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Waveform Inversion with Source Encoding for Breast Sound Speed Reconstruction in Ultrasound Computed Tomography

Kun Wang, Thomas Matthews, Fatima Anis, Cuiping Li, Neb Duric, Mark A. Anastasio

arXiv:1501.00021v1physics.med-ph

TL;DR

USCT breast imaging needs waveform-based reconstruction for improved spatial resolution, but conventional waveform inversion is computationally demanding. This paper introduces WISE, which uses random source encoding and stochastic optimization for sound-speed reconstruction. Simulations and experimental breast-phantom studies suggest that WISE retains high spatial resolution while reducing computational burden.

  • Problem

    Waveform inversion can improve spatial resolution over ray-based USCT methods, but its computational burden has limited widespread breast-imaging application.

    Source: §I · p. 2, §VII · p. 23

  • Method

    WISE encodes emitter measurements with a random vector and estimates sound speed by solving a stochastic optimization problem with stochastic gradient descent.

    Source: §I · p. 3

  • Results

    Simulation and experimental breast-phantom studies suggest that WISE maintains waveform inversion's high spatial resolution while significantly reducing computational burden.

    Source: §VII · p. 23

  • Takeaways & Limitations

    WISE shows potential value for practical USCT breast imaging applications.

    Source: §VII · p. 23

Abstract

from arXiv · show

Ultrasound computed tomography (USCT) holds great promise for improving the detection and management of breast cancer. Because they are based on the acoustic wave equation, waveform inversion-based reconstruction methods can produce images that possess improved spatial resolution properties over those produced by ray-based methods. However, waveform inversion methods are computationally demanding and have not been applied widely in USCT breast imaging. In this work, source encoding concepts are employed to develop an accelerated USCT reconstruction method that circumvents the large computational burden of conventional waveform inversion methods. This method, referred to as the waveform inversion with source encoding (WISE) method, encodes the measurement data using a random encoding vector and determines an estimate of the sound speed distribution by solving a stochastic optimization problem by use of a stochastic gradient descent algorithm. Both computer-simulation and experimental phantom studies are conducted to demonstrate the use of the WISE method. The results suggest that the WISE method maintains the high spatial resolution of waveform inversion methods while significantly reducing the computational burden.

I. INTRODUCTION

USCT can provide breast sound-speed images, but ray-based methods omit diffraction and waveform inversion is computationally burdensome. The paper develops WISE to reduce this burden while retaining waveform-inversion resolution, with simulation and phantom validation.

  • USCT reconstructs breast sound-speed distributions from wavefield measurements acquired with emitters and receivers.Source: §I · p. 2
  • Ray-based methods can be computationally efficient, but their spatial resolution is limited because diffraction effects are not modeled.Source: §I · p. 2
  • Waveform inversion accounts for higher-order diffraction effects and can produce higher spatial resolution than ray-based methods.Source: §I · p. 3
  • Conventional waveform inversion repeatedly solves the wave equation, creating a large computational burden that has hindered widespread USCT breast-imaging use.Source: §I · p. 3
  • WISE combines emitter pulses and measured wavefields with a random encoding vector, potentially reducing wave-equation solves to as few as twice per iteration.Source: §I · p. 3
  • The study develops WISE for circular-array breast imaging, using stochastic gradient descent, time-domain modeling, and broad-band measurements, then evaluates simulations and breast phantoms.Source: §I · p. 3

II. BACKGROUND: USCT IMAGING MODELS

The USCT imaging model represents acoustic pulses, wave propagation, and receiver sampling to estimate the breast sound-speed distribution. Continuous quantities are discretized for numerical waveform-based reconstruction.

A. USCT imaging model in its continuous form

B. USCT imaging model in its discrete forms

III. WAVEFORM INVERSION WITH SOURCE ENCODING FOR USCT

Sequential waveform inversion estimates sound speed by minimizing data misfit with regularization, but repeatedly solving the wave equation for all acquisitions makes each iteration computationally expensive.

B. Stochastic optimization-based waveform inversion with source encoding (WISE)

WISE reformulates waveform inversion using random source encoding and stochastic optimization, reducing wave-equation solves while targeting the same cost function as sequential inversion.

B. Numerical breast phantom

The numerical breast phantom models heterogeneous breast anatomy and acoustic propagation on a fine Cartesian grid, with complete, incomplete, attenuated, and idealized data generated for evaluation.

5) Generation of noisy data:

The study generates noisy and incomplete reconstruction data, uses GPU-based wave simulation and regularized reconstruction, and compares WISE with sequential and bent-ray methods.

5) Bent-ray image reconstruction:

WISE reconstructions achieved similar accuracy to sequential waveform inversion while requiring far less computation, and preserved waveform inversion’s higher spatial resolution over bent-ray reconstruction. Early iterations showed radial streak artifacts that diminished with further optimization.

C. Images reconstructed from non-attenuated data containing noise

Penalty choice produced a resolution–noise trade-off, while WISE remained computationally faster and tolerated some attenuation-related data inconsistency. The method also reconstructed images from incomplete measurements and was validated using a breast phantom and USCT scanner data.

B. Data pre-processing

Experimental preprocessing removed unreliable channels, filtered the measurements, estimated the excitation pulse, and initialized WISE with a bent-ray reconstruction. The resulting WISE images resolved phantom structures better than bent-ray images, with convergence near 200 iterations.

B. Data pre-processing

  • After discarding 48 bad channels, the experimental dataset contained 976 acquisitions, 976 receivers, and 2112 samples per trace.Source: B. Data pre-processing · p. 20
  • A 4 MHz low-pass filter was applied to calibration and measurement traces to mitigate numerical errors from the wave-equation solver.Source: B. Data pre-processing · p. 20
  • Unreliable receiver measurements near the emitter were replaced with computer-simulated water-bath data to reduce model-mismatch effects.Source: D. Synthesis of combined data · p. 21
  • The WISE image showed improved spatial resolution over the bent-ray reconstruction, with clearly defined boundaries for structures A and B.Source: G. Images reconstructed from… · p. 22
  • Two hundred experimental WISE iterations required approximately 14 hours, compared with an estimated month for comparable sequential waveform inversion.Source: G. Images reconstructed from… · p. 23

VII. SUMMARY

The paper develops WISE to reduce waveform inversion’s computational burden in USCT breast imaging while retaining its spatial-resolution advantages. Simulations and phantom experiments support practical value, but artifacts, model assumptions, initialization sensitivity, and limited comparative analysis remain constraints.

  • WISE encodes measurements with a random vector and estimates sound speed by stochastic optimization using stochastic gradient descent.Source: §VII · p. 23
  • The implementation reduced computation time from weeks to hours in computer-simulation and experimental breast-phantom studies.Source: §VII · p. 23
  • Results suggest that WISE maintains waveform inversion’s advantages while reducing computational burden and holds value for practical USCT breast imaging.Source: §VII · p. 23
  • WISE reconstructions can contain artifacts whose causes are difficult to distinguish between imaging-model errors and local optimization minima.Source: §VII · pp. 23–24
  • The current model omits out-of-plane scattering, finite-aperture effects, and acoustic absorption, while performance depends on initialization, medium heterogeneity, and excitation-pulse profile.Source: §VII · p. 24
  • Systematic statistical comparison with sequential waveform inversion remains limited because the latter requires excessively long computation times.Source: §VII · p. 24

APPENDIX A CONTINUOUS-TO-DISCRETE USCT IMAGING MODEL

The USCT imaging model discretizes receiver measurements into data vectors and maps sound speed distributions to predicted data. It supports both waveform-based and bent-ray reconstruction formulations, while sequential waveform inversion requires repeated wave-equation solves and is computationally demanding.

Continuous-to-discrete model

  • Each acquisition forms a data vector from spatially and temporally sampled receiver measurements, with Nrec receivers and L time samples.Receiver and temporal indices identify each sampled datum, and receiver locations may vary by acquisition.Source: Appendix A · p. 24
  • The USCT imaging model maps a sound speed distribution c(r) to a predicted data vector for comparison with measured data.The predicted data are computed using the current sound speed estimate.Source: Appendix A · p. 25, Appendix B · p. 25

Waveform inversion

  • Waveform inversion gradients use an adjoint-state method, with the data-misfit and regularization derivatives combined for iterative reconstruction.The adjoint wave equation supplies the data-misfit gradient, while quadratic smoothness penalties permit direct regularization gradients.Source: Appendix B · p. 25
  • Sequential waveform inversion requires solving the forward and adjoint wave equations M times to compute its gradient, making it computationally demanding even in 2D.This repeated-solve burden motivates accelerated alternatives such as source-encoded reconstruction.Source: Appendix B · pp. 25–26

Bent-ray reconstruction

  • The bent-ray model reconstructs sound speed iteratively from travel times by ray tracing, system-matrix construction, and optimization over slowness.Slowness is converted to sound speed by taking the element-wise reciprocal after optimization.Source: Appendix C · p. 26
  • Bent-ray reconstruction extracts travel-time data from measured pressure using thresholding and a geometry-based offset before solving for slowness.The threshold is set to 20% of each time trace’s peak value, and the estimated slowness is converted to sound speed.Source: Appendix C · p. 26

FIGURES

The figures document the simulated and experimental USCT setups, excitation and pressure data, reconstruction progress, and comparisons among WISE, sequential waveform inversion, and bent-ray methods. They also vary noise, attenuation, penalties, and iteration counts to examine reconstruction behavior.

System and phantom setup

  • The USCT setup uses a circular array of 256 uniformly distributed transducers, with element 0 emitting while the remaining elements receive signals.The displayed acquisition includes a region of interest within the array’s fan region.Source: Figure 1 · p. 34, FIGURES · p. 34

Excitation and data

  • The excitation-pulse figures report normalized temporal profiles and amplitude spectra for computer simulations and experiments, whose center frequencies are 0.82 MHz and 2.09 MHz, respectively.The experimental pulse estimate differs from the computer-simulated excitation pulse.Source: Figure 3 · p. 36, FIGURES · pp. 36, 50, Figure 17 · p. 50

Reconstruction comparisons

  • WISE reconstruction figures track images, profiles, RMSEs, and cost functions across iteration counts and wave-equation-solver runs.The figures compare WISE with sequential waveform inversion and bent-ray reconstruction under specified penalty settings.Source: Figure 5 · p. 38, Figure 6 · p. 39, Figure 7 · p. 40
  • The comparison figures report that waveform-inversion reconstructions are more accurate and higher resolution than the bent-ray reconstruction, while WISE and sequential waveform inversion have similar RMSE accuracy.The comparison is shown for noise-free non-attenuated data and includes solver-run counts and grayscale windows.Source: FIGURES · p. 38

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