2.1.

Construction of the Model

Acoustic imaging is concerned with interpreting the detected induced or scattered waves from the surface of medium. In linear regime, the governing equation for a homogeneous medium satisfies³¹

Fig. 1

The projection of acoustic wave emanated from a source within the FoV on the surface of a transducer. For every subelement, a delay is applied corresponding to the ${\beta }_{-6\mathrm{dB}}$ diverging angle, to simulate the effect of negative focusing.

Expanding the formula for every voxel within the aperture FOV yields a time-discrete matrix representation of the above formula.^34,35

2.2.

Voxel Crosstalk Matrix

The concept of crosstalk has been established in the assessment of system design, mainly to determine the reliability of the estimable coefficients.³⁸ The reliability in this context is the measure of strength and standalone contribution to individual sources. For a shift variant system such as OPUS, the coefficients are voxels energy. Hence, the effect of aperture in spatial blurring, aliasing, and spatial sensitivity is amenable to an analytic description. Voxel crosstalk matrix can be obtained as follows:

The crosstalk matrix $H$ is essentially a square matrix with $i^{\prime }$th diagonal elements ($H_{ii}$) describing the stand alone contribution of corresponding voxel {spatial sensitivity [Fig. 2(c)]} and off-diagonal elements, $H_{ij};i\ne j$, representing the parasitic contribution of neighboring voxels, known as spatial aliasing. The value of $H_{ij}^2/H_{ij}{H}_{ii}$ indicates the resolvability and consequently spatial aliasing between any two voxels. Therefore, a better imaging system is the one with weaker off-diagonal and stronger diagonal elements. The quantitative evaluation is possible through a set of metrics for systematic evaluation of the system voxel crosstalk matrix.³⁷ The utilized metrics in this context are the root mean square error (RMSE) and mean absolute error (MAE). These metrics are complementary to each other, providing a robust systematic approach in quantitative analysis of imaging performance. Together they render information about the variance and degree of error in a model.^39,40

Fig. 2

(a) The transfer function $h^{\mathrm{IR}}$ correlates the pair of source-element for every 200 control points and 128 elements of annular array. (b) The corresponding voxel crosstalk matrix and (c) the spatial sensitivity of system obtained from the diagonal elements. (d) Spatial Fourier transform of eigenvectors of model matrix and (e) the eigenvalue distribution. For every eigenvector bound to a nonzero eigenvalue, there is a spatial frequency. The dashed line segregated the $M_{\text{prime}}$ from $M_{\text{null}}$.

2.3.

Eigenanalysis

As the name implies, eigendecomposition of the square matrix $H$ features prominently the principal characteristics of the matrix by delivering the basis of eigenvectors $q$ scaled by scalar eigenvalues $\lambda$.

$H$ is a Hermitian matrix with real and positive eigenvalues and linearly independent orthogonal eigenvectors $q$, $Q$ (square matrix) is the projection of $H$ in the eigenspace with eigenvectors along its columns, and $\mathrm{\Lambda }$ is a diagonal matrix containing eigenvalues ${\lambda }_i$ along its diagonal such that ${\lambda }_1>{\lambda }_2>\cdots >{\lambda }_n$. The eigendecomposition can be considered as an expansion of the matrix with respect to its basis (i.e., eigenvectors), each with certain weighting coefficient (i.e., eigenvalues). Alternatively, eigenvectors of $H$ can be calculated by singular value decomposition (SVD) of matrix $M$, given by $M=U\mathrm{\Sigma }V$. From the mathematical point of view, $V=Q$ and $\mathrm{\Sigma }$ is a diagonal matrix containing singular values ${\sigma }_i$ along its diagonal such that ${\sigma }_i=\sqrt{{\lambda }_i}$. Regardless, the matrix $H$ can be represented simply as weighted and ordered combination of basis. The eigenvectors of the matrix $H$ depict the principal structures of the detected wave components, reserving the original properties of the original matrix.³²

Notably, the matrix $M$ is a spatiotemporal matrix, therefore SVD decomposes $M$ into the temporal matrix $U$ and spatial matrix $V$. In other words, $V$ is composing of $K$ columns equal to the number of voxels while $U$ is containing $J$ columns equal to the number of recorded samples. Therefore, the frequency distribution of $V$ has physical interpretation w.r.t. the sensitivity of system to certain spatial frequencies for the given voxels. This distribution would enable to extract the primary space $M_{\text{prime}}$ (major components) from the null space $M_{\text{null}}$ (e.g., noise).^26,41 In other words, distinguishing the strengths and deficiencies of these systems. The value of ${\sigma }_i$ can be linked to the energy level or stability parameters of these components. In principle, the eigenvalues (or similarly singular values) are providing a measure of fidelity of these components such that the $i^{\prime }$th eigenvalue exemplifies the $i^{\prime }$th eigenvector’s strength. In general, the most salient features of the matrix are stored in the first few eigenvectors ($M_{\text{prime}}$) where ${\sigma }_i>0$. This boundary is referred to as the rank of matrix $\kappa$. Meanwhile, the sum of the eigenvalues or the nuclear norm of the matrix represents the energy of the matrix. Thus, the spectrum of eigenvalues, which is a generalized form of the Fourier power spectrum, allows discrimination between the primary and null space [Figs. 2(d) and 2(f)].

The system with larger $\kappa$ is imputed to the superior focusing competence²⁶ and more accurate set of acquired data. Eigenvalues with the associated eigenvectors are pointing out the maximum sensitivity of the system toward the direction of incoming wave, hence specifying the object features that are more accurately estimated. For a spatially variant system, this can be estimated by the Fourier of eigenvectors as well. Moreover, these vectors are containing the information about the missing projections, which implies the partial view angle [Fig. 2(d)] and resolvability of the spatial frequency component, hence redundancy of projections.

Subsequently, eigenanalysis estimates whether the formed image misses these angular frequencies, even if OPUS is an ideal reflection mode imaging system (i.e., infinite detectors). The quality and quantity of angular frequencies that an aperture can generate or similarly receive is contained in the model matrix [Fig. 2(d)]. Therefore, eigenanalysis provides a simple but credible mechanism for the comparative appraisal between two or more imaging systems.

2.4.

Array Design

By taking into account the light delivery housing for perpendicular illumination, we are exploring the optimum geometrical properties for the OPUS handheld probe. The design array supposedly provides enough flexibility in elements’ geometry and arrangement such that the periodicity is minimized and the agreement among directivity, sensitivity, and spatial aliasing can be settled. To begin with, segmented annular array of point-like transducers is considered. The elements are distributed equidistantly in a ring fashion around the optical probes, enabling real-time volumetric acquisition. The 11-mm $\oslash$ lumen is considered for the optical probe, which is in agreement with the limited 128 elements of $1\lambda$ width for the central frequency of 5 MHz with 80% fractional bandwidth.²³ Indeed, the transducer frequency response is of crucial importance and has a determinant role in the so called relative spatial resolution; defined as the ratio of the penetration limit to the depth resolution. The aforementioned frequency response would allow the detection of OA sources as deep as 38 mm ex vivo and 20 mm in vivo,^42,43 ideally down to the size of 0.3 mm.⁴⁴

First, we explore the effect of element number, which has a reciprocal relationship to the interelement spacing. The optimization is pursued for the element size and geometrical distribution. The directive angular view ($\beta$) for the element edge $a$ (height or width) is compensated by the negative focusing such that ${\beta }_{-6\mathrm{dB}}=2\arcsin (0.6{\lambda }_c/a)$,⁴⁵ for ${\lambda }_c$ is the wavelength corresponding to the central frequency. We simulated negative focusing for every element in the convex structure as if the focus is behind the transducer (Fig. 1). For the numerical calculation and in silico studies we found the well-known FieldII simulations toolbox⁴⁶ convenient.

3.1.

Voxel Crosstalk Matrix

Herein, the result of the voxel crosstalk analysis that estimates the spatial sensitivity and spatial aliasing for the given model matrix is presented. Figure 3 shows the effect of element number for four annular arrays composed of 32, 64, 128, and 256 point-like elements ($0.5\lambda$ size) and an array of $128\lambda$-width elements. Three axial slices within the FoV of $18\times 18\times 20{\mathrm{mm}}^3$ has been visualized, each composed of $100\times 100$ voxels. The illustrated $x\text{-}y$ planes are parallel to the surface of the aperture, at the depth of, respectively, 1, 2, and 3 cm away from the aperture [Fig. 3(I)]. The spatial aliasing for the most and least sensitive voxels in the FoV is provided in Fig. 3(II). Presented in Fig. 3(III), the two metrics of RMSE and MAE are negatively oriented scores and their parallel variation in the same order suggests the existence of many small errors. The variance of the errors is corresponding to the absolute difference between the values of two metrics.³⁹ The qualitative and quantitative analyses indicate the improvement in sensitivity and aliasing with the increase in transducer counts. However, for the case where the overall active area is the same, the aperture with larger elements showcases a better performance.

Fig. 3

Crosstalk analysis on the effect of element number for segmented annular array. I: 60 dB logarithmic scale (w.r.t. the pressure at the surface of transducers) spatial sensitivity. II: Parasitic contribution to the central voxels of the plane at $z=1\mathrm{cm}$ (upper row) and marginal voxels (lower row) of the plane at $z=3\mathrm{cm}$. III: Quantitative metrics.

Figure 4 presents the results of voxel crosstalk analysis w.r.t. the height of the transducers while the interelement spacing is limited to $\lambda$. The larger elements give rise to higher sensitivity and lesser spatial aliasing. More voxels are contributing to the parasitic signals but in lesser values, which also means lesser sidelobe value. The quantitative analysis confirms the qualitative interpretation [Fig. 4(III)]. As the size of element increases, the values of both metrics decrease but the difference between them increases. Due to its point-like transducers, the aperture with $0.5\lambda$ element’s height²² is exempt and here is set as the reference for quantitative analysis.

Fig. 4

Crosstalk analysis on the effect of element height for segmented annular array. I: 60 dB logarithmic scale spatial sensitivity. II: Parasitic contribution to the central voxel of the plane at $z=1\mathrm{cm}$ (upper row) and marginal voxel (lower row) of the plane at $z=3\mathrm{cm}$. III: Quantitative metrics.

In pursuit of an aperture with lesser periodicity^5,18 (i.e., aliasing), Fig. 5 compares the result of voxel crosstalk analysis for three virtual apertures with rather same element size but different arrangements. The performance of segmented annular array of $1\lambda \times 20\lambda$ element size is shown alongside the annular circular array and annular spiral array of $5\lambda \times 5\lambda$ element size [Fig. 5(IV)]. For the last two, the uniform distribution of energy over the FoV is expected due to the rather sparser geometry. Yet, the lack of uniform overlap among the elements’ view angle costs the energy in the central area, shown in Fig. 5(I). Figure 5(III) outlines the quantitative analysis for the three apertures. The segmented annular array showcases lesser errors and lesser difference between RMSE and MAE, an indication for lesser variance in the error.³⁹

Fig. 5

Qualitative and quantitative interpretation of crosstalk analysis on the effect of elements arrangement. I: 60 dB logarithmic scale spatial sensitivity. II: Parasitic contribution to the central voxel of the plane at $z=1\mathrm{cm}$ (upper row) and marginal voxel (lower row) of the plane at $z=3\mathrm{cm}$. III: Quantitative metrics. IV: Array geometries.

3.2.

Eigenanalysis

In this section, we summarize the effect of aperture partial view on the information loss, using eigenanalysis. The system comparison depends both on the number of effective components and the structure of the singular vectors. As the arrays are circularly symmetrical, we speculate the variations over four 18 mm lines of 200 control points at four different depths. The control points lie on a plane parallel to the aperture. Figure 6 shows the spatial Fourier transform of the right-hand side singular vectors, for four axial planes away from the surface of aperture, respectively, at 1, 10, 20, and 30 mm.

Fig. 6

The spatial Fourier transform of eigenvectors of the model matrix for 200 control points along the central line at four different axial planes. The white-dashed line indicates the borderline between the $M_{\text{prime}}$ and $M_{\text{null}}$.

In general, there are less available spectral components with the depth (columns), recalling that OPUS is a spatially variant imaging system. For the case where the object plane is near the aperture (first row of Fig. 6), the lower spatial frequencies are associated with discrepancies due to the limited directivity of large elements (the left side of the first white-dashed line). On the other side, the higher frequency components in the right side of the second dashed line are not containing information but noise. The rather meaningful information is limited to the area between the two dashed lines, thus less sensitivity to generated OA clutters.⁴⁷ For the other planes, where the control points are farther from the aperture, the lower angular frequency components are stronger and contain resolvable structures. The separating border between the noise and meaningful data is highlighted by the white-dashed line. The adjacent number is pinpointing the turning point of which segregates the resolvable and unresolvable spectral information. One can notice that for the segmented annular array, the larger element, hence the larger aperture is accompanied by more and finer definition of the frequency components where each meaningful singular vector is associated with narrower spectrum.

By analogy to the segmented annular array, both annular circular and annular spiral arrays are containing more robust DC components for the axial planes in the far-field ($\ge 1\mathrm{cm}$). However, the number of valuable components (marked next to the dashed line) remains comparative. In addition, the rippled arms of annular circular and annular spiral arrays are indicating less resolvability for the frequency components and more aliasing. This is not observed in the segmented annular array where the thinner arms are implying a better resolvability for frequency components. Therefore, the imaging performance of segmented annular array can outperform the other layouts, for both OA and ultrasound imaging.