CTh2C.1.pdf
Classical Optics 2014 © 2014 OSA
Angular Momentum, PSF Rotation, and 3D Source Localization: A Statistical Performance Analysis S. Prasad, R. Kumar, S. Narravula University of New Mexico, Albuquerque, NM 87131
[email protected]
Abstract: We present a computational imaging approach for encoding the 3D location of a point source via a rotating PSF. We also discuss statistical upper bounds on the precision of 3D sub-diffractive source localization, based on Bayesian error analysis. © 2014 Optical Society of America OCIS codes: 100.6640, 110.6880, 110.7348, 170.6900
1.
Introduction
Orbital angular momentum (OAM) states of light can be used in a variety of settings from the practical, as for biomolecular rotation in optical tweezers, to the esoteric, as in certain superdense quantum coding protocols for highinformation quantum communication. In a recent paper [1], one of us has showed their usefulness for encoding axial depth in an imaging system for which a point-spread function (PSF) comprised of a linear combination of a number of low-vorticity OAM states that rotates uniformly with changing defocus over a large range can be generated by means of suitable wavefront coding. Specifically, one subdivides the circular aperture of an imager, e.g., a microscope, into a number of Fresnel zones and endows them with a sequence of spiral phase profiles with a common phase dislocation line and integer winding numbers ml , l = 1, . . . , L, that step up in a regular manner from one zone to the next larger zone. For such a spiral phase mask, one may show that with changing defocus, the defocus-dependent quadratic phase lag is almost exactly compensated by an equivalent rotation of the dislocation line by an angle that is proportional to the defocus distance from the plane of best focus. This amounts to a linear defocus-dependent rotation of the off-center PSF, which can be used to encode the axial coordinate of a point source. For L zones in all, the rate of PSF rotation with defocus is proportional to 1/L, while the shape and size of the PSF remain nearly invariant over L waves of defocus phase at the edge of the pupil. The rotating PSF thus maintains its shape and size approximately over one complete rotation about the Gaussian image point before breaking apart. The most compact PSF structures result for L of order 6-8 and ml = l, l = 1, . . . , L. Figure 1 shows a schematic rendition of the phase mask using a refractive material of varying thickness to yield the requisite spiral phases in the different zones. For such a phase encoded pupil, the coherent PSF K, given by the pupil integral 1 K(⃗s; ζ ) = √ π
∫ u≤1
d 2 u exp[i2π⃗u ·⃗s − iζ u2 − iψ (⃗u)],
(1)
can be shown to be approximately equal to √ √ sin[ζ /(2L)] L l K(s, ϕ ; ζ ) ≈ 2 π exp[iζ /(2L)] i exp[i l(ϕ − ζ /L)]Jl (2π l/L s). ∑ ζ l=1
(2)
Here ⃗s is the image-plane position vector⃗r divided by the in-focus diffraction spot radius, s, ϕ its polar coordinates, λ the imaging wavelength, and ⃗u the pupil-plane position vector ⃗ρ normalized by the pupil radius, ⃗u = ⃗ρ /R. The defocus phase, ζ , is related to the object defocus distance, δ z, from the plane of best focus, the latter a distance l0 away from the pupil, πδ z R2 ζ= . (3) λ l0 (l0 + δ z) The rotation of K with changing defocus is evident in expression (2).
CTh2C.1.pdf
Classical Optics 2014 © 2014 OSA
Fig. 1: A schematic of the optical element with its spiral phase retardation
For arbitrary ψ (⃗u), the coherent PSF obeys a simple conservation law, rather analogous to the probability flux conservation law in quantum mechanics,
∂h 1 == − 2 ⃗∇ · (h⃗∇ψK ), ∂ζ 2π
(4)
⃗ where √ ∇ denotes the two-dimensional gradient operator in the image plane, ΨK (⃗s) the phase of K, defined via K = h exp(iΨK ), and h⃗∇ψK represents a PSF-flux density. For the rotating PSF (2), the flux density vector h⃗∇ΨK is dominated by its azimuthal component, proportional to ∂ ΨK /∂ ϕ , for which the PSF merely circulates without spreading as the defocus, ζ , is varied. By contrast, for the ideal conventional imager, ΨK is azimuthally symmetric, and hence the PSF flux density purely radial, causing significant radial spreading and loss of sensitivity with increasing ζ . 2.
Statistical Bayesian Analysis of Source Super-localization
We analyze here certain statistical upper bounds on the performance of a rotating-PSF-based imager for a complete 3D super-localization and super-resolution of point sources and compare its performance, via these upper bounds, to that of a conventional imager. Two kinds of Bayesian estimators, specifically the mean and mode of the posterior probability density function (PDF), are adopted for our calculations. The first is associated with the minimum meansquared error (MMSE) and the latter with the minimum probability of error (MPE) in a multi-hypothesis testing (MHT) based Bayesian inference. The two error bounds provide somewhat different quantitative metrics of performance, but are closely related at high SNR [2]. The problem of localizing a point source to sub-diffractive uncertainties in 3D may be phrased in terms of the 2 × M possible sub-voxels into which a nominal voxel, minimum error in localizing the source to within one of M⊥ ∥ corresponding to the diffraction-limited resolution volume, is subdivided uniformly. The integers M⊥ and M∥ represent the transverse and axial localization enhancement factors, respectively. The problem can be posed either as a spatial error bound problem, described by the MMSE metric, or as an MHT problem, described by the MPE metric. The 2 × M sub-voxels. Bayesian prior in each case must be chosen as being uniform over the M⊥ ∥ Some preliminary results [3, 4] for the two approaches under a combination of sensor read noise and signal and background based shot noise sources are presented in Figs. 2-3 below. For the MMSE based description of transverse (2D) source localization, we find that at sufficiently high SNR, defined here as the source flux-to-noise ratio (FNR), the 2 MMSE is reduced well below its zero-SNR value, namely the prior variance. A reduction of the MMSE by a factor M⊥ represents an M⊥ -fold transverse super-localization, with the associated FNR providing a lower bound on the FNR needed to achieve it. The conventional imager at best focus yields a rapid decrease in 2D-localization MMSE with increasing FNR, but its behavior is reversed at large defocus, e.g., at ζ = 16 radians for which the MMSE shows little reduction even at FNR = 40 dB. By contrast, the rotating-PSF imager has a rather robust 2D-localization performance, with all of the MMSE vs. FNR curves closely bunched, over any defocus phase between 0 and 16 radians. Figure 3 captures similar 2D-localization trends for the two imagers from the MPE perspective. For N pixels of image data, when N >> 1 it is possible to develop an asymptotic analysis of the MPE for an image-based Bayesian
CTh2C.1.pdf
Classical Optics 2014 © 2014 OSA
2D, pseudoGaussian noise, 16x resolution 20
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Fig. 2: MMSE vs. FNR for conventional and rotating-PSF based imagers for 2D source localization, at different background levels.
Rotating PSF Based 2D Source Localization
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Fig. 3: Plots of MPE vs. FNR for the rotating-PSF imager for two values of defocus, (a) ζ = 0; (b) ζ = 16. (c) The same plot for the conventional imager for the same two values of ζ . The asymptotic MPE values are shown by dashed line segments without any marker symbols.
MHT problem that involves complimentary error functions [4]. The agreement between the exact and asymptotic MPE values, as the figure shows, is quite good at large FNR. We have also confirmed a cross-over behavior of the MPE vs. defocus for 2D localization between the two imagers, since the conventional imager has a better performance at the plane of best focus but rapidly degrades with increasing defocus while the rotating-PSF imager has a rather constant performance across a large range of defocus. This behavior is consistent with the fact that with increasing defocus the rotating PSF maintains its size and shape, but the conventional PSF spreads rapidly and thus loses its sensitivity to localize a source in the transverse plane. Finally, we have also computed the exact and asymptotic values of the MPE for full 3D source super-localization. An interesting competition between transverse and axial (depth) localization enhancement is seen here, with the latter providing the limiting behavior of the MPE in the limit of high FNR. Also, the conventional imager performs poor depth localization at the plane of best focus where it has vanishing first order derivative relative to defocus and thus no first-order sensitivity to depth localization. These and other details of full 3D localization will be presented. This work has been supported by AFOSR under grant numbers FA9550-11-1-0194 and FA9550-09-1-0495. References 1. S. Prasad, “Rotating point spread function via pupil-phase engineering,” Opt. Lett., vol. 38, pp. 585-587 (2013). 2. S. Prasad, “New error bounds for M-testing and estimation of source location with subdiffractive error,” J. Opt. Soc. Am. A 29, 354-366 (2012). 3. S. Narravula, R. Kumar, and S. Prasad, in preparation. 4. S. Prasad, “Asymptotics of Bayesian error probability and rotating-PSF-based source super-localization in three dimensions,” submitted to Opt. Express (2014).
