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

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 ?

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.

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

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

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

Solution (contd.) To focus on v, find p v to minimize the norm of ( H y T p v * - e v ) e v has 1 in the v th position and zero otherwise. H y is the Green’s function. To get a null on v, find p v to minimize the norm of ( H y T p v * - [s 0 t ] T ) where s and t is our choice.

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

Some simple equations Cost function Q(r)=|H T ya  * yv (r)H * y D * H H y  yv (r)z * | 2 Gradient descent algorithm r k+1 = r k -  r r Q(r k ) r Q(r k ) is obtained directly by differentiating the above equation

Plots

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

Momentum Filter      

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’

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

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 !

Questions