
Dr. Robert Marks
Dr. Byron Newberry

36. Thank you for attending
Questions?