A system and method involve acoustic source localization using passive sonar and capitalizing on the sparse nature of a source location map (SLM). Two types of sparsity are exploited, namely sparsity in the support of the SLMs and sparsity in the innovations across consecutive SLMs. The first type is motivated by the desire to construct SLMs whose non-zero entries corresponded to locations where sources are present. The second type of sparsity is motivated by the observation that few changes occur in the support of consecutive SLMs. Per time instant, an SLM is obtained as the solution of a regularized least-squares problem, where the regularization terms are chosen to encourage the desired sparse structures in each SLM and the innovations. Each SLM may be obtained via a specifically-tailored, computationally-efficient proximal gradient algorithm.
Please help me write a proper abstract based on the patent claims. CLAIM: 1. A method comprising the steps of: determining a grid of G tentative locations rg to localize more than one acoustic sources; using an acoustic model to compute replicas that correspond to simulated acoustic sources at locations in , wherein the replicas comprise model-predicted Fourier coefficient vectors at F frequencies {ωf}f=1F corresponding to the acoustic pressure field as sampled by an acoustic sensor array having N sensors; collecting, using the acoustic sensor array, time-series data of actual acoustic measurements at each sensor of the acoustic sensor array caused by the acoustic sources; using a short-time Fourier transform (STFT) on the collected time series data partitioned into m blocks to compute Fourier coefficient estimates at frequencies {ωf}f=1F for all N sensors; constructing a set of Fourier coefficient vectors Ym:=[y1m, . . . , yFm] ∈ N×F, where [Ym]n,f ∈ denotes the Fourier coefficient corresponding to ωf for the n-th sensor in the m-th STFT block; modeling Fourier coefficient vectors at ωf for the m-th measurement block of the collected time series data as yfm=Σg=1Gsg,fmpg,f+εm,f, ∀f, where sg,fm denotes the unknown Fourier coefficient associated to the acoustic signature at frequency ωf for a source located at rg, pg,f ∈ N is the replica for ωf corresponding to a source located at rg normalized so that ∥pg,f∥2=1, and εm,f denotes the Fourier coefficients at ωf corresponding to the noise in the m-th block; setting to zero all values of sg,fm associated to tentative locations rgwhere acoustic sources are presumed to be absent; obtaining, at time τm, an estimate Sm as the solution to min S ∈ ℂ G × F  1 2  ∑ f = 1 F    y F m - P f  s f  2 2 + ∑ g = 1 G   [ μ   ϛ g  2 + λ   ϛ g - ϛ g m - 1  2 ] , where S:=[s1, . . . , sF] ∈ G×F, ′g is the g-th row of S, Pf:=[p1,f, . . . , pG,f] ∈ N×G is the matrix of replicas for ωf, and μ, λ>0 are tuning parameters; and generating one or more source location maps (SLMs) over per frequency ωf using Sm, wherein each location on a particular SLM is associated with the magnitude of its corresponding acoustic gain estimate |ŝg,f], wherein estimates of the actual locations of the K acoustic sources, {{circumflex over (r)}k(τm)}k∈κ, correspond to the locations of the K-largest coefficients |ŝg,f| depicted in the SLM. 2. The method of claim 1, wherein the step of determining a grid of tentative locations of an acoustic source includes the step of estimating the number of acoustic sources. 3. The method of claim 1, wherein the step of generating SLMs over per frequency ωf comprises using each column sf of Sm to construct the SLMs per frequency ωf. 4. The method of claim 1, wherein the step of generating SLMs over per frequency ωf comprises generating a broadband SLM over comprising all frequencies ωf used to compute Sm using a whole row of Sm for each rg ∈ . 5. The method of claim 1, wherein the step of generating SLMs over per frequency ωf comprises generating a broadband SLM over using Sm by plotting the pairs (rg, ∥gm∥2) for all rg ∈ , where gm:=[sg,1m, . . . , sg,Fm]′ ∈ F comprises the entries of the g-th row of Sm. 6. The method of claim 1, wherein the estimates of the actual locations of the K acoustic sources {{circumflex over (r)}k(τm)}k∈X indexed by the index-set X ⊂ {1, . . . , K} are given by the locations that correspond to the K-largest entries in the SLM, X ∈ max|x|=K Σκ∈X∥κm∥2. 7. The method of claim 1, wherein the SLMs are generated using an iterative problem solver based upon a proximal gradient method. 8. The method of claim 1, wherein the acoustic sources are selected from surface-based acoustic sources and underwater acoustic sources. 9. The method of claim 8, wherein the acoustic model is an underwater acoustic model and the acoustic sensor array is a hydrophone array. 10. A system comprising: a processor operatively connected to an acoustic sensor array, wherein the processor is configured to perform the steps of: determining a grid of G tentative locations rg to localize more than one acoustic sources; using an acoustic model to compute replicas that correspond to simulated acoustic sources at locations in , wherein the replicas comprise model-predicted Fourier coefficient vectors at F frequencies {ωf}f=1F corresponding to the acoustic pressure field as sampled by an acoustic sensor array having N sensors; collecting, using the acoustic sensor array, time-series data of actual acoustic measurements at each sensor of the acoustic sensor array caused by the acoustic sources; using a short-time Fourier transform (STFT) on the collected time series data partitioned into m blocks to compute Fourier coefficient estimates at frequencies {ωf}f=1F for all N sensors; constructing a set of Fourier coefficient vectors Ym:=[y1m, . . . , yFm] ∈ N×F, where [Ym]n,f ∈ denotes the Fourier coefficient corresponding to ωf for the n-th sensor in the m-th STFT block; modeling Fourier coefficient vectors at ωf for the m-th measurement block of the collected time series data as yfm=Σg=1Gsg,fmpg,f+εm,f, ∀f, where sg,fm denotes the unknown Fourier coefficient associated to the acoustic signature at frequency ωf for a source located at rg, pg,f ∈ N is the replica for ωf corresponding to a source located at rg normalized so that ∥pg,f∥2=1, εm,f denotes the Fourier coefficients at ωf corresponding to the noise in the m-th block; setting to zero all values of sg,fm associated to tentative locations rgwhere acoustic sources are presumed to be absent; obtaining, at time τm, an estimate Sm as the solution to min S ∈ ℂ G × F  1 2  ∑ f = 1 F    y F m - P f  s f  2 2 + ∑ g = 1 G   [ μ   ϛ g  2 + λ   ϛ g - ϛ g m - 1  2 ] , where S:=[s1, . . . , sF] ∈ G×F, ′g is the g-th row of S, Pf:=[p1,f, . . . , pG,f] ∈ N×G is the matrix of replicas for ωf, and μ, λ>0 are tuning parameters; and generating one or more source location maps (SLMs) over per frequency ωf using Sm, wherein each location on a particular SLM is associated with the magnitude of its corresponding acoustic gain estimate |ŝg,f|, wherein estimates of the actual locations of the K acoustic sources, {{circumflex over (r)}k(τm)}k∈κ, correspond to the locations of the K-largest coefficients |ŝg,f| depicted in the SLM. 11. The system of claim 10, wherein the step of determining a grid of tentative locations of an acoustic source includes the step of estimating the number of acoustic sources. 12. The system of claim 10, wherein the step of generating SLMs over per frequency ωf comprises using each column sf of Sm to construct the SLMs per frequency ωf. 13. The system of claim 10, wherein the step of generating SLMs over per frequency ωf comprises generating a broadband SLM over comprising all frequencies ωf used to compute Sm using a whole row of Sm for each rg ∈ . 14. The system of claim 10, wherein the step of generating SLMs over per frequency ωf comprises generating a broadband SLM over using Sm by plotting the pairs (rg, ∥gm∥2) for all rg ∈ , where gm:=[sg,1m, . . . , sg,Fm]′ ∈ F comprises the entries of the g-th row of Sm. 15. The system of claim 10, wherein the estimates of the actual locations of the K acoustic sources {{circumflex over (r)}k(τm)}k∈X indexed by the index-set X ⊂ {1, . . . , K} are given by the locations that correspond to the K-largest entries in the SLM, X ∈ max|X|=KΣκ∈X∥κm∥2. 16. The system of claim 10, wherein the SLMs are generated using an iterative problem solver based upon a proximal gradient method. 17. The system of claim 10, wherein the acoustic sources are surface-based acoustic sources. 18. The system of claim 10 wherein the acoustic sources are underwater acoustic sources. 19. The system of claim 10, wherein the acoustic model is an underwater acoustic model and the acoustic sensor array is a hydrophone array. 20. A system comprising: a processor operatively connected to a hydrophone array having N hydrophones, wherein the processor is configured to perform the steps of: determining a grid of G tentative locations rg to localize more than one underwater acoustic sources; using an underwater acoustic model to compute replicas that correspond to simulated acoustic sources at locations in wherein the replicas comprise model-predicted Fourier coefficient vectors at F frequencies {ωf}f=1F corresponding to the acoustic pressure field as sampled by the N hydrophones; collecting, using the hydrophone array, time-series data of actual acoustic measurements at each hydrophone of the hydrophone array caused by the underwater acoustic sources; using a short-time Fourier transform (STFT) on the collected time series data partitioned into m blocks to compute Fourier coefficient estimates at frequencies {ωf}f=1F for all N hydrophones; constructing a set of Fourier coefficient vectors Ym:=[y1m, . . . , yFm] ∈ N×F, where [Ym]n,f ∈ denotes the Fourier coefficient corresponding to ωf for the n-th sensor in the m-th STFT block; modeling Fourier coefficient vectors at ωf for the m-th measurement block of the collected time series data as yfm=Σg=1Gsg,fmpg,f+εm,f, ∀f, where sg,fm denotes the unknown Fourier coefficient associated to the acoustic signature at frequency ωf for a source located at rg, pg,f ∈ N is the replica for ωf corresponding to an underwater acoustic source located at rg normalized so that ∥pg,f∥2=1, and εm,f denotes the Fourier coefficients at ωf corresponding to the noise in the m-th block; setting to zero all values of sg,fm associated to tentative locations rg where underwater acoustic sources are presumed to be absent; obtaining, at time τm, an estimate Sm as the solution to min S ∈ ℂ G × F  1 2  ∑ f = 1 F    y F m - P f  s f  2 2 + ∑ g = 1 G   [ μ   ϛ g  2 + λ   ϛ g - ϛ g m - 1  2 ] , where S:=[s1, . . . , sF] ∈ G×F, ′g is the g-th row of S, Pf:=[p1,f, . . . , pG,f] ∈ N×G is the matrix of replicas for ωf, and μ, λ>0 are tuning parameters; and generating one or more source location maps (SLMs) over per frequency ωf using Sm, wherein each location on a particular SLM is associated with the magnitude of its corresponding acoustic gain estimate |ŝg,f|, wherein estimates of the actual locations of the K underwater acoustic sources, {{circumflex over (r)}k(τm)}k∈κ, correspond to the locations of the K-largest coefficients |ŝg,f| depicted in the SLM.