Partially coherent phase recovery
A system and method for incorporating partially coherent illumination models into the problem of phase and amplitude retrieval from a stack of intensity images. The recovery of phase could be realized by many methods, including Kalman filters or other nonlinear optimization algorithms that provide least squares error between the measurement and estimation.
1. A system for phase recovery of an intensity image with partially coherent illumination, the system comprising:
(a) a computer processor; and
(b) a memory storing instructions executable on the processor;
(c) the instructions, when executed, performing the steps comprising:
(i) mathematically modeling partial coherence with respect to the intensity image;
(ii) formulating a statespace Kalman filter from the partial coherence model; and
(iii) applying a standard Kalman filter to the statespace Kalman filter to estimate a phase with respect to the intensity image;
(iv) applying a sparse model to the Kalman filter to reduce computational complexity;
(v) wherein the sparse model optimizes according to:
â n =K H {circumflex over (b)} n
Q n ={circumflex over (Q)} n −( {circumflex over (Q)} n +{circumflex over (P)} n )( {circumflex over (Q)} n +{circumflex over (P)} n +( {circumflex over (Q)} n )*+( {circumflex over (P)} n )*+ qS −2 ) −1 ( {circumflex over (Q)} n +( {circumflex over (P)} n )*)
P n ={circumflex over (P)} n −( {circumflex over (Q)} n +{circumflex over (P)} n )( {circumflex over (Q)} n +{circumflex over (P)} n +( {circumflex over (Q)} n )*+( {circumflex over (P)} n )*+ qS −2 ) −1 ( {circumflex over (P)} n +( {circumflex over (Q)} n )*)
b n ={circumflex over (b)} n +( Q n +P n ) K diag( â n ) C n R −1 ( I n −γ|â n | 2 );
where â n is the state predicted from the previous n−1 observations, I n and b n are discretized column vectors, C n describes convolution of scaled source intensity distribution, and K is the discrete Fourier transform (DFT) matrix.
2. A system as recited in claim 1 , wherein the image comprises a stack of defocused intensity images; and
wherein a propagation transfer function is applied between captured intensity images.
3. A system as recited in claim 1 , wherein the instructions are configured for estimating a phase with respect to a partially coherent intensity image generated from one or more of: optical microscopes, electron imaging systems, tabletop X-ray imaging systems, synchrotron imaging systems, and lithography aerial imaging systems.
4. A system as recited in claim 1 , wherein the instructions are configured for estimating phase for an image having an arbitrary shape of Köhler illumination.
5. A system as recited in claim 1 , wherein the image comprises a stack of intensity images; and
wherein phase or amplitude filters or other linear optical systems are applied as a transfer function between captured intensity images.
6. A method for phase recovery of an intensity image with partially coherent illumination, the method comprising:
(a) mathematically modeling partial coherence with respect to the intensity image;
(b) formulating a state-space Kalman filter from the partial coherence model; and
(c) applying a standard Kalman filter to the state-space Kalman filter to estimate a phase with respect to the intensity image;
(d) wherein said method is performed by executing programming on at least one computer processor, said programming residing on a non-transitory medium readable by the computer processor;
(e) applying a sparse model to the Kalman filter to reduce computational complexity;
(f) wherein the sparse model optimizes according to:
â n =K H {circumflex over (b)} n
Q n ={circumflex over (Q)} n −( {circumflex over (Q)} n +{circumflex over (P)} n )( {circumflex over (Q)} n +{circumflex over (P)} n +( {circumflex over (Q)} n )*+( {circumflex over (P)} n )*+ qS −2 ) −1 ( {circumflex over (Q)} n +( {circumflex over (P)} n )*)
P n ={circumflex over (P)} n −( {circumflex over (Q)} n +{circumflex over (P)} n )( {circumflex over (Q)} n +{circumflex over (P)} n +( {circumflex over (Q)} n )*+( {circumflex over (P)} n )*+ qS −2 ) −1 ( {circumflex over (P)} n +( {circumflex over (Q)} n )*)
b n ={circumflex over (b)} n +( Q n +P n ) K diag( â n ) C n R −1 ( I n −γ|â n | 2 );
where â n is the state predicted from the previous n−1 observations, I n and b n are discretized column vectors, C n describes convolution of scaled source intensity distribution, and K is the discrete Fourier transform (DFT) matrix.
7. A method as recited in claim 6 , wherein the image comprises a stack of defocused intensity images; and
wherein a propagation transfer function is applied between captured intensity images.
8. A method as recited in claim 6 , wherein the method is configured for estimating a phase with respect to a partially coherent intensity image generated from one or more of: optical microscopes, electron imaging systems, tabletop X-ray imaging systems, synchrotron imaging systems, and lithography aerial imaging systems.
9. A method as recited in claim 6 , wherein the method is configured for estimating phase for an image having an arbitrary shape of Köhler illumination.
10. A system as recited in claim 6 , wherein the image comprises a stack of intensity images; and
wherein phase or amplitude filters or other linear optical systems are applied as a transfer function between captured intensity images.
11. A system for phase recovery of an intensity image with partially coherent illumination, the system comprising:
(a) a computer processor; and
(b) a memory storing instructions executable on the processor;
(c) the instructions, when executed, performing the steps comprising:
(i) mathematically modeling partial coherence with respect to one or more intensity image having partially coherent illumination;
(ii) applying a nonlinear least square error function to fit over the one or more measured intensity images; and
(iii) estimating a phase with respect to the one or more intensity images;
(iv) wherein applying a nonlinear least square error function comprises:
finding an optimal estimation to minimize an average square error between measurement and predicted measurement (cost function) with respect to the one or more intensity images;
minimizing the cost function using nonlinear least square error method; and
deriving a gradient and Hessian;
wherein the gradient and Hessian are computed according to:
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12. A system as recited in claim 11 , further comprising:
computing a multiplication of the Hessian with vector with Fast Fourier transform (FFT).
13. A system as recited in claim 11 , further comprising solving the linear equations system:
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14. A system as recited in claim 13 , wherein the linear equation is solved as a function of a conjugate gradient method iteratively by computing the gradient and the multiplication of the Hessian with a vector with FFT.