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Traditional fluorescence microscopy is constrained by inherent trade-offs among resolution, field-of-view, and system complexity. To navigate these challenges, we introduce a simple and low-cost computational multi-aperture miniature microscope, utilizing a microlens array for single-shot wide-field, high-resolution imaging. Addressing the challenges posed by extensive view multiplexing and non-local, shift-variant aberrations in this device, we present SV-FourierNet, a novel multi-channel Fourier neural network. SV-FourierNet facilitates high-resolution image reconstruction across the entire imaging field through its learned global receptive field. We establish a close relationship between the physical spatially-varying point-spread functions and the network's learned effective receptive field. This ensures that SV-FourierNet has effectively encapsulated the spatially-varying aberrations in our system, and learned a physically meaningful function for image reconstruction. Training of SV-FourierNet is conducted entirely on a physics-based simulator. We showcase wide-field, high-resolution video reconstructions on colonies of freely moving C. elegans and imaging of a mouse brain section. Our computational multi-aperture miniature microscope, augmented with SV-FourierNet, represents a major advancement in computational microscopy and may find broad applications in biomedical research and other fields requiring compact microscopy solutions.

Ultrafast 3D imaging is indispensable for visualizing complex and dynamic biological processes. Conventional scanning-based techniques necessitate an inherent trade-off between acquisition speed and space-bandwidth product (SBP). Emerging single-shot 3D wide-field techniques offer a promising alternative but are bottlenecked by the synchronous readout constraints of conventional CMOS systems, thus restricting data throughput to maintain high SBP at limited frame rates. To address this, we introduce EventLFM, a straightforward and cost-effective system that overcomes these challenges by integrating an event camera with Fourier light field microscopy (LFM), a state-of-the-art single-shot 3D wide-field imaging technique. The event camera operates on a novel asynchronous readout architecture, thereby bypassing the frame rate limitations inherent to conventional CMOS systems. We further develop a simple and robust event-driven LFM reconstruction algorithm that can reliably reconstruct 3D dynamics from the unique spatiotemporal measurements captured by EventLFM. Experimental results demonstrate that EventLFM can robustly reconstruct fast-moving and rapidly blinking 3D fluorescent samples at kHz frame rates. Furthermore, we highlight EventLFM's capability for imaging of blinking neuronal signals in scattering mouse brain tissues and 3D tracking of GFP-labeled neurons in freely moving C. elegans. We believe that the combined ultrafast speed and large 3D SBP offered by EventLFM may open up new possibilities across many biomedical applications.

Imaging through scattering is a pervasive and difficult problem in many biological applications. The high background and the exponentially attenuated target signals due to scattering fundamentally limits the imaging depth of fluorescence microscopy. Light-field systems are favorable for high-speed volumetric imaging, but the 2D-to-3D reconstruction is fundamentally ill-posed, and scattering exacerbates the condition of the inverse problem. Here, we develop a scattering simulator that models low-contrast target signals buried in heterogeneous strong background. We then train a deep neural network solely on synthetic data to descatter and reconstruct a 3D volume from a single-shot light-field measurement with low signal-to-background ratio (SBR). We apply this network to our previously developed Computational Miniature Mesoscope and demonstrate the robustness of our deep learning algorithm on scattering phantoms with different scattering conditions. The network can robustly reconstruct emitters in 3D with a 2D measurement of SBR as low as 1.05 and as deep as a scattering length. We analyze fundamental tradeoffs based on network design factors and out-of-distribution data that affect the deep learning model's generalizability to real experimental data. Broadly, we believe that our simulator-based deep learning approach can be applied to a wide range of imaging through scattering techniques where experimental paired training data is lacking.

Fluorescence microscopy is essential to study biological structures and dynamics. However, existing systems suffer from a tradeoff between field-of-view (FOV), resolution, and complexity, and thus cannot fulfill the emerging need of miniaturized platforms providing micron-scale resolution across centimeter-scale FOVs. To overcome this challenge, we developed Computational Miniature Mesoscope (CM$^2$) that exploits a computational imaging strategy to enable single-shot 3D high-resolution imaging across a wide FOV in a miniaturized platform. Here, we present CM$^2$ V2 that significantly advances both the hardware and computation. We complement the 3$\times$3 microlens array with a new hybrid emission filter that improves the imaging contrast by 5$\times$, and design a 3D-printed freeform collimator for the LED illuminator that improves the excitation efficiency by 3$\times$. To enable high-resolution reconstruction across the large imaging volume, we develop an accurate and efficient 3D linear shift-variant (LSV) model that characterizes the spatially varying aberrations. We then train a multi-module deep learning model, CM$^2$Net, using only the 3D-LSV simulator. We show that CM$^2$Net generalizes well to experiments and achieves accurate 3D reconstruction across a $\sim$7-mm FOV and 800-$\mu$m depth, and provides $\sim$6-$\mu$m lateral and $\sim$25-$\mu$m axial resolution. This provides $\sim$8$\times$ better axial localization and $\sim$1400$\times$ faster speed as compared to the previous model-based algorithm. We anticipate this simple and low-cost computational miniature imaging system will be impactful to many large-scale 3D fluorescence imaging applications.

In this paper, we investigate the $H^p(G) \rightarrow L^p(G)$, $0< p \leq 1$, boundedness of multiplier operators defined via group Fourier transform on a graded Lie group $G$, where $H^p(G)$ is the Hardy space on $G$. Our main result extends those obtained in [Colloq. Math. \textbf{165} (2021), 1--30], where the $L^1(G)\rightarrow L^{1,\infty}(G)$ and $L^p(G) \rightarrow L^p(G)$, $1< p <\infty$, boundedness of such Fourier multiplier operators were proved., Comment: 20 pages

Let $M^{(k)}$, $k=1,2,\ldots, n$, be the boundary of an unbounded polynomial domain $\Omega^{(k)}$ of finite type in $\mathbb C ^2$, and let $\Box_b^{(k)}$ be the Kohn Laplacian on $M^{(k)}$. In this paper, we study multivariable spectral multipliers $m(\Box_b^{(1)},\ldots, \Box_b^{(n)})$ acting on the Shilov boundary $\widetilde{M}=M^{(1)} \times\cdots\times M^{(n)}$ of the product domain $\Omega^{(1)}\times\cdots\times \Omega^{(n)}$. We show that if a function $F(\lambda_1, \ldots ,\lambda_n)$ satisfies a Marcinkiewicz-type differential condition, then the spectral multiplier operator $m(\Box_b^{(1)}, \ldots, \Box_b^{(n)})$ is a product Calder\'on--Zygmund operator of Journ\'e type.

Let $(M,\rho,\mu)$ be a metric measure space satisfying the doubling, reverse doubling and non-collapsing conditions, and $\mathscr{L}$ be a self-adjoint operator on $L^2 (M, d\mu)$ whose heat kernel $p_t (x,y)$ satisfy the small-time Gaussian upper bound, H\"{o}lder continuity and Markov property. In this paper, we give characterizations of inhomogeneous "classical" and "non-classical" Besov and Triebel-Lizorkin spaces associated to $\mathscr{L}$ in terms of continuous Littlewood-Paley and Lusin area functions defined by the heat semigroup, for complete range of indices. This extends related classical results for Besov and Triebel-Lizorkin spaces on $\mathbb{R}^n$ to more general setting, and extends corresponding results in [Trans. Amer. Math Soc. 367 (2015), 121-189] to complete range of indices., Comment: 31 pages

Let $L_{1}$ and $L_{2}$ be non-negative self-adjoint operators acting on $L^{2}(X_{1})$ and $L^{2}(X_{2})$, respectively, where $X_{1}$ and $X_{2}$ are spaces of homogeneous type. Assume that $L_{1}$ and $L_{2}$ have Gaussian heat kernel bounds. This paper aims to study some equivalent characterizations of the weighted product Hardy spaces $H^{p}_{w,L_{1},L_{2}}(X_{1}\times X_{2})$ associated to $L_{1}$ and $L_{2}$, for $p \in (0, \infty)$ and the weight $w$ belongs to the product Muckenhoupt class $A_{\infty}(X_{1} \times X_{2})$. Our main result is that the spaces $H^{p}_{w,L_{1},L_{2}}(X_{1}\times X_{2})$ introduced via area functions can be equivalently characterized by Littlewood-Paley $g$-functions, Littlewood-Paley $g^{\ast}_{\lambda_{1}, \lambda_{2}}$-functions, and Peetre type maximal functions, without any further assumptions beyond the Gaussian upper bounds on the heat kernels of $L_{1}$ and $L_{2}$. Our results are new even in the unweighted product setting., Comment: 18 pages

We prove that the classical one-parameter convolution singular integrals on the Heisenberg group are bounded on multiparameter flag Hardy spaces, which satisfy `intermediate' dilation between the one-parameter anisotropic dilation and the product dilation on $\mathbb{C}^{n}\times \mathbb{R}$ implicitly.