Enhancement of extended depth of field by digital focusing

With the use of a Bessel-like illumination beam produced by an axicon lens an extended depth of field can be achieved. The resulting illumination pattern exhibits a strong central lobe extending over 300um along the optical axis without diffraction. Here we present a fast digital focusing algorithm to correct for remaining out of focus artifacts, manifest in sidelobes for out of focus scatterers, due to the decoupled Gaussian detection. The waist of the detection in the focal plane is 2.1um, with a Rayleigh-range of ~35um. The FDOCT data is Fourier transformed in the lateral spatial directions. Passing from the spectral (K) to the spatial (Z) domain, the correction term is implemented in a chirped z- transform algorithm, performing a resampling in the K domain. Finally the data is transformed back in the lateral direction to spatial coordinates. The illumination scheme provides a tightly focused spot over the whole axial extent of the illumination needle, providing the high signal to noise ratio necessary to correct for the diffraction effects of the detection. Signals from scatterers multiple Rayleigh distances away from the focal plane are recovered with minimal loss in resolution. With this digital focusing algorithm the performance of the physically extended depth of field is further enhanced, presenting a uniform high lateral resolution of 1.3um, corresponding to an effective numerical aperture of NA=0.24, over more than 300um in the axial direction.

Diffractionless Fourier Domain Microscopy

Cédric Blatter, Theo Lasser, Rainer Leitgeb, Tilman Schmoll

Several methods have been proposed and realized to increase the lateral resolution in microscopy and ultimately to beat the actual diffraction limit as stated by Abbe. Structured illumination for example enhances the lateral resolution by virtually extending the spatial frequency support of the system transfer function. Higher spatial frequencies are in fact mapped into the range covered by the detection transfer function. The set of spatial frequencies can be further enhanced by decoupling illumination and detection and by employing concepts similar to synthetic apertures. We present a novel method for diffractionless microscopic imaging – Fourier Domain or k-Microscopy[1] (FDM). Again we decouple illumination from detection to achieve a lateral resolution that is independent of the detection numerical aperture. Similar to structured illumination we produce a set of fringes covering a band of spatial frequencies on the sample. However, instead of detecting with an area sensor we use a simple PIN detector: By tuning the spatial frequency of the illumination we obtain the Fourier coefficient from the measured total backscattered or transmitted intensity for the particular spatial frequency in the sample dimension normal to the fringe orientation. The fringe pattern is obtained from two collimated coherent beams that are superimposed. The spatial frequency of the pattern is controlled via tuning the wavelength. In order to obtain an unambiguous Fourier representation of the one dimensional sample structure it is necessary to record at least two π/2 phase shifted copies of the frequency response signal. The lateral resolution is then determined by the tuning bandwidth of the laser as well as the subtended angle and the central wavelength of the laser. This relation is in fact very similar to the concept of Fourier Domain Optical Coherence tomography, only that it acts on the transverse instead of the longitudinal resolution. We give first experimental proof of the concept in one dimension using a slowly tuning broad- bandwidth Ti:Sapph laser. We resolve the defined structure of a resolution test target. A particular feature of the method is that the reconstructed sample will appear spatially band-pass filtered due to the finite spectral support given by the tuned fringe frequencies. Hence only the edges of the test target lines are visible. We believe that the concept of FDM will greatly benefit from new developments in rapidly- wavelength tuning source technology that recently entered the field of OCT. 1. M. Geissbuehler, T. Lasser, and R. A. Leitgeb, "K- Microscopy - resolution beyond the diffraction limit," in Progress in Biomedical Optics and Imaging - Proceedings of SPIE(2008).

2011

Coherent transfer functions and extended depth of field

Theo Lasser, Rainer Leitgeb, Christophe Pache, Martin Villiger

To preserve the speed advantage of Fourier Domain detection in Optical Coherence Microscopy (OCM), extended depth of field is needed. With a narrow probing volume that extends over a long axial range, tissue could be measured in vivo and at cellular resolution. To assess and improve the DOF and the lateral resolution, we analyzed the coherent transfer function (CTF) of OCM. Both the illumination and detection optics contribute equally to the overall imaging performance. In the Fourier domain detection, each pixel of the spectrometer has its specific CTF, sampling a different region of the object’s spatial frequency spectrum. For classical optics and increasing numerical apertures these regions start to overlap and bend, which limits the depth of field. Annular apertures, created with Bessel-like beams produced by axicon lenses or phase filters, circumvent these detrimental effects, but introduce strong side lobes. Decoupling the detection and the illumination apertures is needed to provide the flexibility in engineering a CTF that optimizes the lateral resolution and the DOF at the same time all while reducing these side lobes. We evaluated different combinations of Gaussian and Bessel-like illumination and detection optics, both theoretically and experimentally. Using Bessel-like beams as well in the illumination as in the detection paths, but with annular apertures of different lobe radii, we obtained a lateral resolution of 1.3 μm and an extended depth of field of more than 300 μm, which was completely decoupled from the numerical aperture and scalable to high lateral resolution.

The International Society for Optical Engineering Spie-Int Soc Optical Engineering2010

Fourier Sampling in Signal Processing and Numerical Linear Algebra

Amir Zandieh

This thesis focuses on developing efficient algorithmic tools for processing large datasets. In many modern data analysis tasks, the sheer volume of available datasets far outstrips our abilities to process them. This scenario commonly arises in tasks including parameter tuning of machine learning models (e.g., Google Vizier) and training neural networks. These tasks often require solving numerical linear algebraic problems on large matrices, making the classical primitives prohibitively expensive. Hence, there is a crucial need to efficiently compress the available datasets. In other settings, even collecting the input dataset is extremely expensive, making it vital to design optimal data sampling strategies. This is common in applications such as MRI acquisition and spectrum sensing. The fundamental questions above are often dual to each other, and hence can be addressed using the same set of core techniques. Indeed, exploiting structured Fourier sparsity is a recurring source of efficiency in this thesis, leading to both fast numerical linear algebra methods and sample efficient data acquisition schemes. One of the main results that we present in this thesis is the first Sublinear-time Model-based Sparse FFT algorithm that achieves a nearly optimal sample complexity for recovery of every signal whose Fourier transform is well approximated by a small number of blocks (e.g., such structure is common in spectrum sensing). Our method matches in sublinear time the result of Baraniuk et. al. (2010), which started the field of model-based compressed sensing. Another highlight of this thesis includes the first Dimension-independent Sparse FFT algorithm that, computes the Fourier transform of a sparse signal in sublinear runtime in any dimension. This is the first algorithm that just like the FFT of Cooley and Tukey is dimension independent and avoids the curse of dimensionality inherent to all previously known techniques. Finally, we give a Universal Sampling Scheme for the reconstruction of structured Fourier signals from continuous measurements. Our approach matches the classical results of Slepian, Pollak, and Landau (1960s) on the reconstruction of bandlimited signals via Prolate Spheroidal Wave Functions and extends these results to a wide class of Fourier structure types. Besides having classical applications in signal processing and data analysis, Fourier techniques have been at the core of many machine learning tasks such as Kernel Matrix Approximation. The second half of this thesis is dedicated to finding compressed and low-rank representations of kernel matrices, which are the primary means of computation with large kernel matrices in machine learning. We build on Fourier techniques and achieve spectral approximation guarantees to the Gaussian kernel using an optimal number of samples, significantly improving upon the classical Random Fourier Features of Rahimi and Recht (2008). Finally, we present a nearly-optimal Oblivious Subspace Embedding for high-degree Polynomial kernels which leads to nearly-optimal embeddings of the high-dimensional Gaussian kernel. This is the first result that does not suffer from an exponential loss in the degree of the polynomial kernel or the dimension of the input point set, providing exponential improvements over the prior works, including the TensorSketch of Pagh (2013) and application of the celebrated Fast Multipole Method of Greengard and Rokhlin (1986) to the kernel approximation problem.