Review of Effective Ways for Computing Point Spread Functions

Session

Poster Session 2

Authors

Ratsimandresy Holinirina Dina Miora (1, 2), Gurthwin Bosman (1), Erich Rohwer (1), Rainer Heintzmann (2, 3)

Affiliations

1. Department of Physics, University of Stellenbosch, Private Bag X1, Matieland 7602, South Africa
2. Institute of Physical Chemistry and Abbe Center of Photonics, Friedrich Schiller University Jena, Helmholtzweg 4, 07743 Jena, Germany
3. Leibniz Institute of Photonic Technology, Albert-Einstein Str. 9, 07745 Jena, Germany

Keywords

PSF, super-resolution, fluorescence microscopy, computational method

Abstract text

Any image of a point source in a diffraction-limited system will result in a blurred pattern, the point spread function (PSF). In the case of fluorescence microscopy, the incoherent imaging can be described by a convolution of the object with the PSF, a common approach to significantly improve the image quality tries to undo this convolution. This “deconvolution” can in some cases even beat the diffraction limit as it makes plausible assumptions about the object such as positivity of fluorescence. A successful deconvolution, requires a good model of the PSF. 

A practical way to obtain a PSF is by measuring it experimentally and averaging over images of multiple beads. This approach can be used to estimate the aberration present in the system, which is generally difficult to model.  However, the noise distribution level and small depth of field in the region of interest limits the use of the approach. Studies have been conducted for computing PSFs, but a comprehensive overview of those different techniques is still missing. In this work, we aim to compare various theoretical techniques for computing PSFs and present novel approaches. Not only an accurate and realistic estimation of the PSF is needed, but its computation should also be fast, and memory efficient as experimental data can be large. A good PSF calculation routine will have a big impact in image processing and deconvolution. 

As described by McCutchen [1], the 3D diffraction pattern, obtained by imaging a point source by a lens, is the 3D-Fourier transform of a generalisation of the lens aperture. Our techniques for computing PSFs therefore consist of Fourier based techniques. The fast Fourier-transformation (FFT) is a very handy tool to speed up PSF calculations, but its pitfalls must carefully be circumvented [2]. Mathematical operators such as Chirp-Z transform can be used to tackle some of the pitfalls of the FFT [3]. Four techniques were developed, and they all start by constructing the generalized lens aperture. The amplitude spread function (ASF) is the 3D-Fourier transform of the generalized aperture and the PSF consists of the absolute square of the ASF. We compare the PSF models with the results obtained from the work of Richards and Wolf [4, 5] with excessive oversampling, as a slow but precise gold standard. 

Each technique has its own pros and cons. As expected, Fourier based techniques are fast to compute but the wrap-around effect of the Fourier transform operator cause large errors at higher depth if the window size of the image is too small. The Chirp-Z transform prevents this wrap-around effect adding computational costs but gaining precision. Non-FFT-based calculations, using the Bessel series for instance, may not suffer from inaccuracies due to wrap-around, but often need excessive sampling to not violate energy conservation around the focus. 

This overview and comparison allow us to conclude which calculation technique is best suited for a given size and sampling requirement.

References

[1] McCutchen CW. Generalized aperture and the three-dimensional diffraction image. JOSA. 1964 Feb 1;54(2):240-4.

[2] Goodman JW. Introduction to Fourier optics. Roberts and Company Publishers; 2005.

[3] Rabiner L, Schafer RW, Rader C. The chirp z-transform algorithm. IEEE transactions on audio and electroacoustics. 1969 Jun;17(2):86-92. 

[4] Wolf E. Electromagnetic diffraction in optical systems-I. An integral representation of the image field. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. 1959 Dec 15;253(1274):349-57.

[5] Richards B, Wolf E. Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. 1959 Dec 15;253(1274):358-79.