PaperlayerView exact revision on arXiv

Source-linked AI summary

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.

  • Method

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

  • Results

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

  • Takeaways & Limitations

    WISE shows potential value for practical USCT breast imaging applications.

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.
  • Ray-based methods can be computationally efficient, but their spatial resolution is limited because diffraction effects are not modeled.
  • Waveform inversion accounts for higher-order diffraction effects and can produce higher spatial resolution than ray-based methods.
  • Conventional waveform inversion repeatedly solves the wave equation, creating a large computational burden that has hindered widespread USCT breast-imaging use.
  • WISE combines emitter pulses and measured wavefields with a random encoding vector, potentially reducing wave-equation solves to as few as twice per iteration.
  • 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.

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: The continuous imaging model maps each source pulse through the acoustic wave operator and restricts the resulting pressure field to the measurement aperture.
  • A. USCT imaging model in its continuous form: The continuous USCT problem estimates the sound-speed distribution c(r) from acoustic pulses and recorded pressure wavefields.
  • B. USCT imaging model in its discrete forms: A discrete-to-discrete model represents the sound speed and source pulses as finite-dimensional vectors with spatial and temporal samples.
  • B. USCT imaging model in its discrete forms: The pseudospectral k-space method numerically propagates the discretized fields on Cartesian grid points.
  • B. USCT imaging model in its discrete forms: A sampling matrix extracts predicted pressure data at receiver locations to model the USCT acquisition process.

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.

  • Sequential waveform inversion minimizes a regularized nonlinear objective consisting of summed squared ℓ2 data residuals and a penalty term.The objective uses data fidelity F(c), regularization R(c), and parameter β.
  • The method accumulates gradients from all M data acquisitions before updating the sound speed estimate once per iteration.
  • Unlike sequential waveform inversion, Kaczmarz reconstruction updates the sound speed estimate multiple times within one algorithmic iteration.
  • Hc is the computationally burdensome wave-equation solver and appears in forward, adjoint, and line-search calculations.
  • At least (2M + 1) wave-equation solver runs are required per iteration, with line search generally requiring additional runs.

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.

  • WISE addresses the large computational burden of sequential waveform inversion by reformulating data fidelity as an expectation over a random encoding vector.
  • The encoded formulation is mathematically equivalent to the conventional formulation when the encoding vector has zero mean and identity covariance.
  • The WISE method seeks to minimize the same cost function as sequential waveform inversion while using encoded data and sources.
  • WISE solves the resulting stochastic optimization problem using stochastic gradient descent with independently drawn Rademacher encoding elements.
  • 3 wave-equation solver runs are the per-iteration lower bound for WISE, compared with (2M + 1) for sequential waveform inversion.
  • Two-dimensional computer simulations were conducted with a circular 256-transducer USCT geometry to demonstrate WISE's computational advantage.

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.

  • The 98 mm numerical phantom contains eight structures representing adipose tissue, parenchyma, cysts, benign tumors, and malignant tumors.
  • Acoustic attenuation is modeled by a power law with fixed exponent y = 1.5, alongside spatially varying sound speed and attenuation slope distributions.
  • The phantom is sampled on a uniform Cartesian grid with spacing ∆s = 0.25 mm, and the finest structure has diameter 3.75 mm.
  • Wave propagation is simulated in acoustically absorbing media using three coupled first-order partial differential equations and a pseudospectral k-space solver.
  • A complete data set contains M = 256 sequential acquisitions, while incomplete acquisitions retain Nrec = 100 receivers opposite each emitter.
  • Incomplete data emulate unreliable near-emitter receivers and rotating arc-shaped transducer arrays, making image reconstruction challenging.

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.

  • Electronic measurement noise is modeled as additive Gaussian white noise with standard deviation set to 5% of a reference signal amplitude.
  • Reconstruction wave propagation uses a second-order pseudospectral k-space method on a 1024×1024 grid with 0.5 mm spacing.
  • One reconstruction forward simulation requires approximately 7 seconds on the stated GPU platform.
  • Sequential waveform inversion was evaluated only preliminarily on noise-free, non-attenuated data because of its extreme computational burden.
  • WISE reconstructions use quadratic and total-variation penalties, whose effects were explored by sweeping regularization parameters.
  • Two completion strategies fill missing measurements using homogeneous-medium pressure data or simple replacement, while the first assumes weak backscatter in a matched water bath.

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.

  • WISE and sequential waveform inversion produced more accurate, higher-resolution images than bent-ray reconstruction.
  • WISE and sequential waveform inversion achieved similar RMSEs of 1.08 × 10^-3 and 1.19 × 10^-3, respectively.
  • WISE required 1.4 hours versus 81.4 hours for sequential waveform inversion, using 1018 versus 57088 wave equation solver runs.
  • Radial streak artifacts appeared before 100 iterations and were mitigated with additional iterations, improving quantitative accuracy.
  • WISE required more iterations to reach the same RMSE, but reduced computation time by approximately two orders of magnitude.

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.

  • A quadratic penalty reduced image noise at the expense of spatial resolution, whereas a TV penalty retained similar resolution with lower noise.
  • WISE with penalties reduced computation time by approximately two orders of magnitude relative to sequential waveform inversion.
  • The current WISE implementation assumes an absorption-free acoustic medium, although TV-penalized reconstructions tolerated attenuation-related inconsistencies for feature detection to a certain level.
  • Filling missing measurements with water-bath pressure data enabled highly accurate reconstruction, whereas zero-filled data produced strong artifacts.
  • Experimental data came from a breast phantom containing benign and cancerous masses embedded in glandular tissue with a subcutaneous fat layer.

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.
  • B. Data pre-processing: A 4 MHz low-pass filter was applied to calibration and measurement traces to mitigate numerical errors from the wave-equation solver.
  • B. Data pre-processing: Unreliable receiver measurements near the emitter were replaced with computer-simulated water-bath data to reduce model-mismatch effects.
  • B. Data pre-processing: The WISE image showed improved spatial resolution over the bent-ray reconstruction, with clearly defined boundaries for structures A and B.
  • B. Data pre-processing: Two hundred experimental WISE iterations required approximately 14 hours, compared with an estimated month for comparable sequential waveform inversion.

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.
  • The implementation reduced computation time from weeks to hours in computer-simulation and experimental breast-phantom studies.
  • Results suggest that WISE maintains waveform inversion’s advantages while reducing computational burden and holds value for practical USCT breast imaging.
  • WISE reconstructions can contain artifacts whose causes are difficult to distinguish between imaging-model errors and local optimization minima.
  • 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.
  • Systematic statistical comparison with sequential waveform inversion remains limited because the latter requires excessively long computation times.

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.
  • Continuous-to-discrete model: 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.
  • 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.
  • Waveform inversion: 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.
  • 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.
  • Bent-ray reconstruction: 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.

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.
  • 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.
  • 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.
  • Reconstruction comparisons: 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.
Loading 1501.00021v1…