K-space interpretation of image-scanning-microscopy

Confocal imaging is a technique that improves axial-sectioning and transverse resolution in optical microscopy compared to conventional “widefield” imaging.^1,2 In widefield imaging, the entire field of view (FOV) is illuminated, and all points in the sample are simultaneously imaged to a detector (camera) plane. Confocal imaging allows an improvement in the transverse and axial resolution by illuminating the sample with a scanned focused spot and detecting the signal emerging from that focal spot with a small “confocal” detector conjugated to the illumination spot. The effective size of the confocal detector, which determines the spatial filtering and collection efficiency, is set by a small “confocal pinhole” placed in front of the detector. While the resolution improvement in confocal-microscopy increases with decreasing pinhole size,³ a smaller pinhole lowers the detection efficiency, resulting in a lower signal-to-noise ratio (SNR).

A recently introduced method termed image-scanning microscopy^4–6 (ISM) or pixel-reassignment⁷ allows full use of all signal photons in a confocal scanning system without sacrificing transverse imaging resolution. Moreover, depending on the imaging point spread function (PSF), ISM can also provide an improvement in imaging resolution.^8,9 ISM achieves this feat by using an array of detectors at the image plane instead of the conventional single detector of confocal systems. The center detector of the ISM array collects the same information as would have been collected by a confocal pinhole. However, the neighboring detectors collect light that is otherwise rejected in a confocal system. To construct the ISM image, at each illumination point, $r i n$⁠, the signals from each detector position, $r out$⁠, are reassigned to the midpoint between illumination and detection positions:^2,4 $r ISM = 1 2 ( r i n + r out )$⁠. The reassigned signals are summed over all scan positions, forming an ISM image. ISM was first implemented in incoherent microscopy modalities^4,10,11 and was recently adapted to coherent imaging modalities.^12–15 

Here, we analyze coherent-ISM in the spatial Fourier domain. Utilizing the projection-slice theorem,¹⁶ we show that ISM is equivalent to spotlight synthetic-aperture radar (SAR)^17,18 (Fig. 1), a well-established beam-scanning imaging technique, which utilizes a single detector. As a direct result, we numerically demonstrate that ISM can be performed with a single detector and leverage the k-space analysis to highlight the close connection to oblique-illumination microscopy.¹⁹ 

We begin our analysis by considering the case of aberration-free, single-scattering imaging scenario. We represent the signals collected in coherent-ISM using the reflection matrix formalism.^14,20 In this formalism, the fields that are collected at position $r out$ at the detection plane when the illumination is focused at $r i n$ in the object plane are given by the matrix element $R ( r i n , r out )$ [see coordinate notation in Fig. 1(b)]. To simplify the mathematical derivations and without loss of generality, we consider below one transverse dimension, whose coordinate is given by x.

Note that the general case of non-identical PSFs is analyzed in the supplementary material, Sec. C.

In this matrical representation of Eq. (1), the ISM summation can be interpreted as a sum over the anti-diagonal elements of the reflection matrix [Fig. 1(a)].

This mathematical equivalence implies that the same ISM image can be acquired in two different forms. One, conventionally, utilizes a scanning array in the spatial domain, detecting the reflected fields from a spatially focused scanning illumination beam [Fig. 1(b)]. Alternatively, the sample can be scanned by plane waves at different angles of incidence (given by k_in), and a single detector placed in the far field of the sample that detects the reflected plane wave at the same angle, i.e., the single Fourier component k_out = k_in. This second approach, illustrated in Fig. 1(c), is known as spotlight synthetic aperture radar (SAR) imaging.^18,24

The scanning plane wave illumination of k-space ISM is closely related to oblique-illumination microscopy.¹⁹ In oblique-illumination microscopy (originally proposed by Abbe²⁵), the sample is illuminated by different angled plane-waves, and a camera is measuring the resulting image. In the k-space, the plane wave illumination is manifested as a shift of the high frequencies angular-spectrum information of the sample into the passband of the system, allowing its detection.²⁶ In k-space ISM, each illumination is identical to the plane wave illumination of oblique-illumination microscopy. The difference from oblique illumination microscopy is that only a single Fourier component is detected in each illumination in k-space ISM [Eq. (2)]. Similar to oblique illumination microscopy, k-space ISM measurements, thus, yield extended k-space support. The differences between the techniques are the use of a single detector in k-space ISM rather than the detector-array of oblique-illumination microscopy, which requires a larger number of illuminations, and the angular-spectrum reweighting that is required in oblique illumination microscopy.^19,27

To support and demonstrate our mathematical derivation, we numerically simulated the coherent imaging of a test sample with conventional ISM [Fig. 2(a)] and k-space ISM [Fig. 2(b)]. While the two approaches differ in both their illumination scheme (focused vs plane wave) and detection scheme (array detection in the object plane vs a single detector in the k-space), the resulting images are identical and present an improvement over confocal imaging.

The simulated conventional ISM setup [Fig. 2(a)] consists of a 4-f imaging system, where the scanning laser and the detection array (CAM) are both imaged onto the object plane. A focused illumination beam raster scans the target object plane (x, y). At each illumination position, $( x i n , y i n )$⁠, the reflected field across the image plane, $( x out , y out )$⁠, is measured. The simulated k-space ISM setup [Fig. 2(b)] is based on the Fourier transform of a point illumination (LASER) and confocal detection, from the detector plane (DET) to the object plane using a lens. The point illumination is, thus, converted to a tilted plane wave in the object plane, and the detector measures a single Fourier component, k_out = k_in. Observing Fig. 2(b) reveals that k-space ISM is, in fact, confocal coherent imaging performed in the k-space domain. See simulation parameters in the supplementary material, Sec. A.

In Fig. 3, we compare the two ISM processing approaches and confocal imaging using an experimental ultrasound echography dataset. The dataset represents measurements performed on an acoustic phantom with multi-plane wave transmission.²⁸ Data were acquired using a Verasonics P4–2v probe at a center frequency of $f 0 = 4.8 MHz$⁠, having 64 elements with a total aperture size of $D = 19.2 mm$⁠. The imaged target is composed of five pins [Fig. 3(a), red X's] having a diameter of $0.1 mm$ at a depth of 60 mm. Data were post-processed with the proper phases to perform coherent compounding for either confocal, x-space ISM,¹⁴ or k-space ISM [Figs. 3(a), 3(b), 3(c), respectively]. These experimental results agree with the analytic and numerical investigations [Fig. 3(d)]. See a discussion of the data processing in the supplementary material, Sec. B.

In conclusion, by leveraging the Fourier slice theorem, we show that it is possible to perform coherent ISM with a single detector located at the Fourier plane of the object. This allows the same resolution improvement gained in ISM over confocal imaging without requiring a detector array. This may be interesting for utilizing ISM in other modalities where detector arrays are not accessible. Nevertheless, in this work, we have not analyzed imaging scenarios that include multiple-scatterings or the presence of aberrations. Tackling such propagation effects is an important concern in many imaging applications. While aberrations and scattering can be analyzed using the presented mathematical framework, by considering a distorted PSF, the full analysis of this subject is left for future work. In addition, the k-space acquisition and simple (confocal-like) analysis reduce the memory requirements and may be important also for the reduction of computational burden.

We note that a single detector approach will result in a lower SNR than the detector-array approach in many practical scenarios, where the same total illumination power is used, since only a fraction of the reflected field is detected. On the contrary, the plane wave illumination reduces the power density concentrated on the sample, as compared to focused scanned illumination, and, thus, may allow higher total illumination power in the case where damage threshold of the sample is the limiting factor for the used power.