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

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

⁢

S

λ

⁡

[

z

⁡

(

i

)

]

=

{

1

z

⁡

(

i

)

>

λ

0

z

⁡

(

i

)

≤

λ

;

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

⁢

S

λ

⁡

[

z

⁡

(

i

)

]

=

{

1

z

⁡

(

i

)

>

λ

0

z

⁡

(

i

)

≤

λ

;

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.