Patent Description:
Some challenges in radar imaging using distributed sensing platforms is being not able to identify locations of platforms due to inaccurate calibration or various position perturbations. For example, for a vehicle mounted radar system, as the vehicle is moving along some predesigned trajectory, position perturbations are introduced due to non-smooth road surface or varying driving velocity and direction. These position perturbations can be as large as several wavelengths of the radar center frequency. Although modern navigation systems such as the Global Positioning System (GPS) can measure positions, the possible position errors due to position perturbations are beyond the scope of high-resolution distributed radar imaging.

<NPL> describes an approach for a radar coincidence imaging with array position error. The mathematical model for radar coincidence imaging in the presence of array position error is established. A non-linear relationship between the received signal and an array position error is approximated linearily by the first-order tailor expansion.

<NPL>, DOI: <NUM>/SAM. <NUM> describes a sparsity-driven radar auto-focus imaging under over-wavelength position perturbations. The publication considers a 2D imaging problem, where a perturbed mono-static radar is used to detect localized targets situated in a region of interest. In order to deal with position-induced out-of focus, a sparsity-driven auto-focus imaging approach is proposed in which each radar measurement is modeled as a super position of waited and delayed target signatures scattered from the corresponding target phase centers.

Therefore, there is a need for radar imaging systems and methods for distributed arrays that can perform autofocusing in order to compensate for unknown over-wavelength position perturbations.

Embodiments of the present disclosure relate to systems and methods for radar imaging using distributed arrays of moving radar platforms that perform autofocusing in order to compensate for unknown position perturbations.

Some embodiments of the present disclosure relate to systems and methods for radar imaging using distributed arrays of moving radar platforms for detecting targets in a region of interest (ROI), where the radar sensors of the moving radar platforms are perturbed with location errors. The present disclosure provides for an improved radar imaging performance, based on data coherence and compressive sensing that simultaneously compensates for position-induced phase errors and performs focused imaging, among other things.

Specifically, the systems and methods of the present disclosure assume one transmitting / receiving radar platform and multiple receiving radar platforms are moving towards the ROI with position perturbations, so as to detect targets inside the ROI. The multiple distributed arrays of moving radar platforms receive reflections reflected by the targets located in the ROI. The multiple arrays of moving radar platforms are uniform linear arrays randomly distributed with different locations and orientations at a same side of the area of interest. Despite the image resolution of each array is low, due to the small aperture size, a high resolution is achieved by combining signals received by all distributed arrays using a sparsity-driven imaging method if the positions of the distributed radar platforms are known exactly. However, due to inaccurate positioning and motion errors, the actual multiple arrays of moving radar platforms positions are perturbed up to several times the central radar wavelength, causing out-of-focus imaging results if the position perturbations are not well compensated.

Based upon a realization of making an assumption of a sparse scene, auto-focus imaging of the sparse scene can be realized iteratively by solving a series of optimization problems for compensating position-induced phase errors, exploiting target signatures, and estimating antenna positions. Specifically, in order to improve the imaging performance, the auto-focusing systems and methods are based on position error correction by exploiting data coherence and the spatial sparsity of the imaged area to concurrently perform focused imaging and estimate unknown antenna positions.

Because of the above realizations the present disclosure exhibits significant advantages in dealing with antenna array with position errors up to several wavelengths of the radar center frequency, taking antenna radiation pattern and target signature into consideration. Imaging results with simulated noisy data demonstrate that the systems and methods of the present disclosure significantly improved performance in imaging localized targets with only several iterations. In particular, the present autofocusing systems and methods form sharp images of targets situated in the ROI, even for position errors that are ten wavelengths large.

According to an embodiment of the present disclosure, a method for fusing a radar image in response to radar pulses transmitted to a region of interest (ROI). A transmitter transmits the radar pulses toward the ROI and a set of distributed receivers to receive a set of reflections from the ROI. The method including receiving the set of reflections from at least one target located in the ROI, each reflection is recorded by a receiver from the set of distributed receivers at a corresponding time and at a corresponding coarse location, wherein the coarse locations include unknown position errors. Aligning the set of reflections on a time scale using the corresponding coarse locations of the set of distributed receivers to produce a time projection of the set of reflections for the at least one target. Fitting a line into data points formed from radar pulses in the set of reflections received from the set of distributed receivers, the pulses corresponding to the same target that the set of reflections are aligned in time assuming that the average of the position errors is zero. Determining a distance between the fitted line and each data point. Adjusting the coarse location of the set of distributed receivers on a virtual array using the corresponding distance between the fitted line and each data point. Finally, fusing the radar image using the set of reflections received at the adjusted coarse location of the set of distributed receivers, wherein steps of the method are performed by a processor.

According to another embodiment of the present disclosure, a method a non-transitory computer readable storage medium embodied thereon a program executable by a processor for performing a method. The method for fusing a radar image in response to radar pulses transmitted to a ROI. A transmitter transmits the radar pulses toward the ROI and a set of distributed receivers to receive a set of reflections from the ROI. The method including storing, by the processor, the set of reflections received from at least one target located in the ROI. Wherein each reflection is recorded by a receiver from the set of distributed receivers at a corresponding time and at a corresponding coarse location, wherein the coarse locations include unknown position errors. Aligning, by the processor, the set of reflections on a time scale using the corresponding coarse locations of the set of distributed receivers to produce a time projection of the set of reflections for the at least one target. Fitting, by the processor, a line into data points formed from radar pulses in the set of reflections received from the set of distributed receivers, the pulses corresponding to the same target that the set of reflections are aligned in time, assuming that the average of the position errors is zero. Determining, by the processor, a distance between the fitted line and each data point. Adjusting, by the processor, the coarse location of the set of distributed receivers on a virtual array using the corresponding distance between the fitted line and each data point. Finally, fusing, by the processor, the radar image using the set of reflections received at the adjusted coarse location of the set of distributed receivers.

According to another embodiment of the present disclosure, a system for fusing a radar image in response to radar pulses transmitted to a region of interest (ROI). A transmitter transmits the radar pulses toward the ROI. The system including a set of distributed receivers to receive a set of reflections from the ROI corresponding to the transmitted radar pulses. A processor operatively connected to a memory and in communication with the set of distributed receivers. The processor is configured to: receive the set of reflections from at least one target located in the ROI, each reflection is recorded by a receiver from the set of distributed receivers at a corresponding time and at a corresponding coarse location, wherein the coarse locations include unknown position errors. The processor aligns the set of reflections on a time scale using the corresponding coarse locations of the set of distributed receivers to produce a time projection of the set of reflections for the at least one target. The processor fits a line into data points formed from radar pulses in the set of reflections received from the set of distributed receivers, the pulses corresponding to the same target that the set of reflections are aligned in time, assuming that the average of the position errors is zero. The processor determines a distance between the fitted line and each data point. The processor adjusts the coarse location of the set of distributed receivers on a virtual array using the corresponding distance between the fitted line and each data point. Finally, the processor fuses the radar image using the set of reflections received at the adjusted coarse location of the set of distributed receivers.

The embodiments of our present disclosure include coherent distributed radar imaging by allowing location ambiguities, and on autofocusing for a single sensor array by distributed sensing with multiple sensors. In particular, a multi-static radar imaging approach where one transmitting/receiving radar platform and multiple receiving radar platforms are moving towards a region of interest (ROI) with position perturbations. The embodiments of our present disclosure detect targets inside the ROI. Due to inaccurate positioning and motion errors, the actual array positions are perturbed up to several times a central radar wavelength. Although the image resolution of each sensor array may be low due to its small aperture size, a high-resolution image can be formed by jointly processing the outputs of all distributed arrays with well-compensated position errors. The embodiments of our present disclosure assume a sparse scene, and is realized iteratively by solving series of optimization problems for compensating position-induced phase errors, exploiting target signatures, and estimating antenna positions.

The embodiments of our present disclosure also provide for auto-focus radar imaging for generating a radar image of targets situated in an area of interest using a single moving transmit radar platform or combination of transmitter/receiver along with multiple spatially distributed moving radar receiver platforms or receivers. The moving radar receivers are perturbed with unknown position errors up to several radar wavelengths.

<FIG> is a schematic illustrating a distributed moving radar imaging system <NUM> having distributed arrays of moving radar platforms <NUM> for detecting targets <NUM> in a region of interest (ROI) <NUM>, according to embodiments of the present disclosure. In particular, the distributed radar imaging system <NUM>, can be an airborne platform or vehicle mounted platform, etc, that includes at least one moving transmit/receive platform or transmitter/receiver <NUM>, and a set of M distributed moving similar receiver platforms or receivers <NUM>, <NUM>, <NUM>. It is contemplated that the set of M distributed receivers may be five or more, <NUM> or more or <NUM> or more. Radar pulses <NUM> are transmitted from the at least one transmitter/receiver <NUM>, to illuminate targets <NUM> situated in an area of interest or region of interest (ROI) <NUM>, and the corresponding reflected radar reflections <NUM> are recorded by the multiple distributed receivers <NUM>, <NUM>, <NUM> and <NUM>. The reflections <NUM> can be characterized as a weighted combination of delayed pulses, where complex weights depend on specific target reflectivities and antenna patterns. Given the pulses and reflections, radar images can be generated in a range-azimuth plane according to corresponding weights and delays. The azimuth resolution of the radar images depends on a size of an array aperture,and a range resolution depends on a bandwidth of the pulses.

<FIG> is a block diagram illustrating some steps of a method, according to embodiments of the present disclosure. The method includes step <NUM> of acquiring a set of radar reflections received by a set of receivers from a target in the region of interest (ROI). The set of radar reflections correspond to a transmitted radar signal from a transmitter directed toward the ROI. The set of radar reflections or reflections can be stored in a memory of processor for each receiver and communicated to a central controller for processing.

Step <NUM> of <FIG> includes aligning in time the set of radar reflections, to compensate for a deviation of coarse positions of the set of receivers from positions of the set of receivers forming a virtual array. The coarse positions of the set of receivers are given by real-time GPS signals or by pre-designed stationary positions or moving trajectories. The radar receivers can be stationary or moving along a pre-designed trajectory, and the effective position of each radar receiver where pulse reflections are received forms a virtual array. The virtual arrays are positioned at the same side of the area of interest, where targets are situated. The deviation is determined by subtracting the "coarse position of each receiver in the set of receivers" from the position of each receiver in the set of receivers forming the virtual array. The position deviation is caused by calibration error of stationary positions, or inaccurate GPS. If the deviation, which can be as large as several radar central frequency wavelengths, is not well compensated, the generated radar image will be out of focus. If the deviation is well compensated, the subtraction of the receiver coarse position from the receiver virtual array position should be zero and the corresponding fused radar image is well focused. With proper distance compensation, the radar reflections are aligned in time such that they can add up spatially at the target position to form a focused image of the target in radar imaging process.

Step <NUM> of <FIG> includes fitting a line into pulses corresponding to the same target that the set of reflections are aligned in time. The fitting line is determined by GPS positions or predesigned positions assuming that the average of position errors is zero.

Step <NUM> of <FIG> includes adjusting positions of each receiver on the virtual array, using a corresponding distance, between the "pulse of the corresponding reflection" and the "fitted line". The corresponding distance shows how much the propagation distance of the transmitted radar pulse, i.e., the total distance from the transmitter to the target and back to the receiver, is different from the coarse located transmitter to the target and back to the coarse located receiver.

Step <NUM> of <FIG> includes fusing the radar image from set of reflections using the adjusted positions of the receivers on the virtual array. For each transmitter-receiver pair, a radar image is generated by delay-and-sum radar imaging using the adjusted positions. The fused radar image is generated by coherently sum all radar images of all possible transmitter-receiver pairs.

In <FIG>, parts (a) and (b) are schematics illustrating step <NUM> and step <NUM> of <FIG>. The part (a) is a schematic of step <NUM> of <FIG> illustrating the aligning in time of the set of radar reflections according to their coarse positions of the set of receivers 102x, 103x, 104x, 105x, so as to later determine a deviation of coarse positions of the set of receivers from positions of the set of receivers forming a virtual array.

The part <FIG> is a schematic illustrating step <NUM> of <FIG>. The part (b) shows line t<NUM> is fitted into pulses corresponding to the same target that the set of reflections 102x, 103x, 104x, 105x are aligned in time. The fitting line is determined by GPS positions or predesigned positions assuming that the average of position errors is zero.

<FIG> is a schematic illustrating step <NUM> of <FIG>. <FIG> is a schematic illustrating adjusting positions of each receiver 102t<NUM>, 103t<NUM>, 104t<NUM>, 105t<NUM> for the virtual array, using a corresponding distance, between the "pulse of the corresponding reflection" and the "fitted line" t<NUM>. As noted above, the reason each receiver position is adjusted on the virtual array is because the corresponding distance shows how much the propagation distance of the transmitted radar pulse, i.e., the total distance from the transmitter to the target and back to the receiver, is different from the coarse located transmitter to the target and back to the coarse located receiver.

<FIG> refers to step <NUM> of <FIG>. <FIG> is illustrating that the method can fuse the radar image from the set of reflections using the adjusted positions of the set of receivers on the virtual array upon computing deviations (or adjustments) of the set of receivers, and using the deviations of the set of receivers 102t<NUM>, 103t<NUM>, 104t<NUM>, 105t<NUM> to position the reflections of the set of receivers 102x, 103x, 104x, 105x to the fitted line.

<FIG> is a schematic illustrating a distributed moving radar imaging system <NUM> of <FIG>. <FIG> shows the radar receivers receiving radar pulse reflections 102A, 103A, 104A, 105A, that form the virtual array <NUM>. The radar pulse reflections are from emitted radar pulses from a transmitter toward the ROI and reflected from the ROI toward the receivers to form the virtual array <NUM> of receivers. The distributed arrays of moving radar platforms <NUM> includes at least one radar platform having an antenna cell which is connected to a radar transmitter <NUM> that generates the radar pulses toward the ROI. As noted above, the radar transmitter <NUM> is combined with a receiver <NUM>. The radar receivers <NUM>, <NUM>, <NUM>, <NUM> acquire reflections reflected by targets <NUM> in the area of interest (ROI) <NUM>.

Still referring to <FIG>, the radar receivers <NUM>, <NUM>, <NUM>, <NUM> are moving along a pre-designed trajectory, and the effective position of each radar receiver where the pulse reflections are received 102A, 103A, 104A, 105A, forms the virtual array <NUM> of a set of Nm (m = <NUM>,. , M) elements. The virtual arrays <NUM> are positioned at the same side of the area of interest <NUM>, where targets <NUM> are situated. The radar receivers <NUM>, <NUM>, <NUM>, <NUM> are perturbed with random position errors greater than the radar source pulse center wavelength, but within a predetermined range. The multiple radar receivers <NUM>, <NUM>, <NUM>, <NUM> form multiple distributed non-uniform arrays.

<FIG> is a block diagram of the distributed radar imaging system showing radar platforms <NUM> that communicate to each other and are in communication with a computer 300A. The radar platforms <NUM> are synchronized and can store collected data in a memory <NUM> that is processed by an auto-focus imaging processor <NUM> of the computer 300A. The auto-focus imaging processor <NUM> can perform the radar imaging method to produce an auto-focused high resolution two-dimensional (2D) radar image. The imaging result can be shown in an user interface <NUM> of the computer 300A.

<FIG> is a block diagram of a computer system of the distributed radar imaging system contemplated by the present disclosure, in accordance with some embodiments of the present disclosure. The computer system 300B is in communication with the synchronized radar platforms <NUM> and can store the collected data in the memory <NUM> that is processed by the auto-focus imaging processor <NUM> of the computer 300B. The computer system 300B includes a human machine interface or user interface <NUM> that can connect the computer system to a keyboard <NUM> and display device <NUM>. The computer system 300B can be linked through the bus <NUM> to a display interface <NUM> adapted to connect the system 300B to a display device <NUM>, wherein the display device <NUM> can include a computer monitor, camera, television, projector, or mobile device, among others.

The computer system 300B can include a power source <NUM>, depending upon the application the power source may be optionally located outside of the computer system. The auto-focus imaging processor <NUM> maybe one or more processors that can be configured to execute stored instructions, as well as be in communication with the memory <NUM> that stores instructions that are executable by the auto-focus imaging processor <NUM>. The auto-focus imaging processor <NUM> can be a single core processor, a multi-core processor, a computing cluster, or any number of other configurations. The auto-focus imaging processor <NUM> is connected through a bus <NUM> to one or more input and output devices. The memory <NUM> can include random access memory (RAM), read only memory (ROM), flash memory, or any other suitable memory systems.

Still referring to <FIG>, the computer system 300B can also include a storage device <NUM> adapted to store supplementary data and/or software modules used by the auto-focus imaging processor <NUM>. For example, the storage device <NUM> can store historical data relating to predesigned radar platform trajectories, radar operating frequency bandwidth, transmitted waveform, estimated signal-to-noise ratio, image data relating to target recognition, imaging results using simulated noisy data with different methods dealing with position errors, among other things. The storage device <NUM> can include a hard drive, an optical drive, a thumb-drive, an array of drives, or any combinations thereof.

Still referring to <FIG>, a printer interface <NUM> can also be connected to the computer system 300B through the bus <NUM> and adapted to connect the computer system 300B to a printing device <NUM>, wherein the printing device <NUM> can include a liquid inkjet printer, solid ink printer, large-scale commercial printer, thermal printer, UV printer, or dye-sublimation printer, among others. A network interface controller (NIC) <NUM> is adapted to connect the computer system 300B through the bus <NUM> to a network <NUM>. The image data or related image data, among other things, can be rendered on a display device, imaging device, and/or printing device via the network <NUM>.

Still referring to <FIG>, the image data or related image data, among other things, can be transmitted over a communication channel of the network <NUM>, and/or stored within the computer's storage system <NUM> for storage and/or further processing. Further, the image data or related image data may be received wirelessly or by wire from a receiver <NUM> or transmitted via a transmitter <NUM> wirelessly or wire, the receiver <NUM> and transmitter <NUM> are both connected to the computer system 300B through the bus <NUM>.

The computer system 300B may be connected to external sensors <NUM>, one or more input devices 341a, other computers <NUM> and other devices <NUM>. The external sensors <NUM> may include motion sensors, inertial sensors, a type of measuring sensor, etc. The external sensors <NUM> may include sensors for, speed, direction, air flow, distance to an object or location, weather conditions, etc. The input devices 341a can include, for example, a keyboard, a scanner, a microphone, a stylus, a touch sensitive pad or display.

<FIG> illustrates radar reflections in block <NUM> collected by distributed platforms incorporated with transmitted pulses in block <NUM>, such that the radar reflections <NUM> are representative of graph 120B and the transmitted pulses <NUM> are representative of graph 110B. The radar reflections <NUM> collected by distributed platforms incorporated with transmitted pulses <NUM> are processed using iterative sparsity-driven procedures with data coherence analysis in block <NUM> (see <FIG> regarding the data coherence analysis block <NUM>).

The first step is to compress in block <NUM>, the received reflections <NUM> using the transmitted pulse <NUM>, and initialize the compressed reflections as input of block <NUM>. The compressed received reflections <NUM> in block <NUM> are shown in graph 401B as the output of block <NUM>.

<FIG> refer to the steps of block <NUM> of <FIG>. For example, block <NUM> of <FIG> illustrates the second step that performs data coherence analysis to align signals using cross-correlation such that signals are coherent to each to the maximum extent.

<FIG> refers to the steps of block <NUM> of <FIG> to compute time shifts of different antennas according to data coherence. <FIG> refers to the steps of computing distance shift between corresponding transmitter and receiver antennas.

<FIG> refers to the details of extracting the coherent signal 504B. If this coherent signal 504B corresponds to a target in graph <NUM> of <FIG>, then the target signal 504B and target location 504C is saved in block <NUM> of <FIG> and to be projected to a uniform linear array in block <NUM> of <FIG> to generate graph signal <NUM>, which is then to be processed with delay-and-sum radar imaging <NUM> of <FIG>. The target signal 504B in <FIG> is then subtracted from residual data <NUM>, the updated residual data <NUM> will be examined for a next strongest target detection. If the coherent signal 504B of <FIG> does not corresponds to a target in graph <NUM> of <FIG>, the system stops searching targets, and performs imaging <NUM> on projected data <NUM> of <FIG> to form a focused image <NUM> of <FIG>. The focused image <NUM> of <FIG> is illustrated in graph <NUM> of <FIG>.

We consider a two-dimensional (2D) radar imaging problem in which a total of D distributed radar platforms are moving towards a ROI to detect targets within. As noted above, <FIG> illustrates distributed arrays of moving radar platforms <NUM> for detecting targets <NUM> in the ROI <NUM>. Each radar platform forms a forward-looking virtual array. We use p(t) and P(ω) to denote the time-domain source pulse <NUM> and its corresponding frequency spectrum, respectively, where <MAT>.

The scattered field at location r' due to the target with a phase center at l and the excitation pulse originating from r, can be approximated with the first-Born approximation as <MAT> where S(ω,l) is a complex-valued function of frequency, which accounts for all the terms due to the asymptotic approximation; G(ω, l, r) accounts for propagation from r to l and can be represented by <MAT> where a(r,l) represents the overall magnitude attenuation due to the antenna beampattern and the propagation between r and l, and <MAT> is the phase change term of the received signal relative to the source pulse after propagating distance ∥r - l∥ at speed c. For simplicity, we have omitted the noise term from Eq. (<NUM>).

Without a loss of generality, assume that there are up to K targets, each with a phase center located at a pixel in the ROI image. Let ik ∈ {<NUM>,. ,I} be the pixel index of the kth target and lik be the corresponding location. Let rd,n be the ideal location of the nth element of the dth array, where n ∈ {<NUM>,<NUM>,. ,N} and d ∈ {<NUM>,<NUM>,. Due to position perturbations, the actual measurements are taken at r̃d,n = rd,n + εd,n, where εd,n stands for corresponding unknown position perturbation with <NUM> ≤ |εd,n| ≤ <NUM>λ, and λ is the wavelength of the radar central frequency. The overall signal received by the perturbed array is then a superposition of scattered waves from all targets in the ROI. For the source signal transmitted by the d<NUM> th array, where d<NUM> ∈ {<NUM>,<NUM>,. , D} , we consider measurements at a discrete frequency ωm, where m = <NUM>,<NUM>,. The received reflections <NUM> can be represented by an M × D × N data cube <MAT>, whose entry (m, d, n) is <MAT>.

After range compression using the source pulse P(ω), we end up with an M × D × N data cube <IMG>, whose entry (m, d, n) is <MAT>.

To simplify notation, we define a scalar <MAT> an M × <NUM> unit vector <MAT> and an M × <NUM> exponential vector <MAT>.

The vector <MAT> can then be written in a matrix-vector form as <MAT> where the symbol ∘ represents element-wise product. Here, Γ̃(d,n) = [ϕ<NUM> ∘ <MAT> is an M × I projection matrix of the nth antenna position in the d th array, <MAT> is a I × <NUM> vector of target scattering coefficients. It is important to note that ϕik is a target signature vector independent of antenna positions, which is extracted from measured data efficiently during the imaging formation.

Since the antenna positions r̃d,n are not known exactly, image formation that treats the perturbed array as a uniform array generally yields a de-focused image with its quality related to the position perturbations. In order to perform imaging with autofocus, we solve the following sparsity constrained optimization problem <MAT> where Γ̃= {Γ̃(d,n)}d,n and x = {x(d,n)}d,n. The above optimization problem is similar to the group sparsity formulation that is often used in compressive sensing imaging [<NUM>]. Specifically, it relies on the fact that all unknown vectors share the same non-zero support but have generally different values within the support. However, the autofocusing problem formulated in eq. (<NUM>) is more general than the group sparsity problem since the projection matrices are not identical across all antennas. They share the same target signature vector ϕik, but are different in the unknown exponential term <MAT>.

Motivated by the orthogonal matching pursuit algorithm, we solve (<NUM>) iteratively with maximum of K iterations. At the kth iteration, given the residual data <MAT>, which is initialized as measured data, and updated at each iteration by removing the signals of all the detected targets, we have a degenerated problem <MAT>.

Note that the ℓ<NUM> -norm of vectors { <MAT>} is <NUM>, where the only non-zero component corresponding the strongest target phase center of each iteration. Let the image reconstructed by the residual data <MAT> be x̂res,k. A target is then detected at location lik where the maximum absolute value of x̂res,k is observed as follows <MAT>.

To determine Γ̃, we stack <MAT> to form an M × ND matrix <MAT>.

Similarly, vectors <MAT> were also stacked into an M × ND matrix. The stacked matrix <MAT> is then re-organized as <MAT> where <MAT> is an M × DN rank-one matrix, whose dominant left singular vector is exactly ϕik, and <MAT> is an M × DN exponential matrix parameterized by the distance between the kth target and the perturbed distributed arrays. Based on (<NUM>) and (<NUM>), and given x, Γ̃ can be determined by solving <MAT> where the subscript F represents the Frobenius norm of the matrix. Equation (<NUM>) is then solved by an inner loop in which we alternately update Ψ̃ik by data coherence analysis, described in <FIG> to the steps of block <NUM> of <FIG>, and Eik by dominant target signature analysis, as described later in <NUM>.

<FIG> in block <NUM> shows the data coherence analysis using cross-correlation such that the signals measurement by distributed platforms are correlated to each other to the maximum extent.

<FIG> illustrates the details of block <NUM> in <FIG> specific to the estimate time shift for alignment for each individual potential target. To estimate time lags, we use the cross-correlation (CC) of signals. Specifically, given Ỹres,k and Eik, we compute the time-delay parameter Ψ̃ik by finding the delay corresponding to the maximum of the CC function. However, CC is not concave and, thus, may have multiple local maxima. To reduce ambiguity in the CC function, we extract the k th target response using time gating. Assume that at the k th iteration, we reconstruct an image x̂res,k using residual data <MAT>. With the target location, the residual signal is gated in time as <MAT> where <MAT> is the time-domain residual signal, and <MAT> lik∥ + ∥rd,n - lik∥)/c. Note that the time-gating boundary (<NUM>λ)/c is determined by the maximum position perturbation. It can be tightened by considering the smooth trajectory of each radar platform. Let <MAT> be the time domain signal of the dominant vector ϕik of Eik. We then take <MAT> as a reference, and estimate the time shift of <MAT> in (<NUM>) as <MAT>.

Let <MAT> represent the unknown pulse propagation time from r̃d<NUM>,n to r̃d,n via lik. Based on (<NUM>), and assuming the total propagation time is the same as that of the ideal distributed uniform array, we have the following equations to solve <MAT> for all d ∈ [<NUM>,<NUM>,. ,D] and n ∈ [<NUM>,<NUM>,. ,N], such that the signals in (<NUM>) are coherent at lik after back-propagation, <MAT>.

With the solution <MAT> of (<NUM>), <MAT> is computed using (<NUM>).

Given Ψ̃ik, we determine Eik using singular value decomposition (SVD) of <MAT> [<NUM>]: <MAT> where the superscript * represents the phase conjugate and the superscript H represents the Hermitian transpose. Based on the SVD, we have <MAT> where σk<NUM>, is the largest singular value of Yres,k representing the strength of the k th target, <MAT> is the corresponding right singular vector representing the antenna pattern, and uk<NUM> is the corresponding left singular vector representing target signature, <MAT>.

Since the largest singular value σ<NUM>,ik is related to the target strength, we terminate our algorithm based on the target strength relative to the background noise. Specifically, we terminate when <MAT> is satisfied, where σk<NUM> is the second largest singular value of Yres,k, and ε is a threshold with value <NUM> < ε < <NUM>.

<FIG> is a block diagram of step <NUM> of the iterative method of block <NUM> of <FIG> that includes details of estimating propagation distance between antennas and targets, i.e. the estimated position shift for each antenna, according to embodiments of the present disclosure;
Referring to <FIG>, based on the propagation time between each antenna and all k detected targets after the k th iteration, we estimate the array element positions (<FIG>) by minimizing the cost functions <MAT> and <MAT> for d ≠ d<NUM>. Each of the cost functions above is composed of two parts. The first part minimizes the azimuth discrepancy between the perturbed antenna and its ideal position. The second part restricts the distance in the range direction according to the propagation time. We use normalized target strength σk'<NUM>/ ( <MAT>) to weight the contribution of targets according to their scattering strength. While the cost functions in the optimization (<NUM>) and (<NUM>) are not convex, it might be possible to computationally find their global optimal solutions using the algorithm with a proper initial value of r.

<FIG> is a block diagram of step <NUM> of the iterative method of block <NUM> of <FIG> that includes details of computing time shift of different antennas, according to embodiments of the present disclosure. Note that since the antenna locations are determined based on distance measurements, which are translation and rotation invariant, we assume in our simulations that the mean and the dominant orientation of the perturbed array are the same as the ideal uniform array. To remove the translation and the rotation effects of the perturbed antennas and keep the distance between the perturbed antennas and targets unchanged, a linear transform on both the antenna locations and the target locations is necessary.

Referring to <FIG>, the updated antenna positions are then used to estimate the next target position using the residual data, as schematically illustrated in <FIG>.

Given the estimated projection matrix <MAT> scattering coefficients are computed using least squares, <MAT> where <MAT> is a k × <NUM> vector representing scattering coefficients of the k detected targets and the superscript † denotes the Penrose-Moore pseudoinverse.

A sparse image x̂s,k of the ROI is then reconstructed by assigning <MAT> to the corresponding pixel locations as follows <MAT> for all k' ∈ [<NUM>,. For the purpose of target recognition, a dense image preserving target signature information can also be reconstructed by incorporating the target signatures. We first project data of an ideal side-looking uniform array using k detected target signature dictionaries <NUM> as follows <MAT> where <MAT> has the same expression as <MAT> except using the ideal uniform element position rd,n. Based on the reconstructed data, we then perform delay-and-sum imaging to reconstruct a dense image <NUM> <MAT> where Ψ(d,n) is an M × I exponential matrix related to the ideal uniform array and the whole ROI.

<FIG> is a block diagram of step <NUM> of the iterative method of the detailed review of block <NUM> of <FIG> (i.e. block <NUM> is the auto-focus imaging processor (<NUM>) of the computer 300A of <FIG> and 300B of <FIG>), that includes details of projecting the signal of each detected target to the signal received by a uniform array. Given the input coherent signals <NUM> of detected targets, the projection processed is performed target by target. For example, the coherent signal 402A of target #<NUM> is projected, see details illustrated in <FIG>, according to the difference of propagation distance from the target to each of the receiver antennas and from the target to the corresponding projected uniform distributed antenna, the coherent signal are shifted in time to generate the projected coherent signal 404A. The projected signals of targets are summed together to form projected signal <NUM> of the projected uniform linear array.

<FIG> are schematics illustrating antenna positions before (<FIG>) and after (<FIG>) signal projection. <FIG> illustrates the actual antenna positions before the signal projection. <FIG> illustrates the antenna positions after the signal projection, wherein the receiver antennas are uniformly distributed in a straight line with the same aperture size as the distributed array.

<FIG> is a block diagram of signal projection steps 600a, 600b, 600c, for each target. The projection process is performed for each target and for each antenna position. For each target signal 402A, 402B, 402C, measured by an antenna 402e, the propagation distance is computed given the transmit antenna position 402d, receiver antenna positions 402e and the target position 402f, the propagation distance 700a is computed. Given the target position 402f and projected uniform linear array position <NUM>, the projected propagation distance 700b is computed. The time shift 700c is then computed based on the difference of 700a and 700b. The coherent signal of target 402A, 402B, 402C is then separately shifted in time 700d to form projected coherent signal 404A, 404B, and 404C.

<FIG> is schematic illustrating determining an antenna location based on the distances from the antenna to all targets. Given target locations 130a, 130b and 130c, and the distances 131a, 131b and 131c from an antenna <NUM> to the targets 130a, 130b, and 130c, circles 132a, 132b, and 132c are drawn. The intersection indicates the location of the antenna <NUM>.

<FIG> is schematic illustrating determining a target location based on the distances from the target to all antennas. Given antenna positions <NUM>, <NUM>, <NUM>, and <NUM>, and the distances 702a, 703a, 704a, and 705a, from a target <NUM> to the antennas <NUM>, <NUM>, <NUM>, and <NUM>, circles 702b, 703b, 704b, and 705b are drawn. The intersection of circles 702b, 703b, 704b, and 705b indicates the location of the target <NUM>.

Claim 1:
A method for fusing a radar image in response to radar pulses transmitted to a region of interest (ROI), a transmitter (<NUM>) transmits the radar pulses toward the ROI (<NUM>) and a set of distributed receivers (<NUM>-<NUM>) to receive a set of reflections from the ROI, the method comprising:
receiving (<NUM>) the set of reflections from at least one target (<NUM>) located in the ROI, each reflection is recorded by a receiver from the set of distributed receivers at a corresponding time and at a corresponding coarse location, wherein the coarse locations include unknown position errors;
aligning (<NUM>) the set of reflections on a time scale using the corresponding coarse locations of the set of distributed receivers to produce a time projection of the set of reflections for the at least one target;
fitting (<NUM>) a line into data points formed from radar pulses in the set of reflections received from the set of distributed receivers, the pulses corresponding to the same target that the set of reflections are aligned in time, assuming that the average of the position errors is zero;
determining (<NUM>) a distance between the fitted line and each data point;
adjusting (<NUM>) the coarse location of the set of distributed receivers on a virtual array using the corresponding distance between the fitted line and each data point; and
fusing (<NUM>) the radar image using the set of reflections received at the adjusted coarse location of the set of distributed receivers, wherein steps of the method are performed by a processor (<NUM>).