$\texttt{GEM}$: 3D Gaussian Splatting for Efficient and Accurate Cryo-EM Reconstruction

## Abstract

Abstract Cryo-electron microscopy (cryo-EM) has become a central tool for high-resolution structural biology, yet the massive scale of datasets (often exceeding 100k particle images) renders 3D reconstruction both computationally expensive and memory intensive. Traditional Fourier-space methods are efficient but lose fidelity due to repeated transforms, while recent real-space approaches based on neural radiance fields (NeRFs) improve accuracy but incur cubic memory and computation overhead. Therefore, we introduce GEM , a novel cryo-EM reconstruction framework built on 3D Gaussian Splatting (3DGS) that operates directly in real-space while maintaining high efficiency. Instead of modeling the entire density volume, GEM represents proteins with compact 3D Gaussians, each parameterized by only 11 values. To further improve the training efficiency, we designed a novel gradient computation to 3D Gaussians that contribute to each voxel. This design substantially reduced both memory footprint and training cost. On standard cryo-EM benchmarks, GEM achieves up to 48 × 48\times faster training and 12 × 12\times lower memory usage compared to state-of-the-art methods, while improving local resolution by as much as 38.8 % 38.8\% . These results establish GEM as a practical and scalable paradigm for cryo-EM reconstruction, unifying speed, efficiency, and high-resolution accuracy.

### 1 Introduction

Cryo-electron microscopy (cryo-EM) (Bai et al., 2015a; Milne et al., 2013b) is a transformative biological imaging technique that enables the determination of macromolecular structures at near-atomic resolution, and its significance was recognized with the 2017 Nobel Prize in Chemistry (Pre,). In cryo-EM, biological samples are rapidly frozen in vitreous ice and imaged under an electron microscope, producing thousands to millions of noisy two-dimensional projection images of individual particles. Each projection corresponds to a different orientation of the same underlying macromolecule. By aggregating these projections, the three-dimensional electron density of the protein can be reconstructed (Murata & Wolf, 2018). The cryo-EM imaging model naturally aligns with the Fourier slice theorem, where each projection corresponds to a central slice of the 3D Fourier transform of the density, enabling efficient reconstruction in Fourier space (Zhong et al., 2021; Punjani et al., 2017; Scheres, 2012; Tang et al., 2007).

Despite its efficiency, the reliance on Fourier transforms introduces information loss in the traditional methods, limiting achievable resolution. Recent works propose adapting neural radiance fields (NeRFs) to cryo-EM by modeling protein densities directly in real-space (Huang et al., 2024; Qu et al., 2025; Liu et al., 2023), using representations such as MLPs (Liu et al., 2023), Transformers (Huang et al., 2024), or multi-resolution hash encoding (Qu et al., 2025). These approaches remove the dependence on Fourier transforms in the Fourier slice theorem (see Appendix [A.1](https://arxiv.org/html/2509.25075v2#A1.SS1) for more details) and demonstrate improved resolution compared to traditional Fourier-space pipelines. However, direct application of NeRF to cryo-EM remains impractical. Unlike view synthesis tasks, cryo-EM requires reconstruction of the full 3D density rather than rendering sparse pixels. And each training step involves applying the contrast transfer function (CTF) across entire projection images, resulting in cubic scaling of memory and computation. Even for small proteins, this requires millions of sampled points and gradients, making NeRF-based cryo-EM reconstruction prohibitively slow and memory-intensive, especially on commodity GPUs.

Figure: Figure 1: GEM achieves lower memory usage, faster speed, and higher reconstruction resolution compared to existing approaches.
Refer to caption: x1.png

To address these challenges, we propose GEM, a novel framework that introduces 3D Gaussian Splatting (3DGS) (Kerbl et al., 2023; Zha et al., 2024) for cryo-EM reconstruction. Unlike NeRF-based approaches that model density implicitly, GEM adopts an explicit representation using a set of 3D Gaussians, each parameterized by only $11$ values. This representation is both compact and efficient: ($i$) 3D Gaussians are instantiated only at nonzero-density regions, and during projection, ($ii$) each 3D Gaussian contributes to multiple pixels and can be rasterized in parallel. Furthermore, ($iii$) gradient computation is restricted to the subset of Gaussians influencing a given pixel, in contrast to NeRF methods that require activating the entire network for each pixel. This design eliminates the cubic memory footprint of NeRFs and yields substantial acceleration in training.

We validate GEM on four widely used cryo-EM datasets under diverse evaluation protocols. As shown in Figure [1](https://arxiv.org/html/2509.25075v2#S1.F1), GEM achieves substantially faster training and lower memory usage than the real-space NeRF-based pipeline CryoNeRF, while also surpassing both the Fourier-space baseline CryoDRGN and the real-space baseline CryoNeRF in reconstruction quality. Across all datasets, GEM consistently attains gold-standard Fourier shell correlation (GSFSC) resolutions better than $3$ Å, outperforming existing methods. Additional evaluations further confirm its resolution advantage, and compared to CryoNeRF (Qu et al., 2025), GEM not only improves quality but also delivers significantly higher efficiency, even surpassing the Fourier-based CryoDRGN.

Our main contributions are summarized as follows:

- •
We introduce GEM, a novel framework that incorporates 3D Gaussian Splatting into cryo-EM reconstruction, providing an explicit, efficient, and accurate representation of protein density that avoids the information loss of Fourier-based methods and the cubic overhead of NeRF-based real-space approaches.
- •
By explicitly modeling protein density with 3D Gaussians placed only in nonzero-density regions and enabling parallel rasterization across pixels, GEM achieves up to $48\times$ faster training speed compared to existing methods.
- •
We design an efficient gradient computation strategy that restricts updates to only the 3D Gaussians contributing to each pixel, thereby eliminating the cubic memory footprint and computation overhead of prior real-space methods.
- •
Extensive experiments on four widely used cryo-EM datasets under diverse evaluation protocols demonstrate that GEM consistently outperforms existing approaches, often reaching resolutions close to the instrumental limitation.

### 2 Related Work

##### Cryo-EM reconstruction.

Classical single-particle cryo-EM pipelines formulate reconstruction in Fourier space via the Fourier slice theorem, solving for the 3D density with iterative refinement (Zhong et al., 2021; Punjani et al., 2017; Scheres, 2012; Tang et al., 2007). This formulation enables efficient global solvers and robust regularization schemes and has underpinned much of the field’s progress (Milne et al., 2013a; Murata & Wolf, 2018; Bai et al., 2015b). Quality is typically assessed using gold-standard Fourier shell correlation (GSFSC) (Van Heel & Schatz, 2005), local resolution (Glaeser et al., 2021) and directional consistency analyses (Huang et al., 2024). Despite their maturity, Fourier-space methods involve repeated FFTs and interpolation steps and operate on band-limited representations, which can incur information loss. Our work targets these pain points by adopting 3D Gaussian Splatting as an explicit, efficient real-space representation while remaining compatible with standard cryo-EM pipelines and evaluation protocols.

##### Real-space cryo-EM reconstruction.

A growing line of work reconstructs density directly in Euclidean space using neural implicit fields and related parameterizations. Neural-field approaches, which often use MLPs with Fourier or hash encodings, or Transformer-based fields, optimize a continuous density so that its simulated projections match observed images, thereby avoiding frequency-domain interpolation and enabling real-space regularization (Huang et al., 2024; Qu et al., 2025; Liu et al., 2023; Herreros et al., 2025; Palmer & Aylett, 2022; Lu et al., 2022). These methods have demonstrated improved resolution on several benchmarks. However, directly training neural fields for cryo-EM is computationally demanding: unlike view synthesis, we must reconstruct full volumes rather than sparse rays, apply CTFs per micrograph, and backpropagate through dense sampling, leading to memory/time that scale roughly with the volume (and number of samples) instead of the number of visible pixels. This cubic-like footprint makes large proteins and commodity GPUs challenging and motivates representations whose gradients remain localized to the subset of primitives that actually contribute to each pixel.

##### 3D Gaussian splatting.

3D Gaussian Splatting (3DGS) (Kerbl et al., 2023; Zha et al., 2024; Wu et al., 2024; Fei et al., 2024; Yu et al., 2024; Ye et al., 2025) represents scenes as explicit sets of anisotropic Gaussians rendered by fast rasterization, yielding state-of-the-art real-time view synthesis with memory that scales with the number of Gaussians rather than the discretized volume (Kerbl et al., 2023). For biological settings, recent work proposes rectified radiative Gaussian splatting to improve fidelity under line/slice integrals (Zha et al., 2024; 2022; Yu et al., 2025), suggesting that splatting is a promising alternative to implicit fields when projections are the fundamental measurements. Adapting 3DGS to cryo-EM introduces two key advantages: ($i$) sparsity, since Gaussians need only occupy non-empty regions of the macromolecule, and ($ii$) locality, since gradients for a pixel involve only the Gaussians that cover it, avoiding the cubic memory of dense neural fields. Our framework instantiates these ideas for single-particle cryo-EM by coupling a CTF-aware, differentiable projection of Gaussians with joint optimization over density and imaging latents.

### 3 Method

#### 3.1 Cryo-EM Reconstruction

Cryo-EM reconstruction (Elmlund & Elmlund, 2015) aims to recover the 3D structure of a protein from a large collection of noisy 2D particle images. In practice, each image is acquired as a projection of a protein particle illuminated by an electron beam, modulated by the microscope’s contrast transfer function (CTF) and corrupted by background noise.

##### Image Formation Model.

Let $\mathcal{I}=\{I_{i}\}_{i=1}^{N}$ denote a dataset of $N$ particle images and $\mathcal{V}=\{V_{i}\}_{i=1}^{N}$ the corresponding particles. Each image $I_{i}\in\mathbb{R}^{d\times d}$ in Figure [3](https://arxiv.org/html/2509.25075v2#S3.F3) is a projection of an underlying 3D density map $V_{i}\in\mathbb{R}^{d\times d\times d}$ following the imaging model that can be expressed as

$$ $I_{i}=C_{i}*\mathrm{Proj}\left(V_{i};\phi_{i},t_{i}\right)+\eta_{i},$ (3.1) $$

where $C_{i}$ is the CTF determined by microscope settings and $*$ is the convolution operation. $\mathrm{Proj}(V)=\int_{-\infty}^{0}Vdz$ denotes the projection operator with rotation angles $\phi_{i}$ and translation $t_{i}$, and $\eta_{i}$ represents additive noise.

Figure: Figure 2: Cryo-EM imaging process. Randomly oriented proteins are illuminated by the electron beam, leaving traces on the sensor as particle images $I_{1}$, $I_{2}$, and $I_{3}$. In this work we focus on homogeneous reconstruction, where all particles $V_{1}$, $V_{2}$, and $V_{3}$ share the same underlying 3D structure and differ only in orientation (i.e., multiple copies of the same object).
Refer to caption: x2.png

##### Homogeneous Reconstruction.

In this work, we focus on homogeneous reconstruction, where all particles, e.g. $V_{1}$, $V_{2}$, and $V_{3}$ in Figure [3](https://arxiv.org/html/2509.25075v2#S3.F3), share the same underlying structure:

$$ $V_{i}=V,\ \forall\ V_{i}\in\mathcal{V}.$ (3.2) $$

The goal is then to recover the shared 3D density $V$ from all images and their associated parameters:

$$ $\widehat{V}=\mathrm{Reconstruct}\left(\mathcal{I};\Phi,\mathcal{T},\mathcal{C}\right),$ (3.3) $$

where $\Phi=\{\phi_{i}\}_{i=1}^{N}$, $\mathcal{T}=\{t_{i}\}_{i=1}^{N}$, and $\mathcal{C}=\{C_{i}\}_{i=1}^{N}$ denote the set of rotations, translations and CTFs, which are known parameters before the reconstruction.

Fourier Slice Theorem for Reconstruction:
Fourier-space methods leverage the Fourier slice theorem and therefore only need to predict a slice in the Fourier domain, i.e., $\mathrm{Slice}(\mathcal{F}(\widehat{V}_{i});,\phi_{i},t_{i})$, without instantiating the full protein density $\widehat{V}$.
Real-Space Cryo-EM Reconstruction:
In contrast, real-space methods instantiate the entire volume $\widehat{V}$ and perform projection following Equation [3.1](https://arxiv.org/html/2509.25075v2#S3.E1), which requires cubic memory to store both the predicted values and their gradients.

#### 3.2 3D Gaussian Splatting for Reconstruction

We present the GEM training framework in Figure [4](https://arxiv.org/html/2509.25075v2#S3.F4). First, protein density can be represented as a set of 3D Gaussians (Figure [3(a)](https://arxiv.org/html/2509.25075v2#S3.F3.sf1)). To optimize the randomly initialized 3D Gaussian-based representation, we can project 3D Gaussian representations to 2D space. The 3D Gaussian representations can then be optimized by minimizing the differences between the projected 2D images and the captured 2D images by the real microscope. During training, we also introduced novel efficient gradient computation to reduce memory cost. After training, the reconstructed protein density can be queried from the trained 3D Gaussians (Figure [3(b)](https://arxiv.org/html/2509.25075v2#S3.F3.sf2)).

##### Protein Density represented by 3D Gaussians.

Following previous 3DGS literature (Kerbl et al., 2023; Zha et al., 2024), we model the protein density as a set of 3D Gaussians
$\mathcal{G}=\{G_{j}\}_{j=1}^{M}$, where each Gaussian is parameterized by its center $\mathbf{p}_{j}$, covariance matrix $\mathbf{\Sigma}_{j}$, and density coefficient $\rho_{j}$. Each kernel $G_{j}$ defines a local Gaussian-shaped density field:

$$ $G_{j}(\mathbf{x}\,|\,\rho_{j},\mathbf{p}_{j},\mathbf{\Sigma}_{j})=\rho_{j}\cdot\exp\!\left(-\frac{1}{2}(\mathbf{x}-\mathbf{p}_{j})^{\top}\mathbf{\Sigma}_{j}^{-1}(\mathbf{x}-\mathbf{p}_{j})\right),\qquad\mathbf{\Sigma}_{j}=\mathbf{R}_{j}\mathbf{S}_{j}\mathbf{S}_{j}^{\top}\mathbf{R}_{j}^{\top}.$ (3.4) $$

where the covariance $\mathbf{\Sigma}_{j}$ is further decomposed into a rotation $\mathbf{R}_{j}$ and a scale matrix $\mathbf{S}_{j}$. The overall density of protein $V_{i}$ at position $\mathbf{x}$ is then given by the summation of all Gaussian kernels:

$$ $\hat{V}_{i}(\mathbf{x})=\sum_{j=1}^{M}G_{j}(\mathbf{x}\,|\ \rho_{j},\mathbf{p}_{j},\mathbf{\Sigma}_{j}).$ (3.5) $$

Compared to the original 3DGS formulation, which incorporates view-dependent effects for neural rendering, our variant adopts an isotropic, density-based, density-based formulation consistent with the cryo-EM imaging model, where reconstruction relies on where reconstruction relies on direct density integration along the electron beam.

##### Fast 2D Projection of 3D Gaussian representations via Explicit Integration for .

Owing to the property of 3D Gaussians, the projection can be expressed in a closed form, avoiding querying density values and then aggregating them. Let $\widehat{V}_{i}(\mathbf{x})$ denote the protein density for view $i$. The volume is defined as a sum of 3D 3D Gaussians following Equation [3.5](https://arxiv.org/html/2509.25075v2#S3.E5), and the integration along the optical axis $z$ yields a 2D Gaussian mixture:

$$ $\displaystyle\widehat{I}_{i}(\mathbf{r})=\sum_{j=1}^{M}\rho_{j}\,\frac{1}{2\pi\,\bigl|\widehat{\bm{\Sigma}}_{j}\bigr|^{1/2}}\exp\!\left(-\tfrac{1}{2}\,(\widehat{\mathbf{x}}-\widehat{\mathbf{p}}_{j})^{\!\top}\widehat{\bm{\Sigma}}_{j}^{-1}(\widehat{\mathbf{x}}-\widehat{\mathbf{p}}_{j})\right)=\sum_{j=1}^{M}G_{j}^{2},$ (3.6) $$

where $\widehat{\mathbf{x}}\in\mathbb{R}^{2}$, $\widehat{\mathbf{p}}_{j}\in\mathbb{R}^{2}$, and $\widehat{\bm{\Sigma}}_{j}\in\mathbb{R}^{2\times 2}$ are the in-plane (mean, covariance) obtained by dropping the third row/column of their 3D counterparts $\mathbf{x}$, $\mathbf{p}_{j}$, and $\bm{\Sigma}_{j}$, respectively. This closed-form marginalization enables efficient CUDA kernels and removes the need to explicitly query $\widehat{V}_{i}$ on 3D grids before the projection, substantially reducing both compute and memory. Please see Appendix [A.2](https://arxiv.org/html/2509.25075v2#A1.SS2) for more detail of the derivation of the projection. As shown in Figure [4](https://arxiv.org/html/2509.25075v2#S3.F4), Step1 applies Equation [3.6](https://arxiv.org/html/2509.25075v2#S3.E6) to generate the projection from the protein density.

##### Optimizing 3D GS representations

Given the projected image $\widehat{I}_{i}$, we apply the CTF $C_{i}$ in Fourier space and compare to the noisy measurement $I_{i}$ via an $\ell_{2}$ loss:

$$ $I_{i}^{\mathrm{pred}}=\mathcal{F}^{-1}\!\bigl(\mathcal{F}(C_{i})\cdot\mathcal{F}(\widehat{I}_{i})\bigr),\qquad\mathcal{L}_{i}=\bigl\|I_{i}^{\mathrm{pred}}-I_{i}\bigr\|_{2}^{2},$ (3.7) $$

where $\mathcal{F}$ and $\mathcal{F}^{-1}$ are Fourier and inverse Fourier transforms, respectively. In Equation [3.7](https://arxiv.org/html/2509.25075v2#S3.E7), Fourier transform is utilized as the convolution in Equation [3.1](https://arxiv.org/html/2509.25075v2#S3.E1) equals to the multiplication in Fourier space (McGillem & Cooper, 1991). Although 3DGS is typically trained on clean natural images, we observe that the standard MSE loss is sufficient for noisy cryo-EM images. Equation [3.7](https://arxiv.org/html/2509.25075v2#S3.E7) corresponds to Step 2 and 3 in Figure [4](https://arxiv.org/html/2509.25075v2#S3.F4), where the CTF is first applied to the projected image, followed by computing the standard MSE loss between the CTF-corrected image and the experimental ground-truth image.

##### Efficient Gradient Computation to Reduce Memory Overhead.

While our rasterization implementation follows the principle of Equation [3.6](https://arxiv.org/html/2509.25075v2#S3.E6), directly evaluating all 3D Gaussians remains inefficient. To address this, we introduce a thresholding strategy that selects only the Gaussians with non-negligible contributions to a given ray $\mathbf{r}$. Formally,

$$ $\widehat{I}_{i}(\mathbf{r})=\sum_{j=1}^{K}G_{j}^{2},\quad\text{where }G_{j}^{2}>\tau,$ (3.8) $$

where $\tau$ is a predefined threshold. This pruning avoids unnecessary computation from Gaussians far from the ray, since their contribution decays quadratically with distance to the Gaussian center $\mathbf{p}_{j}$.

After selection, the remaining 3D Gaussians are sorted along the $z$-axis, and the accumulation in Equation [3.8](https://arxiv.org/html/2509.25075v2#S3.E8) is performed from the lowest $z$-value (closest to the cryo-EM sensor) to the highest (farthest from the sensor). This ordering prioritizes the most influential 3D Gaussians during rendering and further improves computational efficiency.

Figure: (a) Protein density is represented by 3D Gaussians. Before training, the parameters of 3D Gaussians are randomly initialized.
Refer to caption: x3.png

Figure: Figure 4: The overview of GEM training. The training begins with randomly initialized Gaussians. The 3D Gaussians are projected following Equation [3.6](https://arxiv.org/html/2509.25075v2#S3.E6). Then the CTF is applied (Equation [3.7](https://arxiv.org/html/2509.25075v2#S3.E7)) to the projection and being compared with the noisy experimental image to calculate the loss (Equation [3.7](https://arxiv.org/html/2509.25075v2#S3.E7)).
Refer to caption: x5.png

### 4 Experiment

In this section, we evaluate the proposed method GEM using four widely adopted cryo-EM datasets: EMPIAR-10005 (TRPV1, Liao et al. (2013)), EMPIAR-10028 (Plasmodium falciparum 80S ribosome, Wong et al. (2014)), EMPIAR-10049 (Synaptic RAG1-RAG2 complex, Ru et al. (2015)), and EMPIAR-10076 (L17-depleted 50S ribosomal intermediates, Davis et al. (2016)).

To demonstrate the effectiveness of GEM, we compare it with two representative cryo-EM reconstruction approaches: (i) the conventional voxel-based method CryoSPARC (Punjani et al., 2017), and (ii) the neural network-based CryoDRGN (Zhong et al., 2021). Both of the methods reconstruct cryo-EM densities in Fourier space levaraging the Fourier slice theorem (Radon, 2005; Garces et al., 2011), in contrast to our GEM that operates in 3D Euclidean space. For CryoSPARC, we first perform ab initio reconstruction, followed by homogeneous refinement. For CryoDRGN, we employ the train_nn function, which is designed for homogeneous reconstruction without modeling heterogeneity. In both cases, we use their default parameters. In the following sections, we first evaluate the reconstruction efficiency, and then examnine the reconstructed protein densities from GEM, CryoSPARC and CryoDRGN with three mainstream evaluation protocols: gold-standard Fourier shell correlation (Harauz & van Heel (1986), Section [4.2](https://arxiv.org/html/2509.25075v2#S4.SS2)), local resolution estimation (Adams et al. (2010), Section [4.3](https://arxiv.org/html/2509.25075v2#S4.SS3)), and Fourier slice correlation (Huang et al. (2024), Section [4.4](https://arxiv.org/html/2509.25075v2#S4.SS4)).

#### 4.1 Efficiency Comparison

**Table 1: Efficiency comparison of deep learning-based cryo-EM reconstruction methods. Speed (it/s) measures the number of images processed per second, an Memor (GB) reports the maximum GPU memory consumption during training. All experiments are conducted on a single NVIDIA RTX A6000 (48 GB). OOM indicates out-of-memory errors. Our method achieves the highest training throughput while requiring substantially less memory across all datasets.**
|  | EMPIAR-10028 | EMPIAR-10049 | EMPIAR-10076 | EMPIAR-10005 |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
|  | Speed (it/s) $\uparrow$ | Memory (GB) $\downarrow$ | Speed (it/s) $\uparrow$ | Memory (GB) $\downarrow$ | Speed (it/s) $\uparrow$ | Memory (GB) $\downarrow$ | Speed (it/s) $\uparrow$ | Memory (GB) $\downarrow$ |
| CryoDRGN | 47.62 | 8.81 | 142.86 | 2.58 | 58.82 | 8.79 | 100.00 | 4.93 |
| CryoNeRF | - | OOM | 9.13 | 9.94 | 2.10 | 37.75 | 4.03 | 20.50 |
| Ours | 94.10 | 1.54 | 160.49 | 0.50 | 96.60 | 0.74 | 108.11 | 0.51 |

We first evaluate the efficiency of GEM against deep learning-based cryo-EM reconstruction methods. Specifically, we compare with CryoDRGN (Fourier-based) and CryoNeRF (NeRF-based). All experiments are conducted on a single NVIDIA RTX 6000 GPU, and we report both training throughput (images per second) and peak GPU memory usage. As shown in Table [1](https://arxiv.org/html/2509.25075v2#S4.T1), GEM consistently achieves the best efficiency, with substantially lower memory usage and faster speed than the baselines. CryoNeRF suffers from cubic memory and computation overhead inherent to NeRF, resulting in very slow training and frequent out-of-memory (OOM) failures. CryoDRGN, which leverages the Fourier slice theorem with quadratic complexity, is faster than CryoNeRF but still significantly outperformed by GEM. Moreover, GEM exhibits more stable memory usage, as it depends primarily on the number of 3D Gaussians used to represent the protein density.

Figure: Figure 5: GSFSC of our GEM and baselines. The horizontal axis denotes resolution in ångströms (Å), and the vertical axis denotes the FSC value. The two dashed horizontal lines are FSC thresholds of $0.5$ and $0.143$. The final resolution is defined as the resolution at which the GSFSC curve first drops below the $0.143$ threshold (smallest possible resolution if no intersection). Intersection points further to the right corresponds to better reconstruction quality.
Refer to caption: x6.png

**Table 2: Final resolutions (in Å) obtained by each method according to the GSFSC $0.143$ threshold. Lower values indicate better resolution.**
|  | EMPIAR-10028 | EMPIAR-10049 | EMPIAR-10076 | EMPIAR-10005 |
| --- | --- | --- | --- | --- |
| CryoSPARC | 4.12 Å | 4.53 Å | 3.39 Å | 4.50 Å |
| CryoDRGN | 3.81 Å | 4.02 Å | 3.66 Å | 3.27 Å |
| Ours | 2.98 Å | 2.56 Å | 2.62 Å | 2.43 Å |

Figure: Figure 6: Local resolution maps of reconstructions from GEM, CryoSPARC, and CryoDRGN. Each map is colored according to the estimated local resolution, with the color scale ranging from 2 Å (blue, higher resolution) to 5 Å (red, lower resolution). Better reconstructions are indicated by larger regions in blue-green.
Refer to caption: x7.png

**Table 3: Mean local resolution (in Å) of reconstructions on the four cryo-EM datasets. Lower values correspond to higher resolution.**
|  | EMPIAR-10028 | EMPIAR-10049 | EMPIAR-10076 | EMPIAR-10005 |
| --- | --- | --- | --- | --- |
| CryoSPARC | 3.55 Å | 3.57 Å | 3.16 Å | 5.54 Å |
| CryoDRGN | 3.39 Å | 3.22 Å | 3.22 Å | 3.38 Å |
| Ours | 2.87 Å | 2.62 Å | 2.99 Å | 2.58 Å |

#### 4.2 Gold-Standard Fourier Shell Correlation

Gold-standard Fourier shell correlation (GSFSC) (Van Heel & Schatz, 2005) is a commonly used evaluation to quantify the reconstruction quality of a cryo-EM reconstruction method on a dataset.

Given a cryo-EM particle image dataset $\mathcal{I}=\left\{I_{i}\right\}_{i=1}^{N}$, to evaluate a method with GSFSC, the dataset $\mathcal{I}$ is first randomly splitted into two non-overlapping splits $\mathcal{I}_{1}$ and $\mathcal{I}_{2}$. Then the method to test is applied on the two splits to produce two reconstructions $X_{1}$ and $X_{2}$. The FSC value at spatial frequency shell $r_{i}$ in the Fourier space is defined as:

$$ $\mathrm{FSC}(r_{i})=\frac{\sum_{r\in r_{i}}F_{1}(r)\cdot F_{2}(r)^{\ast}}{\sqrt{\sum_{r\in r_{i}}\left|F_{1}(r)\right|^{2}\cdot\sum_{r\in r_{i}}\left|F_{2}(r)\right|^{2}}},$ (4.1) $$

where $F_{1}(r)$ and $F_{2}(r)$ denote the Fourier coefficients of $X_{1}$ and $X_{2}$ at frequency $r$, and ^∗ indicates complex conjugation. The summations are computed over all voxels in the shell $r_{i}$. As in the GSFSC protocol the particle images are split into two halves $\mathcal{I}_{1}$ and $\mathcal{I}_{2}$ and reconstructed independently, it ensures that the correlation reflects reproducible structural information rather than overfitting.

Since the spatial frequency $r$ is inversely related to the 3D Euclidean space resolution, the GSFSC values can be plotted as a curve of FSC values versus resolutions (in Å). The resolution estimate is then determined by the point at which the GSFSC curve drops below a predefined threshold (commonly $0.143$). This crossing point represents the highest spatial frequency (lowest resolution value) at which the reconstruction remains reliable, thereby providing a quantitative resolution assessment.

As shown in Figure [5](https://arxiv.org/html/2509.25075v2#S4.F5) and Table [2](https://arxiv.org/html/2509.25075v2#S4.T2), ❶ GEM consistently achieves higher resolutions (i.e., lower Å values at the intersection with the $0.143$ FSC threshold) than the baselines across all datasets. It is noteworthy that, according to the Nyquist-Shannon sampling theorem (Shannon, 2006), the ultimate attainable resolution is limited to twice the voxel size of the protein density map, corresponding to $2.46$ Å and $2.62$ Å for EMPIAR-10049 and EMPIAR-10028, respectively. ❷ On these two datasets, GEM achieves resolutions that approach this theoretical upper bound, highlighting its strong ability to recover fine structural details from cryo-EM images.

Figure: Figure 7: FSLC maps of reconstructions from GEM, CryoSPARC, and CryoDRGN across four datasets. Each map depicts the correlation values of slices sampled across different elevation and azimuth angles, with the color scale ranging from low correlation (purple) to high correlation (yellow). More uniform yellow regions indicate higher and more isotropic directional resolution.
Refer to caption: x8.png

**Table 4: FSLC results on four cryo-EM datasets. Each row reports the mean FSLC value with its standard deviation across randomly sampled orientations. Higher mean values indicate stronger directional consistency, and lower standard deviation reflects greater stability.**
|  | EMPIAR-10028 | EMPIAR-10049 | EMPIAR-10076 | EMPIAR-10005 |
| --- | --- | --- | --- | --- |
| CryoSPARC | 0.961 ± 0.030 | 0.348 ± 0.160 | 0.978 ± 0.012 | 0.886 ± 0.084 |
| CryoDRGN | 0.989 ± 0.005 | 0.814 ± 0.109 | 0.976 ± 0.020 | 0.974 ± 0.025 |
| Ours | 0.998 ± 0.002 | 0.971 ± 0.022 | 0.992 ± 0.010 | 0.982 ± 0.019 |

#### 4.3 Local Resolution Estimation

We then evaluate the reconstructions from all methods using the standard protocol of local resolution estimation. While the FSC provides a single global resolution value, the same principle can be localized to regions within the protein density, providing more detailed estimation of the quality of the reconstruction. Specifically, given two half maps reconstructed from independent data splits, a cubic subregion is extracted at the same location from each map. The FSC between the two cubes is then computed following Equation [4.1](https://arxiv.org/html/2509.25075v2#S4.E1), and the local resolution is determined from the frequency at which the FSC curve falls below the $0.143$ threshold. This estimated resolution is assigned to the voxel at the cube center. Repeating this process across the entire map yields a resolution map that depicts the spatial variability of the reconstruction quality.

As shown in Figure [6](https://arxiv.org/html/2509.25075v2#S4.F6) and Table [3](https://arxiv.org/html/2509.25075v2#S4.T3), GEM consistently achieves superior local resolution across all four datasets. The EMPIAR-10005 dataset poses a significant challenge to existing approaches due to its high symmetry and structural complexity, leading CryoSPARC and CryoDRGN to achieve average local resolutions of approximately 5 Å and 3 Å, respectively. In contrast, GEM attains an average local resolution of 2.28 Å, which approaches the Nyquist-Shannon limit of 2.43 Å, thereby demonstrating its strong capability to recover fine structural details.

#### 4.4 Fourier Slice Correlation

We further validate the reconstructions using a recently proposed evaluation protocol, Fourier slice correlation (FSLC, Huang et al. (2024)). FSLC serves as an intermediate-level metric that quantifies directional resolution anisotropy by measuring the similarity between corresponding slices extracted from two reconstructed protein densities (Tan et al., 2017; Huang et al., 2024). In practice, slices are sampled at uniformly distributed elevation angles ($0\degree-180\degree$) and azimuth angles ($0\degree-360\degree$).

As shown in Figure [7](https://arxiv.org/html/2509.25075v2#S4.F7) and Table [4](https://arxiv.org/html/2509.25075v2#S4.T4), GEM consistently achieves higher FSLC values, indicating stronger agreement across a wide range of rotation angles. While all methods exhibit relatively lower correlation around the azimuthal regions of the elevation axis, GEM still outperforms the baselines, demonstrating superior robustness in challenging orientations. Moreover, our method attains the lowest standard deviation across the four datasets, further highlighting its stability and consistency with respect to directional variations.

**Table 5: Ablation study on different parameters of 3DGS. "No Rotation" means the rotation of all 3D Gaussians are fixed at 0. "Isotropic Scale" indicates the scale matrix $\mathbf{S}_{i}$ is identical across the three axis of the 3D Gaussians, resulting in a spherical shape of the 3D Gaussians.**
|  | EMPIAR-10028 | EMPIAR-10049 | EMPIAR-10076 | EMPIAR-10005 |
| --- | --- | --- | --- | --- |
| ❶ No Rotation | 3.71 Å | 4.23 Å | 5.99 Å | 4.15 Å |
| ❷ Isotropic Scaling | 4.02 Å | 5.53 Å | 12.80 Å | 4.21 Å |
| ❶ $+$ ❷ | 10.67 Å | 6.81 Å | 13.54 Å | 7.95 Å |
| Original | 2.98 Å | 2.56 Å | 2.62 Å | 2.43 Å |

#### 4.5 Design Choices of 3DGS for cryo-EM

While 3DGS was developed for reconstruction of natural scenes, it is unclear which set of parameters are essential for cryo-EM reconstruction. We therefore ablate per-Gaussian rotation and anisotropic scaling. Concretely, we evaluate: ($i$) No Rotation, fix $\mathbf{R}_{i}=\mathbf{I}$; ($ii$) Isotropic Scaling, enforce $\mathbf{S}_{i}=s_{i}\mathbf{I}$, where $s_{i}$ is the distance of the $i$-th 3D Gaussian to its nearest neighbor; ($iii$) No Rotation and Isotropic Scaling (both constraints).

As shown in Table [5](https://arxiv.org/html/2509.25075v2#S4.T5) (lower Å is better), ❶ all ablations degrade resolution across datasets. ❷ Enforcing isotropic scaling is consistently more harmful than removing rotation (e.g., on EMPIAR-10076, $2.62$ Å to $12.80$ Å vs. $2.62$ Å to $5.99$ Å), and combining both constraints cause the most harm to the performance. These results indicate that both per-Gaussian rotation and anisotropic scaling are critical for accurately modeling cryo-EM densities, validating our choice to retain the full 3DGS parameterization in GEM.

### 5 Conclusion

In this work, we introduced GEM, a novel framework that leverages 3D Gaussian Splatting (3DGS) for cryo-EM reconstruction. By explicitly representing protein densities with 3D Gaussians, GEM avoids the information loss of Fourier-space methods and the cubic memory and computation overhead of NeRF-based real-space approaches, thereby achieving high resolution and efficiency. Through comprehensive experiments, GEM shows that it delivers up to $48\times$ faster training, $12\times$ lower memory consumption, and up to $38.8\%$ improvement in local resolution over prior state-of-the-art methods, often approaching the physical resolution limit of the microscope. Overall, GEM establishes 3DGS as a practical and scalable paradigm for cryo-EM, unifying efficiency with high-resolution accuracy and offering broad benefits to the structural biology community.

### Acknowledgement

This research was partially funded by the National Institutes of Health (NIH) under award 1R01EB037101-01. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the NIH. Tianlong Chen was also partially supported by the Amazon Research Award (Spring 2025) and the Gemma Academic Program GCP Credit Award.

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## Appendix

### Appendix A Appendix

#### A.1 Description of Baseline Methods

##### Fourier Slice Theorem for Reconstruction.

A common strategy in cryo-EM reconstruction (Zhong et al., 2021; Punjani et al., 2017) is to leverage the Fourier slice theorem (Bracewell, 1956), which states that the Fourier transform of a 2D projection of a 3D density $V_{i}$ corresponds to a central slice of its 3D Fourier transform perpendicular to the projection direction. Formally,

$$ $\mathcal{F}\!\left(\mathrm{Proj}(V_{i};\,\phi_{i},t_{i})\right)=\mathrm{Slice}\!\left(\mathcal{F}(V_{i});\,\phi_{i},t_{i}\right),$ (A.1) $$

where $\mathcal{F}(\cdot)$ denotes the Fourier transform and $\mathrm{Slice}(\cdot;\phi_{i},t_{i})$ extracts the 2D slice orthogonal to the projection direction specified by $(\phi_{i},t_{i})$. This theorem implies that reconstructing the protein density reduces to learning to predict individual slices in Fourier space, thereby significantly lowering the memory footprint during training.

##### Real-Space Cryo-EM Reconstruction.

The reconstruction formulation in Equation [3.3](https://arxiv.org/html/2509.25075v2#S3.E3) closely parallels 3D reconstruction problems in computer vision, which has motivated recent work to perform cryo-EM reconstruction directly in real space using neural radiance fields (NeRFs) (Huang et al., 2024; Qu et al., 2025). Unlike Fourier-based methods, this approach avoids repeated Fourier transforms and can in principle achieve higher resolution. However, it requires predicting the full density $\widehat{V}_{i}$ before supervision can be applied, since the contrast transfer function (CTF) $C_{i}$ must operate on the complete projection image due to the convolution operation.

Concretely, given a set of predefined coordinates $D$, each particle’s pose is applied as

$$ $D_{i}=\mathrm{Rotate}(D;\,\phi_{i})+t_{i},$ (A.2) $$

and the resulting coordinates are used to query the NeRF, producing the predicted density volume $\widehat{V}_{i}$. The model is then trained by projecting $\widehat{V}_{i}$ under the same pose and computing the loss between the simulated projection $\widehat{I}_{i}$ and the experimental image $I_{i}$:

$$ $\mathcal{L}_{i}=\mathrm{MSE}\!\left(I_{i},\mathrm{Proj}\!\left(\widehat{V}_{i};\,\phi_{i},t_{i}\right)\right).$ (A.3) $$

Because the entire density $\widehat{V}_{i}$ must be generated at each iteration, this method incurs cubic memory and computation cost, making training slow and resource-intensive despite its resolution advantage.

#### A.2 Derivation of 3DGS Projection

The $j$-th 3D Gaussian is defined as:

$$ $G_{j}(\mathbf{x}\,|\,\rho_{j},\mathbf{p}_{j},\mathbf{\Sigma}_{j})=\rho_{j}\cdot\exp\!\left(-\frac{1}{2}(\mathbf{x}-\mathbf{p}_{j})^{\top}\mathbf{\Sigma}_{j}^{-1}(\mathbf{x}-\mathbf{p}_{j})\right).$ (A.4) $$

Therefore for a query position $\mathbf{x}$ in the protein density, its value is defined as

$$ $\hat{V}_{i}(\mathbf{x})=\sum_{j=1}^{M}G_{j}(\mathbf{x}\,|\ \rho_{j},\mathbf{p}_{j},\mathbf{\Sigma}_{j}).$ (A.5) $$

To avoid the query of the entire protein density before the projection, we aim to explicitly derive the expression of the projected image $\hat{I}_{i}$. Since each pixel in the image $\hat{I}_{i}$ corresponds to an electron beam $\mathbf{r}$, we denote the projection of this specific pixel as $\widehat{I}_{i}(\mathbf{r})$.

Following the definition, the projection of the pixel can be defined as:

$$ $\displaystyle\widehat{I}_{i}(\mathbf{r})=\int\widehat{V}_{i}(\mathbf{x})\ dz$ $\displaystyle=\int\sum_{j=1}^{M}G_{j}(\mathbf{x}\mid\rho_{j},\mathbf{p}_{j},\bm{\Sigma}_{j})\ dz$ (A.6) $\displaystyle=\sum_{j=1}^{M}\int G_{j}(\mathbf{x}\mid\rho_{j},\mathbf{p}_{j},\bm{\Sigma}_{j})\ dz.$ (A.7) $$

From the standard 3DGS for natural images, we have

$$ $\hat{I}_{r}(\mathbf{r})\approx\sum_{j=1}^{M}\int G_{j}(\mathbf{x}|\rho_{j},\mathbf{p}_{j},\underbrace{\mathbf{J}_{j}\mathbf{W}\mathbf{\Sigma}_{j}\mathbf{W}^{\top}\mathbf{J}_{j}^{\top}}_{\widetilde{\mathbf{\Sigma}}_{i}})\ dz,$ (A.8) $$

where $\widetilde{\mathbf{\Sigma}}_{i}$ is the new Gaussian covariance matrix controlled by local approximation matrix $\mathbf{J}_{j}$ and viewing transformation matrix $\mathbf{W}$ that only corresponds to the rotation $\phi$. The projection of natural images follows the pinhole camera model, while in cryo-EM the projection rays are parallel, therefore the Jacobian matrix satifies $\mathbf{J}_{j}=\mathbf{I}$, which gives $\tilde{\mathbf{\Sigma}}_{j}=\mathbf{\Sigma}_{j}$. Thus following (Zwicker et al., 2002; 2001) we have

$$ $\displaystyle\widehat{I}_{i}(\mathbf{r})$ $\displaystyle=\sum_{j=1}^{M}\rho_{j}(2\pi)^{\frac{3}{2}}\left|\tilde{\bm{\Sigma}}_{j}\right|^{\frac{1}{2}}\int\frac{1}{(2\pi)^{\frac{3}{2}}\left|\tilde{\bm{\Sigma}}_{j}\right|^{\frac{1}{2}}}\exp\left(-\frac{1}{2}\left(\mathbf{x}-\widetilde{\mathbf{p}}_{j}\right)^{\top}\tilde{\bm{\Sigma}}_{i}^{-1}\left(\mathbf{x}-\mathbf{p}_{j}\right)\right)\ dz$ (A.9) $\displaystyle=\sum_{j=1}^{M}\rho_{j}(2\pi)^{\frac{3}{2}}\lvert\bm{\Sigma}_{i}\rvert^{\frac{1}{2}}\int\frac{1}{(2\pi)^{\frac{3}{2}}\lvert\bm{\Sigma}_{i}\rvert^{\frac{1}{2}}}\exp\left(-\frac{1}{2}\left(\mathbf{x}-\mathbf{p}_{i}\right)^{\top}\bm{\Sigma}_{i}^{-1}\left(\mathbf{x}-\mathbf{p}_{i}\right)\right)dz$ (A.10) $\displaystyle=\sum_{j=1}^{M}\rho_{j}\,\frac{1}{2\pi\,\bigl|\widehat{\bm{\Sigma}}_{j}\bigr|^{1/2}}\exp\!\left(-\tfrac{1}{2}\,(\widehat{\mathbf{x}}-\widehat{\mathbf{p}}_{j})^{\!\top}\widehat{\bm{\Sigma}}_{j}^{-1}(\widehat{\mathbf{x}}-\widehat{\mathbf{p}}_{j})\right).$ (A.11) $$

#### A.3 Description of Datasets

The characteristics of datasets used in this paper are summarized as follows:

- •
EMPIAR-10005 (TRPV1): A well-characterized protein in the vertebrate TRP family, and is frequently used to investigate fundamental TRP channel functions and structures.
- •
EMPIAR-10028 (Pf 80S ribosome): Contains a complex composition of the antibiotic and Pf ribosome and is commonly used for evaluation of the reconstruction resolution.
- •
EMPIAR-10049 (Synaptic RAG1–RAG2 complex): Exhibits substantial compositional and conformational heterogeneity and is therefore challenging for high-resolution reconstruction.
- •
EMPIAR-10076 (L17-depleted 50S ribosomal intermediates): Comprises complex intermediate assembly states that can cause reconstruction quality degradation.

## Peer Reviews

confidence: 5
contribution: 3
initial_rating: 6
presentation: 3
questions: ### Computational Complexity
Please provide a theoretical and empirical analysis of the time and memory complexity of GEM. Specifically, how does the number of Gaussians (M) scale with the size and resolution of the map, and how does the overall complexity compare to the O(N^3) NeRF methods and O(NlogN) Fourier methods?

### Gaussian Initialization 
How sensitive is the final reconstruction to the initialization of the 3D Gaussian set? What method is used to control the density (number of Gaussians) during optimization, and how is the over-splitting or under-splitting of Gaussians prevented? Showing a result of reconstruction quality vs. number of Gaussians would be highly informative.

### Handling of Anisotropy and Local Resolution 
Cryo-EM maps often exhibit anisotropic resolution. How does the adaptive nature of the anisotropic 3D Gaussians explicitly help in modeling regions with poorer local resolution (e.g., flexible domains) versus rigid, high-resolution core regions? Can GEM provide a per-Gaussian confidence metric analogous to local resolution maps?

### Contrast and Background Modeling 
How is the background solvent and noise modeled in this 3DGS representation? Since the Gaussians represent the object density, is there a risk that low-contrast regions of the molecule are poorly represented or pruned during optimization?
soundness: 3
strengths: It proposes GEM, a novel application of 3D Gaussian Splatting (3DGS)—a technique recently popularized in computer graphics and vision for novel view synthesis—to the challenging domain of 3D cryo-EM reconstruction. Adapting this explicit, point-based rendering system to the parallel-ray projection geometry of cryo-EM is a highly creative methodological step. The significance is substantial: the paper directly addresses the major limitations of current state-of-the-art reconstruction methods. Traditional Fourier-space methods are fast but often sacrifice fidelity due to repeated transforms, while recent real-space methods (like those based on Neural Radiance Fields, NeRF) achieve high accuracy but incur cubic complexity (O(N^3)) in both memory and computation, making them impractical for the massive datasets common in cryo-EM. GEM promises to break this accuracy-efficiency deadlock. The quality of the method, particularly the adaptation of the 3DGS projection model to cryo-EM's parallel rays, appears sound, and the results on datasets like EMPIAR-10005 are promising. The clarity of the paper is good, clearly outlining the mathematical basis for the projection.
summary: GEM introduces a novel method for 3D cryo-EM reconstruction by adapting the 3D Gaussian Splatting (3DGS) technique. This explicit, point-based representation offers high accuracy while significantly improving efficiency and memory consumption compared to O(N^3) real-space methods like NeRF, making large-scale structural biology more feasible.
weaknesses: A substantive assessment of the weaknesses of the paper. Focus on constructive and actionable insights on how the work could improve towards its stated goals.

Limited Experimental Scope for Generalization: The primary weakness is the limited breadth and depth of the experimental evaluation. While results on a well-known dataset like TRPV1 (EMPIAR-10005) are shown, the evaluation must be expanded to demonstrate the generalizability of GEM. Specifically, the paper is missing results on:

Complexes with different symmetry types (e.g., helical, octahedral) beyond simple C-symmetry or no symmetry.

Datasets spanning a wider range of resolutions (from very low to near-atomic).

Very large complexes to definitively prove the claimed computational advantage over O(N^3) methods.

Explicit Representation Limitations: 3DGS is an explicit, discrete representation using a set of anisotropic Gaussians. The paper does not sufficiently discuss how this representation handles the continuous nature of the electron density map, especially at sub-nanometer resolutions where fine-grained features like side-chain density need to be accurately modeled. This discrete nature may introduce discretization artifacts or fail to capture the smoothness of the underlying density compared to implicit representations like NeRFs.

Comparison Rigor: The comparison against existing methods needs to be more robust. The paper should include detailed comparative metrics, not just for final resolution (FSC), but also for reconstruction speed (wall-clock time), memory footprint, and local resolution variability compared to both a leading Fourier-based method (e.g., RELION, cryoSPARC) and the state-of-the-art NeRF-based approaches.

---

confidence: 5
contribution: 2
initial_rating: 2
presentation: 3
questions: - Just for clarification purpose, for both cryoDRGN and GEM experiments, the poses of the particles are given, correct? Where do they come from?
soundness: 1
strengths: - The paper is well written and the figures are very clear and concise.
- The introduction of 3DGS into cryo-EM reconstruction is interesting and the improvement of the computational efficiency is promising.
summary: This paper introduced a new way, using Gaussian splatting, to reconstruct 3D density maps from cryo-EM particles with given poses. The authors demonstrated better computational efficiency and better metrics (mostly GS-FSC) using 3DGS compared to other NeRF-based methods.
weaknesses: - The major weakness of this paper is the evaluation. To evaluate and show that GEM can achieve better results compared to other methods, the paper heavily used metrics (including GSFSC, local resolution, FSLC) that assess the self consistency between the two half maps. This is wrong. Here the methods themselves, will lead to spurious correlation between the half maps due to the induction bias of the methods. Therefore, GSFSC, FSLC and local resolution are all hacked and unreliable. This can be reflected in the results by: 1) GSFSC curves never reaches zero and even increase by a lot at very high frequency (Fig. 5), 2) Final resolution (0.143 cutoff) almost reach Nyquist while the density maps do not support such high resolution, 3) Local resolutions are severely overestimated while the maps lack structural details compared to other methods (Fig. 6), 4) even when the particle poses are the same, FSLC are quite different (Fig. 7). All of these show that half maps based metrics are all over estimated.

- The author should also compare 3DGS with the traditional reconstruction method using (weighted) back-projection (WBP) in Fourier space. The comparison with cryoSPARC is unfair, since both ab initio and homogenous refinement in cryoSPARC refine particle poses, while other methods do not. The easiest way to compare is reconstruct only with given poses. In RELION, you can use the `relion_reconstruct` command. Since WBP is simply performing back-projection with accumulation, computation can be easily paralleled across particles and it is usually very fast even without a GPU.

- Given the previous points, the motivation of using 3DGS into cryo-EM, although interesting and insightful, seems very unclear to me. 

Minor: In Fig. 6, the threshold level for EMPIAR-10005 map is too low (probably also 10049 as well).

---

confidence: 4
contribution: 2
initial_rating: 4
presentation: 2
questions: I have raised my questions in the main weakness sections.
soundness: 3
strengths: 1. This paper presents an interesting and timely contribution, offering solid theoretical derivations for the application of 3D Gaussian Splatting (3DGS) to cryo-EM homogeneous reconstruction. The motivation for adopting 3DGS in this domain is sound: it provides an explicit and efficient real-space representation that avoids the information loss of Fourier-based methods and the heavy computational cost of neural-field models. This is a promising and original direction that meaningfully bridges computer graphics advances and structural biology, enabling high-quality protein reconstruction without sacrificing training speed.

2. The experimental protocol is comprehensive and carefully designed. The authors benchmark GEM across multiple standard cryo-EM datasets and report not only global resolution metrics (GSFSC) but also local resolution and Fourier Slice Correlation (FSLC) analyses. This multi-faceted evaluation demonstrates the method’s robustness, efficiency, and generalizability.
summary: The paper presents GEM, a new framework for efficient and accurate 3D reconstruction in cryo-electron microscopy (cryo-EM) using 3D Gaussian Splatting (3DGS). Unlike traditional Fourier-based methods that lose information during repeated transformations, and neural field–based real-space methods that suffer from cubic memory and computation costs, GEM explicitly models protein density as a compact set of 3D Gaussians, each parameterized by 11 values. This representation enables localized gradient updates, parallel rasterization, and selective computation of only contributing Gaussians, dramatically reducing resource demands.
weaknesses: ## Major Weaknesses

1. **Clarity and Writing Quality.** The paper’s writing is often rough, with numerous grammatical errors and typos that hinder readability and clarity. Several technical explanations are hard to follow, particularly the definitions of rays and the sentence spanning L204–L205, which require clearer formulation. A thorough language revision is essential to improve accessibility and ensure technical accuracy.
2. **Unclear Necessity of Gaussian Sorting.** The decision to sort 3D Gaussians along the projection axis (Sec. 3.2) is not well justified. In principle, sorting should not affect the final result of the projection integral, yet it introduces nontrivial computational overhead, especially in CUDA kernels. The authors should explain the rationale behind this step—e.g., whether it serves numerical stability, differentiability, or rasterization ordering—and ideally show an ablation demonstrating that removing sorting does not harm reconstruction quality.
3. **Inconsistent Quantitative and Qualitative Results.** The reported quantitative results (Tables 2 and 3) appear unusually strong and potentially inconsistent with the qualitative reconstructions shown in Figures 5 and 6. For example, the sharp improvement in GSFSC for EMPIAR-10076 near 4–5 Å seems atypical and lacks discussion. Meanwhile, the visual reconstructions in Figure 6 appear smoother and less detailed than those from CryoDRGN or CryoSPARC, especially in α-helix regions. This suggests that GEM may underfit the true density due to the strong structural prior imposed by 3DGS (insufficient number of 3DGS may cause this). A useful diagnostic would be to increase the number of Gaussians (e.g., from 1 k to 10 k) and analyze whether this reduces oversmoothing. Additionally, experimenting with more advanced loss functions—such as those used in RELION or other likelihood-based formulations—could help recover finer local features. Overall, the current performance claims are not fully convincing without further analysis of these discrepancies.
4. **Lack of Details on the Number and Impact of 3D Gaussians.** The paper does not specify how many 3D Gaussians are used in training or how this parameter affects speed, memory usage, and reconstruction quality. Since this is central to both the efficiency and fidelity of GEM, the authors should include an ablation or scaling study to clarify this relationship.

## Minor Weaknesses

1. L165: The slice operation in Fourier space still requires instantiating the entire 3D density volume because the slicing orientation is unknown a priori.

2. L192: “density-based, density-based” → should be “density-based.”

3. L195: The phrase “Explicit Integration for XXX” appears incomplete and should be clarified.

4. L204–L205: The sentence is difficult to interpret, and the definition of r is missing, despite being fundamental to the imaging model—please rewrite and define it formally.

5. L243: The phrase “CTF-corrected” should be replaced with “CTF-corrupted” or “CTF-applied,” as no correction occurs in this step.

6. L198: Typo — “3D 3D” → “3D.”

7. L721: Variables x^hat and p^hat are not defined in the Appendix and should be explicitly introduced, as they are used in the projection derivation.

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confidence: 4
contribution: 3
initial_rating: 6
presentation: 2
questions: Please refer to questions embedded in weaknesses.
soundness: 3
strengths: The method is a natural combination of 3D Gaussian Splatting, which works well in computer vision, to cryo-EM homogeneous reconstruction. Quantitative results are encouraging compared to strong baselines, and the implementation is computationally efficient.
summary: The paper adapts 3D Gaussian Splatting to real-space homogeneous reconstruction in cryo-EM. GEM generally exceeds the local and global resolution (measured via Fourier Shell Correlation) of CryoSPARC and CryoDRGN, on representative proteins.
weaknesses: Most of these weaknesses are presentation related; they are written roughly in order of appearance in the paper:
- The input to GEM is not entirely clear until the bottom of page 3, where it is specified that the CTFs, poses, and translations are assumed known. This context should be mentioned in the introduction.
- The description of how the Gaussians are initialized is lacking in details. The paper states that the Gaussians are initialized randomly, but doesn’t specify the sampling distribution for the Gaussian means, rotations, scales, and densities. As 3DGS is sensitive to initialization, this is important information to include.
- Line 100-101 claims that Fourier-space cryo-EM reconstruction methods incur information loss because they “involve repeated FFTs and interpolation steps and operate on band-limited representations.” What is the evidence for this claim of information loss? Since there is presumably limited resolution on the cryo-EM detector leading to an inherent bandlimit in the measurements themselves, it’s not obvious why FFTs with interpolation on a bandlimited representation would incur any further information loss. It would also be good to specify what the “bandlimited representation” is–does this refer to a voxel grid?
- Line 115 argues that a benefit of 3DGS as a representation is that the “gradients remain localized to the subset of primitives that actually contribute to each pixel.” While it is true that Gaussians satisfy this desirable property, a sparse voxel grid would have the same property. The same is true for the properties marked (i) and (ii) on lines 124-126.
Line 128 mentions joint optimization over density and “imaging latents” but doesn’t specify what these latents are. I would expect these might refer to CTF or pose, but line 161 says that both of these are assumed known so it’s not clear what remaining latents need to be jointly optimized over.
- Lines 162-167 explain the difference between Fourier-space and real-space reconstruction. Since this is part of the method section, it is important to specify that the proposed method is real-space; ideally the description of Fourier-space reconstruction would not be in the methods section as it could confuse a reader as to which method is used here.
- There are some typos, mostly repeated and missing words/phrases but also some typos at the letter level. See lines 191, 192, 195, 198, 271, 304, 360.
- Figure 3a is confusing/misleading. It makes it look like you start with a known protein density and use that to initialize the Gaussians. But in reality the Gaussians are initialized randomly and then optimized to fit the density.
- The paragraph surrounding equation 8 describes efficient gradient computation as part of the method. While this is important for computational and memory efficiency, it is not clear what (if any) aspects of this computation are novel to GEM versus reused from 3DGS. For example, equation 8 describes thresholding each Gaussian into an ellipsoid to enable skipping some Gaussians that do not meaningfully influence a given pixel projection; this is exactly the same thresholding that 3DGS does for the same purpose. The only difference I am aware of is that 3DGS uses perspective projection whereas cryo-EM uses orthographic projection, but that does not seem to be the focus of this section so I am left wondering what is the novelty in the efficient computation.
- Lines 415-417, and line 446, describe the Nyquist-Shannon resolution limit as based on the voxel size of the protein density map. I have two questions about this. (1) What voxels are being discussed? The proposed method does not use voxels, nor is there a ground truth voxel grid that is mentioned. (2) Line 446 claims that GEM’s average local resolution is better than the Nyquist-Shannon limit. How is this possible?
- Section 4.4 introduces FSLC, but doesn’t really explain how it differs from GSFSC. From the names I gather that the difference is comparison on Fourier slices instead of Fourier shells, but the reader shouldn’t need to infer.
- Section 4.5 describes an ablation study involving isotropic scaling, in which S_i = s_i I. However, instead of allowing s_i to be optimized over (which would still enforce isotropic scaling since s_i is scalar), this ablation study forces s_i to equal the distance between the i’th Gaussian and its nearest neighbor. I think this would be a more informative ablation study if s_i were allowed to optimize, while still being enforced to be isotropic.


Decision: Reject