System and method for a hierarchical Bayesian-map approach for solving inverse problems

9,613,439

14/884841

October 16, 2015

A method of reconstructing an image of an object, the method including: determining, by a plurality of sensors, a waveform based on the object, wherein the plurality of sensors view the object from a plurality of directions; determining, by a pre-processing module, a plurality of measurements of the object using the waveform, wherein the plurality of measurements are arranged in a vector form; determining, by an option module, a sampling matrix, a dictionary, and a noise factor, wherein the sampling matrix represents a geometric arrangement of the plurality of sensors, and the dictionary is pre-selected by the option module; estimating, by an estimation module, a coefficient vector using the measurements, the sampling matrix, and the noise factor; and reconstructing, by a reconstruction module, the image, using the coefficient vector and the dictionary.

The United States of America, as represented by the Secretary of the Navy (Washington, DC, US)

The United States of America, as represented by the Secretary of the Navy (Washington, DC, US)

1. A method of reconstructing an image of an object, said method comprising: determining, by a plurality of sensors, a waveform based on said object, wherein said plurality of sensors view said object from a plurality of directions; determining, by a pre-processing module, a plurality of measurements of said object using said waveform, wherein said plurality of measurements are arranged in a vector form; determining, by an option module, a sampling matrix, a dictionary, and a noise factor, wherein said sampling matrix represents a geometric arrangement of said plurality of sensors, and said dictionary is pre-selected by said option module; estimating, by an estimation module, a coefficient vector using said measurements, said sampling matrix, said noise factor and a global compound Gaussian prior; and reconstructing, by a reconstruction module, said image, using said coefficient vector and said dictionary.

2. A method of reconstructing an image of an object, said method comprising: determining, by a plurality of sensors, a waveform based on said object, wherein said plurality of sensors view said object from a plurality of directions; determining, by a pre-processing module, a plurality of measurements of said object using said waveform, wherein said plurality of measurements are arranged in a vector form; determining, by an option module, a sampling matrix, a dictionary, and a noise factor, wherein said sampling matrix represents a geometric arrangement of said plurality of sensors, and said dictionary is pre-selected by said option module; estimating, by an estimation module, a coefficient vector using said measurements, said sampling matrix, and said noise factor; and reconstructing, by a reconstruction module, said image, using said coefficient vector and said dictionary, wherein said estimating said coefficient vector comprises computing, by a first variable module: a first variable, using a pre-selected non-linear factor; and a multi-scale Gaussian tree structure, using a quad-tree decomposition of said image, said sampling matrix, said dictionary, and said measurements.

3. The method of claim 2 , wherein said estimating said coefficient vectors further comprises: estimating, by a parameter module, a structural parameter based on a parent-child relationship for each node in said tree structure; repeating, by a loop module: said computing of said first variable and said multi-scale Gaussian tree structure, and said estimating of said structural parameter, across said tree structure, until said structural parameter is lower than a first pre-selected threshold.

4. The method of claim 3 , wherein said estimating said coefficient vectors further comprises: computing, by a second variable module, a second variable based on said first variable, said sampling matrix, a variable selection operator, and a second pre-selected threshold; and computing, by a coefficient module, said coefficient vector based on a Hadamard product of said first variable and said second variable.

5. A method of reconstructing an image of an object, said method comprising: determining, by a plurality of sensors, a waveform based on said object, wherein said plurality of sensors view said object from a plurality of directions; determining, by a pre-processing module, a plurality of measurements of said object using said waveform, wherein said plurality of measeaents are arranged in a vector form; determining, by an option module, a sampling matrix, a dictionary, and a noise factor, wherein said sampling matrix represents a geometric arrangement of said plurality of sensors, and said dictionary is pre-selected by said option module; estimating, by an estimation module, a coefficient vector using said measurements, said sampling matrix, and said noise factor: and reconstructing, by a reconstruction module, said image, using said coefficient vector and said dictionary, wherein said estimating said coefficient vector comprises: initializing, by said estimation module, x 0 =|h −1 (Ψ T y)| and n=0, wherein x 0 is an initial approximation of a temporary parameter, h is a nonlinear factor, Ψ is said sampling matrix, y is a vector comprising said measurements, and n is a loop counter; calculating, by said estimation module, a descent direction d n ; determining, by said estimation module, a step size λ; and computing, by said estimation module, x n+1 =x n +λd n , wherein x n is an n th approximation for said temporary variable and x n+1 is an (n+1) th approximation for said temporary variable.

6. The method of claim 5 , further comprising: incrementing, by said estimation module, said loop counter n; computing, by said estimation module, x*, by repeating said calculating said descent direction d n , said determining said step size λ, said computing x n+1 , and said incrementing said loop counter n until a norm of a steepest descent vector is smaller than a pre-selected threshold, wherein x* is a temporary variable.

7. The method of claim 6 , further comprising: computing, by said estimation module, z*=h(x*); calculating, by said estimation module, =diag(S λ [z*]) and Λ R =diag(z*), where [mathematical expression included] calculating, by said estimation module, u*=L −1 R where L= ({tilde over (Ψ)} T Σ v −1 {tilde over (Ψ)}+Λ L −1 Σ u −1 Λ L −1)Λ R and R= {tilde over (Ψ)} T Σ v −1 where vε m is said noise factor; and calculating, by said estimation module, c*=z*⊙u* where c*ε n is said coefficient vector.

8. The method of claim 7 , wherein said reconstructing said image further comprises: calculating, by said reconstruction module, I=Φc* where Iε n is the reconstructed image and Φε d×n is said pre-selected dictionary.

9. The method of claim 8 , wherein said plurality of sensors are independent from each other, wherein said dictionary comprises at least one class of wavelet dictionaries, and wherein said sampling matrix comprises a sampling operator determined by a transmitted waveform associated with said measurements and said geometric arrangement of said plurality of sensors.

10. An imaging device comprising: a plurality of sensors configured to generate a waveform based on an object; a pre-processor configured to determine a plurality of measurements using said waveform, wherein said plurality of measurements are arranged in a vector form; and a central imaging device comprising: an option module configured to determine a sampling matrix, a pre-selected dictionary, a pre-selected non-linear factor, a first pre-selected threshold, a second pre-selected threshold, and a noise factor, wherein option module determines said sampling matrix using a geometric arrangement of said plurality of sensors; an estimation module configured to estimate a coefficient vector using said plurality of measurements, said sampling matrix, said pre-selected dictionary and a global compound Gaussian prior; and a reconstruction module configured to reconstruct image of said object, using said coefficient vector and said pre-selected dictionary.

11. An imaging device comprising: a plurality of sensors configured to generate a waveform based on an object; a pre-processor configured to determine a plurality of measurements using said waveform, wherein said plurality of measurements are arranged in a vector form; and a central imaging device comprising: an option module configured to determine a sampling matrix, a pre-selected dictionary, a pre-selected non-linear factor, a first pre-selected threshold, a second pre-selected threshold, and a noise factor, wherein option module determines said sampling matrix using a geometric arrangement of said plurality of sensors; an estimation module configured to estimate a coefficient vector using said plurality of measurements, said sampling matrix, and said pre-selected dictionary; and a reconstruction module configured to reconstruct an image of said object, using said coefficient vector and said pre-selected dictionary, wherein said estimation module further comprises: a first variable module configured to: compute a first variable using a pre-selected non-linear factor; and determine a multi-scale Gaussian tree structure using a quad-tree decomposition of said sampling matrix, said pre-selected dictionary, and said measurements.

12. The device of claim 11 , wherein said estimation module further comprises: a parameter module configured to determine a structural parameter using a parent-child relationship for each node in said multi-scale Gaussian tree structure; and a loop module configured to control operation of said first variable module and said parameter module across said multi-scale Gaussian tree structure until said structural parameter is lower than said first pre-selected threshold.

13. The device of claim 12 , further comprising: a second variable module configured to determine a second variable using said first variable, said sampling matrix, said variable selection operator and said second pre-selected threshold; and a coefficient module configured to determine said coefficient vector using a Hadamard product of said first variable and said second variable.

14. An imaging device comprising: a plurality of sensors configured to generate a waveform based on an object; a pre-processor configured to determine a plurality of measurements using said waveform, wherein said plurality of measurements are arranged in a vector form; and a central imaging device comprising: an option module configured to determine a sampling matrix, a pre-selected dictionary, a pre-selected non-linear factor, a first pre-selected threshold, a second pre-selected threshold, and a noise factor, wherein option module determines said sampling matrix using a geometric arrangement of said plurality of sensors; an estimation module configured to estimate a coefficient vector using said plurality of measurements, said sampling matrix, and said pre-selected dictionary; and a reconstruction module configured to reconstruct an image of said object, using said coefficient vector and said pre-selected dictionary, wherein said estimation module is further configured to: initialize x 0 =|h −1 (Ψ T y)| and n=0, wherein x 0 is an initial approximation of a temporary parameter, h is a nonlinear factor, Ψ is said sampling matrix, y is a vector comprising said measurements, and n is a loop counter; calculate a descent direction d n ; determine a step size λ; and compute x n+1 =x n +λd n , wherein x n is an n th approximation for said temporary variable and x n+1 is an (n+1) th approximation for said temporary variable.

15. The device of claim 14 , wherein said estimation module is further configured to: increment said loop counter n; and compute x* by repeating said calculating said descent direction d n , said determining said step size λ, said computing x n+1 , and said incrementing said loop counter n until a norm of a steepest descent vector is smaller than a pre-selected threshold.

16. The device of claim 15 , wherein said estimation module is further configured to: compute z*=h(x*); calculate =diag(S λ [z*]) and Λ R =diag(z*) where [mathematical expression included] calculate u*=L −1 R where L= ({tilde over (Ψ)} T Σ v −1 {tilde over (Ψ)}+Λ L −1 Σ u −1 Λ L −1)Λ R and R= {tilde over (Ψ)} T Σ v −1 where vε m is said noise factor; and calculate c*=z*⊙u* where c*ε n is said coefficient vector.

17. The device of claim 16 , wherein said reconstruction module is further configured to reconstruct said image by calculating I=Φc* where Iε n is the reconstructed image and Φε d×n is said pre-selected dictionary.

18. A non transitory program storage device readable by computer, tangibly embodying a program of instructions executable by said computer to perform a method for reconstructing an image of an object, said method comprising: determining, by a plurality of sensors, a waveform based on said object, wherein said plurality of sensors view said object from a plurality of directions; determining, by a pre-processing module, a plurality of measurements of said object using said waveform, wherein said plurality of measurements are arranged in a vector form; determining, by an option module, a sampling matrix, a dictionary, and a noise factor, wherein said sampling matrix represents a geometric arrangement of said plurality of sensors, and said dictionary is pre-selected by said option module; estimating, by an estimation module, a coefficient vector using said measurements, said sampling matrix, and said noise factor; and reconstructing, by a reconstruction module, said image, using said coefficient vector and said dictionary, wherein said reconstructing said image further comprises using hierarchical Bayesian maximum a posteriori and using a global compound Gaussian model as a statistical prior.

19. A non-transitory program storage device readable by computer, tangibly embodying a program of instructions executable by said computer to perform a method for reconstructing an image of an object, said method comprising: determining, by a plurality of sensors, a waveform based on said object, wherein said plurality of sensors view said object from a plurality of directions; determining, by a pre-processing module, a plurality of measurements of said object using said waveform, wherein said plurality of measurements are arranged in a vector form: determining, by an option module, a sampling matrix, a dictionary, and a noise factor, wherein said sampling matrix represents a geometric arrangement of said plurality of sensors, and said dictionary is pre-selected by said option module; estimating, by an estimation module, a coefficient vector using said measurements, said sampling matrix, and said noise factor; and reconstructing, by a reconstruction module, said image, using said coefficient vector and said dictionary, wherein said estimating said coefficient vector comprises computing, by a first variable module: a first variable, using a pre-selected non-linear factor; and a multi-scale Gaussian tree structure, using a quad-tree decomposition of said image, said sampling matrix, said dictionary, and said measurements.

20. The program storage device of claim 19 , wherein said estimating said coefficient vectors further comprises: estimating, by a parameter module, a structural parameter based on a parent-child relationship for each node in said tree structure; repeating, by a loop module: said computing of said first variable and said multi-scale Gaussian tree structure, and said estimating of said structural parameter, across said tree structure, until said structural parameter is lower than a first pre-selected threshold.

21. The program storage device of claim 18 , wherein said reconstructing said image further comprises using global compound Gaussian model as a statistical prior.

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