Dual deconvolution in multiphoton structured illumination microscopy for deep-tissue super-resolution imaging

Abstract

Imaging in thick biological tissues is often degraded by sample-induced aberrations, which reduce resolution and contrast, particularly in super-resolution techniques. While hardware-based adaptive optics (AO) using wavefront shaping can correct these aberrations, their complexity and cost hinder widespread adoption. Here, we present a computational AO framework for multiphoton structured illumination microscopy, enabling deep-tissue super-resolution imaging with minimal hardware modifications. By replacing the photodetector with a camera from the conventional laser-scanning multiphoton microscope, we capture a sequence of scanned images. Using virtual structured illumination, we develop a dual deconvolution algorithm that independently corrects excitation and emission aberrations, recovering an aberration-free object spectrum with an extended spatial frequency bandwidth. We experimentally validate this framework through two-photon super-resolution imaging, achieving a lateral resolution of 130 nm—one-fourth of the emission wavelength—at a depth of 180 μm in thick mouse brain tissue, where conventional deconvolution fails to maintain super-resolution capability. This approach provides a cost-effective and accessible alternative to hardware-based AO, expanding the potential for high-resolution deep-tissue imaging in biological research.

The authors present a computational adaptive optics framework for multiphoton structured illumination microscopy, enabling deep-tissue super-resolution imaging with minimal hardware modifications. Their dual deconvolution algorithm independently corrects excitation and emission aberrations, and they achieve a lateral resolution one-fourth of the emission wavelength at a depth of 180 µm in mouse brain tissue.

Introduction

Fluorescence imaging is an essential tool in biological research, offering molecular specificity and high contrast. Over the past few decades, significant efforts have been made to overcome the diffraction limit of resolution, leading to the development of various super-resolution (SR) imaging techniques. Stimulated emission depletion (STED) microscopy achieves resolution enhancement by exploiting nonlinear point spread function (PSF) engineering¹, while single-molecule localization microscopy (SMLM) reconstructs high-resolution images by precisely localizing individual fluorescent molecules^2,3. Super-resolution optical fluctuation imaging (SOFI) improves resolution by analyzing higher-order temporal correlations in fluorescence signals⁴. Structured illumination microscopy (SIM) doubles the diffraction-limited resolution using patterned excitation, offering ~150 nm lateral and ~300 nm axial resolution⁵. Although SIM has lower resolution than STED or SMLM, it provides advantages in imaging speed and compatibility with conventional fluorophores.

A key challenge in SR imaging is extending the imaging depth to visualize fine structures within biological tissue⁶. Conventional SIM is generally limited to shallow depths of 10–20 μm in mouse brain tissue. To improve penetration depth, point-scanning SIM techniques such as multifocal SIM (MSIM)⁷ and instant SIM (iSIM)⁸ have been developed, enabling imaging depths of ~50 μm in live zebrafish embryos⁹. By integrating two-photon excitation with line-scanning SIM¹⁰ or point-scanning SIM, imaging depths exceeding 100 μm have been achieved in the eye of a live zebrafish embryo⁸. However, tissue-induced aberrations distort the PSF, degrading both resolution and imaging depth. This issue is particularly critical for SR techniques, which are highly sensitive to PSF distortions^11,12.

To resolve tissue-induced aberrations and further extend SR imaging depth, hardware-based adaptive optics (AO) has been incorporated into SIM^13–16. AO uses wavefront control devices, such as deformable mirrors or spatial light modulators, to correct aberrations in the excitation path^16,17. These aberrations can be measured using direct wavefront sensing (e.g., Shack-Hartmann sensors)^17,18 or sensorless approaches¹³ that iteratively optimize image metrics. Hardware-based AO-SIM has demonstrated imaging depths of ~25 μm in mouse brain tissue and ~100 μm in transparent zebrafish larvae¹³. When combined with two-photon excitation, AO extends imaging depths up to ~250 μm¹⁴. However, because excitation and emission wavelengths differ, they experience distinct aberrations, requiring separate corrections. Recent advancements in two-photon AO-SIM have enabled dual-path (both excitation and emission path) aberration correction, achieving imaging depths up to 500 μm in mouse brain tissue¹⁹. Despite these improvements, hardware-based AO systems introduce significant complexity, require additional calibration steps, and remain inaccessible to many biological researchers due to cost and technical constraints.

Computational AO provides an alternative by correcting aberrations directly from conventionally acquired fluorescence images, without requiring additional optical hardware. However, fluorescence imaging lacks coherent wavefront information, making it challenging to directly retrieve aberrations as in coherent imaging techniques^20–22. Image reconstruction methods, such as deconvolution^23–28 or machine-learning-based AO method^29,30 are commonly used to estimate the most likely PSF and sample image from a single fluorescence image. Recently, computational aberration compensation on the detection path of conventional wide-field fluorescence imaging has been demonstrated by constructing a fluorescence reflection matrix³¹, the incoherent analog of the coherent reflection matrix³². Nevertheless, these methods do not independently account for the excitation and emission PSFs. Instead, they either consider only the emission PSF or adopt a single effective PSF¹⁷, which is defined as the product of excitation and emission PSFs. This results in a narrower spectral frequency bandwidth than that of individual PSFs, leading to the loss of high-frequency information. As a result, conventional deconvolution methods perform well under mild aberrations but struggle in conditions with strong aberrations.

Here, we introduce a computational AO framework for multiphoton SR imaging that overcomes these limitations. Our approach builds upon image scanning microscopy (ISM)³³, replacing the conventional integral detector with a camera in a laser-scanning microscope. On the basis of multiphoton virtual structured illumination³¹, we develop a dual deconvolution algorithm that independently corrects the excitation and emission PSFs. By separately accounting for these two sources of aberration, we reconstruct the aberration-free object spectrum with an extended spatial frequency bandwidth. Unlike conventional deconvolution methods that use vector decomposition to find a single effective PSF, our dual deconvolution technique applies matrix decomposition to correct excitation and emission PSFs independently. This significantly enhances image reconstruction, particularly for high-frequency components, and enables SR imaging even under severe aberrations. We validate our framework in two-photon fluorescence microscopy, demonstrating the visualization of dendritic spines in mouse brain tissue with a lateral resolution of 130 nm, corresponding to a quarter of the emission wavelength, at an imaging depth of 180 μm. By providing a cost-effective and hardware-free alternative to AO, our approach has the potential to broaden access to deep-tissue SR imaging for biological research.

Results

Multiphoton structured illumination imaging in the aberrating medium

The proposed method begins with an imaging configuration where the sample is illuminated with a focused beam, and the generated fluorescence signal is measured by an array detector. Consider a tightly focused excitation laser beam of unit intensity directed to a position ${\mathbf{r}}_{\mathrm{i}}$ in the conjugate plane of the sample plane, which has spatial coordinates $\mathbf{r}$, where the fluorescent objects are embedded within a complex medium (Fig. 1a). The excitation laser beam is distorted by sample-induced aberrations, characterized by the excitation point spread function (PSF), $h_{\mathrm{ex}}\left(\mathbf{r}\right)$. This distorted beam interacts with the fluorescent targets at the sample plane, which have a fluorophore density distribution $\gamma \left(\mathbf{r}\right)$. In the case of multiphoton imaging, the emitted fluorescence from the targets is proportional to $h_{\mathrm{ex}}^n\left(\mathbf{r}\right)$, where $n$ is the photon excitation order (e.g., $n=2$ for two-photon excitation). The emitted fluorescence undergoes further aberration, described by the emission PSF, $h_{\mathrm{em}}\left(\mathbf{r}\right)$. Consequently, the fluorescence intensity map at the detector plane ${\mathbf{r}}_{\mathrm{d}}$ for each point-illumination position ${\mathbf{r}}_{\mathrm{i}}$ is given by:

$$f\left({\mathbf{r}}_{\mathrm{d}},{\mathbf{r}}_{\mathrm{i}}\right)=\int h_{\mathrm{em}}\left(\mathbf{r}-{\mathbf{r}}_{\mathrm{d}}\right)\gamma \left(\mathbf{r}\right)h_{\mathrm{ex}}^n\left(\mathbf{r}-{\mathbf{r}}_{\mathrm{i}}\right)\mathrm{d}\mathbf{r}$$ 1

Fig. 1. Principle of multiphoton virtual structured illumination in an aberrating medium.

a Image scanning: a focused excitation beam is sequentially positioned at ${\mathbf{r}}_{\mathrm{i}}=\left(x_{\mathrm{i}},y_{\mathrm{i}}\right)$ in the illumination plane conjugate to the sample plane $\mathbf{r}$, where fluorophores are embedded, and the emitted fluorescence is recorded by an array detector at the conjugate plane ${\mathbf{r}}_{\mathrm{d}}=\left(x_{\mathrm{d}},y_{\mathrm{d}}\right)$. Both excitation and emission PSFs are distorted by sample-induced aberration and scattering. b Representative fluorescence images $f\left({\mathbf{r}}_{\mathrm{d}},{\mathbf{r}}_{\mathrm{i}}\right)$ for different scan positions ${\mathbf{r}}_{\mathrm{i}}$. c Virtual structured illumination: the modulation $I_{\mathrm{ill}}\left({\mathbf{r}}_{\mathrm{i}}\right)=e^{i{\mathbf{k}}_{\mathrm{i}}\cdot {\mathbf{r}}_{\mathrm{i}}}$, generated by linear superposition of $f\left({\mathbf{r}}_{\mathrm{d}},{\mathbf{r}}_{\mathrm{i}}\right)$, yields structured illumination images $I_{\mathrm{fl}}\left({\mathbf{r}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)$. d Representative spectra $F\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)$ obtained by Fourier transform of $I_{\mathrm{fl}}\left({\mathbf{r}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)$ with respect to ${\mathbf{r}}_{\mathrm{d}}$. Each spectrum is limited by ${\alpha k}_{\mathrm{em}}$, where $\alpha$ is the numerical aperture and $k_{\mathrm{em}}$ the emission wavenumber. Green dots indicate the centers of the object spectra shifted by ${\mathbf{k}}_{\mathrm{i}}$. e Synthesized spectrum after shifting each spectrum by $-{\mathbf{k}}_{\mathrm{i}}$, aligning the spectrum centers and expanding the synthesized bandwidth from $\alpha k_{\mathrm{em}}$ to $\alpha \left(n{k}_{\mathrm{ex}}+k_{\mathrm{em}}\right)$, where $n$ is the photon excitation order and $k_{\mathrm{ex}}$ the excitation wavenumber. Synthesized SIM spectrum is distorted, and high spatial frequencies are attenuated in the presence of strong aberrations. f Same as (e) but with aberration correction by the dual deconvolution algorithm, which restores high spatial frequency components and recovers the synthesized bandwidth.

In this study, we mainly focus on two-photon fluorescence imaging in the epi-detection mode as it is better suited for deep-tissue imaging. Figure 1b shows a set of the representative fluorescence intensity maps on ${\mathbf{r}}_{\mathrm{d}}$ plane taken for various ${\mathbf{r}}_{\mathrm{i}}$ in the two-photon imaging case. Each intensity map corresponds to a double convolution of the object function $\gamma \left(\mathbf{r}\right)$ with $h_{\mathrm{ex}}^2\left(\mathbf{r}\right)$ and $h_{\mathrm{em}}\left(\mathbf{r}\right)$.

From the perspective of measuring $f\left({\mathbf{r}}_{\mathrm{d}},{\mathbf{r}}_{\mathrm{i}}\right)$, the proposed method follows the same initial approach as ISM³³. However, we employ spectral synthesis instead of pixel reassignment for super-resolution imaging³⁴. To this end, we transition from the space domain to the spatial frequency domain. Given $f\left({\mathbf{r}}_{\mathrm{d}},{\mathbf{r}}_{\mathrm{i}}\right)$, we can consider the linear superposition of the focused illumination to computationally synthesize a fluorescence image $I_{\mathrm{fl}}\left({\mathbf{r}}_{\mathrm{d}}\right)$ for a virtual illumination pattern $I_{\mathrm{ill}}\left({\mathbf{r}}_{\mathrm{i}}\right)$: $I_{\mathrm{fl}}\left({\mathbf{r}}_{\mathrm{d}}\right)=\int f\left({\mathbf{r}}_{\mathrm{d}},{\mathbf{r}}_{\mathrm{i}}\right)I_{\mathrm{ill}}\left({\mathbf{r}}_{\mathrm{i}}\right)\mathrm{d}{\mathbf{r}}_{\mathrm{i}}$. As a special case, we consider $I_{\mathrm{ill}}\left({\mathbf{r}}_{\mathrm{i}}\right)=e^{i{\mathbf{k}}_{\mathrm{i}}\cdot {\mathbf{r}}_{\mathrm{i}}}$ with a wavevector ${\mathbf{k}}_{\mathrm{i}}$ and obtain wide-field fluorescence image, $I_{\mathrm{fl}}\left({\mathbf{r}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)$, as illustrated in Fig. 1c. By taking the Fourier transform of each fluorescence image with respect to ${\mathbf{r}}_{\mathrm{d}}$, we can obtain the $F\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)$, the spatial frequency representation of $f\left({\mathbf{r}}_{\mathrm{d}},{\mathbf{r}}_{\mathrm{i}}\right)$. A representative set of $F\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)$ maps, obtained from Fig. 1b, is shown in Fig. 1d.

The spectral bandwidth of each spectrum is given by $\alpha k_{\mathrm{em}}$, where $\alpha$ is the numerical aperture and $k_{\mathrm{em}}={2\pi /\lambda }_{\mathrm{em}}$, with ${\lambda }_{\mathrm{em}}$ as the emission wavelength. Notably, each spectrum is shifted by ${\mathbf{k}}_{\mathrm{i}}$. In the absence of sample-induced aberrations, $F\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)$ is dominated by a shifted copy of the object spectrum, $\Gamma \left({\mathbf{k}}_{\mathrm{d}}+{\mathbf{k}}_{\mathrm{i}}\right)$, where $\Gamma \left(\mathbf{k}\right)$ is the Fourier transform of $\gamma \left(\mathbf{r}\right)$. The maximum magnitude of the spectral shift ${\mathbf{k}}_{\mathrm{i}}$ is given by $n\alpha k_{\mathrm{ex}}$, where $k_{\mathrm{ex}}=2\pi /{\lambda }_{\mathrm{ex}}$, with ${\lambda }_{\mathrm{ex}}$ as the excitation wavelength. The increased spectral shift by a factor of $n$ is a direct consequence of the narrowing of the PSF width in $h_{\mathrm{ex}}^n$ induced by multiphoton excitation.

To obtain the super-resolution image, all $F\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)$ components are synthesized in such a way that the object spectrum is coherently combined. This is achieved by shifting each $F\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)$ by $-{\mathbf{k}}_{\mathrm{i}}$ and summing them, as illustrated in Fig. 1e. This process expands the spectral bandwidth to $k_{\mathrm{c}}=2\alpha \left(k_{\mathrm{em}}+n{k}_{\mathrm{ex}}\right)$. The resulting spatial resolution is ${\lambda }_{\mathrm{ex}}{\lambda }_{\mathrm{em}}/2n\alpha \left({\lambda }_{\mathrm{em}}+{\lambda }_{\mathrm{ex}}/n\right)$. In the case of two-photon excitation with ${\lambda }_{\mathrm{ex}}/2\approx {\lambda }_{\mathrm{em}}$ and $\alpha =1$, the spatial resolution approaches ${\lambda }_{\mathrm{em}}/4$, surpassing the diffraction limit.

This spectral synthesis process is equivalent to conventional structured illumination microscopy (SIM) in one-photon imaging ($n=1$). In conventional SIM, raw images are recorded by illuminating the sample with sinusoidal intensity patterns $I_{\mathrm{ill}}\left({\mathbf{r}}_{\mathrm{i}}\right)=1+\cos \left({\mathbf{k}}_{\mathrm{i}}\cdot {\mathbf{r}}_{\mathrm{i}}+\varphi \right)$ for a few spatial frequencies ${\mathbf{k}}_{\mathrm{i}}$ with known phases $\varphi$. The function $F\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)$ is then obtained via demodulation with respect to $\varphi$ and subsequently used for spectral synthesis. In our approach, we directly extract one of the three spectral components in conventional SIM by considering $I_{\mathrm{ill}}\left({\mathbf{r}}_{\mathrm{i}}\right)=e^{i{\mathbf{k}}_{\mathrm{i}}\cdot {\mathbf{r}}_{\mathrm{i}}}$. However, there is a critical difference between our virtual structured illumination and real structured illumination in multi-photon imaging. Our virtual structured illumination is a mathematical operation designed to extract spectral components from the experimentally measured $f\left({\mathbf{r}}_{\mathrm{d}},{\mathbf{r}}_{\mathrm{i}}\right)$.

As we shall show below, this linearizes the aberration correction problem, thereby playing a crucial role in the computational correction of sample-induced aberrations. This is not achievable with real structured illumination, as a linear relation cannot be established (see Methods for details).

It is noteworthy to compare SIM and ISM in the k-space. Pixel reassignment in ISM is equivalent to sampling the main-diagonal of $F\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)$³⁴, whereas SIM/confocal performs an anti-diagonal projection along ${\mathbf{k}}_{\mathrm{d}}+{\mathbf{k}}_{\mathrm{i}}=\mathbf{k}$ that coherently accumulates equal-k components (see Methods and Supplementary Information for detailed derivation). Essentially, both provide a similar expansion of the spectral bandwidth with different weighting factors. SIM/confocal expands bandwidth by the convolution of excitation and emission OTFs, i.e., $H_{\mathrm{em}}*H_{\mathrm{ex}}$, while ISM does a similar task by doubling each OTF, i.e., $H_{\mathrm{ex}}\left(\frac{{\mathbf{k}}_{\mathrm{d}}}{2}\right)H_{\mathrm{em}}\left(\frac{{\mathbf{k}}_{\mathrm{i}}}{2}\right)$. As such, they provide equivalent resolution in the absence of aberrations.

Dual deconvolution to retrieve excitation and emission OTFs

The presence of sample-induced aberrations compromises the coherent synthesis of the $F\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)$. Each spectrum $F\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)$ derived from the virtual structured illumination is equivalent to taking the Fourier transform of $f\left({\mathbf{r}}_{\mathrm{d}},{\mathbf{r}}_{\mathrm{i}}\right)$ in Eq. (1) with respect to both ${\mathbf{r}}_{\mathrm{i}}$ and ${\mathbf{r}}_{\mathrm{d}}.$ This leads to the following expression (see Methods for detailed derivation):

$$F\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)=H_{\mathrm{em}}\left({\mathbf{k}}_{\mathrm{d}}\right)H_{\mathrm{ex}}\left({\mathbf{k}}_{\mathrm{i}}\right)\Gamma \left({\mathbf{k}}_{\mathrm{d}}+{\mathbf{k}}_{\mathrm{i}}\right)$$ 2

Here, $H_{\mathrm{ex}}$, $\Gamma$, and $H_{\mathrm{em}}$ are Fourier transforms of $h_{\mathrm{ex}}^n$, $\gamma$, and $h_{\mathrm{em}}$, respectively. Therefore, $H_{\mathrm{ex}}$ and $H_{\mathrm{em}}$ correspond to the excitation and emission OTFs, respectively. Sample-induced aberrations modify the phase part of the OTF, known as the phase transfer function (PTF), which distorts PSF shape. They also attenuate the amplitude part of the OTF, termed the modulation transfer function (MTF), especially at high spatial frequencies. Essentially, aberrations transform $H_{\mathrm{ex}}$ and $H_{\mathrm{em}}$ into complex-valued functions with reduced magnitude. As such, for each given ${\mathbf{k}}_{\mathrm{i}}$, $H_{\mathrm{ex}}\left({\mathbf{k}}_{\mathrm{i}}\right)$ gives rise to overall phase shift and amplitude attenuation in $F\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)$, while $H_{\mathrm{em}}\left({\mathbf{k}}_{\mathrm{d}}\right)$ modifies the spectral map. Consequently, individual $F\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)$ components are not coherently summed during the synthesis process, leading to a reduction in contrast and effective spectral bandwidth (Fig. 1e).

Here, we developed a dual deconvolution algorithm that separately identifies $H_{\mathrm{em}}$ and $H_{\mathrm{ex}}$ from a set of $F\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)$. The key principle of dual deconvolution is to iteratively compensate for phase distortion and spectral attenuation in $F\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)$ between excitation basis (${\mathbf{k}}_{\mathrm{i}}$) and emission basis (${\mathbf{k}}_{\mathrm{d}}$) in a way to maximize the correlation among individual $F$ maps. An intuitive explanation is to tune the overall phase of each spectrum in Fig. 1d to maximize the total intensity of the synthesized spectrum in Fig. 1e, which yields an estimate of $H_{\mathrm{ex}}$. A similar operation is conducted on the reciprocal basis (interchanging ${\mathbf{k}}_{\mathrm{i}}$ and ${\mathbf{k}}_{\mathrm{d}}$) to estimate $H_{\mathrm{em}}$. By optical reciprocity, swapping the illumination and detection ports yields the transpose of the response, $F^{\mathrm{T}}\left({\mathbf{k}}_{\mathrm{i}},{\mathbf{k}}_{\mathrm{d}}\right)$, representing the same optics traversed in reverse: the overall phase that previously resided on the illumination side now appears on the detection side. Accordingly, we apply the same spectral synthesis and phase tuning to $F^{\mathrm{T}}$, maximizing the total intensity of the synthesized (transposed) spectrum to obtain the estimate of $H_{\mathrm{em}}$. Iterating these two paths drives the synthesized spectrum toward maximum total intensity (see Methods for detailed mathematical description). This results in enhanced contrast and an expanded spectral bandwidth (Fig. 1f).

As we demonstrate, our dual deconvolution method surpasses conventional blind deconvolution in correcting aberrations and reconstructing the object spectrum, particularly in high-frequency components. This advantage fundamentally stems from the way scanned images are recorded, which inherently captures more information than conventional multiphoton imaging using an integral detector or single-pixel detection in confocal imaging. Our dual deconvolution exploits all the signals in the distorted scanned images to form a sub-diffraction PSF, which is an advanced version of the pixel reassignment working for the aberrated PSF. From a different perspective, the spectrum of a scanned image is expressed as the product of $H_{\mathrm{em}}$ and $H_{\mathrm{ex}}$ in Eq. (2), whereas conventional multiphoton/confocal imaging spectrum is given by their autocorrelations. As a result, significant information loss occurs at the recording stage, especially at high spatial frequencies, in conventional methods. Essentially, our dual deconvolution algorithm restores these relatively well-preserved high-frequency components with high fidelity, which substantially mitigates the intrinsic limitations of computational adaptive optics.

Computational validation of dual deconvolution in multiphoton super-resolution imaging

Figure 2 illustrates the validation of the proposed algorithm with simulated two-photon fluorescence imaging following the image scanning geometry in Fig. 1a (see Methods for details of numerical simulation). We ran the simulation with the emission wavelength of ${\lambda }_{\mathrm{em}}=520$ nm and $\alpha =1$. For simplicity, we assume that the two-photon excitation wavelength is twice the emission wavelength, i.e., ${\lambda }_{\mathrm{ex}}=2{\lambda }_{\mathrm{em}}$ such that the cut-off spatial frequencies for excitation and emission OTFs are the same. Therefore, the theoretically achievable resolution of a reconstructed two-photon SIM (2PSIM) image is a quarter of the fluorescence wavelength, ${\lambda }_{\mathrm{em}}/4=130$ nm. The procedure to generate two-photon scanned images in the presence of excitation and emission aberrations is as follows. We computationally added two independent phase aberrations, ${\phi }_{\mathrm{ex}}\left({\mathbf{k}}_{\mathrm{i}}\right)$ and ${\phi }_{\mathrm{em}}\left({\mathbf{k}}_{\mathrm{d}}\right)$, to excitation and emission pupils, respectively, which were generated by the superposition of Zernike modes numbered up to order 30 with various mode coefficients. The absolute squares of the Fourier transform of the complex pupil functions lead to the one-photon excitation PSF and the emission PSF. To simulate two-photon fluorescence imaging, we obtained the two-photon excitation PSF by taking the square of the one-photon excitation PSF. A set of fluorescence intensity maps $f\left({\mathbf{r}}_{\mathrm{d}},{\mathbf{r}}_{\mathrm{i}}\right)$ was obtained for a given target function, $\gamma \left(\mathbf{r}\right)$, using Eq. (1) (Fig. 1b). Finally, $F\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)$ was constructed by Fourier transforming $f\left({\mathbf{r}}_{\mathrm{d}},{\mathbf{r}}_{\mathrm{i}}\right)$ (Fig. 1d).

Fig. 2. Aberration correction by dual deconvolution of a simulated target.

a 2PFM image reconstructed from simulated scanned images in Fig. 1b. b 2PSIM image obtained from synthesized spectrum in Fig. 1e. The 2PFM and 2PSIM images are the results of conventional blind deconvolution. c AO-2PSIM image recovered by the synthesized spectrum in Fig. 1f after applying the dual deconvolution algorithm. d, e Estimated excitation and emission OTF maps, respectively. The height represents the MTF, while the color represents the PTF. The spectral frequencies, $k_x$ and $k_y$, are in units of $k_{\mathrm{em}}$. f, g Excitation and emission PSFs obtained by the inverse Fourier transform of the estimated excitation and emission OTFs, respectively. h Intensity profiles along the white curves in (a–c). i MTFs obtained from the 2PFM, 2PSIM, and AO-2PSIM images in (a–c). MTF values for each spatial frequency were determined from the contrast values of the intensity profiles in (h), with the radius corresponding to the spatial frequency. The ideal 2PSIM represents a 2PSIM image obtained in the absence of aberrations. Scale bars in (c, g), 5 μm.

We first constructed an image equivalent to the conventional two-photon fluorescence microscopy (2PFM) image (Fig. 2a) by summing all pixel values in each $f\left({\mathbf{r}}_{\mathrm{d}},{\mathbf{r}}_{\mathrm{i}}\right)$, i.e. $I_{2\mathrm{PFM}}\left({\mathbf{r}}_{\mathrm{i}}\right)={\sum }_{{\mathbf{r}}_{\mathrm{d}}}f\left({\mathbf{r}}_{\mathrm{d}},{\mathbf{r}}_{\mathrm{i}}\right)$. A 2PSIM image (Fig. 2b) was obtained by inverse Fourier transforming the aperture-synthesized object spectrum in Fig. 1e. Both the 2PFM and 2PSIM images were blurred due to aberrations.

Next, we applied our dual deconvolution algorithm to $F\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)$, estimating the excitation and emission OTFs, $H_{\mathrm{em}}$ and $H_{\mathrm{ex}}$, and object spectrum $\Gamma$. The aberration-corrected 2PSIM (AO-2PSIM) image obtained from the object spectrum $\Gamma$ recovered by the dual deconvolution algorithm is shown in Fig. 2c, which is in good agreement with its ground-truth with the similarity of 98% of correlation. The originally blurred image was made sharper, with a resolution approaching ${\lambda }_{\mathrm{em}}/4$. The estimated $H_{\mathrm{em}}$ and $H_{\mathrm{ex}}$ are visualized in Figs. 2d and 2e, respectively. The OTF bandwidths were substantially reduced, and the PTFs exhibited complex phase distributions due to aberrations. We obtained excitation and emission PSFs by applying the Fourier transform to the respective OTF maps (Fig. 2f, g).

The performance of our aberration correction algorithm was quantified by comparing the intensity profiles and the MTFs of the reconstructed images in Fig. 2h, i, respectively. The red curve in Fig. 2i represents the MTF of the AO-2PSIM image in Fig. 2c, which closely matches the ideal MTF obtained from an aberration-free 2PSIM image (green curve). In contrast, the MTFs of the 2PFM and 2PSIM images without aberration correction, shown as gray and blue curves, respectively, exhibited a substantial loss of information at high frequencies.

To emphasize the advantages of our methodology, which corrects both excitation and emission PSFs, we compared it to a conventional single-PSF-based blind deconvolution algorithm (Supplementary Fig. S3). The conventional blind deconvolution marginally improved both 2PFM and 2PSIM images, and it was unable to achieve the resolution and contrast enhancement provided by our proposed method.

Experimental validation of multiphoton super-resolution imaging

For the experimental validation of the proposed concept, we constructed a custom-made two-photon fluorescence imaging system equipped with a scientific CMOS camera at the detector plane (see Methods and Supplementary Information for details). A wavelength-tunable femtosecond pulsed laser (INSIGHT X3, Spectra Physics) served as the excitation source, and a high-numerical-aperture objective (N60X-NIR, Nikon, 60×, 1.0 NA) was used to focus the excitation. Image magnification at the camera was adjusted to ensure the camera’s pixel pitch satisfied the Nyquist sampling interval for the emission cut-off frequency, $k_{\mathrm{em}}$. We captured two-photon fluorescence images by 2D-scanning the focused excitation beam. These images were used to obtain $f\left({\mathbf{r}}_{\mathrm{d}},{\mathbf{r}}_{\mathrm{i}}\right)$ and reconstruct 2PFM and AO-2PSIM images. Here, the scanning interval $r_{\mathrm{sc}}$ determines the maximum spatial frequency for excitation according to the Nyquist theorem, $k_{\mathrm{ex}}<2\pi /\left(2r_{\mathrm{sc}}\right)$. Specifically, the interval could be set to $r_{\mathrm{sc}}={\lambda }_{\mathrm{ex}}/2\alpha n$ to achieve the diffraction-limited cut-off frequency for $n$-photon excitation.

To verify two-photon super-resolution imaging capability, we used 100-nm-diameter polystyrene beads labeled with Rhodamine B (see Methods for sample preparation). The excitation and peak emission wavelengths were ${\lambda }_{\mathrm{ex}}=850$ nm and ${\lambda }_{\mathrm{em}}=567$ nm, respectively, yielding a theoretical bandwidth-limited resolution of ~121 nm. The scanning interval was set to $r_{\mathrm{sc}}=131\mathrm{nm}$. The reconstructed 2PFM, 2PSIM, and AO-2PSIM images are shown in Fig. 3a-c, respectively, with corresponding zoom-in regions shown in Fig. 3d. The reconstructed 2PFM and 2PSIM images are the results of conventional blind deconvolution, providing a fair comparison with the AO-2PSIM image. Line profiles along the dashed lines in Fig. 3d are displayed in Fig. 3e. Compared to 2PFM, the 2PSIM image exhibited resolution enhancement and optical sectioning due to aperture synthesis. However, AO-2PSIM demonstrated a substantial resolution improvement, clearly resolving the two beads with a separation of 240 nm. Figure 3f presents the spatial frequency spectra of the reconstructed images, clearly showing the extended spectral bandwidth of AO-2PSIM. Corresponding radially averaged spectra are displayed in Fig. 3g.

Fig. 3. Two-photon super-resolution imaging of 100-nm-diameter Rhodamine B polystyrene beads.

a–c 2PFM, 2PSIM, and AO-2PSIM images. Here, 2PFM and 2PSIM images are the results of conventional blind deconvolution. d Zoomed-in regions of the white squares in (a–c), showing 2PFM (left), 2PSIM (middle), and AO-2PSIM (right) images. e Line profiles along the dashed lines in (d). The AO-2PSIM image clearly resolves two peaks with a measured distance of 240 nm. f Spatial frequency spectra of 2PFM, 2PSIM, and AO-2PSIM images in (d). Two dashed circles indicate bandwidths with radii of $2k_{\mathrm{em}}$ and $4k_{\mathrm{em}}$. AO-2PSIM exhibits the largest frequency bandwidth, close to full bandwidth of $4k_{\mathrm{em}}$. g Radial average of spectra are measured from (f), corresponding to the extended spectral bandwidth of AO-2PSIM. h Excitation OTF (top) and emission OTF (bottom), visualized in 4D. Height represents MTF and colormap represents PTF. Scale bars in (a–c) and (d), 1 μm and 200 nm, respectively.

The excitation and emission OTFs estimated by the dual deconvolution algorithm are shown in Fig. 3h. While PTFs exhibited a relatively flat profile due to the absence of sample-induced aberrations, MTFs demonstrated a decrease at high spatial frequencies. This effect was more pronounced in the emission OTF due to its increased susceptibility to system aberrations caused by its shorter wavelength. By correcting both PTFs and MTFs, our algorithm enabled recovery of the full OTF bandwidth, achieving a resolution close to the theoretical limit.

Experimental validation of multiphoton AO-2PSIM for severe aberrations

We experimentally validated the performance of the proposed multiphoton super-resolution imaging for targets under an aberration layer that introduces substantial aberrations. The first sample, consisting of Alexa 488 stained gold particles with an average diameter of 100 nm, was placed under an artificial aberrating layer (see Methods for sample preparation). The excitation and peak emission wavelengths were ${\lambda }_{\mathrm{ex}}=900$ nm and ${\lambda }_{\mathrm{em}}=520$ nm, respectively, yielding a theoretical bandwidth-limited resolution of ~121 nm. The scanning interval was set to $r_{\mathrm{sc}}=131\mathrm{nm}$. Figure 4a–c show the 2PFM, 2PSIM, and AO-2PSIM images, respectively. The excitation and emission OTF maps, along with the corresponding PSFs, are presented in Fig. 4d, e, respectively. The estimated OTF maps contain both the system aberrations and those induced by the sample. Both excitation and emission MTF maps fall sharply from the center, significantly reducing the effective MTF bandwidths of 2PFM and 2PSIM images. Both the excitation and emission PSFs recovered from the corresponding OTFs show pronounced distortion. In particular, the excitation PSF was split into two spots mainly because the PTF was modulated along the $k_x$ direction. Consequently, each particle appeared as two particles in 2PFM and 2PSIM images, as indicated by the arrowheads in Fig. 4a, b. However, our algorithm was able to correct this artifact and recover a single sharp PSF. Furthermore, we could normalize the sharp drop of MTFs at high spatial frequencies using the identified OTFs, leading to the recovery of spatial resolution. The resolution, estimated as the full width at half maximum (FWHM) of five distinct particles in the AO-2PSIM image, was ~174 nm, which fell short of the theoretical limit of 121 nm. We attribute this to imperfect recovery of the target’s MTF at high spatial frequencies due to insufficient signal-to-noise ratio (SNR). Nevertheless, the achieved resolution still surpassed the diffraction limit of two-photon imaging, ${\lambda }_{\mathrm{ex}}/4=225$ nm, confirming the ability of our AO-2PSIM to achieve super-resolution imaging even under severe aberration conditions.

Fig. 4. AO-2PSIM under aberrating layers.

a–c 2PFM, 2PSIM, and AO-2PSIM images of 100-nm-diameter Alexa 488 conjugated gold particles under an artificial aberrating medium. The 2PFM and 2PSIM images are the results of conventional blind deconvolution. d Excitation OTF (left) and the corresponding PSF (right), recovered by the dual deconvolution algorithm. e Recovered emission OTF (left) and the corresponding PSF (right). f Schematic of imaging a custom-made fluorescent resolution target. The resolution target is placed beneath a scattering medium, introducing substantial aberrations. The target mask is fabricated on a cover glass, and Rhodamine B-ethanol solution is placed underneath the resolution target. g–i 2PFM, 2PSIM, and AO-2PSIM images of the resolution target, respectively. The 2PFM and 2PSIM images are the results of conventional blind deconvolution. The third smallest line spacing of the target (red dotted box in (i)) is 250 nm. j Excitation MTF (left), PTF (middle), and PSF (right), respectively, identified by the dual deconvolution. k Emission MTF (left), PTF (middle), and PSF (right), respectively. Scale bars in the figures are 3  μm.

We also validated the proposed method for the extended targets, not the point particles, under even more pronounced aberrations. A fluorescent resolution target was fabricated by placing a thin metal mask of an etched USAF target pattern onto a Rhodamine B solution (Fig. 4f). A scattering layer was superimposed on top of the metal mask to introduce the aberration (see Methods for sample preparation). Figure 4g–i show 2PFM, 2PSIM, and AO-2PSIM images. The AO-2PSIM image shows a clear, high-contrast structure, whereas the 2PFM and 2PSIM images are severely blurred with multiple ghost artifacts due to strong aberrations. The third smallest line pairs with a spacing of 250 nm at the resolution target were clearly distinguished in the AO-2PSIM image.

The estimated excitation and emission OTFs with corresponding PSFs are shown in Fig. 4j, k, respectively. Both the excitation and emission PSFs are severely blurred and exhibit multiple foci, indicating strong wavefront distortions arising from the scattering medium. These multi-focal PSFs are responsible for the ghost artifacts in the images. Our algorithm retrieves both PTFs and MTFs over high spatial-frequency components and computationally corrects aberration and MTF attenuation. Once again, our dual deconvolution algorithm finds both the excitation and emission OTFs without need for any prior knowledge, thereby correcting both the excitation and emission aberrations. This is a clear advantage, especially in the presence of strong aberrations, in recovering the resolving power compared to conventional single-PSF-based blind deconvolution.

Two-photon super-resolution imaging in cells and tissues

We demonstrate the super-resolution imaging capabilities of our AO-2PSIM method in imaging cells and thick biological tissues (see Methods for sample preparation). Figure 5a shows a conventional 2PFM image of microtubules stained with Alexa 488 within a fixed COS-7 cell, and Fig. 5b shows the corresponding AO-2PSIM image. Excitation and peak emission wavelengths were ${\lambda }_{\mathrm{ex}}=900$ nm and ${\lambda }_{\mathrm{em}}=520$ nm, respectively. The 2PFM image exhibits noise, and the microtubule structures appear blurry. Applying aberration correction by our dual deconvolution method leads to significant improvements in both resolution and contrast. This is evident from the left panel in Fig. 5c, which shows the line profiles along the red dotted lines in the insets of Fig. 5a, b. We estimated the resolution based on the average width of a single microtubule branch, indicated by the dotted yellow boxes in the insets, with the corresponding line profiles shown in the right panel of Fig. 5c. The resolution of the AO-2PSIM image was measured to be 134 ± 6 nm, whereas that of the 2PFM image was 280 ± 12 nm.

Fig. 5. AO-2PSIM imaging of biological samples.

a, b Imaging of microtubules inside a whole COS-7 cell, stained with Alexa 488 fluorescence conjugate. 2PFM and AO-2PSIM images are shown in (a) and (b), respectively. The insets on the bottom right of each figure are the zoomed-in images of dashed rectangular boxes. Complex tubular structures depicted by white arrowheads are well visualized in the AO-2PSIM image. c Line profiles along the red dotted lines in the insets are shown on the left panel. The average intensity profiles of a single branch of microtubule in the yellow dotted boxes are shown on the right panel. d, e Imaging of Thy1-EGFP ex-vivo mouse brain tissue. 2PFM and AO-2PSIM images of dendritic spines at a depth of 130 μ$\mathrm{m}$ are shown in (d) and (e), respectively. Zoomed-in images of dashed rectangular boxes are shown at the bottom left of each figure. f Measured widths of the spine necks labeled as 1-5 in the insets are shown. g–i Same as (d–f), but taken at a depth of 180 μm. Scale bars: Cell images (a, b), 10 μm; The insets of (a, b), 2 μm; Dendrite images (e, h), 5 μm; Insets of (e, h), 2 μm.

Next, we performed two-photon imaging of ex-vivo mouse brain tissues with Thy1-EGFP. The excitation and peak emission wavelengths were 900 nm and 510 nm, respectively, yielding a theoretical bandwidth-limited resolution of ~120 nm. Figure 5d, e display the reconstructed 2PFM and AO-2PSIM images, respectively, at a depth of 130 μm. The 2PFM image (Fig. 5d) reveals somewhat blurred dendritic structures, with the necks either indistinct or invisible. In contrast, the AO-2PSIM image (Fig. 5e) offers a clear view of dendrites and their associated spines. By measuring the widths of the spine necks (Fig. 5f), we confirmed that our AO-2PSIM method can achieve super-resolution exceeding the diffraction limit, even for tissue imaging. Figure 5g, h show the 2PFM and AO-2PSIM images at a depth of 180 μm. The increased depth results in a more blurred 2PFM image compared to that at a depth of 130 μm. However, the AO-2PSIM image maintains high resolution and SNR, clearly visualizing the dendritic spine heads and necks even at this depth. Although the narrowest neck width at this depth is slightly larger than the theoretical super-resolution limit, it remains smaller than the diffraction limit (Fig. 5i).

We also conducted two-photon imaging of the hindbrain of a fixed whole-mount zebrafish to demonstrate the capability of AO-2PSIM in handling spatially varying aberrations¹¹, where the isoplanatic patch size is relatively small. Then, we acquired two-photon fluorescence images of a 10-day-post-fertilization (dpf) zebrafish hindbrain at a depth of 180 μm and applied the dual deconvolution algorithm for aberration correction. Figure 6a shows the reconstructed 2PFM and AO-2PSIM images, visualizing the fine structures of zebrafish oligodendrocyte membranes. To correct spatially varying aberrations, the whole ROI of 145 × 145 μm² was divided into 7 × 7 subregions (20 × 20 μm²), and each subregion was analyzed separately. Some of the identified PSFs show multiple foci that induce ghost artifacts and cause a significant loss of resolution. The AO-2PSIM image (the right panel in Fig. 6a) revealed commissural tracts within the caudal hindbrain, along with additional commissural tracts connected to anterior medial projections at the peripheries, showing the myelination in early development stage of nervous system. In contrast, the 2PFM image (the left panel in Fig. 6a) failed to visualize discernible central nervous structures.

Fig. 6. Two-photon imaging of zebrafish hindbrain.

a 2PFM (left) and AO-2PSIM (right) images of oligodendrocytes in a 10-dpf zebrafish at 180 μm depth. b Excitation (left) and emission (right) PSFs identified by the dual deconvolution algorithm for each isoplanatic patch. Scale bars in (a) and (b) are 20 μm and 3 μm, respectively.

Discussion

In this study, we introduced a multiphoton super-resolution fluorescence imaging technique via dual deconvolution of the scanned images, which offers computational correction of complex sample-induced aberrations and achieves deep-tissue super-resolution imaging. The proposed method allows for the reconstruction of an object image with a spatial resolution twice the diffraction limit by expanding the spatial frequency bandwidth through the spectral synthesis. Our method has advantages compared to hardware AO in that it does not require complex wavefront shaping devices or wavefront sensing systems. It simply replaces a PMT or photodiode in a conventional laser scanning microscope with an array detector, such as a CMOS camera. Moreover, the proposed method does not require guide-stars or any prior knowledge about the sample, making it applicable to a wide range of samples. Additionally, while most hardware AO systems correct aberrations in either the excitation or emission path¹⁶, our method corrects both paths, resulting in better resolution recovery.

The distinctive feature of the proposed dual deconvolution algorithm lies in that it is a matrix decomposition. This enables independent identification and correction of excitation and emission PSFs without any prior knowledge of the PSFs. This is crucial for achieving the best possible resolution in super-resolution imaging when dealing with severe aberrations. Conventional blind deconvolution algorithm is a vector decomposition, which identifies a single effective PSF given by the product of excitation and emission PSFs from a single blurred image. Therefore, it can be effective only for mild aberrations with known PSF shapes, and their performance in resolution recovery is intrinsically lower than our dual deconvolution algorithm (see Supplementary Information for the detailed comparison).

Our proposed computational AO is simpler to implement than hardware AO, but it has drawbacks. Hardware AO increases raw-image SNR by physically concentrating light into a tighter PSF, whereas computational AO cannot increase the photon budget; high-spatial-frequency components of the OTF are attenuated by aberrations and often buried in noise. Although our dual deconvolution approach recovers high-frequency content more effectively than conventional deconvolution, its gains are ultimately bounded by the acquisition SNR. In practice, the two are complementary: hardware AO first removes dominant aberrations and boosts signal, after which spectral synthesis combined with dual deconvolution recovers residual errors and achieves super-resolution. Another drawback is the long acquisition time. Our method relies on using an array detector to acquire images, which is associated with image scanning microscopy. This results in slower image acquisition compared to conventional confocal or multiphoton microscopes. The primary factor influencing imaging speed is the camera exposure time needed to capture the fluorescence maps, typically requiring around 1 ms per scanning point. Thus, acquiring a 100×100-pixel image takes about 10 s. However, employing multifocal illumination or speckle illumination techniques could significantly reduce the acquisition time, potentially enabling real-time imaging for most biological studies. After acquisition, the image processing time needed to process a 100×100-pixel image is less than 1 s on a graphics processing unit (GPU, GeForce RTX 3090, NVIDIA).

Given the simplicity of the hardware and the significant improvements in resolution and imaging depth offered by our proposed image reconstruction algorithm, we anticipate rapid integration of our technique into commercial multiphoton microscopy systems. Future strategies include enhancing image acquisition speed through methods such as parallel imaging with multifocal illumination^7,35–37 using a digital micromirror device or a liquid-crystal spatial light modulator^19,38–40. Multifocal sparse sampling could further accelerate image acquisition in combination with sparsity-based high-resolution image reconstruction from incomplete measurements⁴¹. Additionally, employing SNR-optimized array detectors such as single-photon avalanche diodes (SPAD) array detector⁴² may facilitate capturing raw fluorescence maps at a higher SNR, thereby enabling the recovery of the full bandwidth of OTFs.

As an intriguing future perspective, the dual deconvolution framework could be extended to three dimensions by incorporating depth-dependent PSFs, such as those engineered with astigmatic aberration⁴³ or a double-helix structure⁴⁴. Our dual deconvolution concept is not restricted to SIM. It can support single-molecule localization methods (e.g., STORM/SMLM) and STED microscopy⁴⁵ by estimating broadened, space-varying PSFs for improved localization. Beyond optics, the dual deconvolution concept extends to an inverse problem in systems wherever a linear response function can be measured.

Methods

Mathematical description of multiphoton virtual structured illumination in an aberrating medium

Equation (1) in the main text is valid for multiphoton imaging with focused illumination. To clarify this, let us consider the fluorescence signal $I_{\mathrm{fl}}\left({\mathbf{r}}_{\mathrm{d}}\right)$ detected at ${\mathbf{r}}_{\mathrm{d}}$ under coherent illumination of an electric field $E_{\mathrm{ill}}\left({\mathbf{r}}_{\mathrm{i}}\right)$:

$$I_{\mathrm{fl}}\left({\mathbf{r}}_{\mathrm{d}}\right)=\int h_{\mathrm{em}}\left(\mathbf{r}-{\mathbf{r}}_{\mathrm{d}}\right)\gamma \left(\mathbf{r}\right)I_{\mathrm{ex}}\left(\mathbf{r}\right)\mathrm{d}\mathbf{r}$$ 3

In the case of multiphoton imaging with a photon excitation order $n$, $I_{\mathrm{ex}}\left(\mathbf{r}\right)$ is given by

$$I_{\mathrm{ex}}\left(\mathbf{r}\right)={∣\int E_{\mathrm{ill}}\left({\mathbf{r}}_{\mathrm{i}}^{\prime }\right)h_{\mathrm{ex}}^E\left(\mathbf{r}-{\mathbf{r}}_{\mathrm{i}}^{\prime }\right)d{\mathbf{r}}_{\mathrm{i}}^{\prime }∣}^{2n}$$ 4

Here $h_{\mathrm{ex}}^E$ is the electric field PSF, which is related to $h_{\mathrm{ex}}$ by $h_{\mathrm{ex}}={∣h_{\mathrm{ex}}^E∣}^2$.

For focused illumination at ${\mathbf{r}}_{\mathrm{i}}$, where $E_{\mathrm{ill}}\left({\mathbf{r}}_{\mathrm{i}}^{\prime }\right)=\delta \left({\mathbf{r}}_{\mathrm{i}}^{\prime }-{\mathbf{r}}_{\mathrm{i}}\right)$, $I_{\mathrm{ex}}$ simplifies to $I_{\mathrm{ex}}\left(\mathbf{r}\right)={\mid h_{\mathrm{ex}}^E\left(\mathbf{r}-{\mathbf{r}}_{\mathrm{i}}\right)\mid }^{2n}=h_{ex}^n\left(\mathbf{r}-{\mathbf{r}}_{\mathrm{i}}\right)$. Therefore, Eq. (1) is justified in the case of focused illumination.

In the main text, we consider the computational synthesis of structured illumination for a virtual illumination $I_{\mathrm{ill}}\left({\mathbf{r}}_{\mathrm{i}}\right)=e^{i{\mathbf{k}}_{\mathrm{i}}\cdot {\mathbf{r}}_{\mathrm{i}}}$:

$$I_{\mathrm{fl}}\left({\mathbf{r}}_{\mathrm{d}}\right)=\int f\left({\mathbf{r}}_{\mathrm{d}},{\mathbf{r}}_{\mathrm{i}}\right)I_{\mathrm{ill}}\left({\mathbf{r}}_{\mathrm{i}}\right)\mathrm{d}{\mathbf{r}}_{\mathrm{i}}$$ 5

This synthesis of structured illumination is a purely mathematical operation designed to extract the spectral component contained in $f\left({\mathbf{r}}_{\mathrm{d}},{\mathbf{r}}_{\mathrm{i}}\right)$. If we instead send an incoherent illumination $I_{\mathrm{ill}}\left({\mathbf{r}}_{\mathrm{i}}\right)$ experimentally, the physically obtained structured illumination image differs from Eq. (5). In real physical multiphoton excitation, $I_{\mathrm{ex}}\left(\mathbf{r}\right)$ is given by

$$I_{\mathrm{ex}}\left(\mathbf{r}\right)={\left[\int I_{\mathrm{ill}}\left({\mathbf{r}}_{\mathrm{i}}^{\prime }\right)h_{\mathrm{ex}}\left(\mathbf{r}-{\mathbf{r}}_{\mathrm{i}}^{\prime }\right)d{\mathbf{r}}_{\mathrm{i}}^{\prime }\right]}^n$$ 6

Thus, the detected fluorescence image is given by

$$I_{\mathrm{fl}}\left({\mathbf{r}}_{\mathrm{d}}\right)=\int h_{\mathrm{em}}\left(\mathbf{r}-{\mathbf{r}}_{\mathrm{d}}\right)\gamma \left(\mathbf{r}\right){\left[I_{\mathrm{ill}}\left({\mathbf{r}}_{\mathrm{i}}^{\prime }\right)h_{\mathrm{ex}}\left(\mathbf{r}-{\mathbf{r}}_{\mathrm{i}}^{\prime }\right)d{\mathbf{r}}_{\mathrm{i}}^{\prime }\right]}^n\mathrm{d}\mathbf{r}$$ 7

Due to the cross-terms between $I_{\mathrm{ill}}\left({\mathbf{r}}_{\mathrm{i}}\right)$ at different ${\mathbf{r}}_{\mathrm{i}}$, Eq. (7) differs from Eq. (5). In fact, these cross-terms make it challenging to extract $h_{\mathrm{ex}}^n$ and $h_{\mathrm{em}}$ from $I_{\mathrm{fl}}\left({\mathbf{r}}_{\mathrm{d}}\right)$, as they invalidate the convolution relation. As such, with real structured illumination, we cannot establish a simple linear transfer function relation as in Eq. (2). Thus, virtual structured illumination synthesized with focused illumination is essential for enabling computational adaptive optics processing in multiphoton imaging.

Mathematical derivation of Eq. (2)

We can obtain $F\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)$ using virtual structured illumination with the illumination $I_{\mathrm{ill}}\left({\mathbf{r}}_{\mathrm{i}}\right)=e^{i{\mathbf{k}}_{\mathrm{i}}\cdot {\mathbf{r}}_{\mathrm{i}}}$ and a subsequent Fourier transform with respect to ${\mathbf{r}}_{\mathrm{d}}$. Mathematically, this is expressed as

$$F\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)=\int f\left({\mathbf{r}}_{\mathrm{d}},{\mathbf{r}}_{\mathrm{i}}\right)e^{i\left({\mathbf{k}}_{\mathrm{d}}\cdot {\mathbf{r}}_{\mathrm{d}}+{\mathbf{k}}_{\mathrm{i}}\cdot {\mathbf{r}}_{\mathrm{i}}\right)}d{\mathbf{r}}_{\mathrm{d}}d{\mathbf{r}}_{\mathrm{i}}$$ 8

Using Eq. (1), we derive the following relation:

$$F\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)=\int \gamma \left(\mathbf{r}\right)e^{i{\mathbf{k}}_{\mathbf{d}}\cdot \mathbf{r}}{H}_{\mathrm{em}}\left({\mathbf{k}}_{\mathrm{d}}\right)e^{i{\mathbf{k}}_{\mathbf{i}}\cdot \mathbf{r}}{H}_{\mathrm{ex}}\left({\mathbf{k}}_{\mathrm{i}}\right)\mathrm{d}\mathbf{r}$$ 9

Here $H_{\mathrm{em}}\left({\mathbf{k}}_{\mathrm{d}}\right)=\int h_{\mathrm{em}}\left(\mathbf{r}-{\mathbf{r}}_{\mathrm{d}}\right)e^{-i{\mathbf{k}}_{\mathrm{d}}\cdot \left(\mathbf{r}-{\mathbf{r}}_{\mathrm{d}}\right)}d{\mathbf{r}}_{\mathrm{d}}$ and $H_{\mathrm{ex}}\left({\mathbf{k}}_{\mathrm{i}}\right)=\int h_{\mathrm{ex}}^n\left(\mathbf{r}-{\mathbf{r}}_{\mathrm{i}}\right)e^{-i{\mathbf{k}}_{\mathrm{i}}\cdot \left(\mathbf{r}-{\mathbf{r}}_{\mathrm{i}}\right)}d{\mathbf{r}}_{\mathrm{i}}$. Since $\Gamma \left({\mathbf{k}}_{\mathrm{d}}+{\mathbf{k}}_{\mathrm{i}}\right)=\int \gamma \left(\mathbf{r}\right)e^{i\cdot \mathbf{r}\left({\mathbf{k}}_{\mathrm{d}}\mathit{\mp }{\mathbf{k}}_{\mathrm{i}}\right)}\mathrm{d}\mathbf{r}$, Eq. (9) simplifies to

$$F\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)=H_{\mathrm{em}}\left({\mathbf{k}}_{\mathrm{d}}\right)\Gamma \left({\mathbf{k}}_{\mathrm{d}}+{\mathbf{k}}_{\mathrm{i}}\right)H_{\mathrm{ex}}\left({\mathbf{k}}_{\mathrm{i}}\right)$$ 10

which corresponds to Eq. (2) in the main text.

Dual deconvolution algorithm

The spectrum $F\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)$ is a function of ${\mathbf{k}}_{\mathrm{i}}$ and ${\mathbf{k}}_{\mathrm{d}}$. Therefore, we can construct a spectral matrix $\mathit{F}$, whose elements are given by $F\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)$, with ${\mathbf{k}}_{\mathrm{i}}$ and ${\mathbf{k}}_{\mathrm{d}}$ as the column and row indices, respectively. Then, Eq. (2) can be expressed in the matrix form as:

$$\mathit{F}=\mathit{H}\mathit{\circ }\mathit{\Gamma }$$ 11

Here, $\mathit{H}$ is the effective OTF matrix, whose elements are defined as $H\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)=H_{\mathrm{em}}\left({\mathbf{k}}_{\mathrm{d}}\right)H_{\mathrm{ex}}\left({\mathbf{k}}_{\mathrm{i}}\right)$, and $\mathit{\Gamma }$ is the object spectrum matrix composed of $\Gamma \left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)$. The symbol $\circ$ denotes the Hadamard product. The goal of the dual deconvolution algorithm is to estimate $H_{\mathrm{em}}$, $H_{\mathrm{ex}}$, and $\Gamma$ such that they minimize the squared Frobenius norm of the difference between the measured spectral matrix F and the model: ${\mathrm{minimize}}_{H_{\mathrm{em}},H_{\mathrm{ex}},\Gamma }\mid \mid \mathit{F}-\mathit{H}\mathit{\circ }\mathit{\Gamma }{\mid \mid }^{\mathit{2}}$, subject to $\mathit{\Gamma }$ being a Toeplitz matrix.

The basic principle of our method is based on the iterative Wiener filter method^46,47, but it successively estimates three unknowns: $\Gamma$, $H_{\mathrm{em}}$, and $H_{\mathrm{ex}}$. In each iteration, the algorithm sequentially updates $\Gamma$, $H_{\mathrm{em}}$, and $H_{\mathrm{ex}}$ using Wiener filter-like equations.

The Algorithm for dual deconvolution is given in the following:

Algorithm:

Dual Deconvolution

1: input: spectral matrix $F\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)$

2: initialize: $H_{\mathrm{em}}\left({\mathbf{k}}_{\mathrm{d}}\right)=1$, $H_{\mathrm{ex}}\left({\mathbf{k}}_{\mathrm{i}}\right)=1$, and $H\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)=H_{\mathrm{em}}\left({\mathbf{k}}_{\mathrm{d}}\right)H_{\mathrm{ex}}\left({\mathbf{k}}_{\mathrm{i}}\right)$

3: for, until stopping criterion is met

4:  update $\Gamma$:

5:  $W\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)=\frac{H^{*}\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)}{{∣H\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)∣}^2+e_{\Gamma }^2}$

6:  $\Gamma \left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)=W\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)F\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)$

7:  ${\Gamma }_{\mathrm{syn}}\left(\mathrm{\Delta }\mathbf{k}\right)={\sum }_{{\mathbf{k}}_{\mathrm{i}}}\Gamma \left(\mathrm{\Delta }\mathbf{k}-{\mathbf{k}}_{\mathrm{i}},{\mathbf{k}}_{\mathrm{i}}\right)$

8: $W_{\mathrm{eff}}\left(\mathrm{\Delta }\mathbf{k}\right)=\frac{H_{\mathrm{eff}}^{*}\left(\mathrm{\Delta }\mathbf{k}\right)}{{∣H_{\mathrm{eff}}\left(\mathrm{\Delta }\mathbf{k}\right)∣}^2+e_{\mathrm{eff}}^2}$, where $H_{\mathrm{eff}}\left(\mathrm{\Delta }\mathbf{k}\right)={\sum }_{{\mathbf{k}}_{\mathrm{i}}}W\left(\mathrm{\Delta }\mathbf{k}-{\mathbf{k}}_{\mathrm{i}},{\mathbf{k}}_{\mathrm{i}}\right)H\left(\mathrm{\Delta }\mathbf{k}-{\mathbf{k}}_{\mathrm{i}},{\mathbf{k}}_{\mathrm{i}}\right)$

9:  ${\Gamma }_{\mathrm{syn}}\left(\mathrm{\Delta }\mathbf{k}\right)\leftarrow W_{\mathrm{eff}}\left(\mathrm{\Delta }\mathbf{k}\right){\Gamma }_{\mathrm{syn}}\left(\mathrm{\Delta }\mathbf{k}\right)$

10: $\Gamma \left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)\leftarrow \Gamma \left(\mathrm{\Delta }\mathbf{k}\right)$

11: update $H_{\mathrm{em}}$:

12:  $W_{\mathrm{em}}\left({\mathbf{k}}_{\mathrm{d}}\right)=\frac{{\sum }_{{\mathbf{k}}_{\mathrm{i}}\ne 0}{∣\Gamma \left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)H_{\mathrm{ex}}\left({\mathbf{k}}_{\mathrm{i}}\right)∣}^2}{{\left({\sum }_{{\mathbf{k}}_{\mathrm{i}}\ne 0}{∣\Gamma \left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)H_{\mathrm{ex}}\left({\mathbf{k}}_{\mathrm{i}}\right)∣}^2\right)}^2+e_H^2}$

13:  $H_{\mathrm{em}}\left({\mathbf{k}}_{\mathrm{d}}\right)\leftarrow W_{\mathrm{em}}\left({\mathbf{k}}_{\mathrm{d}}\right){\sum }_{{\mathbf{k}}_{\mathrm{i}}\ne 0}F\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right){\Gamma }^{*}\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)H_{\mathrm{ex}}^{*}\left({\mathbf{k}}_{\mathrm{i}}\right)$

14: update $\Gamma$: repeat line 5-10 with updated $H_{\mathrm{em}}$

15: update $H_{\mathrm{ex}}$:

16:  $W_{\mathrm{ex}}\left({\mathbf{k}}_{\mathrm{i}}\right)=\frac{{\sum }_{{\mathbf{k}}_{\mathrm{d}}\ne \mathit{\ne }\mathbf{0}}{∣\Gamma \left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)H_{\mathrm{em}}\left({\mathbf{k}}_{\mathrm{d}}\right)∣}^2}{{\left({\sum }_{{\mathbf{k}}_{\mathrm{d}}\ne \mathbf{0}}{∣\Gamma \left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)H_{\mathrm{em}}\left({\mathbf{k}}_{\mathrm{d}}\right)∣}^2\right)}^2+e_H^2}$

17:  $H_{\mathrm{ex}}\left({\mathbf{k}}_{\mathrm{i}}\right)\leftarrow W_{\mathrm{ex}}\left({\mathbf{k}}_{\mathrm{i}}\right){\sum }_{{\mathbf{k}}_{\mathrm{d}}\mathit{\ne }0}F\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right){\Gamma }^{*}\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)H_{\mathrm{em}}^{*}\left({\mathbf{k}}_{\mathrm{d}}\right)$

18: end for

Detailed explanation on the essential steps of the dual deconvolution algorithm is given in the following.

1. Initialize $H_{\mathrm{ex}}\left({\mathbf{k}}_{\mathrm{i}}\right)$, $H_{\mathrm{em}}\left({\mathbf{k}}_{\mathrm{d}}\right)$, and $H\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)=H_{\mathrm{em}}\left({\mathbf{k}}_{\mathrm{d}}\right)H_{\mathrm{ex}}\left({\mathbf{k}}_{\mathrm{i}}\right)$.

2. Update $\Gamma$:

   • A.
     Compute the estimate of $\Gamma \left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)$ using a matrix Wiener filter $W\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)=\frac{H^{*}\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)}{{∣H\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)∣}^2+e_{\Gamma }^2}$, where $e_{\Gamma }$ is a regularization parameter: $\Gamma \left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)=W\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)F\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)$.
   • B.
     Compute the synthesized object spectrum ${\Gamma }_{\mathrm{syn}}\left(\mathrm{\Delta }\mathbf{k}\right)$ by summing $\Gamma \left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)$ over ${\mathbf{k}}_{\mathrm{i}}$:

     $${\Gamma }_{\mathrm{syn}}\left(\mathrm{\Delta }\mathbf{k}\right)={\sum }_{{\mathbf{k}}_{\mathrm{i}}}\Gamma \left(\mathrm{\Delta }\mathbf{k}-{\mathbf{k}}_{\mathrm{i}},{\mathbf{k}}_{\mathrm{i}}\right).$$
   • C.
     Deconvolve ${\Gamma }_{\mathrm{syn}}\left(\mathrm{\Delta }\mathbf{k}\right)$ using the effective Wiener filter $W_{\mathrm{eff}}\left(\mathrm{\Delta }\mathbf{k}\right)=\frac{H_{\mathrm{eff}}\left(\mathrm{\Delta }\mathbf{k}\right)}{{∣H_{\mathrm{eff}}\left(\mathrm{\Delta }\mathbf{k}\right)∣}^2+e_{\mathrm{eff}}^2}$, where $H_{\mathrm{eff}}\left(\mathrm{\Delta }\mathbf{k}\right)$ is the effective OTF, given by $H_{\mathrm{eff}}\left(\mathrm{\Delta }\mathbf{k}\right)={\sum }_{{\mathbf{k}}_{\mathrm{i}}}W\left(\mathrm{\Delta }\mathbf{k}-{\mathbf{k}}_{\mathrm{i}},{\mathbf{k}}_{\mathrm{i}}\right)H\left(\mathrm{\Delta }\mathbf{k}-{\mathbf{k}}_{\mathrm{i}},{\mathbf{k}}_{\mathrm{i}}\right)$, and $e_{\mathrm{eff}}$ is another regularization parameter.
   • D.
     Construct the updated $\Gamma \left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)$ by generating a Toeplitz matrix from ${\Gamma }_{\mathrm{syn}}\left(\mathrm{\Delta }\mathbf{k}\right)$ with ${\mathbf{k}}_{\mathrm{d}}={\mathbf{k}}_{\mathrm{i}}+\mathrm{\Delta }\mathbf{k}$.
3. Update $H_{\mathrm{em}}$:

   • A.
     Obtain $H_{\mathrm{em}}\left({\mathbf{k}}_{\mathrm{d}}\right)$ by summing the columns of the component-wise inner product (Frobenius inner product) of $F\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)$ and $\Gamma \left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)H_{\mathrm{ex}}\left({\mathbf{k}}_{\mathrm{i}}\right)$.
   • B.
     Deconvolve $H_{\mathrm{em}}\left({\mathbf{k}}_{\mathrm{d}}\right)$ using a Wiener filter, $W_{\mathrm{em}}\left({\mathbf{k}}_{\mathrm{d}}\right)=\frac{{\sum }_{{\mathbf{k}}_{\mathbf{i}}\ne 0}{∣\Gamma \left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)H_{\mathrm{ex}}\left({\mathbf{k}}_{\mathrm{i}}\right)∣}^2}{{\left({\sum }_{{\mathbf{k}}_{\mathrm{i}}\ne 0}{∣\Gamma \left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)H_{\mathrm{ex}}\left({\mathbf{k}}_{\mathrm{i}}\right)∣}^2\right)}^2+e_H^2}$ with another regularization parameter $e_H$.
4. Update $H_{\mathrm{em}}$:

   • A.
     Obtain $H_{\mathrm{ex}}\left({\mathbf{k}}_{\mathrm{i}}\right)$ by summing along the rows of the component-wise inner product of $F\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)$ and $\Gamma \left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)H_{\mathrm{ex}}\left({\mathbf{k}}_{\mathrm{i}}\right)$.
   • B.
     Deconvolve $H_{\mathrm{ex}}\left({\mathbf{k}}_{\mathrm{i}}\right)$ using a Wiener filter $W_{\mathrm{ex}}\left({\mathbf{k}}_{\mathrm{i}}\right)=\frac{{\sum }_{{\mathbf{k}}_{\mathrm{d}}\ne 0}{∣\Gamma \left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)H_{\mathrm{em}}\left({\mathbf{k}}_{\mathrm{d}}\right)∣}^2}{{\left({\sum }_{{\mathbf{k}}_{\mathrm{d}}\ne 0}{∣\Gamma \left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)H_{\mathrm{em}}\left({\mathbf{k}}_{\mathrm{d}}\right)∣}^2\right)}^2+e_H^2}$.

For Wiener filters, parameters $e_{\Gamma }$, $e_{\mathrm{eff}}$, and $e_H$ are used for Tikhonov regularization depending on the signal-to-noise ratios. The iteration stops when the error of the OTFs between successive iterates converge to a predetermined threshold: $\mathrm{error}=\left[1-{\sum }_{{\mathbf{k}}_{\mathrm{i}}}{H}_{\mathrm{ex}}\left({\mathbf{k}}_{\mathrm{i}}\right)H_{\mathrm{ex}}^{*}\left({\mathbf{k}}_{\mathrm{i}}\right)\right]\cdot \left[1-{\sum }_{{\mathbf{k}}_{\mathrm{i}}}{H}_{\mathrm{em}}\left({\mathbf{k}}_{\mathrm{d}}\right)H_{\mathrm{em}}^{*}\left({\mathbf{k}}_{\mathrm{d}}\right)\right]$. Note that when updating $H_{\mathrm{em}}\left({\mathbf{k}}_{\mathrm{d}}\right)$ and $H_{\mathrm{ex}}\left({\mathbf{k}}_{\mathrm{i}}\right)$ at lines 13 and 17 in the algorithm, the summations over the spatial frequencies exclude the zero-frequency term. For objects with significant DC spectral components (e.g., uniformly distributed fluorescence objects), accurate decomposition of the excitation and emission OTFs can be challenging, potentially slowing algorithm convergence.

Numerical generation of scanned images in Fig. 2

The simulation employed an emission wavelength of 520 nm, a numerical aperture of $\alpha =1$, and a theoretical bandwidth-limited resolution of 130 nm, which is a quarter of the emission wavelength. The full set number of scans was 242 × 242 positions with a scan interval of 130 nm, which covers the region of interest (ROI) of 31.5 × 31.5 μm². The excitation and emission pupil maps were computationally prepared by the random superposition of Zernike modes numbered up to order 30. These pupil aberrations were autocorrelated to derive OTFs, which were subsequently converted into intensity PSFs by fast-Fourier transform. We simulated point-illumination and wide-field detection by convolving the target with the two independently generated excitation and emission PSFs. Each fluorescence image is sampled at the region of detection (ROD) of $100\times 100$ pixels with a pixel resolution of 130 nm. Therefore, the full basis points in our simulation resulted in a raw data set of $\left(100\times 100\right)\times \left(242\times 242\right)$ pixels corresponding to the ROI of $242\times 242$ pixels and ROD of $100\times 100$ pixels.

Comparison between dual deconvolution and conventional blind deconvolution

The conventional blind deconvolution algorithm used in confocal imaging relies on element-wise vector decomposition, whereas our dual deconvolution algorithm utilizes element-wise matrix decomposition. This distinction has a substantial impact on recoverable resolution, especially in the presence of complex aberrations. To make this point clear, let us explain the image formation in confocal imaging. A confocal image can be described as $I_{\mathrm{CON}}\left(\mathbf{r}\right)=f\left(\mathbf{r},\mathbf{r}\right)$:

$$I_{\mathrm{CON}}\left(\mathbf{r}\right)=\int h_{\mathrm{con}}\left({\mathbf{r}}^{\prime }-\mathbf{r}\right)\gamma \left({\mathbf{r}}^{\prime }\right)\mathrm{d}{\mathbf{r}}^{\prime },$$ 11

where $h_{\mathrm{con}}\left(\mathbf{r}\right)$ is the effective PSF given by $h_{\mathrm{con}}\left(\mathbf{r}\right)=h_{\mathrm{em}}\left(\mathbf{r}\right)h_{\mathrm{ex}}^n\left(\mathbf{r}\right)$. The spectrum of the confocal image is given by ${\tilde{I}}_{\mathrm{con}}\left(\mathrm{\Delta }\mathbf{k}\right)=H_{\mathrm{eff}}\left(\mathrm{\Delta }\mathbf{k}\right)\Gamma \left(\mathrm{\Delta }\mathbf{k}\right)$. Here, $H_{\mathrm{eff}}\left(\mathrm{\Delta }\mathbf{k}\right)$ is the effective OTF, given by $H_{\mathrm{eff}}\left(\mathrm{\Delta }\mathbf{k}\right)=\int H_{\mathrm{em}}\left(\mathrm{\Delta }\mathbf{k}-\mathbf{k}\right)H_{\mathrm{ex}}\left(\mathbf{k}\right)\mathrm{d}\mathbf{k}$. To recover the original object image from the acquired image, blind deconvolution such as joint Richardson-Lucy deconvolution^23,24 is commonly employed, which estimates both the unknown $H_{\mathrm{eff}}$ and $\Gamma$ from $I_{\mathrm{con}}$. This vector decomposition is generally underdetermined and works only when prior knowledge of the PSF is available. Furthermore, effective OTF ($H_{\mathrm{eff}}$) can experience significant attenuation at high spatial frequencies during image acquisition, as each element of $H_{\mathrm{eff}}$ is the result of the summation of the product of two complex-valued functions, $H_{\mathrm{em}}$ and $H_{\mathrm{ex}}$, through deconvolution. Consequently, frequency components below the noise level may not be properly restored or could be entirely lost. In contrast, our dual deconvolution algorithm is a matrix decomposition, a well-determined problem. Therefore, it works for arbitrary complex aberrations with no prior knowledge or assumption. Furthermore, effective OTF matrix $H\left({\mathbf{k}}_{\mathrm{d}},{\mathbf{k}}_{\mathrm{i}}\right)$ in the spectral matrix $\mathit{F}$ is simply the product of $H_{\mathrm{em}}$ and $H_{\mathrm{ex}}$, allowing better preservation of high-frequency content. By leveraging this preservation, our dual deconvolution algorithm can recover much higher frequency components compared to conventional blind deconvolution algorithms.

Experimental setup

We built a two-photon microscope with a wavelength-tunable pulsed laser (INSIGHT X3, Spectra Physics). The excitation wavelength was chosen depending on the types of fluorophores. For example, an excitation wavelength of 900 nm was utilized for Alexa 488, GFP, and eGFP, and 850 nm was used for Rhodamine B. A short-pass dichroic mirror (DMSP680B, Thorlabs) was positioned to separate the emitted fluorescence from the excitation laser beam. The excitation laser beam underwent raster-scanning through 2D Galvano mirrors (GVS002, Thorlabs) before being focused onto the sample plane via a high-numerical-aperture objective (N60X-NIR, Nikon, ×60, 1.0 NA). Fluorescence emissions were captured by the same objective lens, de-scanned by the 2D Galvano mirrors, and subsequently traversed through the previous short-pass dichroic mirror, a short-pass filter (FESH0700, Thorlabs), and a band-pass filter to eliminate the stray excitation laser beam. The band-pass filter was chosen according to the peak emission wavelengths of fluorophores. The peak emission wavelength was 530 nm for Alexa 488, GFP, and eGFP, and 565 nm for Rhodamine B. The filtered fluorescence signals were recorded by a scientific CMOS camera (pco.edge 4.2, PCO AG) in the de-scanned frame. A photomultiplier tube (PMT, H13543-20, Hamamatsu) was also installed for fast imaging to identify and localize fluorescence targets and fluorophores. The total acquisition time ranged from 52 to 260 s for an ROI of 30 × 30 μm with an exposure time of 1–5 ms, depending on the degree of aberrations.

Patch dual deconvolution speed

Reconstruction of the large field-of-view zebrafish image shown in Fig. 6 (145 × 145 μm²) took a total of 1886 s in processing 49 patches entirely on the CPU. This corresponds to ~38 s per patch (size of each patch of 20 × 20 μm²). The computations were performed on a workstation equipped with an Intel(R) Core(TM) i9-11900K @ 3.50 GHz processor and 128 GB RAM.

Preparation for test samples

Rhodamine B polystyrene beads and Alexa 488 gold nanoparticles sample preparation

From 10% L-lysine diluted with distilled water, $1\mathrm{ml}$ of L-lysine was carefully dispensed onto a slide glass (P000BMBS, Marienfeld Superior), which was subsequently covered with a cover glass (P000BMCU, Marienfeld Superior) to suppress the droplet for a duration of $30$ seconds. This procedure ensured the even distribution of L-lysine across the entire surface of the slide glass. Following a $30$-minute drying period, the L-lysine- coated slide glass was rinsed with distilled water to remove any residual impurities. Meanwhile, a solution containing 100 μl of either Rhodamine B polystyrene beads (PS100-RB-1, Nanocs) or Alexa 488 conjugated gold nanoparticles (GFL-100, CD Bioparticles) was prepared with $5~10\mathrm{ml}$ of distilled water in a disposable beaker. Subsequently, an appropriate volume of this nanoparticle mixture was applied to the slide glass, dependent on the particle concentration within the defined region of interest. The fluorescent particles underwent a chemical attachment process with the L-lysine on the slide glass surface. After allowing $10$ minutes for the mixture to dry, the slide was rinsed with distilled water and left to air-dry in preparation for the subsequent particle imaging demonstration.

Artificial scattering medium preparation

A pure hardened PDMS layer of 150 µm thickness was placed onto a slide glass, where fluorescence-dyed beads or particles were positioned. A procedure for aberration and scattering, a cleaning polymer solution (FCDFR, First Contact) was subsequently applied over the PDMS layer. The surface of the cleaning polymer was intentionally roughened by scratching it with sandpaper, creating different structures of slow-varying grating patterns to fast-varying randomized patterns. These surface curvatures that create wavefront error are randomly generated as the polymer solution gradually dried in room temperature conditions.

Fluorescence USAF resolution target fabrication

First, we deposited a 40 nm thickness of titanium metal and a 40 nm thickness of gold metal layers on a cover glass with a thickness of 500 μm. The titanium metal serves as an adhesion layer between gold and glass, and both gold and titanium metal act as a metal mask to block the excitation and emission. Next, we created a customized USAF resolution target pattern using the FIB etcher. Through an etching process that removed the entire 80 nm of gold and titanium, the residual metal on the target pattern formed a mask to conceal background fluorescence.

Following this, we prepared a solution of Rhodamine B (MFCD00011931, Sigma-Aldrich) in ethanol, diluted to one-hundredth of the maximum solubility of Rhodamine B in ethanol. The fluorescence solution was then poured into a well created by a double-sided sticker spacer (654002, Grace Bio-Labs). The fabricated target mask was flipped over and stuck on the spacer. Finally, a custom-made scattering medium with a cleaning polymer solution (FCDFR, First Contact) was attached to the cover glass surface. With the metal mask positioned between the scattering medium and the fluorescence solution, the fabricated sample served as a well-defined 2D extended fluorescence target, as illustrated in Fig. 4f.

Preparation of biological samples

COS-7 cell preparation

In the COS-7 cell imaging experiment, AC28806 cells from the Korean Collection for Type Cultures (KCTC) were cultured on cover glasses. The cells were cultured overnight in DMEM (Gibco) with 10% FBS (Thermo Fisher) and 1% penicillin-streptomycin (Thermo Fisher). Before fixation, the cells were rinsed with pre-warmed PBS(Corning) and treated with pre-warmed extraction buffer (37 °C) containing 0.125% Triton X-100 (Sigma-Aldrich) and 0.4% glutaraldehyde (Sigma-Aldrich) in PBS. After rinsing three times with PBS, the cells were fixed using pre-warmed fixation buffer (37 °C) containing 3.2% paraformaldehyde (Biosesang) and 0.1% glutaraldehyde for 10 min at room temperature (RT). Following three PBS rinses, the cells were permeabilized using a solution containing 3% BSA (Sigma Life Science) and 0.5% Triton X-100 in PBS. Subsequently, the cells were incubated with a primary antibody targeting tubulin (ab6046, Abcam), diluted 1000-fold in a blocking buffer (3% BSA and 0.5% Triton X-100 in PBS) for 1 hour at RT. After primary antibody incubation, the cells were rinsed with PBS and exposed to a secondary antibody labeled with Alexa Fluor 488 (A-11006, Thermo Fisher). The secondary antibody, also diluted 1000-fold in the blocking buffer, was applied to the cells for 1 hour at RT with agitation. The cells underwent an additional three PBS rinses and were then stored at 4 °C.

Mouse brain preparation

In the mouse brain imaging study, 12-weeks-old adult Thy1-EGFP line M mice (The Jackson Laboratory, stock #007788) were anesthetized with an intraperitoneal injection of 100 mg/kg ketamine and 10 mg/kg xylazine before decapitation. Their brains were promptly extracted and placed in an ice-cold artificial cerebrospinal fluid (ACSF). The brains were then sliced into 400–500 μm-thick coronal sections with a vibratome (World Precision Instruments, USA) and fixed at 4 °C in 4% paraformaldehyde overnight. For imaging, the fixed brain was washed with PBS three times and then stuck to plastic dish and immersed in PBS.

Zebrafish preparation

For the zebrafish imaging study, embryos of the Tg (claudinK:gal4vp16;uas:mEGFP) (#FRZCC1013) line were cultivated at a temperature of 28 °C in E3 embryo medium. The E3 medium composition consisted of 5 mM NaCl (7548-4405, Daejung), 0.17 mM KCl (6566-4400, Daejung), 0.33 mM CaCl2 (2507-1400, Daejung), and 0.33 mM MgSO4 (21032, Daejung). After hatching, the embryos were transferred to E3 medium supplemented with N-phenylthiourea (P7629, Sigma Aldrich) to inhibit pigmentation. At 10 days post-fertilization, in the early stage of zebrafish larvae, the larvae were anesthetized using tricaine (E10521, Sigma-Aldrich) within the E3 medium. Subsequently, the larvae were fixed in a 4% paraformaldehyde solution for 1 h at room temperature.

All experimental procedures involving animals were approved by the Committee of Animal Research Policy at Korea University (approval number KOREA2021-0037).

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.

Author contributions

S.Y. and W.C. conceived the project. S.L. and S.Y. constructed the experimental setup. S.L. conducted super-resolution acquisition. With guidance of W.C., S.K., J.H.H., and M.K., S.L., S.Y., and K.G. analyzed data. J.H.H., S.K.*, and Y.-H.J. prepared and provided biospecimens and fabricated target samples. S.L., K.G., W.C., and S.Y. prepared the manuscript and all authors contributed to finalizing the manuscript. S.Y. and W.C. supervised the project. S.K. and S.K.* correspond to Sungsam Kang and Suhyun Kim, respectively.

Peer review

Peer review information

Nature Communications thanks Yicong Wu and the other anonymous reviewers for their contribution to the peer review of this work. [A peer review file is available].

Data availability

The main demonstration datasets presented (Figs. 4 and 5) are openly available to the public at: https://github.com/CenterForDeepImaging/Dual-Deconvolution, and 10.6084/m9.figshare.28600847.

Code availability

The data analysis code used to run and visualize the main demonstration datasets (Figs. 4 and 5) is openly available to the public at: https://github.com/CenterForDeepImaging/Dual-Deconvolution, and 10.6084/m9.figshare.28600847.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-026-69798-y.