Ultrasonic Beamforming with the Genetic Algorithm

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Ultrasonic Beam-forming with the Genetic Algorithm

• Andrew Fiss, Vassar College
• Nathan Baxter, Ohio Northern University
• Jerry Magnan , Florida State University

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Abstract

• We used Genetic Algorithm optimization of sparse
  array parameters to produce an ultrasonic beam
  with a higher, narrower main-lobe and lower
  side-lobes. The solutions obtained are neither
  unique nor trivial.

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Background

• Ultrasonic Transducers
• Piezoelectric elements generate sound waves which
  are used in a variety of applicationsnon-destruct
  ive testing and evaluation of material structure,
  sonar, diagnostic and therapeutic medicine, etc
• Arrays of transducer elements provide advantages
  over single elements in forming and controlling
  the ultrasound beam.
• We explore the beam-shaping properties of sparse
  ultrasonic transducers by performing an
  optimization over the arrays structural
  parameters using a Genetic Algorithm.

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Background (cont.)

• Genetic Algorithm
• ? The Genetic Algorithm mimics natural
  evolution.
• ? Real number strings called chromosomes
  contain parameters that can vary to produce
  desired results. Our chromosomes contained 63
  parameters30 positions for the 30 active
  elements, 30 apodizations , x-width of
  the elements, y-width of the elements, and center
  frequency.
• ? Random initial chromosomes are used to
  populate the system. These chromosomes are
  assigned a fitness value based on their
  proximity to an optimal solution and produce
  subsequent child chromosomes through mutation
  or recombination operators, which respectively
  change or rearrange the number strings. Our
  fitness function is represented as follows

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Background (cont.)
main-lobe height side-lobe height width at
3dB ? The Field II program was used to
generate the pressure fields. (see Method
section) ? The children replace the majority of
the previous generation, leaving only the best
parents.
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Objective
We wanted to shape the beam to have lower side
lobes and a higher, narrower main lobe. This
combination produces greater clarity in imaging
via improved directivity of the beam.
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Method
Field II, developed by Jørgen Jensen, was used in
generating the pressure data for the
beam-patterns, which were observed along the
y-axis at x0 and a fixed z. These beam-patterns
were assigned a fitness that rewarded higher
main-lobes, narrower main-lobes, and lower
side-lobes, and the Genetic Algorithm was used as
an optimization technique with regular mutation,
creep mutation, uniform list crossover, and
average crossover operators. The solutions
consisted of 30 active elements in a 30-by-30
array, with width and height constrained to lie
between and , a center frequency
between 1 MHz and 15MHz, and apodization varying
between 0 and 1.
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Results
After 30 generations, the Genetic Algorithm
improved the beam-pattern from a main-lobe height
of 3.94410 -14 barr to 5.595210 -14 barr, a
main-lobe width at -3 dB of 1.95mm to 1.35mm, and
a side-lobe height of 1.384110 -14 barr to
0.5637210 -14 barr (see figures). However, the
algorithm converges quickly for this optimization
problem (see figure), so similar results can be
obtained after just five generations (see
figures). The best configuration of a batch of
5,000 random chromosomes is presented, thus
showing that similar beam-patterns could not be
obtained through random generation. This is not
surprising, since there are 10 55 possible
arrangements of transducer element positions.
Also, one might expect a trivial solution to be a
dense transducer, such as a large, single
element, but that is not the case (see figure).
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Equations
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Equations (cont.)
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Conclusion

• Ultrasonic beams can be effectively shaped
  through use of the Genetic Algorithm, and the
  solution is neither trivial nor unique.

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Future work
In future work, we would like to try asymmetric
beam forming and apodization reconfiguration with
fixed positions in a sparse array.  We would also
try different optimization techniques and more
precise programs for pressure calculations, and
we would like to explore further the
characteristics of the optimization problems
through different Genetic Algorithm structure and
operators.
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References
For more information, see Samuel Temkins
Elements of Acoustics, Jon Mathews and R.L.
Walkers Mathematical Methods of Physics, or
J?rgen Arendt Jensens Calculation of Pressure
Fields from Arbitrarily Shaped, Apodized, and
Excited Ultrasound Transducers published in the
March 1992 edition of the IEEE Transactions on
Ultrasonics, Ferroelectrics, and Frequency
Control.