FIELD

The present disclosure relates generally to radar systems, and more particularly radar imaging by fusing measurements of various antennas whose positions are not accurately known.

BACKGROUND

High-resolution radar imaging is a requirement in a variety of remote sensing applications, including synthetic aperture radar (SAR) and through-the-wall radar imaging (TWI). Whereas the down-range resolution is mostly controlled by the bandwidth of the transmitted pulse, the cross-range (azimuth) resolution depends on the aperture of the radar array. Generating a large physical aperture can be practically achieved by deploying a number of distributed antennas or arrays, each having a relatively small aperture. A distributed setup allows for flexibility of platform placement, reduces the operational and maintenance costs, and adds robustness to sensor failures. Leveraging prior knowledge of the scene, such as the precise knowledge of the antenna positions and a full synchronization of received signals has been shown to significantly improve the radar imaging resolution. However, geographical distribution of an array introduces data coherence problems due to ambiguities in the position of the antennas and/or difficulties in precisely synchronizing the antenna clocks.

For example, the state-of-the-art models antenna position errors as phase and magnitude errors in the received data. Wherein, the data distortion introduced due to position errors may be equivalent to a phase and magnitude distortion in the data. To that end, the conventional methods attempts to estimate and correct the phase errors in the data, in order to apply coherent imaging techniques on the corrected data. However, at least one common issue with those solutions, among many problems, is that the estimation of the phase and magnitude distortion is not straightforward due to the non-linearity of the resulting formulation, and the lack of models to directly compute the effect of position ambiguities to the phase and magnitude of the system. For example, typical phase models in the literature, such as subspace restrictions, under-perform because they fail to capture the true nature of the error. In addition, there is an additional complication that phase is a wrapped quantity, making its estimation more difficult. As a result, those solutions are not practical.

Therefore, there is a need for radar imaging systems and methods for fusing measurements of various antennas with position errors.

SUMMARY

Embodiments of the present disclosure relate to radar systems and methods for radar imaging by fusing measurements of various antennas with position errors. Some embodiments, further relate to radar systems and methods for radar imaging by fusing measurements of various antennas with synchronized or unsynchronized clocks.

The embodiments described below are described as two-dimensional examples. However there is nothing that is specific to two dimensions. Other embodiments rely on the same realizations and algorithms, replacing two-dimensional quantities, such as the image or the unknown shifts and shift kernels, with one or three-dimensional ones, as necessary for the particular application.

Some systems and methods of the present disclosure provide for an optimization-based solution to the problem of distributed radar imaging using antennas with position ambiguities. In distributed settings, it is challenging to know the positions of different antennas to the desired precision required for radar imaging. At least one aspect of the present disclosure assumes that these positions are not known precisely, and attempts to determine the position of each antenna, in addition to recovering the image of the scene the radars are illuminating. This solution further allows, by non-limiting example, modeling and recovering of timing ambiguities in a case the antennas are not perfectly synchronized.

Some of the systems and methods of the present disclosure explicitly and separately model the position uncertainty of the transmitter antenna as an unknown position shift of the transmitted field, and the position uncertainty of the receiver antenna as an unknown position shift of the reflected field. Thus, this approach, by non-limiting example, accurately models each of the shifts as a convolution with a spatial shift operator. This formulation results in an optimization problem which simultaneously recovers all the position errors and the radar image acquired by the radar. In addition, the solution provides the resolution benefits of coherent imaging, even if the measurements are not coherent.

Further, other beneficial aspects of this solution include exploiting the realization that the effect of position errors is different for the transmitting antennas than for the receiving antennas. This solution further relies on a realization that separating the typical radar operator to two separate operators, one describing the incident field and another describing the reflected field, enables the description of the transmitter and receiver errors as simple translation of the two incident and the reflected fields, respectively. Furthermore, this approach exploits the realization that translation in space is equivalent to convolution with a shift kernel operating in space on the fields, and that shift kernels are sparse. Thus, sparse optimization methods can be very effective in recovering the kernel.

In particular, position errors in the transmitting antenna result to a shifting of the field induced onto the scene by the same amount. Similarly, position errors of receiving antenna result to data received as if the reflected field was shifted by the same amount in the opposite direction. Thus, it is possible to formulate a multilinear optimization problem that simultaneously recovers all the antenna position errors, as well as the sparse scene being imaged.

In order to implement some of the above realizations, some embodiments provide for radar imaging from measurements provided by antennas having position ambiguities, where the phrase position ambiguities indicate a true position of the antennas, which may or may not be accurately known, and may differ from an assumed position of the antennas. In order to address when some antennas have position ambiguities, some embodiments, for example, act under an additional assumption that the antennas have clocks that are not perfectly synchronized.

Some embodiments are based on the recognition that a radar-imaging problem for distributed antennas with position ambiguities can be an ill-posed problem with a vast number of unknowns. Specifically, when the radar-imaging problem is formulated as determining a radar image from the measurements related to the radar image through a radar operator having uncertainties encoding the antenna position ambiguities, each measurement of an unknown scene includes an error caused by the position errors of the antennas. Moreover, due to non-linearity of relationships between the measurements and the errors in the position of the antennas, each sample of the measurements from the same antenna can have a different error, thereby increasing a number of unknowns in a model of the radar-imaging problem formed by multiple measurements from multiple antennas. To that end, the formulation of the radar-imaging problem that aims to recover the correct radar image from incorrect measurements caused by the incorrect radar operator is a difficult to accurately model, as an ill-posed problem.

Some embodiments are based on another recognition that the effect of position errors is different for transmitting antennas for receiving antennas. In particular, an effect of a transmitting antenna position error is a shift in the incident field that this transmitting antenna induces to the scene by the same amount as the position error. The scene interacts with the incident field, creating a reflected field. Each receiving antenna measures the reflected field at the position of the receiving antenna. The effect of a position error in the receiving antenna is equivalent to the field being measured at a different point, which, in turn, is equivalent to a field, shifted by the same amount to the opposite direction, being measured by an antenna without position error, i.e., at the assumed position.

Some embodiments are based on another recognition that a problem of radar image recovery under position ambiguity is to determine and unknown scene, a shift of the reflected field generated by the unknown scene and measured by a receiving antenna, and a shift of the incident field induced by a transmitting antenna and interacting with the scene to generate the reflected field, such that the scene and the shifts explain the radar measurements by the receiving antennas. Such a problem transformation allows decoupling the effect of the unknown shift of the transmitting antenna from the effect of the unknown shift of the receiving antenna and from the unknown scene that generates the unknown ideal signal.

In addition, some embodiments are based on a realization that the unknown shift of the incident field, which is an unknown translation of incident field in space, can be represented as a convolution with an unknown two-dimensional convolutional shift kernel, with the further realization that this kernel is a unit impulse in two dimensions. Such a representation is counterintuitive because representing this unknown shift as an unknown two-dimensional kernel expands the dimensionality of the problem. In particular, a two-dimensional translation can be represented using a two-dimensional vector, i.e., two parameters, whereas a two-dimensional kernel has size that depends on the maximum ambiguity of the position error in each dimension, as a function of the size of a representation grid used to represent the kernel. However, the unknown translation affects the data in a nonlinear way, while the convolution by the unknown impulse is a liner operation. In such a manner, the decoupling of the unknowns combined with linearization gained from the convolutional representation result in a multilinear radar image recovery problem. Multilinear problems, although not convex, and not as simple to solve as convex problems, are still easier to solve than more general non-convex problems, using methods such as lifting or alternating optimization.

In addition, some embodiments are based on another realization that the unknown shift of the reflected field, which is an unknown translation of reflected field in space, can also be represented as a convolution with an unknown convolutional shift kernel, with the further realization that this kernel is a unit impulse in two dimensions. In a similar manner, such a representation is counterintuitive because representing this unknown shift as an unknown two-dimensional kernel further expands the dimensionality of the problem. However, the unknown translation affects the data in a nonlinear way, while the convolution by the unknown impulse is a liner operation. In such a manner, the decoupling of the unknowns combined with linearization gained from the convolutional representation result in a multilinear radar image recovery problem.

However, even after decoupling, this multilinear radar image recovery problem is still difficult to solve, because the resulting problem has a large number of solutions, while only one of them is the desired one. For this reason, some embodiments are based on another realization that the unknown convolutional shift kernels can be represented as shifted impulses, which are unknown two-dimensional signals that are one-sparse in space. In effect, these realizations allow transforming the original nonlinear image recovery problem into a multilinear sparse recovery problem, which in turn allows using sparse reconstruction techniques to reduce the size of the solution space of the radar image recovery problem.

To that end, some embodiments solve a multilinear radar image recovery problem to produce a radar image of the scene. The radar image recovery separately recovers the shift of the incident field, the reflectivity of the scene observed by the radar and the equivalent shift of the reflected field that, when combined, describe the measured data. The shifted incident and reflected fields are represented as convolutions in space of upshifted fields with spatial shift kernels, which are one-sparse in space. In some embodiments, the scene is also sparse in space or when transformed by an appropriate transformation, such as wavelets, gradient, or Fourier.

During experimentation, some experimental approaches included explicitly modeling position ambiguities on the distributed antennas as a single unknown shift on the image for each transmitter-receiver antenna pair. However, by modeling these ambiguities as unknown shifts on the image, instead of a separate shift on the incident and the reflective field, these approaches allow for only one shift for each pair of transmitting-receiving antenna.

Effectively, these experimental approaches implicitly assume that the position error is the same for the transmitter and the receiver. In practice, this is a very limiting model, accurate only in a small number of distributed sensing systems. For this model to be accurate, the transmitter and the receiver should be collocated, i.e., located on the same platform, move together, and experience the same position error. Even if they are on the same platform but at different locations on the platform, and the platform rotates or changes direction, this model is not accurate.

In effect, this experimental model is only accurate in practical systems configured such that an antenna is acting as both a transmitter and a receiver, and no other antenna is used as a receiver for the same transmission. Using such a system configuration diminishes the advantages and the effectiveness of the distributed radar system, as it cannot exploit the availability of multiple receiver antennas. To acquire more data and improve system performance, other receiving antennas, at a different location than a transmitting antenna should receive reflections from the scene.

However, these experimental approaches failed to recognize the separation of the incident field with the reflective field in the radar operator. This separation is necessary in order to model the transmitter position error as distinct from the receiver position error. Otherwise, a single unknown translation error describes the position error of each transmitter-receiver antenna pair, which implies that they both exhibit the same common error. Thus, these experimental approaches tried to recover this common position error for each transmitter/receiver pair, modeling the common shift of the position of the transmitter and the receiver in the pair, compared to their assumed position, as a corresponding shift of the scene being imaged by the radar by the same amount in the opposite direction. As described above, under this formulation, the model is accurate only if the transmitter and receiver pairs are collocated for each transmission and reception, which is not a practical assumption in most distributed systems.

In addition, modeling the position error for each antenna pair results in a multiplicative growth of the unknown position errors, as the number of transmitter or receiver antennas grows. In contrast, by recognizing the separation of the incident and the reflected field in the radar operator, and separately modeling the error in the positions of the transmitting antennas and the receiving antennas, the number of unknown position errors only grows linearly with the number of antennas. For example, in a system with 3 transmitting and 5 receiving antennas, in which all receiving antennas receive the reflections from all the transmitting antennas, there are 3×5=15 transmitter-receiver antenna pairs, and therefore, 15 unknown position errors, one for each pair. Increasing the transmitting antennas to 4 would create 5 more pairs, one for each receiving antenna, and therefore 5 more unknown position errors for a total of 4×5=20 unknown position errors. In contrast, there are only 3+5=8 and 4+5=9 unknown position errors, respectively, if the position errors of transmitting antennas are treated separately form the position errors of the receiving antennas. Reducing the number of unknowns significantly reduces the computational complexity of solving the problem, as well as the amount of data necessary to find the correct solution. Both of these issues are important in most practical applications. Therefore, separating the incident and the reflected fields is also preferable from a computational and data acquisition complexity standpoint.

Furthermore, separating the transmitting antenna position errors and the receiving antenna position errors enforces consistency in the solution that is not enforced when the unknown position error is on each transmitter-receiver antenna pair. In reality, at the time of transmission the transmitting antenna is at a single location and has a single unknown position error. Similarly, the receiving antennas are at a single position each, and exhibit a single unknown position error. Modeling the transmitter-receiver pair position error as a single unknown position error implies that the unknown error is a combination of the two individual position errors in the pair, e.g, an average. Given the individual position errors, it may be possible to combine them and determine the position error of the pair in this model. However, given estimates of combined position errors of each pair, there is no guarantee that these pairs are consistent, i.e., that there is a choice of separate position errors for each transmitter and receiver in the pairs, such that when appropriately combined one can obtain the given pairwise errors. Thus, directly estimating the errors for each pair might result in inconsistent estimates. Reconciling these inconsistencies is not straightforward. Instead, separating the transmitter and receiver position errors naturally results in estimates that are consistent.

At least one further realization in some embodiments of the present disclosure that include separating the operator, as noted above, is the realization that separating the typical radar operator to two separate operators, one describing the incident field and another describing the reflected field, enables the description of the transmitter and receiver errors as simple translation of the two incident and the reflected fields, respectively. In particular, a key realization is that a shift of the transmitter results to a shift of the incident field by the same amount in the same direction, whereas a shifted receiver would receive the same data as a receiver in the original position observing a reflected field shifted by the same amount in the exact opposite direction.

Furthermore, this approach exploits the realization that translation in space is equivalent to convolution with a shift kernel operating in space on the fields, and that shift kernels are sparse. While this is a significant expansion in the dimensionality of the unknown parameter, it makes the problem linear in the unknown shift and separates the effect of the shifts from the rest of the operator describing the physics of the system. Furthermore, since the shift kernel is sparse, sparse optimization methods can be very effective in recovering the kernel. Specifically, if an unknown quantity is sparse, then estimating this quantity by enforcing sparsity using sparse optimization improves the quality of the estimated value and reduces the amount of data that should be collected in order to successfully produce this estimate.

In addition, separating the operator has the beneficial aspect, among many benefits, of being able to accurately model a larger variety of systems, and significantly improve the reconstruction performance. This is because this separation in the model capture the physical manifestation of most practical systems, even if the transmitter and the receiver are not co-located. At the same time, it can also model the errors if it is known that the transmitter and receiver are co-located simply by using the same unknown kernel to model the unknown shift. Furthermore, the model is still applicable if a transmitter and receiver are on the same platform, but not the same antenna, and the platform exhibits both translation and rotation.

Other experimental approaches may attempt to correct synchronization errors between the antennas. While in practice accurate synchronization is very difficult to achieve, these experiments demonstrated that it is possible to describe a convex optimization algorithm that accurately recovers timing, i.e., synchronization, errors between the antennas. However, it is not possible to incorporate position errors in this formulation. Thus, these experimental approaches are only applicable if the position of each of the antenna is known exactly, without any errors.

Instead, some embodiments of this present disclosure further recover synchronization errors, in addition to position errors, by formulating a multilinear problem. In particular they exploit the realization that synchronization errors cause unknown notional time delays and advances in the recorded data, which can be modeled as time shifts in the data. Similarly to shifts in position, described above, these time shifts in the data can be modeled as convolutions with a shift kernel in the time domain. Thus, they interact linearly with the sensing model. Since the sensing model includes the unknowns for the position errors of the antennas and the unknown scene, the resulting problem is multilinear.

Some embodiments of the present disclosure rely on the realization that, while multilinear problems are not convex, they are relatively well-studied, well-understood, and straightforward to solve, compared to more general convex problems. In particular, while general non-convex problems can be difficult to solve, multilinear problems exhibit specific structure that makes them amenable to different algorithmic solutions, such as alternating optimization, or lifting to higher dimensions, among others. Thus, they admit algorithms that can be very efficient for practical use. Therefore, giving up convexity for the flexibility of correcting both position and synchronization errors is a favorable trade-off in many applications.

Furthermore, some embodiments of this present disclosure may exploit the realization that a multilinear formulation is better able to enforce timing consistency when recovering synchronization errors. In particular, the convex formulation in the art recovers timing errors between transmitter and receiver pairs. Embodiments of this present disclosure may recover, instead, the timing error of the clock of each transmitter and each receiver relative to an absolute clock.

As an example, consider the clock of one transmitter as the absolute clock, and all clocks are considered running fast or slow as relative this clock. Consider the clock of a second transmitter running 2 time-units slow, the clocks of two receivers running 1 and 3 time-units fast, respectively, while the clock of a third receiver running 1 time-unit slow. Embodiments of this present disclosure only need to recover four absolute clock errors: one for the second transmitter and three for the three receivers. A convex formulation, experimented with during experimentation would need to recover the relative delay for each pair, i.e., 2×3=6 unknown relative delays. In the same example, these delays should be consistent. For example, the relative delays between the clock of the first transmitter and the clocks of the three receivers should be equal to the relative delay of the clock of the second transmitter added to the relative delays between the clock of the second transmitter and the three receivers.

Since the convex formulation in the above experiment recovers six separate unknown relative delays, the resulting estimates of the relative clock errors might not be consistent, especially in the presence of noise. The correct solution would be to recover a time advance of 1 and 3 units and a time delay of 1 unit for the clocks of the three receivers respectively, relative to the clock of the first transmitter, and a time advance of 3, 5, and 1 time-units, respectively relative to the clock of the second transmitter. A recovered time advance of 3, 5, and 2 time units, for the second set of advances provides an inconsistent solution, which is still possible in the above experiment. Using embodiments of the present disclosure that recover the clock errors for each transmitter and receiver separately, such inconsistency is not possible.

At least some benefits of the systems and methods of the present disclosure including overcoming problems with conventional distributed radar imaging methods, that use geographical distribution of an array, which introduces data coherence problems due to ambiguities in the position of the antennas and/or difficulties in precisely synchronizing the antenna clocks. Some embodiments of the present disclosure overcome these problems by separately modeling the position uncertainty of the transmitter antenna as an unknown position shift of the transmitted field, and the position uncertainty of the receiver antenna as an unknown position shift of the reflected field, and further modeling the shifts as a convolution with a spatial shift operator. This, more accurate formulation, also results in an optimization problem, which simultaneously recovers all the position errors and the radar image acquired by the radar.

Another benefit of the present disclosure is overcoming the conventional methods attempts to estimate and correct the phase errors in the data, in order to apply coherent imaging techniques on the corrected data. Some embodiments of the present disclosure overcome these problems by separating the typical radar operator to two separate operators, one describing the incident field and another describing the reflected field, enabling the description of the transmitter and receiver errors as simple translation of the two incident and the reflected fields, respectively. Furthermore, this approach exploits the realization that translation in space is equivalent to convolution with a shift kernel operating in space on the fields, and that shift kernels are sparse. Thus, sparse optimization methods can be very effective in recovering the kernel.

According to an embodiment of the present disclosure, a radar system for generating a radar image of a scene. Wherein, during an operation, the radar system is connected to a set of transmitters configured to transmit radar pulses to the scene and a set of receivers configured to receive reflections of the radar pulses from the scene. The radar system including a memory configured to store predetermined configuration data indicative of propagation of the radar pulses to and from the scene. The configuration data includes positions of the transmitters and positions of the receivers, such that a position of at least one receiver is different from the transmitter positions of all transmitters. Wherein the configuration data defines an incident field generated by the radar pulses transmitted by the transmitters from the transmitter positions. Wherein the configuration data defines a structure of a reflection field generated by reflections of the incident field from the scene and measured by the receivers at the receiver positions. An input interface configured to receive radar measurements of reflectivity of each point in the scene measured by the set of receivers. A hardware processor configured to solve a radar image recovery problem using the configuration data to produce the radar image of the reflectivity of each point in the scene. Based on connecting the received radar measurements to a shift of the reflection field with a receiver unknown position shift. Wherein the receiver unknown position shift defines an error between the receiver positions stored in the memory and actual positions of the receivers. The reflection field is generated by reflecting the transmitted field from the scene in accordance with the reflectivity of each point in the scene. Connect the reflection field to a shift of the incident field with a transmitter unknown position shift. Wherein the transmitter unknown position shift defines an error between the transmitter positions stored in the memory and actual positions of the transmitters. Solve the radar image recovery problem as a multilinear problem of joint estimation of the reflectivity of each point in the scene, the receiver shift, and the transmitter shift. An output interface configured to render one or combination of the radar image of the reflectivity of each point in the scene, the receiver shift, or the transmitter shift.

According to another embodiment of the present disclosure, a method for a radar system to produce a radar image of a region of interest (ROI). The method including measuring, using transmitting antennas at different positions to transmit radar pulses to the ROI. Receiving antennas configured to receive reflections of the radar pulses from the ROI corresponding to the transmitted radar pulses. Such that the receivers measure the reflections of the radar pulses to obtain radar measurements of reflectivity of each point in the scene. Using a hardware processor in communication with the transmitters, receivers and a memory, that is configured to access the memory having stored data including predetermined configuration data indicative of propagation of the radar pulses to and from the ROI. The configuration data includes positions of the transmitters and positions of the receivers, such that a position of at least one receiver is different from the transmitter positions of all transmitters. Wherein the configuration data defines an incident field generated by the radar pulses transmitted by the transmitters from the transmitter positions. Wherein the configuration data defines a structure of a reflection field generated by reflections of the incident field from the scene and measured by the receivers at the receiver positions. Solving a radar image recovery problem using the configuration data to produce the radar image of the reflectivity of each point in the scene. Based on connecting the received radar measurements to a shift of the reflection field with a receiver unknown position shift. Wherein the receiver unknown position shift defines an error between the receiver positions stored in the memory and actual positions of the receivers. The reflection field is generated by reflecting the transmitted field from the scene in accordance with the reflectivity of each point in the scene. Connecting the reflection field to a shift of the incident field with a transmitter unknown position shift. Wherein the transmitter unknown position shift defines an error between the transmitter positions stored in the memory and actual positions of the transmitters. Solving the radar image recovery problem as a multilinear problem of joint estimation of the reflectivity of each point in the scene, the receiver shift, and the transmitter shift. Outputting via an output interface one or combination of the radar image of the reflectivity of each point in the scene, the receiver shift, or the transmitter shift.

According to another embodiment of the present disclosure, a non-transitory computer readable storage medium embodied thereon a program executable by a processor for performing a method, the method for a radar system to produce a radar image of a region of interest (ROI). The method including measuring, using transmitting antennas at different positions to transmit radar pulses to the ROI. Receiving antennas configured to receive reflections of the radar pulses from the ROI corresponding to the transmitted radar pulses. Such that the receivers measure the reflections of the radar pulses to obtain radar measurements of reflectivity of each point in the scene. Using a hardware processor in communication with the transmitters, receivers and a memory, that is configured to access the memory having stored data including predetermined configuration data indicative of propagation of the radar pulses to and from the ROI. The configuration data includes positions of the transmitters and positions of the receivers. Such that a position of at least one receiver is different from the transmitter positions of all transmitters. Wherein the configuration data defines an incident field generated by the radar pulses transmitted by the transmitters from the transmitter positions. Wherein the configuration data defines a structure of a reflection field generated by reflections of the incident field from the scene and measured by the receivers at the receiver positions. Solving a radar image recovery problem using the configuration data to produce the radar image of the reflectivity of each point in the scene. Based on connecting the received radar measurements to a shift of the reflection field with a receiver unknown position shift. Wherein the receiver unknown position shift defines an error between the receiver positions stored in the memory and actual positions of the receivers. The reflection field is generated by reflecting the transmitted field from the scene in accordance with the reflectivity of each point in the scene. Connecting the reflection field to a shift of the incident field with a transmitter unknown position shift. Wherein the transmitter unknown position shift defines an error between the transmitter positions stored in the memory and actual positions of the transmitters. Solving the radar image recovery problem as a multilinear problem of joint estimation of the reflectivity of each point in the scene, the receiver shift, and the transmitter shift. Outputting via an output interface one or combination of the radar image of the reflectivity of each point in the scene, the receiver shift, or the transmitter shift.

