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Three-dimensional image formation in microscopy is greatly enhanced by the use of computed imaging techniques. In particular, Interferometric Synthetic Aperture Microscopy (ISAM) allows the removal of out-of-focus blur in broadband, coherent microscopy. Earlier methods, such as optical coherence tomography (OCT), utilize interferometric ranging, but do not apply computed imaging methods and therefore must scan the focal depth to acquire extended volumetric images. ISAM removes the need to scan the focus by allowing volumetric image reconstruction from data collected at a single focal depth. ISAM signal processing techniques are similar to the Fourier migration methods of seismology and the Fourier reconstruction methods of Synthetic Aperture Radar (SAR). In this article ISAM is described and the close ties between ISAM and SAR are explored. ISAM and a simple strip-map SAR system are placed in a common mathematical framework and compared to OCT and radar respectively. This article is intended to serve as a review of ISAM, and will be especially useful to readers with a background in SAR.

Traditional sensing modalities such as X-ray projection imaging [

The application of computed imaging techniques to the imaging and non-imaging sensor systems listed above has been revolutionary: X-ray projection imaging has evolved into computed tomography [

This article is focused specifically on ISAM imaging technologies. In addition to the broad commonality ISAM has with other computed imaging techniques, it has strong physical and mathematical connections to a family of instruments including SAR, synthetic aperture sonar [

In the following section, OCT, the forerunner of ISAM, is described. In Sec. 3 a general framework for ISAM, OCT, SAR and radar is developed. The distinctions between the ISAM/SAR and OCT/radar models are discussed within this framework in Sec. 4. In Sec. 5 it is shown how the models used lead to a simple Fourier-domain resampling scheme to reconstruct the imaged object from the collected data. Simulated and experimental results are shown in Sec. 6, while alternative ISAM instrument geometries are briefly discussed in Sec. 7. Conclusions and references appear at the end of this article.

An obvious distinction between ISAM and SAR is the spectrum of the electromagnetic field used to probe the sample—ISAM operates in the near infrared (IR), while most SAR systems operate in the radio spectrum. Probing in the near-IR allows the formation of an image with resolution on the order of microns. Additionally, in many biological tissues the near-IR spectral band is primarily scattered rather than absorbed [

OCT combines interferometry, optical imaging, and ranging. Due to its sensitivity to wavelength-scale distance changes, interferometry has been an important tool in physics (e.g., Young's experiment [_{c}_{c}

As described above, depth discrimination in OCT is achieved via coherence gating, while transverse resolution is achieved using focusing optics. Ideal focusing optics would produce a thin collimated beam in the sample, described as a pencil beam in _{0} of the beam. As illustratted in _{0} and _{0}, and the depth of focus are competing constraints in OCT. When the coherence gate is set to image planes outside of the depth of focus, the transverse resolution suffers as the beam becomes wider.

ISAM uses computational imaging to overcome the trade-off between depth of focus and resolution. By accurately modeling the scattering processes and the data collection system, including the defocusing ignored in OCT image formation, the scattering properties of the object can be quantitatively estimated from the collected data. As in SAR, diffraction-limited resolution is achieved throughout the final image. for both ISAM and SAR the key to this capability is the coherent collection of a complex data set.

Interferometric microscopes [

In both SAR and ISAM an electromagnetic wave is used to probe the object, the detection apparatus is scanned in space, and a time-of-flight measurement is used to image an additional spatial dimension. Thus, in a fundamental sense, the connection between the data and the object is determined by the same physical laws in either case. This analogy can also be extended to other wave-based techniques such as ultrasound and seismic migration imaging. In this section a general model for radar, SAR, OCT and ISAM techniques is presented. While there are significant differences in system scale and operation, see

As shown in ^{T}^{T}^{T}^{T}_{0} ·_{0}_{0}, _{0}) in the Fourier domain.

Consider the scattered field returned to the aperture when the aperture is offset from the origin by _{r}_{r}

It is often convenient to represent the temporal convolution seen in

The backscattered field incident on the detecting aperture is represented in _{R}_{r}_{s}

Expressing

While coherent detection of _{s}

The response times of optical detectors are generally of such a scale that the measured data can be considered a long-time average over optical time scales. Assuming that the fields in the system are statistically stationary and ergodic (see [_{T}

Because Γ_{rr}_{ss}_{T}_{sr}_{sr}

Using the definition of Γ_{sr}_{rr}

The identical forms of

The instrument modality described in the section above and illustrated in

In Fourier-domain OCT or ISAM the collected data are,
_{0} represents the fixed delay on the reference arm. Note that the Fourier-domain reference and sample fields appearing above are spectral domain representations of random processes. These Fourier domain representations are assumed to exist, at least in the sense of mean-square stochastic convergence of the Fourier integral [

The first term of

The Fourier spectrum, _{0})_{0} to be shorter than the least time-of-flight in sample arm plus the coherence length _{c}

In this section equivalent detection models have been posed for OCT and radar, as represented by

As shown by

As described in Sec. 3.1, the time-domain kernel _{‖} is the transverse component of _{‖}) describes the width of the illuminating beam, υ (_{‖}) describes the width of the detection sensitivity, and _{d}

The temporal delay _{d}_{‖}) = υ (_{‖}). Appealing to ^{2}(_{‖}) and axial extent determined by Γ_{rr}

It is convenient to take

The computed imaging approaches of SAR and ISAM are based on models that more closely approximate solutions of Maxwell's equations. Contrary to the assumptions made in OCT, the transverse and axial system responses cannot be decoupled accurately, due to the beam-spreading illustrated in both

The kernel ^{2}(

When the same aperture is used for both illumination and detection (as is typically the case), reciprocity can again be invoked to show that the illumination and detection patterns are equal. Furthermore, the illumination field

The angular spectrum of _{‖} and _{‖} are each one-dimensional. However, the electromagnetic fields present in the system spread in three dimensions. In the simple strip-map SAR system considered here, the SAR aperture track and the object are both assumed to lie in the ^{−1/2} decay so that, for the SAR system,

The angular spectra _{‖}, _{s}_{x}_{‖}, _{‖}, _{‖}, _{s}_{x}

The forward model used in SAR and ISAM (

The linear integral equation of

Since the ISAM forward model is well defined, the inverse problem can, in principle, be solved using numerical techniques. However, an approximation to the forward model allows a more elegant, and significantly more efficient [

The angular spectrum representations seen in ^{−1}.

As a first step towards the solution of the inverse problem, it is useful to recognize that the transverse part of the integral appearing in

As illustrated in

Applying the method of stationary phase in two dimensions gives the ISAM result,
_{D}_{‖}, ^{−1} appearing above describes the signal decay away from focus. In SAR, the method of stationary phase in one dimension is applied to a kernel based on the angular spectrum of ^{−3/2}.

As seen in _{‖},

In the focal region, the integrand of

The approximated models described above can be substituted into the data model of _{‖}, _{F}_{‖}, _{‖}, _{D}_{‖}, ^{3/2} in the SAR case) for

In _{‖}, _{‖},

In either case, the remaining integral in

The equivalent Fourier mapping for OCT, found from the kernel of

The relation given in

Starting with the complex data _{‖}, _{‖},

Implement a linear filtering, i.e., a Fourier-domain multiplication of a transfer function with _{‖}, _{‖},

Warp the coordinate space of _{‖},

Take the inverse three-dimensional Fourier transform to get an estimate of

If required, multiply the resulting estimate by

The operations described above are computationally inexpensive and allow a fast implementation of ISAM processing [

In this section ISAM images are compared to those obtained using standard OCT methods. The high quality of the results obtained validates the calculations made above, while also showing that the approximations made to the forward model in Sec. 5, do not introduce significant error in the solution to the inverse problem.

Numerical simulations of the ISAM system are useful for providing a theoretical corroboration of the proposed methods in a tightly controlled and well understood environment. In

The data were produced using the focused vector beam formulation given in [

The magnitude of the spatial-domain OCT data gives a broadly spread and low-amplitude response. Ideally the image would be point-like, corresponding to the point scatterer. The blurring observed is due to the scatterer being in the out-of-focus region. When the OCT image is examined in the Fourier domain, curved phase fronts can be seen. For the offset point scatterer imaged, the Fourier spectrum should have flat phase fronts parallel to the _{x}_{y}

The Fourier resampling of ISAM can be seen to take the curved OCT phase fronts to the expected straight lines. When the ISAM image is represented in the spatial domain, the desired high-amplitude, point-like image is seen. These simulations lend strong support to ISAM, as the detailed, vectorial forward model is inverted accurately by a simple Fourier-domain resampling only.

Beyond simulations, the next step in ISAM validation is to image an engineered object (i.e. a phantom) with known structure. Here the phantom was constructed by embedding titanium dioxide scatterers, with a mean diameter of 1

ISAM processing, including dispersion compensation [

Out of focus blurring is clearly visible in the collected data. This blurring limits the depth of field in OCT. The ISAM reconstruction can be seen to bring the out-of-focus regions back into focus, as evidenced by the point-like features in the image, which correspond to individual titanium dioxide scatterers. It should be noted that the point-like reconstructions observed are produced by the physics-based computational imaging, not by the use of any assumed prior knowledge of the sample, e.g., [

To further illustrate the SAR-ISAM analogy, ISAM and SAR images are compared below. Strip-map radar and SAR images from a linear rail SAR imaging system [

OCT and ISAM are primarily biological imaging methods. As such, the most important capability of ISAM is the imaging of tissue. As described in [

The improvement observed in the ISAM reconstructions has significant consequences in terms of the diagnostic utility of the images. In the out-of-focus OCT images, the cellular structure is almost entirely lost, while in the ISAM reconstructions, significant features can be seen on the micrometer scale. For example, cell membranes can be recognized, and the boundary between the adipose and fibrous tissue can be clearly seen. There is also a strong correspondence to the histological sections, although embedding, sectioning and staining of the tissue disrupt the sample to some extent. ISAM, unlike OCT, can be seen to allow diffraction-limited imaging at all planes within the sample, rather than just at the physical focus. As a result, significantly more information regarding the tissue can be extracted without increasing the measurement duration or scanning the focal plane. In contrast to the histological images, the structure visible in the ISAM images is observed without destruction of the sample. This suggests ISAM may be particularly useful in applications where in vivo imaging over a large tissue volume is preferable to biopsy.

ISAM is a microscopic imaging technique and is implemented on a bench-top scale. This provides significant flexibility in the design of alternative ISAM modalities. In this section some alternative ISAM instruments are briefly discussed.

To achieve a maximum-resolution image it is necessary to use the highest possible numerical aperture objective lens (high-numerical-aperture OCT is often known as optical coherence microscopy [

In the vectorial system, scattering from the object is recognized as being dependent on the polarization state of the relevant fields. As a result, the object is not a scalar function

It can be shown that the data then depend on the scattering tensor as [

It can be seen from

Full-field OCT systems [

The spatial-domain kernel of

Confocal ISAM is analogous to SAR, and both techniques share the Stolt mapping. The full-field ISAM mapping of

Rotationally-scanned ISAM [

The complex analytic signal, _{θ}_{θ}_{θ}_{θ}

In a recent analysis [

ISAM is a computed imaging technique that quantitatively estimates a three-dimensional scattering object in broadband coherent microscopy. The solution of the inverse problem allows the reconstruction of areas typically regarded as out of focus. The result obviates the perceived trade-off between resolution and depth of focus in OCT.

ISAM, like OCT, is a tomographic method, i.e., the images produced are truly three-dimensional. While ISAM addresses an inherent weakness in OCT, namely the need to scan the focus axially to obtain images outside of the original focal plane, ISAM is not merely a method to refocus the field computationally. Refocusing may be achieved from a single interferometric image at a fixed frequency, but the resulting image is still inherently two-dimensional, failing to unambiguously distinguish contributions to the image from various depths. As in other ranging technologies, the broadband nature of ISAM allows a true three-dimensional reconstruction.

ISAM and SAR are closely related technologies, to the point where they can be cast in the same mathematical framework. Both techniques employ a Fourier domain resampling, based on the Stolt mapping, in the inverse processing. While the mathematics of the two systems are closely related, each uses a significantly different region of the electromagnetic spectrum and images objects of com-mensurately different scales. In SAR, translation of the aperture and computational imaging allow the synthesis of a virtual aperture of dimension dependent on the along track path length, rather than the physical aperture size. This larger synthetic aperture produces an image of higher resolution than would otherwise be achievable. In OCT the limitations on the size of the physical aperture (i.e., the objective lens) are not the limiting factor, rather the image acquisition time becomes prohibitively long if the focal plane must be scanned through an object of extended depth. The computational imaging in ISAM gives diffraction-limited resolution in all planes, not just at the physical focus, and hence eliminates the need for focal-plane scanning.

ISAM and SAR are examples in the broad class of modalities known as computed imaging. Like almost all computed imaging modalities in common practice today, they are based on the solution of linear inverse problems. Linear inversion problems offer advantages such as the option to pre-compute and store the elements of an inversion kernel for rapid computation of images from data. Moreover, error and stability may be well understood and there exist a wealth of well-studied methods for regularization (stabilization) of the inversion algorithms. ISAM and SAR are also members of the more restrictive class of problems that may be cast as data resampling. To arrive at the resampling view of these problems, the data must be Fourier transformed and the resampled data Fourier transformed again. Thus the methods take advantage (are even reliant on) one of the greatest advances in applied mathematics in the last half-century, the fast Fourier transform [

The authors would like to thank Gregory L. Charvat for providing the SAR data seen in

A basic illustration of an OCT system. Light traveling in one arm of a Michelson interferometer is focused into the sample. The length of the reference arm can be adjusted using a moveable mirror. The reference light and the light backscattered from the sample interfere at the detector.

Illustration of focusing in OCT and the trade-off between depth of focus and resolution (figure adapted from [_{0} and the depth of focus _{c}

An illustration of the differences between the data acquisition geometries in SAR and ISAM. SAR involves a one-dimensional scan track, while ISAM scans over a plane. Unlike SAR beams, ISAM fields include a region within the object that is in focus. Note that the same aperture is assumed for both transmission and reflection in SAR; similarly the source is imaged onto the detector by the reference arm in ISAM (see

A geometric illustration of the Stolt mapping relating a point [_{‖}, _{‖}, −2_{z}_{‖}/2,

Simulated OCT image from a point scatterer located at (0, 0, 10)

Images of titanium dioxide scatterers—OCT image before dispersion compensation (a), OCT image after dispersion compensation (b), and ISAM reconstruction (c). This figure is adapted from [

Planar

Three-dimensional renderings of the OCT (a) and ISAM (b) images of titanium dioxide scatterers. Out of focus blur can be seen in the OCT image, while the ISAM reconstruction has isotropic resolution. Note that the axial axis has been scaled by a factor of 0.25 for display purposes. This figure is adapted from [

Raw strip-map radar image of a 1:32 scale model of a F14 fighter aircraft before Stolt Fourier resampling (a), and after Stolt Fourier resampling (b).

Breast tissue is imaged according to the geometry illustrated in the rendering in the upper left. Data are shown in the

An illustration of the rotationally-scanned ISAM system. A single-mode fiber delivers light to focusing optics which project the beam into the object. The beam is scanned linearly inside a catheter sheath and is rotated about the long catheter axis. This figure is adapted from [