Combining range-general and range-specific techniques for late-time focused image reconstruction^a)

Synthetic aperture sonar (SAS) systems are commonly used in underwater surveys to identify objects of interest. These sonar systems transmit a signal into the environment and then receive the echoes scattered off of objects and the environment. Then, a model for these echoes (Gough and Hawkins, 1997) is inverted to create an image of the underwater scene. Most SAS image reconstruction only considers direct, two-way wave propagation between the sonar and a target. This image reconstruction assumes the acoustic time-of-flight is only sensitive to the speed of sound in the propagation medium and the geometric relationship between the sonar and the target. While a full treatment of acoustic target scattering is too complicated for binary separation, it is useful to divide the received returns into two categories based on the limitations of traditional SAS reconstruction: geometric returns and late-time returns. Geometric returns are received signals that meet the direct, geometric time-of-flight assumption and therefore are reconstructed compactly within the external outline of the target. This definition is similar to the definition of geometric scattering used in Sec. 3.B of Bucaro (2008).¹ In contrast, late-time returns include returns from all other mechanisms, including those that involve target elasticity, waves coupling to the target, and multiple scattering (España , 2014; Hall and Marston, 2022; Williams , 2010). Such returns stem from the target, but appear to originate later in time than the sonar-target geometry would dictate (Marston , 2010). When reconstructed, these late-time returns appear out-of-focus and away from the target's true location. It is important that late-time components are reconstructed accurately, as visible late-time features may indicate the rough size and material properties of the target, aiding in target classification by both human analysts and machine learning models (Hoang , 2023).

SAS surveys have traditionally used range-general image reconstruction techniques to create imagery from received acoustic data. As the name suggests, range-general techniques attempt to form an image that is focused at all ranges. These range-general methods are best suited for observing bathymetric details of the seafloor or targets that are widely distributed across an imaged scene. Range-general reconstruction algorithms include time-domain implementations such as delay-and-sum (backprojection) reconstruction, and frequency-domain implementations such as ωk reconstruction and range-stacking (Soumekh, 1999).

In contrast, range-specific methods aim to create an image that is focused at a single range, which is generally set to be the range to a distinct target. Such methods enhance acoustic features caused by objects at the focal range in exchange for reduced image quality at ranges other than the focal range. Examples of range-specific reconstruction techniques include a fixed-focus adaptation of ωk reconstruction (Groen , 2009) and quasi-holographic techniques (Baik , 2011; Plotnick and Marston, 2016; Zartman , 2013). The sonar and radar communities have also developed delay-specific imaging techniques that apply additional time delays during image formation to explore the evolution of late-time returns from a target. For example, the work in Park (2021) adds a single, uniform delay during delay-and-sum reconstruction of data from a circular SAS. Another delay-specific technique represents a two-dimensional (2D) scattering scene as a 3D image, where subsequent slices through the 3D image represents reconstruction with an increasing additional delay (McCorkle and Nguyen, 1991).

While range-specific and delay-specific techniques succeed in accentuating some late-time features, they also have several limitations. For example, delay-specific techniques can create a focused 2D image of geometric returns from the target or a focused 2D image of late-time returns, but not both simultaneously. Similarly, previous range-specific methods must be applied to the entire imaged scene, which improves contrast near the focal range, but may degrade image quality at other ranges. Finally, range-specific wavenumber techniques assume the target is in the far-field of the sonar. While this assumption generally valid in long-range sidescan surveys, downward-looking surveys at close imaging ranges are often bistatic and the far-field assumption does not hold. Wavenumber reconstruction at close imaging ranges is still possible, but it requires additional computation and corrections such as the near-field to far-field transformation for circular collections (Plotnick , 2014) and a correction for the phase center approximation in multi-receiver systems (Callow, 2003). These shortcomings motivate a new reconstruction technique that simultaneously focuses both the target and any associated late-time returns in the near-field of the sonar, with the ability to selectively apply late-time focusing to only the area of interest.

The primary contribution of this paper is an image reconstruction technique that allows for the selective application of range-general and range-specific imaging to focus late-time returns. This technique generalizes the concepts of range-specific wavenumber reconstructions in the time-domain, creating imagery that increases late-time return contrast without defocusing the image at ranges leading up to the target. The proposed method inherently captures the bistatic nature of many sonar geometries and allows for imaging in the near-field of the sonar array without additional geometric compensation. Additionally, this paper includes a quantitative analysis of image improvement with the proposed method using data from an in-air, rail sonar experiment. When compared with range-general and range-specific techniques, the proposed method offers increased late-time focusing performance as measured by two image quality metrics. We envision that the proposed reconstruction technique will offer increased utility in real-world, close-range surveys, as it is more robust to errors that may arise in setting the focal range to an unknown target. This analysis shows that late-time focused reconstruction can keep geometric returns from a target in focus when the focal range is incorrectly set beyond the target, unlike range-specific wavenumber techniques.

The formulation of γ and τ is central to the selective application of range-general and range-specific focusing. For ranges leading up to the focal range, the distance calculations in Eqs. (2) and (3) are identical to standard delay-and-sum reconstruction. However, for all ranges beyond the focal range, the distance between the sonar and the pixel is calculated as if the pixel is at the focal range. This piecewise distance calculation allows the width of the range migration curve in Eq. (1) to change with range, until the focal range is reached, at which point the shape of the range migration curve becomes fixed. For ranges closer than the focal range, no additional delay (τ) is applied, making Eq. (1) equivalent to standard delay-and-sum reconstruction. Moving beyond the focal range, τ is set to the two-way travel time between the pixel's true range and the focal range. This formulation of τ allows the fixed-range migration curve calculated at $y P = β$ to be applied to late-time signal components. The inclusion of τ is motivated by Park (2021), where applying an additional time delay allows the late-time returns to be reconstructed at their most-focused location, and by Marston (2010), who found that resonant features are focused by applying the same point-spread function that is applied to the target.

The data in this section was gathered using an in-air synthetic aperture sonar developed at Penn State University (Blanford , 2022). A 19.05 cm (7.5 in.) long, 6.35 cm (2.5 in.) diameter aluminum cylindrical shell with 1.65 mm (0.065 in.) thick walls was used as the target for this experiment. The target was positioned at a slant-range of 78 cm, suspended from overhead supports with thin nylon thread. The cylinder's axis was oriented vertically, perpendicular to the horizontal scan line of the synthetic aperture. The sonar transmitted a down-chirping, 1 ms duration LFM from 30 to 5 kHz and windowed with a 10% Tukey window. The array was moved in a “stop-and-hop” manner with a 5 mm advance per ping across a 3 m pass centered on the target (Dalton 2023). A photo of the experimental setup can be found in the supplementary material.

The array consists of four GRAS 46AM microphones and one Peerless OX20SC00-04 tweeter, mounted on a 5 m linear actuator. For this experiment, four receivers were placed 5, 25, 35, and 45 cm from the transmitter to examine increasingly bistatic observation angles. Only for the 5 cm transmitter-receiver pair may the target be considered in the far-field of the physical array where a monostatic approximation would hold. The differentiation of the near-field and far-field of the array is given by the Fraunhofer distance: $F = 2 D 2 / λ$⁠, where D is the distance between the transmitter and the receiver, and λ is the wavelength at the center frequency of the transmitted signal.

The data collected in Sec. 3 was reconstructed using a range-general, delay-and-sum reconstruction, a fixed-focus implementation of the ωk algorithm, and the proposed late-time focusing technique to showcase the qualitative and quantitative differences between the reconstructions. Two image quality metrics were used to assess a 40 cm along-track by 1 m cross-track region of interest around the target. The first metric is a cross-track (column-by-column) sum of pixel amplitudes in the region of interest. The goal of the late-time focusing method is to increase the difference in amplitude between columns containing the target and columns without the target, effectively increasing the local signal-to-noise ratio in the area near the target. The second quality metric is lacunarity (L), which quantifies the spatial variation of pixel amplitude and is defined as $L r = σ 2 / μ 2$⁠, where r is a vector containing all pixel amplitudes belonging to region r of the image, $σ 2$ is the variance of the pixel amplitudes, and μ is the mean of the pixel amplitudes. For this experiment, lacunarity was calculated for regions of 6 rows of pixels at a time, with a 50% overlap between adjacent regions. Regions with low spatial variability, such as those containing unfocused returns and background reverberation, will have low lacunarity, while regions containing a target or well-focused late-time returns will have higher lacunarity (Williams, 2015). In our context, a larger lacunarity indicates that late-time returns are more spatially condensed and suggests the focused late-time returns are easier to differentiate from background reverberation and noise.

The images in Fig. 1(a) compare range-general reconstruction (left), fixed-focus ωk reconstruction (center), and the late-time focused reconstruction (right) using data from the receiver spaced 45 cm from the transmitter. For the fixed-focus and late-time focused images, the focal range (β) has been set at the center of the target. In the fixed-focus reconstruction, the data received a transmit-receive correction to account for the bistatic nature of the array geometry (Callow, 2003). Additionally, all three reconstructions were filtered in the wavenumber space to ensure only spatial frequencies supported by the sonar configuration were reconstructed. All three images are on a 60 decibel scale and normalized to the maximum intensity in each image.

The semi-circular bands of late-time energy visible behind the target in the range-general reconstruction are focused to a spatially tighter region behind the target in both the fixed-focus and late-time focused reconstructions. The plot in Fig. 1(b) compares the cross-track sum of pixels in the images in Fig. 1(a). The late-time focused reconstruction performs best as it maximizes the difference between the columns containing the cylinder and the columns containing background reverberation. The plot in Fig. 1(c) compares the lacunarity for each six-row region of the images in Fig. 1(a). Again, the late-time focused reconstruction performs best as it consistently offers a higher lacunarity, indicating that the late-time returns are more spatially focused. Notice that the range-general and late-time focused traces in Fig. 1(c) (solid blue and dashed yellow) are identical for ranges before the focal range. This is expected as the two reconstructions are mathematically identical for regions leading up to the focal range [see Eq. (1)].

The results in Fig. 1 show the impact of late-time focusing when the target's range is known and the focal range (β) can be set properly. However, in many field surveys, the exact range of the target is not known beforehand. After an initial image reconstruction, a human operator or automatic target detector must detect the target and estimate the target's location. This estimation has some inherent error, and it is likely that β will not be set at the exact range of the target. Figure 2 shows the qualitative impact of an incorrect focal range by comparing range-general reconstruction (left), fixed-focus ωk reconstruction (center), and the late-time focused reconstruction (right) for $β = 0.9$ m. The range-general image is identical to that in Fig. 1(a), as range-general reconstruction is independent of β. Notice that the geometric returns from the target are distorted in the fixed-focus image while the target remains in focus in the late-time focused reconstruction. In both the fixed-focus and late-time focused reconstructions, late-time returns are focused to a more narrow region behind the target, though not as well as the $β = 0.78$ m case depicted in Fig. 1.

The focal range was swept in 1 cm increments from 0.6 to 1.1 m to observe the performance of the fixed-focus reconstruction and the proposed late-time focused reconstruction as the focal range deviates from the target. For the fixed-focus reconstruction, focusing performance drops as the difference between the target and the focal range grows. This is demonstrated by the hourglass shape in Fig. 3(a), showing that returns from the target spread over a wider range of columns as β deviates from the target's range. It appears that the fixed-focus ωk reconstruction achieves its best performance at a focal range near 0.82 m. After further investigation, at this focal range the fixed-focus reconstruction is condensing geometric returns from the leading edge of the cylindrical target down to a single point near the center of the cylinder. While this leads to a high contrast between columns containing target returns and columns containing only background reverberation (seemingly beneficial according to the image quality metric), the image no longer provides an accurate depiction of target's shape and size.

Much like the fixed-focus reconstruction, late-time focused reconstruction defocuses given focal ranges shorter than the range to the target. However, unlike the fixed-focus technique, the late-time technique brings the target into focus when β is at the target's range and is able to keep the target in focus when β is set at ranges beyond the target as shown in Fig. 3(b). This is due to the formulation of Eq. (1), where the late-time focused reconstruction is equivalent to the range-general reconstruction for ranges up to the focal range. Consequently, the longer focal range causes the late-time focused reconstruction to look more like the range-general reconstruction.

Late-time returns can be useful in discriminating man-made objects from naturally occurring clutter. The range-general image reconstruction techniques commonly used in SAS surveys do not properly model late-time returns from man-made objects, leaving such signals out of focus. This work presents an image reconstruction method for focusing late-time returns that combines elements of range-specific and delay-specific methods. The proposed method does not require far-field assumptions, yielding improvement over common range-specific wavenumber methods in bistatic geometries. It was shown that late-time focused reconstruction outperforms range-general and range-specific reconstruction methods in two image quality metrics using in-air sonar data of a target observed in a bistatic array geometry. Additionally, late-time focused reconstruction was able to keep the target in focus when the focal range was incorrectly set beyond the target, while fixed-focus reconstruction was not.

Future work will apply the late-time focusing technique to downward looking sonar surveys for buried objects in shallow water. By implementing the late-time focused reconstruction with the focal range set at the depth of a buried object, any returns from the water column and water-sediment interface are imaged in a range-general manner, while only the region beneath the buried object receives range-specific reconstruction to accentuate any late-time returns. Later stages of this work will also examine the impact of late-time focused imagery on the performance of automatic target recognition (ATR) networks that leverage late-time returns, such as the network in Hoang (2023). It is hypothesized that with little to no modification to the network itself, using late-time focused imagery will increase ATR performance.

See the supplementary material for a photograph of the experimental setup used in Sec. 3 (Dalton et al., 2023).

This research was supported by the Walker Graduate Fellowship from the Applied Research Laboratory at the Pennsylvania State University. This research was also supported in part by the U.S. Department of Defense, through the Strategic Environmental Research and Development Program (SERDP), under the munitions response portfolio of Dr. Herbert Nelson and Dr. David Bradley. This material is based upon work supported by the Humphreys Engineer Center Support Activity under Contract Nos. W912HQ-22-C-0011 and W912HQ-22-C-0011. Finally, the data collection was funded under Grants Nos. N00014-22-1–2607 and N00014-23-1–2846 from the Office of Naval Research (ONR).

The authors have no conflicts of interest to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.