Late return focusing algorithm for circular synthetic aperture sonar data Open Access

Sonar and radar systems are used in remote sensing applications to characterize the environment and to identify objects. The sensor platforms travel in a predefined (in many instances nominally linear or circular) track and the measurements are made at a regular spatial interval along this track (Soumekh, 1999). Both linear and circular synthetic aperture collection geometries are utilized in radar or sonar surveys. Typically, linear collection is used for wide area coverage, and circular collection is used for detailed investigation. Circular synthetic aperture sonar (CSAS) provides full azimuthal coverage of an object while linear synthetic aperture sonar (SAS) collection has limited azimuthal coverage (Kennedy and Marston, 2018; Marston et al., 2011). The raw measurements are processed into various data representations such as imagery for the human operator or scientists to extract information from and interpret them.

A spatial image representation, due to its similarity with optical photographs, is intuitively useful for human interpretation. Objects that appear in spatial imagery with bright pixels create sharp edges with high contrast to the background, which allows for easier extraction of shape features. Automatic object detection and classification algorithms utilize specific spatial organization of high-intensity pixels. When recognizable shape features are sparsely isolated in the imaging scene without interference from other objects that are not of interest, or clutter, it is straightforward to detect and localize them.

Image formation models and algorithms are used to estimate the scattering cross section of the location being imaged (Hansen and Kolev, 2011; Hawkins, 1996). The primary scattering mechanism that conforms to the image formation model is geometric scattering. This initial geometric response takes the shortest path to return to the sensor, and is typically the strongest. Other components, e.g., multiple scattering and resonant scattering, return later in time relative to the initial geometric scattering due to additional propagation paths. The mismatch between the inversion model and data causes this late arriving energy to appear out of focus when it is present. In this paper, it will be demonstrated that additional processing steps can help focus these late return responses.

The remainder of this paper will discuss traditional SAS image reconstruction algorithms and the simplistic target scattering assumptions they exploit. Following a discussion of wave scattering mechanisms, we will introduce a complementary image formation algorithm that can improve the focus of acoustic energy that would not appear in-focus using traditional imaging algorithms. Finally, this technique will be demonstrated using data collected from an in-air circular SAS laboratory experiment.

A standard approach to synthetic aperture image reconstruction is to spatio-temporally organize the recorded data from a series of transmissions to form a grid of pixels for the survey scene, or the imaging scene. This approach, implemented in the time domain, is known as “delay-and-sum” or “backprojection” reconstruction. For a set of N signals recorded by a monostatic sensor, the image estimate for a pixel at a position (x, y) is given by

where $pn(t)$ is the signal recorded on the nth measurement, r(x, y) is the pixel position in the scene, and $rn′$ is the sensor position over the N transmissions (Hansen and Kolev, 2011). The additional term τ is included to introduce a fixed temporal offset for all pixels reconstructed in the scene, and in the case of standard backprojection τ = 0. Cases where $τ≠0$ will be discussed later in the paper.

Consider a circular collection geometry with fixed sensors and rotating imaging scene on a turntable. With the origin at the center of the turntable, let an ideal point scatterer located at (x, y), or $(ρ cos ϕ,ρ sin ϕ)$ in polar coordinates. As the table rotates by θ in the same direction as $ϕ$⁠, the total distance from the transmitter to the scatterer and back to the co-located receiver changes sinusoidally as

where H is the height of the co-located sensors, and R is the ground distance between the sensors and the center of the turntable. Each of the point scatterers in the image scene exhibits this sinusoidal distance in the raw data, and the image formation algorithm can be thought of as integrating the scattering energy over different paths of sinusoidal propagation distance, to estimate each of the pixel intensity in f(x, y) as in Eq. (1).

The image reconstruction algorithm described in Eq. (1) inverts a forward model that includes only geometric, single scattering. This algorithm excludes multiple scattering interactions and non-geometric (e.g., elastic) scattering. This model is equivalent to assuming geometric-only scattering under the first Born approximation, which is accurate when the scattered field is small compared to the incident field and ignores multiple scattering (Born and Wolf, 1999).

It is important to note that these underlying data model assumptions are not unique to the delay-and-sum algorithm described by Eq. (1). The same approximations are also made by a number of other reconstruction techniques such as the $ω−k$ (Hawkins, 1996) algorithm and the polar format algorithm (PFA) (Doerry, 2012; Ferguson and Wyber, 2005, 2009). In particular, PFA is used for data collected from a circular geometry, in which the raw time series is transformed with 1D Fourier transform into the $(θ,f)$ domain, then interpolated into the spatial wavenumber domain (k_x, k_y), followed by a 2D inverse Fourier transform to reconstruct the estimated image, f(x, y). In this paper PFA with bilinear interpolation yielded similar image quality as the time domain backprojection method, and allowed for significant speed advantage, which made it practical to reconstruct images with many values of τ in Eq. (1).

While the time domain backprojection method can naturally handle the imaging scene being in the nearfield of the synthetic aperture, wavenumber domain-based methods are based on the farfield assumption. Therefore, a Hankel function based nearfield to farfield conversion algorithm is applied to the $(kθ,f)$ domain data before interpolating the data into the spatial wavenumber domain, (k_x, k_y), as described in detail in Plotnick et al. (2014).

For many real scenes, the backscattered signal recorded by the sensor will include a range of scattering mechanisms, including those not considered in the standard image reconstruction techniques. Those field components that are described by the propagation and scattering model inverted by Eq. (1) are accurately reconstructed and produce well-focused energy. Geometric single scattering typically represents the dominant source of energy in the scattered field; therefore, this resulting imagery is adequate for most purposes. Kargl, Williams, and Thorsos provide a thorough discussion of modeling and imaging simple target near boundaries where both multiple scattering and elastic scattering produce visible imaging artifacts (Kargl et al., 2012). The multiple and/or non-geometric scattering mechanisms are delayed in time due to additional propagation paths compared to direct backscattering off of the surface of the object or scene (Plotnick and Marston, 2016; Williams and Marston, 1985). Those additional field components appear in the image as defocused energy, which degrades the apparent focus quality of the image.

There are several mechanisms that are associated with additional propagation paths relative to the initial surface scattering point, which include multiple scattering, acoustic coupling, and subsequent surface wave propagation and re-radiation. Several processing approaches have been applied to SAS data from linear collection geometries in order to improve the focus of some of these phenomena. For example, fixed focusing algorithm enhanced the shadow quality (Groen et al., 2009), and quasi-holographic processing and coherence-based wavelet shrinkage method have shown to improve elastic responses (Hunter and van Vossen, 2014; Plotnick and Marston, 2016; Zartman et al., 2013). Range-specific fixed focusing methods can significantly improve the focus of shadow and resonant response for an extended time, but the focus quality at other ranges are degraded. In this work it is shown that for a circular collection geometry, the application of a simple bulk delay can produce well-focused late returns regardless of the spatial location within the imaging scene at a specific τ value, while the initially focused scatterers with τ = 0 go out of focus everywhere, as will be shown later in Fig. 2 and in the supplementary media.¹ The example investigated in this paper is internal multiple scattering of an object that has a cavity, in which the additional propagation times are relatively consistent over a range of incident angles.

A laboratory scale in-air measurement apparatus has been established as a cost-effective alternative to underwater surveys that allows for controlled and repeated experiments for algorithms development and investigation of object scattering (Blanford et al., 2019a; Blanford et al., 2019b). Underwater circular rail-based experiments (Kennedy et al., 2014) or outdoor ground-based efforts for synthetic aperture radar (SAR) (Cohen et al., 1998) add more practical value, but at the expense of time and cost. This measurement setup is comprised of a co-located transmitter and receiver directed at a turntable that can be rotated at a desired increment operated by a stepper motor. This emulates making measurements from a circular geometry.

The operating frequency range used for the measurements is 10–30 kHz and the data are digitized at a sample rate of fs = 100 kHz. The ground distance between the center of the imaging scene and the sensors is 0.85 m, and the vertical distances are 0.224 m, 0.184 m for the transmitter and the receiver, respectively. The system is housed in an indoor anechoic chamber with a typical temperature of 20 °C. Assuming a nominal sound speed of c = 343 m/s, the acoustic wavelength at the center frequency, fc = 20 kHz, is $λ=c/fc=1.715$ cm. The extent of the imaging scene is D = 60 cm and the far-field of the scene is established beyond the Fraunhofer distance $r=2D2/λ≈42$ m, which is much greater than the range from the sensors to the imaging scene center, approximately 90 cm. Since the data collection geometry violates the farfield assumption made by the standard image reconstruction algorithms, a nearfield to farfield conversion technique (Plotnick et al., 2014) was applied to the data prior to image formation with the polar format algorithm.

Circular SAS data were collected at 0.5° angular increments on an empty 1 liter plastic soda bottle—once with the bottle capped and again with the cap removed, with the bottle opening facing the sensors at 165°. The bottle is 28 cm long from mouth to base and 7.5 cm at the widest point. With the cap on, the bottle exhibits only geometric scattering from the exterior. The open bottle, however, allowed for the incident wave to travel through the neck of the bottle, reflect inside the bottle, then re-radiate out at a later time.

The magnitude images of the matched filtered time series are shown in Figs. 1(a) and 1(b), where the sinusoidal signature of initial geometric scattering is centered about 5 ms. Between angles $100°<θ<230°$⁠, the late returns are visible from 5.8 ms and later. These late returns do not conform to the image formation model provided by Eq. (1), with τ = 0, so they do not focus well in the standard CSAS image. However, the curvature of the return is maintained at a delayed time. The initial delay and subsequent delays are roughly 1.65 ms apart, and given a nominal speed of sound in air of 343 m/s, the travel path length is consistent with the bottle length being approximately 28 cm.

Figures 1(c) and 1(d) show CSAS images of the bottle. In Fig. 1(c) the cap is open and in Fig. 1(d) it is closed. The exterior of the object is clearly focused in both images. However, in Fig. 1(c) there are several low-intensity arcs around the base of the bottle. These arcs are associated with the late returns seen in Fig. 1(a), which are due to the additional propagation of sound that entered the bottle and exited after internal multiple scattering.

While the exterior of the bottle is sparsely represented with high contrast, the late returns from multiple scattering are defocused and dispersed to a lower intensity such that they would be obscured by even a low level of noise. A preprocessing step is taken before applying the image formation algorithm in order to improve the focus of the late return energy for robust identification of its presence. Shifting the entire time series to an earlier time by τ, then applying the image formation algorithm results in focusing the late return energy, as the original image becomes radially defocused from the scene center.

Although the curvature of the late returns are not exactly the same as that of the initial return since the additional wave propagation in the object structure is dependent on θ due to difference in incident angle of the sound entering and exiting the bottle, they consistently focus at the point of re-radiation, as shown in Figs. 2(a) and 2(c). Note that the late returns come into, and out of focus, not at the scene center, but at the mouth of the bottle.

To demonstrate the focus of the late return from the open bottle CSAS images were generated with $τ=1.65$ ms, which corresponds to extra time of sound traveling across the length of the bottle. The bottle with the cap closed in Fig. 2(b) does not show any late return focus, but it is clear in Fig. 2(a) late return is being focused near the mouth of the open bottle, at $(x,y)=(0.035$ m, $−0.125$ m). Using $τ=3.4$ ms for the open bottle, Fig. 2(c) shows the second late return being focused again at the same position.

To characterize the temporal progression of the late return focus, the delay parameter τ is varied from 0 to 10 ms in 50 μs steps. Late returns from the same source of re-radiation that occur at a particular delay will focus at a single point. For comparison, two positions are chosen from the image scene in Fig. 1, $P1=(x1,y1)=(0.035$ m, –0.125 m) for the cap (or mouth) of the bottle, and $P2=(x2,y2)=(−0.185$ m, –0.125 m) for a position on the table, 0.44 m left from the cap. The point at the mouth of the bottle is the source of late-energy re-radiation and is the pixel where late returns from this source will focus. The pixel intensity at (x, y) over τ is defined as $I(x,y,τ)=20 log10|f(x,y,τ)|$⁠. The pixel intensity at the cap of the closed bottle, $IC(P1,τ)$⁠, is initially high, then decays as the image becomes out of focus, as shown in Fig. 3(a). On the other hand, the pixel intensity at the same position of the open bottle, $IO(P1,τ)$⁠, fluctuates as packets of late returns come in and out of focus as τ increases. In contrast, $IC(P2,τ)$ and $IO(P2,τ)$ at the other position the progress between the two functions maintain their similarity for the entire range of τ values, as shown in Fig. 3(b).

A summary measure that captures how much late-energy focus occurs at each pixel over the image sequence is defined as $S(x,y)=(1/T)∑τI(x,y,τ)dτ$⁠, where $dτ=50 μs$⁠, T = 10 ms. The resulting summary images are $SO(x,y)$ and $SC(x,y)$ for open and closed. The difference between the two functions is defined as $D(x,y)=SO(x,y)−SC(x,y)$⁠, and is shown in Fig. 4. It is clear the late acoustic returns are compactly focused at the mouth of the open bottle, which is the focal point of re-radiation after the internal scattering. This is in contrast to the dispersed energy shown in Fig. 1(c). This comparison demonstrates that additional signal processing steps can enhance the phenomenological representation of acoustic scattering differences between the two objects, e.g., open and closed bottles. Furthermore, the point of re-radiation is sparsely identified, clearly localizing the source of the late returns.

Synthetic aperture processing algorithms are used to generate high-resolution imagery using acoustic and electromagnetic waves in various remote sensing and object detection applications. SAS imagery can represent geometric scattering of objects, but there are additional scattering components that return at delayed times. These late returns do not conform to the standard SAS imaging model and appear as defocused energy. In this paper, it was demonstrated how preprocessing steps can align some of the late returns to the image formation model, which improve the focus of these late returns. An in-air circular SAS experiment was used to collect and process the data from a controlled experiment that compares the structural acoustic responses of 1 liter plastic bottles with, and without the cap closed. The results imply alternative processing approaches can be used to accentuate additional information embedded in the raw data, that can help better characterize the environment and objects of interest. The application of this algorithm could aid target detection and identification in cases where the geometric scattering returns are obscured by the environment or the defocused, late returns are difficult to see above background speckle. Furthermore, it may be possible to invert for material properties or the internal dimensions of a hollow target using a model for acoustic propagation internal to the target and the values of τ at which late energy is focused. The spatial points and temporal delays where the late returns focus can provide detail about the material and structure of the target and aid data interpretability.

This work was supported by the Office of Naval Research.