Adaptive autocalibration of unknown transducer positions in time reversal imaging

1. Adaptive autocalibration of unknown transducer positions in time reversal imaging Raghuram Rangarajan EE Systems University Of Michigan, Ann Arbor

2. Concept of time reversal • Equation invariant under time reversal i.e., if P(r,t) satisfies the above equation, so does P(r,-t) ! • P(r, t) wave coming from the source ; P(r,-t) wave focusing on the source • Usefulness ?

3. Focusing at arbitrary location • Three step process • Transmit time varying signals to illuminate region of interest. • Record backscattered field from the medium (say has a point reflector ). • Time reverse and retransmit to refocus on desired location. • Think of it as a matched filter representation.

4. Outline of current research work • Analyze performance of time reversal method (TRM) for imaging unknown scattering environments. • Illuminate specific voxels ( specific locations ) and extract scattering coefficients ( a measure of reflectivity ) – detect targets. • Find CRBs to explore advantages of TRM over conventional methods. • Study performance in the presence of transducer noise and unknown transducer position

5. Main focus of this project • Auto calibration algorithm to calibrate unknown sensor positions • TRM and conventional beam forming methods. • Compare estimates with CR lower bound. • Simultaneous estimation – A possibility • voxel scattering coefficient • unknown sensor positions

6. Solution • Let unknown antenna location be ra. • Cost function Q(r) • argmin Q(r) = ra • Basic Idea • To focus on voxel v, use projection operator pyv . • So to create a null at voxel v, take any operator orthogonal to pyv.

7. Solution (contd.) • To focus on v , find pv to minimize the norm of (HyTpv* - ev) • ev has 1 in the vth position and zero otherwise. • Hy is the Green’s function. • To get a null on v, find pv to minimize the norm of (HyTpv* - [s 0 t ]T ) where s and t is our choice.

8. Solution (contd.) • Two reasonably good choices. • The choices might not be optimal. • [s 0 t ] = (1 – ev) • [s 0 t ] = (ev’) for any v’ (see its advantage later.)

9. Some simple equations • Cost function • Q(r)=|HTyap*yv(r)H*yD*HHypyv(r)z*|2 • Gradient descent algorithm • rk+1 = rk - mrr Q(rk) • r Q(rk) is obtained directly by differentiating the above equation

10. Plots

11. Problems • m needs to be carefully chosen. • Good initial estimate. • Reasonable Assumption. • Problems of local minima(very likely). • Solution ?

12. Momentum Filter

13. More Preliminary Results • Comparison with CRB • Setting • Set of antennas and voxels. Assume one coordinate of an antenna is unknown. • SNR=10dB. • Run over all initial positions in a region R (in the vicinity of other antennas ) and varying noise. • Focus on some voxel v’

14. Contd. • CRB = 4.54 • Exact Location = 180 • Sample Average = 180.1 • Average Error = 4.2 • Increase region R, increase in error

15. Future work • Compare estimators with the CRB. • Comparing estimates from TRM with conventional beam forming methods. • Interesting idea • Use focusing on voxel v’ to calibrate antenna position and estimate scattering coefficient simultaneously. • That would be great !

16. Questions