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During the last decades, several resolution enhancement methods for optical microscopy beyond diffraction limit have been developed. Nevertheless, those hardware-based techniques typically require strong illumination, and fail to improve resolution in deep tissue. Here we develop a high-speed computational approach, three-dimensional virtual spatial overlap modulation microscopy (3D-vSPOM), which immediately solves the strong-illumination issue. By amplifying only the spatial frequency component corresponding to the un-scattered point-spread-function at focus, plus 3D nonlinear value selection, 3D-vSPOM shows significant resolution enhancement in deep tissue. Since no iteration is required, 3D-vSPOM is much faster than iterative deconvolution. Compared to non-iterative deconvolution, 3D-vSPOM does not need a priori information of point-spread-function at deep tissue, and provides much better resolution enhancement plus greatly improved noise-immune response. This method is ready to be amalgamated with two-photon microscopy or other laser scanning microscopy to enhance deep-tissue resolution.
© 2016 Optical Society of America
1. Introduction
Optical microscopy is key to biological researches since it allows non-invasive and very specific imaging modalities in imaging cells and tissues both in vitro and in vivo. In most of the cases in vivo imaging demands deep tissue investigations, such as in skin [1], cancer [2–4] or brain imaging [5, 6]. Nowadays many deep tissue imaging methods have been developed [7, 8]. However, these methods have to compromise with resolution due to scattering. The resolution of optical microscopy can be illustrated by the full-width-at-half-maximum (FWHM) of the point spread function (PSF). When the imaging depth is less than tens of micrometers, the main mechanism of resolution limitation is diffraction. As a result, the best resolution is about half of wavelength. Nevertheless, for deep-tissue imaging with depth larger than one transport mean free path (tMFP), scattering effect escalate in turbid biological tissues, resulting in broadening of the PSF, as well as image blurring [8, 9].
To overcome scattering, while simultaneously avoid absorption, one of the most-adopted strategies nowadays is to use near-infrared wavelength in the optical window. For example, when using 1300 nm or 1700 nm light, the imaging depth of multi-photon microscopy reaches more than 1 mm [10, 11]. However, with increased imaging depth in non-homogeneous tissues, the resolution still significantly degrades due to multiple refraction and scattering.
Although recently several groundbreaking techniques has revolutionized optical imaging by pushing resolution beyond diffraction limit [12], their abilities are very limited, if any, in deep tissue imaging. For example, stimulated emission depletion (SsecTED) enhances resolution by using a donut shaped stimulation beam [13], but the donut is easily distorted due to increased depth [14, 15]. Saturated excitation (SAX) enhances resolution by saturation of fluorescence [16], but the power loss with increasing depth makes it difficult to achieve saturation threshold [17]. Photo-activated localization microscopy (PALM) [18] and stochastic optical reconstruction microscopy (STORM) [19] are wide-field resolution enhancement microscopies based on blinking of fluorescnece, but its wide-field detection prevents effective optical sectioning for deep tissue observation. Although confocal correlated PALM has been developed recently, imaging depth is still limited to 10 micrometers [20]. Structured illumination microscopy (SIM) achieves resolution enhancement by expanding spatial frequency with structured illumination, which relies on wide-field detection and also degrades in deep tissue [21]. Recently, spatial overlap modulation (SPOM) microscopy was proposed to obtain high resolution image in thick tissue over hundreds of micrometer depth [22]. In SPOM, two laser focuses are overlapped with each other while one of them is spatially modulated across the other at high frequency. By retrieving fluorescence at the second harmonic of the modulation frequency, and combining with two-photon excitation, resolution is effectively enhanced by a factor of two inside a deep tissue. However, it is very slow, and is complicated in terms of instrumentation. Note that most of the resolution enhancement techniques require high power excitation, so photodamage and phototoxicity are important considerations when applying them to fragile biological samples.
Other than pushing spatial resolution experimentally by high-power lasers and novel setups, there are continuous efforts to improve resolution computationally, with the advantages of avoiding long acquisition time and minimizing potential photodamage. In addition, computational methods are compatible to many different forms of microscopy. For example, one of the best known computational resolution enhancement method is deconvolution [23], which is a purely mathematical tool that is able to recover the spatial information by analyzing an over-sampled image. Typically, deconvolution requires a priori information of PSF, and assumes a uniform PSF across the imaging volume. Unfortunately, scattering and refraction cause significant PSF variation in deep tissue. To overcome this issue, one method is blind deconvolution [24], which can reconstruct PSF at each depth by iteration. However, it is extremely time-consuming to converge when dealing with 3D imaging in a large tissue.
Another computational method, virtual spatial overlap modulation microscopy (vSPOM), is proposed recently [25]. It’s a mathematical method based on the principle of SPOM, yet only one beam is necessary. With over-sampling, vSPOM cleverly mimics the spatial modulation in Fourier domain, and provides two-fold resolution enhancement, similar to SPOM. Since no iteration is required, the major advantage of vSPOM is significantly enhanced speed, compared to not only SPOM, but also blind deconvolution. Nevertheless, although vSPOM has been reported to enhance resolution for tissue observation, it was limited to 2D image only. In this work, we will first extend vSPOM toward 3D image reconstruction, and then show its superior performance when imaging deep into the tissue by comparing with deconvolution. Furthermore, the feasibility for deep-tissue resolution enhancement is demonstrated by combination of vSPOM and two-photon microscopy in a brain sample.
2. Mathematical principles of deconvolution and vSPOM
2.1 Deconvolution
In the context of incoherent imaging, it is well known that image is the convolution between the sample and the PSF, as shown in Eq. (1):
, where O is the object, P is the PSF of the imaging system and N is the noise. and are the coordinates of the image and object space, respectively.Now, for a given I, the object O can be recovered through deconvolution. Intuitively, the deconvolution can be simply performed by dividing I with PSF in Fourier domain. However, when approaching the cutoff frequency, the value of P becomes very small, so the noise will be amplified due to small value of divisor. A common method to solve this issue is Tikhonov regularization [11, 26], i.e. to minimize the function in Eq. (2):
, where O′ is the solution of deconvolution, is the regularization parameter, and H is a high-pass filter. Equation (2) contains two terms: the first term is to minimize the difference between P*O′ and the image I, while the second term is to suppress noise by avoiding small division. O′ can be derived by minimizing Eq. (2) in the frequency domain [11, 26], and the result is, where ^ signs denotes expressing a function in frequency domain. u is the coordinates of the frequency domain. is the deconvolution filter to recover the point spreading. If the PSF P is known, O′ is expected to be very similar to the original “object function” O.Although deconvolution with regularization is straightforward and has been demonstrated to enhance image resolution [27], it requires a priori information of PSF, and it assumes that PSF is uniform across the imaging space. Nevertheless, in the context of deep-tissue imaging, the PSF is typically not uniform and is difficult to determine due to multiple scattering and refractions [8, 28]. To relieve the prerequisite knowledge of PSF, blind deconvolution is proposed to enhance resolution by iterations of an assumed PSF [24]. However, it needs a smart initial PSF guess, and usually requires thousands of iteration steps. Hence, blind deconvolution is very time consuming and not suitable for real-time biological applications.
2.2 vSPOM
As we mentioned earlier, SPOM enhances spatial resolution by using two lasers: one is fixed and the other modulates back and forth across the fixed beam. By detecting at the second harmonic frequency of the modulation, SPOM achieves 1D two-fold resolution enhancement along the modulating direction [22]. For 2D imaging, SPOM takes two images along orthogonal directions to reconstruct the image by taking the minimal value at each pixel.
vSPOM, on the other hand, enhances resolution based on virtual two-beam modulation by convoluting an over-sampling image I with a Gabor filter F1, as shown below [25]:
, where S represents the vSPOM image and only one dimension x is considered here. Gabor filter is selected due to its superior performance in resolution enhancement [25]. Figure 1(a) shows the intensity function of the 1D filter F1, where w is the half width at of the virtual fixed laser beam, and R is the displacement of the virtual modulation beam. Following [25], the vSPOM filter only exhibits value within R. It has been shown that the 1D vSPOM filter can effectively enhance 1D resolution as much as the real SPOM. Similar to deconvolution, to enhance resolution, vSPOM needs at least two-fold oversampling than conventional scanning microscopy, so photobleaching speed may increase proportionally.
Fig. 1 (a) 1D, (b) 2D, and (c) 3D vSPOM filters. Ratio of the negative to positive region increases gradually from (a) to (c).
For 2D imaging with vSPOM, instead of reconstructing the images along orthogonal directions as SPOM, vSPOM directly reconstructs filters by rotating the 1D filter on the x-y plane:
where wx and wy are the widths of fixed beam of respective axes; Rx and Ry are the modulation displacements of respective axes. In most of the cases, imaging system is symmetric in lateral dimensions (x and y), so vSPOM filter can be expressed in polar coordinate:The intensity distribution of the 2D filter is shown in Fig. 1(b). whose value also exists only within R. With the convolution in Eq. (4), 2D vSPOM has shown resolution enhancement with a factor of 1.8 [25], by narrowing down the FWHM with the negative amplitude of the cosine term of the filter. To avoid information lost, e.g. a PSF is canceled completely by the negative part of its neighborhood, the integral of the vSPOM filter should be kept positive.
With a fluorophore closely surrounded by many others, the amplitude of the vSPOM signal would not be linearly proportional to the amount of fluorophores in the sample, since the neighboring PSF may cancel each other. However, when dealing with sparse samples (isolated fluorophores, for example), the vSPOM signal can be quantitative.
Compared to deconvolution, vSPOM is not only much faster than blind deconvolution since no iteration is required, but also better than deconvolution with regularization for deep tissue imaging since it is less sensitive to PSF variation. In the next section, we will discuss how to extend vSPOM from 2D to 3D.
2.3 From 2D to 3D vSPOM
In principle, vSPOM should be able to extend into a 3D version in a similar fashion that extends 1D to 2D. Since the PSF is not symmetric along z-axis, the 3D vSPOM filter ought to be expressed in cylindrical coordinates:
where wz is the half width at of the fixed beam and Rz is the modulation displacement, both along the z-axis. The schematic intensity distribution of the filter is shown in Fig. 1(c). By combining Eq. (8) and Eq. (4), the best resolution enhancement ratio of this 3D vSPOM process is only 1.18. The reason toward the reduced enhancement in direct 3D filter is that the negative value of the 3D filter should be less than the 2D filter, since the integral of each filter needs to be positive to avoid information loss.In the quest for a more effective mathematical vSPOM tool to achieve resolution enhancement in 3D, we have designed a novel nonlinear method based on 3D minimal value selection. More specifically, the first step is to decompose a 3D image into three image stacks along xy, yz, and xz planes, respectively. Then, in each stack, a 2D vSPOM filter, based on the PSF along its plane, is applied on all slices. Finally, a 3D image (see Fig. 2) is reconstructed by selecting the minimum intensity of each pixel in all stacks. In this way, much better resolution enhancement is obtained, as will be demonstrated in section 3.2.
Fig. 2 Schematic representation of 3D vSPOM process. (a) Decompose a 3D image into image stacks along xy, yz, and xz planes. (b) Apply the respective 2D vSPOM filter for the plane on all slices in each stack. (c) Select the minimum of each pixel in all stacks and reconstruct a 3D image.
3. vSPOM for deep tissue and 3D resolution enhancement
3.1 vSPOM is scattering- and noise-immune for deep-tissue imaging
As we mentioned earlier, when imaging in a deep tissue, the most deteriorative effect for PSF blurring is scattering. In Fig. 3(a), the first column shows the variation of PSF with increasing depth, with the aid of Monte Carlo simulation [9, 28]. The sampling size (pixel size) is 200nm × 200nm. The axial distance is characterized by transport mean free path (tMFP), which is about 1 mm for mouse brain with infrared radiation [29]. The PSF line profile at 1.7 tMFP is given as the black line in Fig. 3(b), showing combination of one narrow peak from ballistic photons and the other broad peak due to scattering.
Fig. 3 Comparison of deconvolution and vSPOM. (a) Depth dependence of PSF. The first column shows PSF broadening by scattering, simulated by Monte Carlo method. The second and third columns show the resolution enhancement by regularized deconvolution and vSPOM, respectively. The axial position (Z) is in the unit of tMFP. (b) The corresponding profiles at 1.7 tMFP depth. (c) The variation of PSF FWHM versus imaging depth. (d) Comparison of deconvolution (red) and vSPOM (blue) at 1.7 tMFP depth with added 8% noise on the original PSF comparing with in-focus signal.
The second and third columns are the resolution enhancement based on regularized deconvolution and vSPOM, respectively. In both processes, the priori information of PSF is set as the PSF at surface (z = 0). When the imaging depth is not large (< 1 tMFP), the result of deconvolution is slightly better than vSPOM. However, when imaging depth go beyond 1.3 tMFP, vSPOM starts to show its advantage of significantly suppressed scattering background. The advantage is further illustrated by Fig. 3(b), which shows the profiles of the reconstructed PSFs at 1.7 tMFP, and by Fig. 3(c), which gives comparison of FWHM of PSFs versus imaging depth. Apparently, when considering scattering-induced image blurring, vSPOM shows better capability to reconstruct a sharp PSF in deep tissue.
The advantage of vSPOM over deconvolution is even more dramatic when noise is included into consideration. In the simulation of Fig. 3(d), noise with 8% peak-to-peak amplitude is added into the original PSF at 1.7 tMFP depth, i.e. the black line of Fig. 3(b). The upper and lower panels show the deconvolution and vSPOM result respectively, whose intensities correspond to the black line in Fig. 3(b). Obviously, only the latter provides a distinguishable narrow PSF, while the former is governed completely by noise and scattering.
The reason why vSPOM outperforms deconvolution can be understood in the Fourier domain. By converting the vSPOM filter of Eq. (7) into the Fourier space, it becomes a doughnut-shape bandpass filter, whose radius is 1/R, corresponding to the size of PSF before scattering. Accordingly the filter amplifies signal only around its radius 1/R but attenuates all other signals, including both high-frequency noise and low-frequency blurring due to scattering, as shown in Fig. 3(d). Please note that due to the broadband nature of noise, including shot noise from the background light, thermal noise from the detection system, etc, part of the noise still bleeds through the vSPOM filter, and affects the final image.
From Fig. 3, vSPOM apparently provides much better noise and scattering immunity compared to regularized deconvolution. Please note that if the PSF is known at each depth, regularized deconvolution can also achieve similar or even superior results. However, since practically it is difficult to get the information of PSF at each depth, the advantage of current method is that by using the same vSPOM filter throughout a deep-tissue sample, we can effectively suppress the contribution of scattering, while maintaining the capability to enhance spatial resolution.
3.2 Demonstration of 3D resolution enhancement by vSPOM
Figure 4 shows simulated images of two neighboring PSFs along (a) x- or (b) z-axis, that are separated by one FWHM (center to center), making them unresolvable. While simulating the PSFs, two-photon excitation was considered with a 910nm laser focused by a water immersion objective of numerical aperture (NA) = 1.0. The voxel size is 19.5nm × 19.5nm × 78.1nm. The FWHMs of the resultant PSF are 300nm and 1320nm along lateral and axial directions, respectively. After vSPOM, the FWHM is narrowed down to 160nm and 720nm, respectively, as shown in Figs. 4(c) and 4(d). This is equivalent to 1.8 times enhancement in both lateral and axial directions. The enhancement factor is similar to the previous 2D SPOM and vSPOM results [22, 25], but we have successfully pushed toward 3D volume imaging.
Fig. 4 Schematic representation of resolution enhancement in the x-z plane with simulated samples. Two PSFs are originally unresolvable along (a) x and (b) z directions, and become resolvable after applying vSPOM filters, as shown in (c) and (d). (e) and (f) presents the visibility versus separation distance, it’s clear that the resolutions are enhanced in both directions after vSPOM process.
However, it should be noted that reduction of FWHM would not lead to resolution enhancement by the same factor. Figures 4(e) and 4(f) presents visibility of two neighboring PSF versus their distance. Resolution threshold should be defined as the distance when the visibility drops to zero. It is clear that with this definition, resolution is still greatly improved in both x and z axes. Nevertheless, the optimal factor of enhancement becomes 1.4 (assuming a Gaussian PSF).
Figure 5 demonstrates the practical application of 3D vSPOM on brain imaging. The sample is a fixed drosophila brain whose antennal lobe is labeled with green fluorescence protein. In the experiment we used a 910-nm laser and a water immersion objective of NA = 1.0. Size of the image is 24μm × 24μm × 12.5μm, sampling size (voxel size) is 47nm × 47nm × 250nm, and total acquisition time is about 90 seconds. Figure 5(a) shows lateral (xy) and cross-sectional (yz) images of the antennal lobe at ~100μm depth. Figure 5(b) presents the same images after 3D vSPOM process. It’s clear that not only lateral but also axial resolutions are enhanced by 3D vSPOM in real sample. For better comparison, the z-profiles alone the white lines are shown in Figs. 5(c) and 5(d). After 3D vSPOM, three fluorescent cellular groups are identified (red arrows), that are completely invisible in the original image. In addition, in the center of Fig. 5(c), there are three noise peaks (green circle), whose widths are narrower than diffraction limit. With the annular frequency filter design of 3D vSPOM, these noise peaks vanish in Fig. 5(d), manifesting the highly desirable noise-immune response.
Fig. 5 Deep-tissue resolution enhancement by 3D vSPOM in a biological tissue. (a) Two-photon images of antennal lobe in a drosophila brain. (b) 3D vSPOM processed images. Intensity profile of (c) original and (d) 3D vSPOM image along the white lines in Fig. 5(a) and (b).
One general issue for resolution enhancement by filtering is that low-frequency parts could lose. However, this does not occur in Fig. 5. As given in Eq. (4) in the text, vSPOM image comes from the convolution between an oversampled image and the vSPOM filter. As long as we keep the integral of vSPOM filter to be positive, the low-frequency information loss can be avoided.
4. Summary
One of the major limitations of current hardware-based resolution enhancing techniques is the capability to work in deep tissue environment, which imposes severe scattering background. In this work, we demonstrated a computational method, 3D vSPOM, to reduce FWHM by 1.8-fold, and enhance resolution by 1.4-fold, in both lateral and axial directions. This method do not need complicated setup or high-intensity laser excitation, and can minimize the effect of scattering as well as noise under deep tissue imaging. The deep-tissue resolution enhancing capability, along with two-photon excitation, has been verified both in theory and in experiments. Our work paves the way toward deep-tissue high-resolution imaging, and has great potential to combine with other live, real-time, imaging modalities.
Acknowledgments
This study was supported by the Ministry of Science and Technology, Taiwan, under grant MOST-101-2923-M-002-001-MY3, and MOST-102-2112-M-002-018-MY3. SWC acknowledge the generous support from the Foundation for the Advancement of Outstanding Scholarship.
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