# > Computational Optics

# Low-Complexity Robust Phase Imaging from Multiple Intensity Images

Time Line

Ongoing since 2011

Team

JD, Zhong Jingshan, Manuel Vazquez (Universidad Carlos III de Madrid), Rene Claus (UC Berkeley), Lei Tian (UC Berkeley), Laura Waller (UC Berkeley)

Problem

When light propagates through a medium other than vacuum, its amplitude and phase changes as a result of the interaction with the medium. These amplitude and phase perturbations contain important information about the optical properties of that medium. However, the phase cannot be measured directly, and it needs to be reconstructed computationally from the intensity images. The intensity images are typically noisy, and therefore, we are developing novel efficient inference methods for recovering the phase from a set of intensity images captured along the optical axis. Such methods can be useful for applications in medical imaging, neuroscience, and materials science.

Contribution

Waller et al. apply the augmented complex extended Kalman filter (ACEKF) to recover phase from a series of noisy intensity images. They first turn the wave propagation and non-linear observation model in optics into an augmented state space model. The complex field is then inferred with complex Kalman fitering. The algorithm is robust to noise, however, it is computationally complex. On a standard CPU, it takes multiple hours to reconstruct a single phase image.

First, we have developed two novel phase reconstruction algorithms that yield more accurate phase reconstructions, require far less memory, are more than four orders of magnitude less computationally complex, and hence are potentially applicable in real-time:

(1) Diagonalized CEKF: In order to alleviate issues of high computational complexity and storage requirement, we propose a diagonalized complex extended Kalman filter (diagonalized CEKF). The diagonalized CEKF is iterative: it needs to cycle through the set of intensity images repeatedly, yielding more accurate phase reconstruction after each cycle.

(2) Sparse ACEKF: The sparse ACEKF uses an augmented state space model to represent the optics model. We store the covariance in two sparse matrices, and introduce a few additional assumptions to simplify the Kalman filter update equations in ACEKF. It eventually results in a low-complexity noise-robust phase reconstruction algorithm.

We have proved that those two algorithms converge, as long as the initial error is sufficiently small. In Figure 1 and 2, we show reconstruction results and the associated computational complexity respectively.

Figure 1: Results for phase reconstruction. (From left to right) True image, recontruction by method of Waller et al, proposed diagonalized CEKF, and proposed sparse ACEKF.

Figure 2: Computational complexity of phase reconstruction. Recontruction by method of Waller et al. (ACEKF), proposed diagonalized CEKF, and proposed sparse ACEKF.

Second, we introduce an exponential spacing measurement scheme to efficiently reduce the number of defocused intensity images. We perform Gaussian process regression over the exponentially spaced intensity images to estimate the axial derivative in the transport intensity of equation (TIE) phase retrieval. It alleviates the nonlinearity error in the derivative estimation by using the prior knowledge of how intensity varies with defocus propagation in the spatial frequency domain.

Third, we consider the phase recovery of partially coherent illumination created by any arbitrary source shape in Köhler geometry. By extending the Kalman filtering algorithms to the partially coherent case, we recover not only the phase but also an estimate of the unknown illumination source shape (see Figure 3).

In this project, we consider the issues of noise, intensity measurement scheme, nonlinearity error in TIE and partially coherent illumination. Since the measurement of defocused intensity images is experimentally simple and flexible, our methods should find use in optical, X-ray and other phase imaging systems.

Figure 3: Experimental results by coherent Kalman filter and partially coherent Kalman filter with known and unknown sources. (Top) Source shapes measured by imaging the condenser aperture onto a second camera. The red line shows the estimated source size (NA) by the partially coherent Kalman filter with unknown source size. (Second row) The recovered phase by the coherent Kalman filter is blurred for less coherent sources. (Third row) Recovered phase by the sparse partially coherent Kalman filter (with known source size). (Bottom) Recovered phase by the partially coherent Kalman filter with unknown source size.

References

L. Waller, M. Tsang, S. Ponda, and G. Barbastathis, Complex-field estimation by Kalman filtering, Opt. Express 19, 2805–2815 (2011).

Zhong J., Dauwels J, Vázquez M.A., Waller L., Sparse ACEKF for phase reconstruction, Optics Express, 2013 Jul 29;21(15):18125-37. [ PDF ]

Zhong J., Claus R.A., Dauwels J., Tian L., Waller L., Transport of intensity phase imaging by intensity spectrum fitting of unequally spaced defocus planes, Optics Express, May 2014, 22:9, pp. 10661-10674. [ PDF ]

Zhong J., Dauwels J, L. Tian, Waller L., Partially Coherent Phase Imaging with Unknown Source Shapes, Biomedical Optics Express, Vol. 6, Issue 1, pp. 257-265 (2015). [ PDF ]

Zhong Jingshan, Justin Dauwels, Manuel A. Vazquez, Laura Waller, Efficient Gaussian Inference Algorithms for Phase Imaging, ICASSP 2012, Mar 25-30, 2012, Kyoto, Japan. [ PDF ]

J. Zhong, R. Claus, J. Dauwels, L. Tian, L. Waller, Non-uniform sampling and Gaussian process regression in transport of intensity phase imaging, ICASSP 2014, May 4-9, 2014, Florence, Italy, in press. [ PDF ]

Zhong Jingshan, Justin Dauwels, Manuel A. Vazquez, Laura Waller, Low-complexity noise-resilient recovery of phase and amplitude from defocused intensity images, Computational Optical Sensing and Imaging (COSI), 24 June - 28 June 2012, Monterey Plaza Hotel, Monterey, California, United States. [ PDF ]

Zhong Jingshan, Lei Tian, Rene Claus, Justin Dauwels, Laura Waller, Partially coherent phase recovery with Kalman filters, OSA Frontiers in Optics conference, paper FW6A.9, October 2013, Orlando, FL. [ PDF ]

Zhong Jingshan, Lei Tian, Justin Dauwels, Laura Waller, Partially coherent phase microscopy with arbitrary illumination source shape, Computation Optical Sensing and Imaging (COSI 2014), June 22-26, Hawaii, USA. [ PDF ]