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Laplace-Domain Beamforming for Ultrafast Plane-Wave Imaging
Martin F. Schiffner
TL;DR
Linear-array Fourier-domain beamforming relies on two-dimensional, lossless models that underestimate attenuation. Laplace-domain beamforming generalizes the Fourier diffraction theorem to account for three-dimensional diffraction, finite element heights, and tissue loss. In a phantom experiment, it improved uniformity, resolution, and contrast compared with FBP.
Problem
Fourier-domain methods for linear arrays rely on two-dimensional, lossless models that underestimate attenuation, despite attenuation limiting imaging depth and degrading resolution with depth.
Source: §I · p. 2Method
Laplace-domain beamforming generalizes the Fourier diffraction theorem to model three dimensions, finite array-element heights, and lossy tissue.
Source: §I · p. 2Results
Compared with FBP, wire lateral and axial FWHMs fell by up to 12% and 25%, respectively, and anechoic-region gCNR improved by up to 9%.
Source: §IV · p. 5Takeaways & Limitations
The proposed method improves image uniformity, resolution, and contrast while retaining the computational efficiency of Fourier-domain methods.
Source: §I · p. 2, §IV · p. 5
Abstract
from arXiv · showhide
Fourier-domain beamforming methods, such as filtered backpropagation (FBP), are beneficial in ultrafast plane-wave imaging due to their computational efficiency and high image quality. These methods use the Fourier diffraction theorem (FDT) for steered plane waves to exactly invert linear scattering models. However, these models rely on simplifying assumptions that degrade image quality. The use of linear transducer arrays, in particular, requires: (i) a two-dimensional space, and (ii) lossless tissues. Herein, Laplace-domain beamforming is proposed. This method retains the computational efficiency of the Fourier-domain methods while improving image quality for linear arrays. Specifically, the method uses a generalization of the FDT, here called Laplace diffraction theorem, to account for three spatial dimensions, finite array element heights, and lossy tissues. A phantom experiment showed that the proposed method improves image uniformity, spatial resolution, and contrast compared to FBP. Using three steering angles, the lateral and axial -6 dB-widths of wires reduced by up to 12% and 25%, respectively, while the contrast of anechoic regions improved by up to 9%.
I. INTRODUCTION
Fourier-domain beamforming efficiently inverts linear-scattering models, but the assumptions used for linear arrays—two-dimensional space and lossless tissue—underestimate attenuation. Laplace-domain beamforming generalizes the Fourier diffraction theorem to model three dimensions, finite element heights, and tissue loss.
I. INTRODUCTION
- Linear-array Fourier-domain models assume two-dimensional space and lossless tissue, underestimating attenuation and degrading image quality.The two-dimensional model omits elevational variations and assumes effectively infinite element and tissue-fluctuation heights; the lossless model omits absorption and dispersion.Source: §I · p. 2
- Laplace-domain beamforming retains Fourier-domain computational efficiency while accounting for three dimensions, finite element heights, and lossy tissue.These changes generalize the Fourier diffraction theorem as the Laplace diffraction theorem.Source: §I · p. 2
- The Laplace diffraction theorem relates scattered-field transverse spectra to tissue properties at complex frequencies, yielding a linear system solvable with standard regularization.The tissue properties in the field of view are modeled as a Fourier sum.Source: §I · p. 3
II. THEORY
The theory models a linear array imaging a heterogeneous object in a lossy three-dimensional half-space. Its incident field is a steered plane wave whose reference position is selected to ensure causal emission.
II. THEORY
- The model places a heterogeneous object in a lossy fluid occupying the positive half-space, with a linear array in an infinite rigid baffle at z = 0.The derivation uses three-dimensional positions and finite array-element height.Source: §II · p. 3
- The derivation assumes bandpass-bandlimited acoustic fields, with energy confined to frequencies between flb and fub.Unless indicated otherwise, the theory considers a single frequency and omits the frequency argument.Source: §II · p. 3
A. Incident Wave
- The incident field is a steered plane wave with steering angle ϑ ∈ (0, π).Its pressure field is specified using a complex amplitude, complex wavenumber, propagation direction, and reference position.Source: A. Incident Wave · p. 3
- The reference position’s lateral component is placed at the corresponding edge of the array aperture to ensure causal wave emission.The sign of the lateral component changes for steering angles at or below π/2 versus above π/2.Source: A. Incident Wave · p. 3
B. Laplace Diffraction Theorem for Steered Plane Waves
The Laplace diffraction theorem characterizes which object-function frequencies each pulse-echo measurement captures. It links scattered pressure on the array aperture to tissue properties through complex angular spatial frequencies and a spatial Laplace transform.
B. Laplace Diffraction Theorem for Steered Plane Waves
- Each pulse-echo measurement acts as a bandpass filter for the object function, with the Born-approximated scattered field carrying information about specific complex angular spatial frequencies.The scattered pressure field is considered on the aperture plane, z = 0.Source: B. Laplace Diffraction Theorem for… · p. 3, Figure 2 · p. 3
- For a sequence of three steered plane waves across the frequency band, the complex angular spatial frequencies form the illustrated combined structures.The figure describes the real and imaginary parts of those frequencies separately.Source: Figure 2 · p. 3
- The scattered field is expanded as a superposition of steered plane waves, whose real transverse frequencies determine a complex axial frequency.Plane waves with large transverse frequencies are evanescent and may be neglected; those within the disk of radius σ are propagable.Source: B. Laplace Diffraction Theorem for… · p. 3
- The expansion relates the scattered field to the spatial Laplace transform of the object function.The complex angular spatial frequencies are illustrated by a line in the real-frequency plane and a hemisphere-like imaginary component for one incident wave.Source: Figure 2 · p. 3, B. Laplace Diffraction Theorem for… · p. 4
C. Element Height
Finite element height is incorporated by integrating the scattered pressure over the elevational interval. The resulting integral is expressed as an inverse Fourier transform.
C. Element Height
- The scattered acoustic pressure is integrated over y ∈ [−he/2; he/2] to incorporate element height he.Source: C. Element Height · p. 4
- The elevational integral equals an inverse Fourier transform.Source: C. Element Height · p. 4
- The expression defines ky,lb and ky,ub as opposite limits and uses sinc(x) = sin(x)/x.Source: C. Element Height · p. 4
D. Spatial Discretization
The object function in the bounded field of view is approximated by a Fourier sum. Its passband-dependent index bounds determine the number of coefficients.
D. Spatial Discretization
- The object function inside the field of view is approximated by a Fourier sum because the field of view and passbands are bounded.Source: D. Spatial Discretization · p. 4
- The field of view is box-shaped and transversally centered in front of the array, with specified lateral and axial bounds and an elevational slice thickness.Source: D. Spatial Discretization · p. 4
- Passband-dependent lateral and axial index bounds set the number of Fourier coefficients Nc.Source: D. Spatial Discretization · p. 4
1) Laplace Transform:
The Laplace transform expression is obtained by inserting the Fourier sum. The accompanying notation defines the field-of-view volume and axial center.
1) Laplace Transform:
- Inserting the Fourier sum into the Laplace transform yields its discretized expression.Source: 1) Laplace Transform · p. 4
- The field-of-view volume is ∆V = Lx∆yLz.Source: 1) Laplace Transform · p. 4
- The axial center is zc = zlb+Lz/2, and sinhc is the hyperbolic version of sinc.Source: 1) Laplace Transform · p. 4
2) Lateral Spectrum:
The lateral-spectrum expression is obtained by inserting the discretized Laplace transform.
2) Lateral Spectrum:
- Inserting the discretized Laplace transform (9) into the lateral spectrum (7) gives the resulting spectrum expression.Source: 2) Lateral Spectrum · p. 4
E. Image Formation by Inversion
Image formation estimates Fourier-sum coefficients from a discretized lateral spectrum. The resulting linear system is ill-conditioned and is regularized to account for model mismatch and noise.
E. Image Formation by Inversion
- Image coefficients ξν,µ are computed from the discretized lateral spectrum across temporal-frequency and lateral-frequency samples.The spectrum combines Nr discrete (f, kx) combinations across multiple pulse-echo measurements.Source: E. Image Formation by Inversion · p. 4
- The Nr × Nc system matrix contains products of coefficients (10b) and (10c) and is ill-conditioned.Source: E. Image Formation by Inversion · p. 4
- The optimization estimates Fourier coefficients ξ̂ from measured lateral-spectrum samples b, with ϵ accounting for model mismatch and noise.Source: E. Image Formation by Inversion · p. 4
III. METHODS
The methods compare beamforming approaches using a multi-tissue phantom scan and assess image quality through wire widths and anechoic-region contrast.
III. METHODS
- A commercial multi-tissue phantom was scanned offline with a linear array using plane waves steered at 85°, 90°, and 95°.The excitation was a single cycle at 4 MHz, with frequency bounds of 2.25–6.75 MHz.Source: §III · p. 4
A. Image Formation
- Image formation compared FBP with four optimization-based combinations of wavenumber and spatial dimensionality.Laplace-domain variants used the complex time-causal wavenumber; Fourier-domain variants used the standard real wavenumber.Source: A. Image Formation · p. 4
- Only up to 0.3% of the 283,197 × 237,760 system matrix entries were relevant, enabling efficient matrix-vector products.The stored matrix occupied 4.9 GB.Source: A. Image Formation · p. 5
B. Evaluation Metrics
- Evaluation measured wire full widths at half maximum and generalized contrast-to-noise ratios for anechoic regions.Table I reports the wire FWHMs and anechoic-region gCNRs.Source: B. Evaluation Metrics · p. 5, Table I · p. 5
IV. RESULTS
In the phantom experiment, Laplace-domain beamforming improved image quality relative to FBP. The conclusion attributes the approach's scope to modeling three-dimensional diffraction and power-law absorption while retaining computational efficiency.
IV. RESULTS
- Median lateral and axial FWHMs fell by up to 6.4% and 10.8%, respectively; the top-right wires showed reductions of 12% and 25%.Anechoic-region gCNRs improved by up to 9%.Source: §IV · p. 5
- With dimensionality fixed, the complex wavenumber improved median axial FWHMs and large-region gCNR; three dimensions helped when paired with that wavenumber.Source: §IV · p. 5
- Compared with FBP, lateral and axial resolution improved by up to 12% and 25%, respectively, while contrast improved by up to 9%.Source: §V · p. 5
V. CONCLUSION
- The proposed method accounts for three-dimensional diffraction and power-law absorption while retaining Fourier-domain methods' computational efficiency.Detailed theory, including computation of axial coefficients (10c), was left to another publication.Source: §V · p. 5