CryoSplat: Gaussian Splatting for Cryo-EM Homogeneous Reconstruction
Authors: Suyi Chen
Haibin Ling
Venue: ICLR 2026 Poster (2026)
Keywords: Cryo-EM
3D Reconstruction
Gaussian Mixture Model
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)

Abstract: As a critical modality for structural biology, cryogenic electron microscopy (cryo-EM) facilitates the determination of macromolecular structures at near-atomic resolution. The core computational task in single-particle cryo-EM is to reconstruct the 3D electrostatic potential of a molecule from noisy 2D projections acquired at unknown orientations. Gaussian mixture models (GMMs) provide a continuous, compact, and physically interpretable representation for molecular density and have recently gained interest in cryo-EM reconstruction. However, existing methods rely on external consensus maps or atomic models for initialization, limiting their use in self-contained pipelines. In parallel, differentiable rendering techniques such as Gaussian splatting have demonstrated remarkable scalability and efficiency for volumetric representations, suggesting a natural fit for GMM-based cryo-EM reconstruction. However, off-the-shelf Gaussian splatting methods are designed for photorealistic view synthesis and remain incompatible with cryo-EM due to mismatches in the image formation physics, reconstruction objectives, and coordinate systems. Addressing these issues, we propose **cryoSplat**, a GMM-based method that integrates Gaussian splatting with the physics of cryo-EM image formation. In particular, we develop an orthogonal projection-aware Gaussian splatting, with adaptations such as a view-dependent normalization term and FFT-aligned coordinate system tailored for cryo-EM imaging. These innovations enable stable and efficient homogeneous reconstruction directly from raw cryo-EM particle images using random initialization. Experimental results on real datasets validate the effectiveness and robustness of cryoSplat over representative baselines.
The code will be released at https://github.com/Chen-Suyi/cryosplat.

Published as a conference paper at ICLR 2026
CRYOSPLAT: GAUSSIAN SPLATTING FOR CRYO-EM HOMOGENEOUS RECONSTRUCTION
Suyi Chen
Haibin Ling"
Department of Computer Science
Department of Artificial Intelligence
Stony Brook University
Westlake University
{suychen] @cs. www.sytrbooo.ke...o edu
[1 (linghaibing @west la lake. edu.cn
ABSTRACT
As a critical modality for structural biology, cryogenic electron microscopy (cryo- EM) facilitates the determination of macromolecular structures at near-atomic resolution. The computational task in single-particle cryo-EM is to recon- struct the 3D electrostatic potential of a molecule from noisy 2D projections acquired unknown orientations. Gaussian mixture models (GMMs) provide a continuous, compact, and physically interpretable representation for molecu- lar density and have recently gained interest in cryo-EM reconstruction. How- ever, existing methods rely on external consensus maps or atomic models for initialization, limiting their use in self-contained pipelines. In parallel, differ- entiable rendering techniques such as Gaussian splatting have demonstrated re- markable scalability and efficiency for volumetric representations, suggesting natural fit for GMM-based cryo-EM reconstruction. However, off-the-shelf Gaussian splatting methods are designed for photorealistic view synthesis and remain incompatible with cryo-EM due to mismatches in the image formation physics, reconstruction objectives, and coordinate systems. Addressing these is- sues, we propose eryoSplat, GMM-based method that integrates Gaussian splat- ting with the physics of cryo-EM image formation. In particular, we develop an orthogonal projection-awee aware Gaussian splatting, with adaptations such as a view-dependent normalization term and FFT-aligned coordinate system tailored for cryo-EM imaging. These innovations enable stable and efficient homoge- neous reconstruction directly from raw cryo-EM particle images using random initialization. Experimental results on real datasets validate the effectiveness and robustness of cryoSplat over representative baselines. The code will be released at h/https://githhi cwwc Chen- Suyi/cryosplat
INTRODUCTION
Single particle cryogenic electron microscopy (cryo-EM) has emerged as transformative tool in structural biology, enabling visualization macromolecular complexes at atomic or near-atomic resolution without crystallization (Kühibrandi) 2014; Nogales) 2016 Renaud et al.) 2018). Central to cryo-EM is the computational lask of reconstructing a 3D volume from a large collection of 2D projection images, each corresponding different, unknown viewing direction of identical particles embedded vitreous ice.
This inverse problem is inherently ill-posed and computationally challenging. First, cryo-EM im- ages are severely corrupted by noise due to the low electron dose required to prevent radiation dam- age. For experimental datasets, the signal-to-noise (SNR) could be as low around 20 dB (Ben-) dory et al.) 2020 Bepler et al., (202)). Second, the orientations (poses) of individual particles are unknown and must be inferred jointly with the 3D structure. Third, many biological samples exhibit structural heterogeneity, with multiple conformational states coexisting in the same dataset.
These difficulties underscore two central objectives in cryo-EM image analysis: ab initio recon- struction, which aims to estimate both the 3D structure and particle poses directly from raw data
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without prior models, and heterogeneous reconstruction, which seeks to disentangle and reconstruct multiple structural states from the dataset. Both objectives fundamentally rely on the availability of robust and efficient homogeneous reconstruction method, which assumes all particles correspond to single structure and serves as a building block for more complex inference.
Approaches homogeneous reconstruction include methods based backprojection, iterative inpectation-maxiiizaiinn with voxel-based volumes (Tang et al.] 2007] Scheres 2012 Punjani
al. 2017; Shekarforoush et al., 2024), and more recently, neural representation learning (Zhong
et al. a b). which models the 3D volume using coordinate-basee networks. In parallel, Gaus-
sian mixture models (GMMs) have received attention for their continuous, compact, and physically interpretable parameterization of molecular density (Chen & Ludtke| [ CCenn et al. (2023a). No-
tably, GMMs offer a natural connection to atomic models and can represent fine structural details using fewer parameters (Chen et al. 2023b Li et al.
2024 [Scwwab et al. 2024 Chen
2025
Despite their conceptual appeal, existing GMM-based methods (Chen Ludtke, 2021 Chen al,) 2023a by [i et al.) [024] (Schwab al.) 2024 Chen 2025) for cryo-EM reconstruction rely on non- trivial prerequisite steps. They typically rely on consensus volumes from external pipelines, or even atomic models, for initialization, and have not demonstrated stable convergence when directly optimizing from experimental images. In fact, no prior method achieves reliable GMM-based re- construction even under known particle poses, due the inherent difficulty of optimizing mixture parameters in extreme noise. As a result, GMMs lack a self-contained and stable formulation that serve backbone for broader reconstruction workflows.
In this work, we propose cryoSplat, a GMM-based homogeneous reconstruction method that fills this foundational gap. Given known partiele poses, cryoSplat performs stable and efficient recon- struction directly from raw cryo-EM projection images, starting from random initialization and re- quiring no external priors. Inspired by recent advances in 3D Gaussian Splatting (3DGS, Kerbl et al. (023)), we model the 3D density as a mixture of anisotropic Gaussians and project them into 2D using a novel differentiable orthographic splatting algorithm consistent with cryo-EM physice To support practical scalability and training efficiency, we develop CUDA-accelerated real-space renderer that enables fast rasterization and optimization of the GMM.
Our contributions can be summarized as follows:
A self-contained GMM-based reconstruction method: We present cryoSplat the first method capable of performing cryo-EM homogeneous reconstruction from a randomly ini- tialized Gaussian mixture model without external prior, thereby establishing the missing foundation needed to develop GMMs into standalone reconstruction tools.
A physically accurate projection model: We design splatting algorithm under orthogo- nal projection tailored to cryo-EM image formation, enabling differentiable projection of anisotropic Gaussians in real space.
An efficient implementation: We adapt the CUDA tile-based framework of 3DGS cryo- EM imaging, introducing modified forward equations and re-derived gradients, which en- ables fast optimization of GMMs with tens of thousands of Gaussians.
Experimental validation: We demonstrate the effectiveness of cryoSplat on real datasets, showing that it converges reliably from random initialization and achieves reconstruction quality outperforming state-of-the-an methods.
2
RELATED WORK
2.1 VOLUME REPRESENTATION IN CRYO-EM
In cryo-EM experiments, purified biomolecules are rapidly frozen in a thin layer of vitreous ice, where each particle adopts a random orientation. A high-energy electron beam passes through the specimen, interacts with the electrostatic potential of the particles, and recorded on a detector as a 2D projection image (Singer & Sigworth 2020). The goal of cryo-EM reconstruction is to recover the 3D electrostatic potential, i.c., the volume, from a large set of such noisy and randomly oriented 2D projections. Central to cryo-EM reconstruction is the choice of volume representation.
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2.1.1 VOXEL-BASED REPRESENTATION
Voxel-based representations are the most widely used in conventional cryo-EM software, e.g., RE- LION (Scheres 2016b), cryoSPARC (Punjani et al., 2017) and EMAN2 (Tagg et al. 2007). The 3D volume is discretized into regular grid of density values, enabling fast projection and reconstruc- tion via FFT-based algorithms. Despite their practical success, voxel grids are memory-intensive, which limits their compatibility with modern learning- based analysis frameworks.
2.1.2 NEURAL FIELD
Neural fields represent the volume as a continuous function parameterized by neural networks. These methods (Zhong et al.) 202 by Levy et al.) [022a b 2025) offer differentiability, implicit smoothness, and natural compatibility with learning-based heterogeneous analysis. However, the implicit nature of neural fields often comes at the cost of interpretability, and such models typi- cally slow to train and difficult to constrain with biological priors.
2.1.3 GAUSSIAN MIXTURE MODEL.
Gaussian mixture models have a long history in structural biology, with early uses for molecular ap- proximation (Grant & Pickup 1993 Grant al. 1996 Kawabata 2008). E2GMM (Chen & Ludtke
2021 was among the first to apply GMMs to cryo-EM heterogeneous reconstruction. Like neural fields, GMMs can approximate any smooth density function and support differentiable optimization. More importantly GMMs provide an explicit and interpretable representation that naturally links to
atomic structures. Recent studies (Chen et al. 2023a et. 2024 Ducroeq et al. 2024]] [Schwab
et al. 2024| Chen 2025 Shekarforoush et al. (2025) have shown that GMMs can capture molecu- lar flexibility by modeling atomic motion directly, making them highly suitable for heterogeneous reconstruction and downstream structural analysis.
However, existing GMM-based methods typically require initialization from an externally recon- structed consensus map or even an atomic model. Without such guidance, random initialization leads to unstable optimization and poor reconstruction quality. Our work addresses this limitation by introducing a GMM-based reconstruction pipeline that can be stably trained from scratch.
2.2 GAUSSIAN SPLATTING
3DGS (Kerbl et al. 2023) is recent differentiable rendering technique developed for real-time novel view synthesis. represents 3D scene as collection of anisotropic Gaussians and ren- ders images via rasterization-batt based accumulation and alpha blending (Zwicker et al. 2002). While 3DGS achieves high visual fidelity in synthetic and real-world RGB datasets, as a volume rendering method, it is not physically accurate model of natural image formation (Huang et al. 2024).
Although the original 3DGS formulation is not physically consistent with natural image formation, its volume rendering framework closely aligns with the cryo-EM imaging model, where each image arises from an orthographic line integral of electrostatic potential modulated by the contrast transfer function (CTF). Leveraging this alignment, jection of anisotropic Gaussians under cryo-EM physics, replacing heuristic alpha blending with physically accurate line integrals and incorporating CTF modulation.
adapt splatting cryo-EM by rederiving the pro-
METHOD
3.1 OVERVIEW
Our goal is to achieve physically accurate and computationally efficient cryo-EM reconstruction by leveraging a Gaussian Mixture Model (GMM). To this end, we propose cryoSplat, differentiable framework that represents the 3D electrostatic potential of specimen as of anisotropic Gaus- sians and directly simulates the cryo-EM image formation process in real space, faithfully adhering to the physics of transmission electron microscopy.
Building upon recent progress in differentiable volume rendering, particularly the Gaussian splatting framework by Kerbl et al. (202)) we adopt tile-based rasterization strategy for scalable and effi-
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(4) Volume
(b) Image Formation
Micrograph
Particle
(c) Fourier Slice Thearem
V: IR
Thansfor
Founter
3D Space
Vorelization
Fast Differentiable Rasterization
Orthogonal Projection
Modulation
View
Projection
Simulated Projection
Gaussian Mixture Model
(W.t)
Gradient Flow
(d) CryoSplat
CTF
Figure 1: Cryo-EM reconstruction aims recover 3D volume (a) from large of 2D particle images (b). (b) Purified biomolecules with random orientations are embedded in a thin layer of vitreous ice. The electrostatic potential of the sample interacts with transmitted electrons, forming a micrograph that contains 2D projections of the molecules. Individual particle images are ex- tracted from the micrograph; they extremely noisy and modulated by highly oscillatory CTFs. (c) Fourier slice theorem: the 2D Fourier transform of particle image corresponds to central slice of the 3D Fourier transform of the volume. (d) Overview of cryoSplat. An anisotropic GMM repre- senting the 3D volume is transformed to the projection direction, orthogonally projected onto 2D image plane using a fast differentiable rasterizer, and modulated by the oscillatory CTF simulate physically accurate projection. The GMM parameters are optimized via gradients propagated from the discrepancy between the simulated and observed particle images. The resulting GMM can be voxelized to obtain the final 3D volume.
cient computation. However, the original 3DGS formulation is not directly applicable cryo-EM due to several fundamental mismatches: (i) it employs perspective projection consistent with pin- hole camera model, in contrast the orthographic projection in cryo-EM imaging; (ii) it tailored for novel-view synthesis (i.e., photorealistie 2D appearance) rather than for physical 3D density re- construction required in cryo-EM; and (iii) its image-centered coordinate system is incompatible with the FFT-aligned conventions assumed in cryo-EM reconstruction.
To address these issues, cryoSplat introduces several key adaptations: we replace heuristic alpha blending with physically grounded line integrals to reflect the transmission nature of electron imag- ing; (ii) fix the normalization between 3D-to-2D transformation and apply consistent learning rates across all parameters to ensure stable optimization; and (iii) we align the rasterization coordi- nate system with the FFT grid, allowing accurate gradient propagation through CTF modulation.
These modifications collectively enable cryoSplat to perform stable, end-to-end differentiable re- construction from raw cryo-EM particle images, starting from random initialization without relying on externally provided volumes atomic models.
3.2 IMAGE FORMATION
As shown in Fig. [b), electrons traverse vitrified specimen, and the transmitted wavefronts dergo phase shifts due to the specimen's 's electrostatic potential (Singer & Sigworth 2020). Under the weak phase approximation, the phase shifts linearly related the 3D potential (volume), and the image formed at the detector is line integral (projection) of this potential along the beam direction, further convolved with H: : R 2 IR, the point spread function (PSF) of the imaging system.
In homogeneous reconstruction, it is assumed that all particle images Y R22 + IR correspond to identical copies of single 3D volume V R - R. and that any conformational or compositional heterogeneity is negligible. Under this assumption, the image formation model can be expressed as:
Y(ra, = H(ra,ry) f V(WT t)dr. +
(1)
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where r |ras" Ty. r=]" are the 3D Cartesian coordinates in real space, W SO(3) is the 3D pose of the particle, and t [t,,, ty. 0 is the in-plane translation, accounting for imperfect centering during particle cropping. The noise term e modeled as independent, zero-mean Gaussian noise.
3.3 CRYOSPLAT
3.3.1
ANISOTROPIC GMM
Anisotropic GMMs are developed represent the volume, which can be written in the form
V(r) = Σ A,G,(r), ==1
(2)
where N denotes the Guassian count and A; is the amplitude of the i-th normalized Gaussian Ge(r).
By substituting Eq. T. we obtain the full forward process of cryoSplat. Specifically, we apply a viewing transformation to align the GMM the target orientation, orthographically project each Gaussian along the z-axis to form 2D image, and convolve the result with the PSF:
Σ A { G:(WT t)dr.
X ry) H(rr
Since the integral is linear, each Gaussian contributes independently to the final image. We thus focus on a single Gaussian and omit the subscript in the following discussion. A normalized 3D Gaussian is defined as:
exp (- -
G(rlp, (2x) E
(4)
where 1a 3R and Σ E 3x3 denote the mean (position) and the covariance matrix (shape), respectively. The determinant | ensures proper normalization. Following Kerbl et al. (2023) to guarantee the positive semidefinite property, we construct the covariance matrix as:
(5) where S = diag(s) is diagonal scaling matrix and R E SO(3) is rotation matrix. In our implementation, we store the scaling vector [Sz, Sy: sy] and parameterize R using quaternion 14w> qm 4y, 9e] _T_ To ensure positivity and stable gradients during optimization, we apply softplus function to both the amplitude A and the scaling vector s. The quaternion q is normalized to ensure represents a valid rotation. Altogether, each anisotropic Gaussian is parameterized by
Σ = RSS R
the 1 11-dimensional set {May My, Mas 8z, Sy, 82, qwo Qu Qy Q2, A).
3.3.2 VIEWING TRANSFORMATION
The viewing transformation is the first step in simulating image formation, aligning each Gaus- sian with a given projection direction. Since the parameters la and Σ describe Gaussians in world coordinates, we must transform them into the image-space coordinates before projection.
According to the derivation in Zwicker et al. (2002), applying affine transformation results in another Gaussian with appropriately transformed parameters. In our case, the transforma- tion consists of rotation W SO(3) and 2D in-plane translation i R3 leading to:
Gaussian
G(rlj, Σ = G(W + tlps. E),
(6)
(μ W(u where the transformed mean and covariance are given by j = W
t) and Σ WEWT.
3.3.3 ORTHOGONAL PROJECTION
The orthogonal projection closely aligns with the physical principles of cryo-EM. Mathematically, it corresponds to line integral of 3D Gaussian along the z-axis, resulting in 2D Gaussian, hereafter referred to as splat, G(Fl E):
G(F(Ā,)) GGrrlp E)dr.
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This operation effectively integrates the 3D Gaussian along the projection axis, preserving its Gaus- sian form in 2D. The resulting closed-fore form expression is:
1 2 (F (F- D ri))
G( Σ) 2T E
exp
(8)
where F [/z. ry] denotes the 2D Cartesian coordinates in real space.
In prior works, such as 3DGS, the normalization term 1/(2T] Σ is often omitted, as their primary focus is photorealistic novel view synthesis rather than the physical fidelity of the underlying 3D representation. However, in cryo-EM reconstruction, the ultimate goal is to recover the correct 3D volume. Omitting this view-dependent normalization introduces bias in amplitude and leads to incorrect reconstructions. Therefore, unlike 3DGS, we retain the normalization term to preserve the quantitative correctness of the model.
After projection, the final image is constructed by summing the weighted contributions of all splats and applying the PSF:
X(rasTy)= H(T,, A,G.(().
3.3.4 FAST DIFFERENTIABLE RASTERIZATION
We adopt the efficient tile-based rasterization framework from Kerbl et al. (2023), which enables scalable and differentiable processing of tens thousands of Gaussians via per-tile accumulation. Unlike 3DGS, which uses alpha blending for photorealistic rendering, we modify the rasterization to directly sum contributions of splats, in accordance with the physical transmission model in cryo-EM.
For an image X E DxD, the original 3DGS implementation places the continuous coordinate cen- ter at [(D - 1)/2, (D 1) i.c., halfway between two discrete pixels. contrast, FFT-based image formation assumes the origin is located at the integer grid point D//2], [D/2]]T To ensure compatibility with FFT-based forward and backward modeling. we shift the rasterization coordinates by half pixel that the image center aligns with the FFT grid. This alignment eliminates phase inconsistencies and enables accurate electron projection simulation, while preserving the compu- tational efficiency of the 3DGS architecture. Let X, E R Px be the matrices representing the GMM-based projection X and the observed image Y after rasterization, respectively.
3.3.5
LOSS FUNCTION
Unlike previous GMM- EMM-d methods that rely on specially designed losses with complex regular- ization constraints to ensure stable optimization, adopt much simpler formulation. Specifi- cally, we directly apply the mean squared error (MSE) loss between the GMM-based projection X and the observed image Y: = X YI Despite its simplicity. this loss formulation leads to stable and fast convergence in practice, without requiring additional regularization terms.
4 EXPERIMENT
4.1 EXPERIMENTAL SETTINGS
Datasets. We evaluate our method on four publicly available cryo-EM datasets from the Electron Microscopy Public Image Archive (EMPIAR) (Tudin et al.) 2016): EMPIAR-10028 (PF80S ribo- some) WWong et al. 2014), EMP1AR-10049 (RAG complex) (Ru al. 2015). CEMPIAR-1007) (E.
coli LSU assembly) (Davis et al. 2016). and EMPIAR- 10 (pre-catalytic spliceosome) (Plaschkal al. 2017). These datasets span range of structural complexity and image quality, from rigid assemblies with high contrast to highly heterogeneous macromolecular machines. For each dataset, use the provided particle images, consensus pose estimates, and CTF parameters. All reconstruc- tions are performed under the homogeneous assumption.
Evaluation metrics. To evaluate reconstruction quality real datasets without ground truth vol- umes, we adopt the gold standard Fourier Shell Correlation (FSC) (Van Heel & Schatz, 2005). fol- lowing established protocols. Each dataset is randomly split into two halves, and the reconstruction

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algorithm is applied independently to each subset. Let the resulting volumes be (k) and Vo (k), representing their Fourier transforms. The FSC is computed as a function of frequency & using the following formula:
Σ
(a(k) Po(K))
||A| 2
FSC(K)
(10)
Σ |Allz=k
(k))
Σ [B(()))2)) |
This metric quantifies the correlation between two independently reconstructed volumes within con- centric shells in Fourier space. The spatial resolution is defined as the frequency where the FSC curve drops below the 0.143 threshold, indicating the limit of reproducible structural detail.
Implementation details. For all experiments, particle images from EMPIAR-1002S, 10076, and 10180 are downsampled to 256 x 256, while AMPIARR-1045 is used at its original 192 192 resolution. Published particle translations are applied to the observed images via phase shifting in Fourier space prior to reconstruction, rather than through the GMM viewing transform. We do not apply any windowing to the observed particle images during preprocessing. The 3D volume is defined over the domain [-E, E|S, and each 2D projection is assumed lie within [-E, EJ in the image plane, where E = 0.5 defines the spatial extent. The values used in initialization are fixed but grounded in straightforward statistical intuition. We observe that most particles are con- centrated within a spherical region of radius E/2. To reflect this prior and accelerate convergence, we initialize the Gaussian means within this region. More specifically, based on the three-sigma rule for Gaussian distributions (M,, o²), where 99.7% 7% of samples fall within [μ - -3r, μ 30r], we obtain o E/6 from 3o = E/2. To slightly tighten the spread, we apply a scaling factor and use 0.9 E/6 0.075 to initialize the means. The initial scale of each Gaussian is set to 0.1 x 0.075 = 0.0075, encouraging localized support. Finally, to maintain consistent overall energy across varying numbers of Gaussians, total number of Gaussians. All parameters are trainable. In the original 3DGS (Kerbl et al.) 2023). different learning rates are assigned to different types of Gaussian parameters (means, scales, rota- tions, opacities). While this works well in novel view synthesis, it introduces instability in cryo-EM reconstruction. Let the full parameter vector be 0 Mu My P2g 8 Sy S2. Swo 92 Qu Iz. A]". In gradient descent optimization, the direction of parameter updates is determined by the gradient CeL. Unequal learning rates distort this direction by scaling different components unequally, which can lead to divergence. We observe that such practice causes Gaussians to spread uncontrollably in early iterations and finally diverge. To avoid this, we adopt a single unified learning rate across all param- eter types, preserving the intended descent direction and ensuring stable convergence. Accordingly, we use Adam (Kingma & Ba (014)) with batch size 1, learning rate 0.001, and exponential decay (1 0.1) at each epoch. All GMMs are trained for epochs. For the voxel-based baseline, run cryoSPARC's "Homogeneous Reconstruction Only" job using the same poses and CTF parameters. For neural representation learning, we follow cryoDRGN's default configuration: three 1,024-node layers, trained for 50 epochs. All experiments are run on a single NVIDIA GeForce RTX 3090.
initialize the amplitude A 1/(2N). where N is the
4.2 EVALUATION ON REAL DATASETS
We evaluate the performance of different volume representations on real cryo-EM datasets under a homogencous reconstruction setting. To ensure a fair comparison focused solely on the choice of volume representation, all methods reconstruct consensus maps using the same set of particle poses published by Zhhong et al. (2021a). without performing pose estimation. Our evaluation focuses on two aspects: (i) the ability reconstruct high-quality consensus maps, and (ii) robustness to noise and imperfect pose assignments. Since related methods (Zhong et al.) [021a b) Levy et al.) 2022a bJ (225)) adopt cryoDRGN's neural field implementation and differ mainly in pose estimation, focus our comparison on cryoDRGN. Accordingly, we evaluate three representative approaches: voxel-based backprojection cryoSPARC (Punjani et al.) 2017), the neural representation method cryoDRGN (Zhong al. 20211), and proposed GMM-based method cryoSplat. Visualizations of the reconstructed volumes are shown in Fig. [] and spatial resolution is quantified using gold- standard FSC curves. Each volume of cryoSplat is represented using 30,000 Gaussians.
The Pf80S ribosome (EMPIAR- is relatively easy to reconstruct due to its high-contrast im- ages and structurally stable particles. All methods achieve high-resolution results (3.80 Å) and
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CryoSPARC
CryoDRGN
CryoSplat
1.0
EMPIAR AAR--0028
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3.80
3.80 A
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1/resolution (Á-))
CryoSPARC
CryoDRGN
CryoSplat, Epoch=1
CryoSplat, Epoch-2
CryoSplat, Epoch=3
CryoSppiat, Epoch:4 CryoSplat, Epoch-5
Figure 2: Qualitative and quantitative comparison of voxel- neural, and GMM- represen- tations. (Lefft) Final 3D reconstructions on four real datasets visualized with ChimeraX (Pettersen al. 202T). (Right) FSC curves are plotted for quantitative evaluation. Gray dashed lines indicate the standard resolution thresholds of 0.5 and 0.143, reported in Angstroms (Á). CryoSplat consis- tently achieves higher resolution across datasets.
strong FSC agreement across the spectrum. cryoDRGN yields slightly higher FSC values inter- mediate frequencies, while cryoSplat outperforms all baselines at high spatial frequencies, demon- strating its ability to recover fine structural details.
The RAG complex (EMPIAR-10049) poses greater challenges due to symmetry-induced pose de- generacy and flexible regions such the DNA elements and the nonamer binding domain (NBD), indicated by arrows. CryoSplat outperforms the baselines with higher FSC, achieving a resolution of 2.49 A. Unlike the baselines, cryoSplat reconstructs the DNA elements and the NBD with mini- mal density fragments. Its FSC curve remains consistently above those of other methods across all spatial frequencies, highlighting its robustness to pose degeneracy and structural variability.
This LSU assembly dataset REMPIAR-10076 contains substantial compositional and conforma- tional heterogeneity, making consensus reconstruction particularly challenging. Both FSC analysis and visualization show that cryoSplat is more resilient under such conditions, achieving resolution of 3.30 with fewer fragments than voxel-based or neural methods, as indicated by the red circle.
The spliceosome dataset (EMPIAR-1018)) features large-scale motions of the SF3b indicated by the red circle, making consensus reconstruction particularly challenging. The reconstructions from cryoSPARC and cryoDRGN show pronounced high-frequency spurious spikes in this region, while
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cryoSplat is more robust to such motions and achieves resolution of 4.26 Á. FSC analysis further confirms that cryoSplat significantly outperforms the baselines across the frequency range.
CryoSplat consistently converges within 5 epochs, with FSC curves from the 4th and 5th epochs tightly overlapping, indicating stable optimization and improved generalization.
EMPIAR-10028
EMPIAR-10049
EMPIAR-10076
EMPIAR-10180
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10.3
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1/reciltion (A-))
Vresolution (A-))
1/reclution (A-))
1/recilution (A-))
CrysSRARC
CrycDRoN
CrysSpian.nno Crysspennnng.0aa N-2,06
Crycdpax, A-3.072
Dryolgtet, A-1,220
DrysSpin 10-10.000
DrycSpiat N-320,00
Figure 3: Reconstruction performance with varying numbers of Gaussians. Increasing the number improves accuracy and robustness.
400
405.56
350
Backprijctiio, D-182
Backprojaction,
300
CryoSRGN, D=192
Frame Second
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CryoORoN, D-258
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Oryošplat D=258
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102.50
154-97
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105:11
100
2.048
3,072
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Number Gaussians
Figure 4: Runtime efficiency across reconstruction methods different resolutions (D 192 and
D = 256). Frame rates (FPS) are measured under increasing numbers of Gaussians (log-scaled).
4.3 ABLATION STUDIES
This section reports ablation studies of our approach. More results can found in Appendix D
Number of Gaussians. Fig. 3 shows the FSC curves for cryoSplat with varying numbers of Gaus- sians. In general, increasing the number of Gaussians leads to improved FSC, as a denser GMM provides greater representational capacity. While eryoSplat performs well most datasets, its rel- ative performance varies due to differences in structural complexity and dataset-specific challenges. On EMPIAR- cryoSplat reaches resolution of 3.8 Á under all settings. While configura- tions with fewer than 10,000 Gaussians exhibit lower FSC values than cryoDRGN and cryoSPARC across most frequencies, the curves intersect at the highest frequency, indicating comparable final resolution. For EMPIAR- 10049, all cryoSplat settings significantly outperform both cryoSPARC (4.23 Å) and cryoDRGN (4.07 Å), achieving resolution 2.49 Å. Moreover, the FSC curves of all cryoSplat variants remain consistently above those of the two baselines across the entire frequency range. For HAPIAR--0076 the S0.ussanan model clearly outperforms other settings; even with fewer Gaussians, cryoSplat still surpasses the baselines, reaching 3.3 Á. For EMPIAR-10180 the models with 10,000 and 30,000 Gaussians achieve the best FSC, reaching 4.3 Å, while sparser GMMs remain competitive at high spatial frequencies. Overall, observe that using 10.000 Gaus- sians is sufficient to provide a robust improvement in FSC-derived resolution metrics over baseline methods across most datasets. Associated qualitative comparisons provided in Appendix D.I

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Table 1: GPU memory usage across reconstruction methods resolutions (7)
192, D 256).
GPU Mem. (MiB)
Methods
Params
Settings
Batch Size
D 192
D 256
Backprojection
508
396
CryoDRGN Zhong ul. 2021a
(6- [D/2] -
C= = 1,024
1
680
1,008
+L. +2
3
8
2,560
4,906
N = 2,048
344
346
N 3,072
314
3-88
CryoSplat (Ours)
11
N 5,120
1
346
348
10,000
348
350
30,000
Runtime efficiency. We compare the runtime efficiency of cryoSplat with other representation base- lines, as shown in Fig. Backprojection is the fastest, as it generates projections by directly index- ing and interpolating from dense voxel grid, but its cubic scaling makes it unsuitable for modern non-linear heterogeneous analysis. For such tasks, neural representations and GMMs offer greater flexibility. Under commonly used settings in heterogeneous reconstruction (e.g., 2,048-3,07 Gaus- sians (Chen & Ludtke, [021] Chen al. (202a bb). cryoSplat achieves 2-3x higher FPS than cry- oDRGN. Moreover, cryoSplat typically converges within 5 epochs, compared to 50 epochs required by cryoDRGN, providing an overall speedup up to 30x. As discussed above, using 10,000 Gaus- sians allows cryoSplat to consistently outperform baseline methods in FSC across most datasets, while still maintaining a higher FPS than cryoDRGN. Even with an extremely large number of Gaussians (e.g., 30,000), cryoSplat provides reasonable runtime performance for orthogonal projec- tion operations. Overall, as shown in Fig. 4 cryoSplat demonstrates sub-linear time complexity with respect the number of Gaussians, offering a favorable trade-off between accuracy and efficiency.
Memory usage. Tab. compares GPU memory usage across different reconstruction methods. Cry- oDRGN (Zhong
2021a) exhibits the highest memory footprint, exceeding 2.5 GiB D 192 and approaching GiB at D 256, primarily due to its deep neural decoder and a larger batch size 256, which
of 8. Interestingly, backprojection consumes more memory D 192 than at factors such as padding overhead or kernel-level opti-
may be attributed to implementattion-speiie mizations favoring power-of-two dimensions. This anomaly appears method-specific and does not reflect a general trend. In contrast, cryoSplat demonstrates consistently low memory usage across all configurations. Even with as many as 30,000 0000 Gaussians, cryoSplat maintains a footprint below 380 MiB, with negligible variation across resolutions. This efficiency underscores the scalability and suitability cryoSplat for large-scale or memory- memory-constrained cryo-EM reconstruction scenarios.
CONCLUSION
We present cryoSplat, a novel GMM-based framework that integrates Gaussian splatting with the physics of cryo-EM image formation. CryoSplat enables stable and efficient homogeneous recon- struction directly from raw cryo-EM particle images, starting from random initialization without relying on consensus volumes. By operating on an anisotropic Gaussian representation under a principled formulation, eryoSplat mitigates instability issues commonly encountered in GMM-based optimization and provides fully differentiable reconstruction framework. Experimental results on real datasets demonstrate the effectiveness, robustness, and faster convergence of cryoSplat com- pared to representative baselines.
Limitation and future work. The current formulation of cryoSplat assumes known particle poses and therefore does not yet constitute an ab initio reconstruction method. While this limits its ap- plicability in fully unsupervised settings, cryoSplat establishes a principled and stable foundation for integrating GMMs into cryo-EM reconstruction under realistic imaging physics. In particular, the proposed framework improves optimization stability and initialization robustness for Gaussian- based models. Promising future directions include jointly optimizing poses and Gaussian param- eters, extending the framework to heterogeneous reconstruction, and integrating cryoSplat into ab initio end-to-end cryo-EM pipelines.
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APPENDIX
A DETAILS OF METHOD
A.1 REAL SPACE RECONSTRUCTION
According to the Fourier slice theorem [Bracewell] 1956). illustrated in Fig. Cc). the 2D Fourier transform of a projection corresponds to central slice of the 3D Fourier transform of the volume, orthogonal to the projection direction and passing through the origin. Based on this property, an alternative and widely adopted formulation models reconstruction directly in the Fourier domain, where the image formation model becomes:
Y hig) (K,, ky) (( k) - 22ikk T +2.
(11)
where & [Kig kys 0] denotes the Cartesian coordinates in Fourier space, and the 2D spectrum Y, the CTF H and the 3D spectrum V denote the Fourier transform of y, H and V, respectively. The noise term ē is similarly modeled independent, zero-mean Gaussian noise in the Fourier domain.
In this work, departing from most existing approaches that adopt Eq. [1]), we instead build our pipeline on Eq. T performing homogeneous reconstruction directly in real space.
A key reason we choose to operate in real space is that it allows us to fully exploit the fast rasteriza- tion strategy from 3DGS. In high-resolution reconstructions, individual Gaussians in real space have small spatial scales and affect only a few nearby tiles, as shown in Fig, 5ia). This locality means that each GPU thread is responsible for single pixel and only needs process small subset of all Gaussians. In contrast, Gaussians in Fourier space become broad as resolution increases, leading to near-gle clar-global support. As result, each pixel in the frequency domain must aggregate contributions from nearly all Gaussians, making fast rendering impractical.
(a)
(b)
Real Space
Fourier Space
Image-centered
FFT-based
Figure 5: Rasterization details. (a) A Gaussian with small spatial scales in real space during high- resolution reconstruction overlaps at most four tiles, while Fourier space it exhibits nearly global support. Tile boundaries are indicated by lines. (b) For 4 x 4 image, the origin of the continu- ous coordinate system during rasterization defined differently: FFT-based coordinates place it at [2, 2] whereas image-centere coordinates place it at [1.5, 1.5] The origin is marked by dot. Image-centered coordinates induce a phase error - up to in 2 at the Nyquist frequency, degrading reconstruction at high frequency. The effect diminishes with larger D but never vanishes.
A.2 COMPUTATIONAL DETAILS
Section 3.3.3 describes how 3D Gaussian is projected along the z-axis to form splat. As discussed in Zwicker et al. (2002), the splat can be computed analytically by removing the 2-axis components from the mean and covariance:
Pa Mys P=]T
=
[iku May
ōzy
0.z
Σ
day
ò oyz
Txx
ōay
=
(12)
Ózz
à yz
d=2]
thereby enabling efficient computation of Eq.
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As discussed in Sec. 3.3.4 and shown in Fig. ())), when continuous image X R² R is rasterized onto pixels X R Dx the origin of the continuous coordinate system should be aligned with [[D/2]]
[D/2/] D/2]]
to match the FFT-based coordinate convention used in cryo-EM. Formally,
2E
25
X1, = x
(13)
where X,, denotes (th entry of matrix X. Note that the row and column indices correspond to the y- and r-axes, respectively, with the y-axis flipped during this rasterization.
In practice since the PSF corresponds to a large convolution kernel, we apply the contrast transfer function (CTF) in the Fourier domain after rasterization for efficiency:
X1, x1 F- H (HOF Σ A,C, (G - - (6-
(14)
where F() and F (.) denote the Fourier and inverse Fourier transform operators, respectively. is the rasterized CTF H and @ denotes element-wise (Hadamard) product.
Before deriving the gradients, we first define
N
(g) = (F).
(15)
which is the pre-rasterization continuous image. For clarity, we omit the Gaussian index in the following derivations, as the gradients are computed independently for each Gaussian. We denote by P the set of 2D coordinates corresponding to the centers of rasterized pixels. When F E P. the coordinate F [rz, ryl refers to a discrete sampling location in the image plane. The gradients used in the backward pass can be summarized as
Oc 0Q(F)
0A
OQ(F) OA
PEP
OC
V
VuC
(Q(F) Vp((F)
FEP
(16)
OL
2Σ
V C
VzQ(F)
(Q(())
0s
FEP
OL (Q(
2Σ
V f Σ PEP
QzQ
oq
where o denotes the composition Jacobian operators (chain rule). The derivation of gradients with respect to the amplitude A and mean f is trivial which can be given directly by
aQ = G(F).
(17)
0A
and
= = AG j
(18)
V = W(V Q To
where T T] embeds the 2D gradient into 3D space by padding the z-component with zero
For completeness, we provide the derivation of the covariance gradients QzQ, noting that our for- mulation retains the normalization term, which is omitted in 3DGS (Kerbl et al. 2023 Remember
G( Σ)
cxp(
2 (F ( E (f - p))
2T Σ
(19)
Σ
1
exp(
2 (f- - pi) T (F p)).
2T
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We can first compute
V Aexp( 7(0 (1) - - - (i))
_ 1
1
4T
+ A 1 exp(- 1
exp(-2(1 ii) - i))( 2 m) T)
(20)
= AF) - -( (f- - -)
and then V=Q=-E-Te
FFinally,
V&Q 0 O 0
V V
(21)
The subsequent derivations of V v and V follow exactly the formulation in Kerbl et al. (2023)
B DATASET DETAILS
We provide detailed statistics and characteristics of the cryo-EM datasets used in our experiments:
EMPIAR-10028 (Plasmodium falciparum 80S (PF80S) ribosome) (Wong et al. 2014 105,247 particle images of size 360 360 pixels at sampling rate of 1.34 A/pixel. This is a widely used benchmark with high-contrast images and a static structure.
EMPIAR-10049 (RAGI-RAG2 complex) (Ru al. 2015): 108,544 particles of size 192 192 pixels at 1.23 A/pixel. This dataset is considered more challenging due to its lower contrast and flexibility in some regions.
EMPIAR- 10076 (E. coli large ribosomal subunit undergoing (LSU) assembly) (Davis et al., 2016 131, ,989 particles of size 320 x 320 pixels at 1.31 A/pixel. This dataset contains substantial conformational and compositional heterogeneity, which poses challenge to homogeneous modeling.
EMPIAR- 10110 (Pre-catalytic spliceosome) (Plaschka et al. 2017): 327,490 particles of size 320 x 320 pixels at 1.69 A/pixel. It samples continuum of conformations, particularly involving large-scale motions of the SF3b subcomplex.
Synthetic 80S ribosome: We construct synthetic dataset of the 80S ribosome with 100,000 particles using Relion (Scheres (0) following the protocol Levy et al. (2022a). The electron scattering potential is derived ChimeraX (Pettersen et al. 2021) at resolution of 6.0 A/pixel, based on two atomic models: the small subunit (PDB 3J7A) and the large subunit (PDB 3J79) (Wong et al. 2014). Each particle image 128 x 128 pixels with pixel size of 3.77 A/pixel. Orientations are uniformly sampled over SO(3), and all images are centered without translations. Defocus values for the CTF are randomly drawn from log-normal distributions following[Leve al. (2022a). and zero-mean white Gaussian noise with varying signal-to-noie noise ratios (SNRs) is added.
C MORE IMPLEMENTATION DETAILS
C.1 OPTIMIZATION ALGORITHM
Our optimization algorithm is summarized in Algorithm 1. Unlike Kerbl al. (2023). which uses gradient magnitude as the criterion for splitting and cloning Gaussians, we observe that gradients are not a reliable indicator for densification cryo-EM reconstruction. Furthermore, elaborate densification schemes are generally unnecessary, as our method seldom suffers from significant local minima owing to close consistency with cryo-EM imaging physics. Nevertheless, we retain simple densification option to balance efficiency and resolution: fewer Gaussians enable faster training, whereas more Gaussians yield higher-resolution reconstructions, as demonstrated in Fig
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Algorithm Optimization and Densification
N: number of Gaussians
D: side length the observed particle images
θ InitAtrrbutess(N)
D Positions, Scales, Quaternions, Amplitudes Epoch Count
5 0
while not converged do
for (Y, W, t, H) in Dataloader() do
> Observed Image, Rotation, Translation, CTF
Y < FourierShift(Y t)
D Center Alignment
Q Rasterize @ W, D)
Algorithm 2 Apply CTF Loss P Backprop and Step
X + (pplyCTF(Q. H)
+- Loss(X, Y)
C Ad)
end for if (sDoubleGaussianss() then for all Gaussian(ı,, q, A) in @ do SplitGaussian(4, 8, q. A)
D (Optional) Densification
end for end if 2++ end while
Algorithm 2 CUDA-accelerated Rasterization
O: Gaussian parameters
W: viewing transformation matrix
D: side length of the observed particle images
function Rasterize(@,, W, D)
μ. Σ, f- BuildGaussians(()
n. Σ f- ViewingTrransfrmiie Σ, W)
Viewing Transformation
μ. > +- Projection(ji, ji, Σ)
b Orthogonal Projection
T + Create'Tiles(D) (D)
>> Tile Count
L. K + DuplicateWitkKeyssq.....
List of Indices and Keys
SorlByKeys(K,, L) + AllentifyTileRRaness(" K)
Q
Init Canvas
for all Tile in Q do for all Pixel p do
r + GetTileRange R, t)
Q(p) + SumSplats(p, L. K, μ. Σ. A)
end for end for
return Q
end function
C.2 DETAILS THE RASTERIZER
The details of the rasterizer summarized in Algorithm 2. We follow the tile-based rasterization framework of Kerbl et al. 2003),, where the output image is divided into 16 x 16 pixel tiles, and each splat instantiated in every tile it overlaps. The splat instances are then assigned keys for sorting, after which each tile can be processed efficiently by locating the corresponding ranges in the sorted list. Since pixels are computed in parallel, the runtime is primarily determined by the maximum number of Gaussians within any tile. For more details, we refer the reader to Kerbl et al. (023).
C.3 DETAILS OF REPORTED METRICS
When computing FSC curves for baselines, a spherical mask is applied to suppress background noise; cryoSplat uses the unmasked FSC because its Gaussian representation naturally suppresses
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noise outside the signal region. For runtime and memory comparisons, the backprojection implementation by Zhong et al. (2021a) instead of cryoSPARC, whose packaged environment intro- duces additional subprocesses and overhead that hinder fair measurement.
D ADDITIONAL EXPERIMENTS
D.1 NUMBER OF GAUSSIANS
EMPIAR- EMPIAR-10028 10028
EMPIAR 10049
EMPIAR EMPIAR-10076 10076
EMPIAR-101188 10180
N=2,048
N=3,072
N=5,120
N=10,000
N=30,000
Figure 6: Qualitative evaluation of reconstruction performance with different numbers of Gaussians. Increasing the number of Gaussians leads to visibly improved reconstructions, with finer structural details and enhanced sharpness. Red arrows mark representative regions that highlight the qualitative differences for clearer comparison across settings.
We present visual comparisons of reconstruction results using different numbers of Gaussians. As shown in Fig. 6 increasing the number components yields progressively sharper and more de- tailed structures. These qualitative observations align with the quantitative improvements in FSC curves reported in Fig. 3 Red arrows highlight representative regions where the differences in re- construction quality are especially pronounced, facilitating direct visual comparison across settings.
D.2 ISOTROPIC vS. ANISOTROPIC
CryoSplat represents 3D volumes using anisotropic Gaussians while remaining fully compatible with the isotropic formulation widely adopted in prior works (Chen Ludtke| 2021 Chen et al. 2023a b. Schwab et al. 2024 Chen 2025). When the scaling is isotropic, i.e., 8z 8y 82
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Isotrapic
Anisotropie
lsotroiic
Anisotropic
000
00
3124 1827
Wrsstatton (4-1)
Nesseuutio 14-3
EMPIAR-10028
EMPIAR-10049
Isctrapic
Anisotropic
Isotropic
Aniastropic
Ankastopi
00
06
1015 FI2 Mesceutton 18-3
Viostation (4-1
EMPIAR-10180
Figure 7: Qualitative and quantitative comparison of isotropic and anisotropic GMMs (N 30,000) on four real datasets. FSC curves show that anisotropic Gaussians consistently achieve higher cor- relations across spatial frequencies, indicating improved reconstruction accuracy. Volume visual- izations further reveal that anisotropic GMMs better recover fine structural details and elongated features, whereas isotropic Gaussians tend to fragment such regions.
the anisotropic Gaussian exactly reduces to the standard isotropic form:
111
G(rlu, = (2
exp(
(22)
20
allowing direct integration into existing isotropic GMM-based pipelines.
We investigate the impact of isotropic versus anisotropic Gaussians reconstruction quality. As shown in Fig. 7 anisotropic GMMs achieve higher FSC scores across spatial frequencies and pro- duce sharper, more detailed structures. Subjectively, isotropic Gaussians struggle to capture elon- gated features and are often captured by noise, which may contribute the unstable convergence from random initialization reported in previous methods. These results highlight the improved rep- resentational capacity and reconstruction robustness enabled by anisotropic modeling.
D.3 SIGNAL-TO-NOISE RATIO
Spau-toriaso Vato
Figure 8: Reconstruction performance under varying SNRs. (top) Ground-truth (GT) and recon- structed volumes different SNR levels. (bottom) Example synthetic particle images correspond- ing to each SNR. (right) FSC curves between GT and reconstructed volumes across SNRs.
We study the effect of SNR levels on cryoSplat with 5,120 Gaussians using the synthetic 80S dataset described in Appendix B Figure shows example synthetic particles, reconstructed volumes, and FSC curves under varying SNRs. FSCs are computed between the ground truth (GT) and recon- structed volumes. Overall, cryoSplat shows strong noise robustness: SNRs above 0 dB have little impact on reconstruction; high resolution is preserved even under severe noise at -15 dB, and re- constructions remain satisfactory at 20 dB, despite particles being barely visible.
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No Mask
Sphenical Mask
Loose Mask
Tight Mask
Corrected
1.0
1.0
1.0
0.8
0.8
0.6
0.8
0.8
FSC
0.4
0.4
0.4
0.2
0.0
0.0
0.0
1/12.4
1/8.2
14.1
1/3.1
1/25
1/1.44
1/8.2
1/4.1
1/3.1
1/2.5
1/124
1/8.2
1/4.1
1/3.1
1/2.5
1/resolution (A-))
1/resolution (A-1)
1/resolution (A-))
V=
v=0.2
v=0.5
Figure 9: Reconstruction using 30,000 Gaussians under different exponential decay parameters y- For each setting, the half-map FSC, masked FSC, and corrected FSC curves are plotted together for better comparison. When y 0.1, the half-map FSC exceeds the corrected FSC, indicating an overestimation of resolution due to strong self-consistency. Increasing 7 reduces this discrepancy, and at 7 0.5 the corrected FSC closely matches the malf-map FSC, suggesting that no detectable artificial bias is introduced.
D.4 CORRECTED FSC
Half-map FSC fundamentally a measure of ceff-consistency rather than visible structural detail. It is interpreted as a proxy for resolution under the assumption that (i) the SNR decreases at high frequencies, and therefore (ii) two independently reconstructed half maps should lose consistency in the high-resolution regime. If method maintains strong self-consistency even under low SNR, whether due to genuine robustness or to an inherent bias, the half-map FSC may overestimate the true resolution. For example, if method consistently overfits random noise into reproducible artificial patterns, the two half maps may show spurious agreement.
Corrected FSC (Chen al. 2013 is specifically designed to detect such artificial bias. It does so by randomizing Fourier phases: half maps with randomized phases should share no meaningful consistency. Any remaining agreement is therefore interpreted as bias and subtracted from the FSC curve. In our results shown in Fig. D we indeed observe a discrepancy between the half-map FSC and the corrected FSC, indicating that the half-map FSC tends overestimate cryoSplat's resolu- tion. Importantly, however, this discrepancy can be removed by slightly increasing the exponential decay parameter ",. When 7 0.5, the corrected FSC closely follows the original half-map FSC, suggesting that no detectable artificial bias is introduced by cryoSplat.
We also observe that FSC curves computed under different masking levels remain tightly aligned. This indicates that cryoSplat suppresses noise effectively outside the signal-support region: the noise level so low that applying a mask has virtually no effect on the FSC, consistent with our synthetic-data experiment in Appendix(D.. showing the strong denoising capability of GMM-based representations.
Having ruled out detectable artificial bias via corrected FSC, we next analyze why the half-map FSC may still overestimate the resolution for cryoSplat. We attribute this to the GMM's strong ability to maintain self-consistency during optimization. This property is largely driven by the in- herently low-pass nature of Gaussian kernels. On the one hand, the low-pass behavior encourages the model fit low- www-freuency components more readily, yielding smoother volumes. As we show later in Appendix [D3] this effect can be substantially mitigated by increasing the number Gaus- sians, which restores high-frequency detail. On the other hand, the same low-pass property also
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contributes to excellent self-consistency: both quantitative metrics and qualitative assessment show that this consistency does not manifest as harmful artifacts. However, because the self self-consistency is exceptionally strong, conventional FSC-based resolution estimation become overly optimistic. As a result, qualitative assessment and domain-expert evaluation remain the most trustworthy way to evaluate the effective resolution produced by cryoSplat. Developing rigorous and objective quality metrics tailored to GMM-based cryo-EM reconstruction remains an important open question.
D.5 RECONSTRUCTION BEHAVIOR IN THE ULTRA-HIGH GAUSSIAN COUNT REGIME
N-32,768
M-65,536
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N-262,144
N-524,288
Traodden
Figure 10: Qualitative and quantitative evaluation of reconstruction performance with ultra-high Gaussian counts. (Left) Reconstructed 3D volumes. (Right) FSC curves plotted for quantitative evaluation. Gray dashed lines indicate the standard resolution thresholds of 0.5 and 0. 0.143.
To further examine the representational capacity and optimization behavior of GMM-based repre- sentations, we increase the Gaussian count from 32,768 up to 524,288, as shown in Fig. [0] To ensure that the increased capacity can fully utilized, we raise the exponential decay parameter to y 0.5 and train for 10 epochs.
(a) CryoSPARC
(b) CryoDRGN
(c) CryoSplat
(d) w/ Model Overlay
Figure 11: Feature comparison at an a-helix the RAGI-RAG2 core region cropped from the reconstructed volume. CryoSplat with 524,288 Gaussians produces sharp and continuous helical density that is comparable to cryoSPARC, while cryoDRGN shows breaks along the backbone and retains visible noise. Overlay with the atomic model demonstrates that cryoSplat accurately recovers the backbone trace and resolves larger side-chain features, including aromatic rings.
Increasing the number of Gaussians consistently improves both quantitative and qualitative recon- struction quality. We observe monotonic increases in FSC scores as the Gaussian count grows, together with visibly sharper structural details. At the highest resolution regime (524k Gaussians), many fine-scale features such as a-helices become clearly resolved, as shown in Fig. II
This is surprising given the common expectation that larger models are harder to optimize and more prone to unstable convergence. Instead, the opposite effect is observed: larger GMMs exhibit better consistency between half-maps and converge to higher FSC, indicating more stable optimization dy- namics the ultra-high-capp high-capacity setting. This suggests that additional Gaussians provide finer local modeling flexibility, allowing the representation to better accommodate noise and subtle structural variability without overfitting.
Finally, even at 524.288 Gaussians, the model remains substantially more compact than voxel-based grids, since 524,288 x 11 < 2563. However, this hhuh-ihggGGaaiiinn regime currently incurs a sig- nificantly increased computational cost, making impractical for routine use. We view these results
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primarily exploration of the upper limit of GMM-based representations; future improvements in optimization and implementation may help reduce the computational burden.
D.6 RECONSTRUCTIO BEHAVIOR IN THE LOW GAUSSIAN COUNT REGIME
2022
USA (82 INT (31 122
N-64
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N--12 N-1,024
Figure 12: Qualitative and quantitative evaluation of reconstruction performance with low Gaussian counts. (Left) Reconstructed 3D volumes. (Right) FSC curves are plotted for quantitative evalua- tion. Gray dashed lines indicate the standard resolution thresholds of 0.5 and 0.143.
We additionally investigate how GMM- reconstructions behave when the number of Gaussians severely limited, reducing the count from 1, 1,024 down to 64, as shown Fig. 12 As the Gaussian count decreases, both quantitative and qualitative reconstruction quality degrade consistently. The reconstructed densities become progressively blurred and blob-like, and once the count falls below 1,024, the maps begin to exhibit clearly visible Gaussian ellipsoids in place of coherent structural details. This reflects the insufficient spatial degrees freedom available to represent localized struc- tural details. The FSC curves also reveal distinctive failure pattern when the number of Gaussians becomes extremely small. Instead of exhibiting a smooth decay, the curves dip at intermediate fre- quencies and then rise again at high frequencies. This rise does not reflect genuine fighfrequency agreement; it occurs because the figh-free components are largely absent, and the Fourier am- plitudes of both half-maps approach zero in these bands, which leads to unreliable consistency and inflated FSC values. Such Now-Gaussian-coon
configurations are not practical for real reconstruc- tion tasks. Although computational cost is reduced, the representation becomes too unser-resolved to yield reliable maps, and both qualitative appearance and quantitative metrics lose interpretabilityy
D.7 ABLATION ON INITIALIZATION STRATEGY
1.0
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Figure 13: Using 2,048 Gaussians, the RAGI-RAG2 complex (EMPIAR-10049) is reconstructed under 10 random seeds (0-9). The figure shows the mean half-map FSC with the mini- mum-maximum envelope across these runs. The narrow band indicates that cryoSplat converges to highly consistent results across initializations, demonstrating strong robustness and stability.
We also examine how sensitive the method is to the choice of initialization. First, we fix the GMM configuration and only vary the random seed from 0 to 9. For each run, we compute the half-map FSC curve and then aggregate the results into mean curve with upper and lower envelope all seeds. As is shown in Fig. [3] the envelopes are tight across all frequencies, indicating that both convergence behavior and final result are highly robust to random seed choices.
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Reconstructed Volume
1.0
0-0.0375
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0-0.1255
FSC
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1/16.5
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a=0.1125
o=0.075
0=0.0375
Figure 14: Initialization ablation on EMPIAR- 10076 using 2,048 Gaussians. Reconstructed vol- umes (top-left), initial Gaussian projections (bottom-left), and FSC curves (right) under three initial- ization spreads E {0.0375, 0.075, 0.1125), Both visual reconstructions and FSCs remain nearly identical across all settings, indicating that cryoSplat is highly robust to the choice initialization.
A more challenging setting arises when the underlying structure exhibits substantial heterogene- ity, introducing ambiguity during early optimization. EMPIAR- a representative example, containing pronounced compositional variability across the complex. To evaluate whether such vari- ability affects the stability of the initialization, we vary the spatial spread of the initial Gaussian lo- cations by sampling with {0.0375, 0.075, 0.1125). which approximately correspond to placing Gaussians within spherical regions radii 3F/2 22 F F/2. and E/4, respectively. The resulting initial projections under these settings, shown in Fig. 14 clearly illustrate the differences. Notably, the smallest spread (a = 0.0375) does not cover the full region where the signal is present and therefore represents a substantially under-dispersed initialization. Despite this, all configurations converge to nearly identical reconstructions on this heterogeneous dataset: both the FSC curves and the visual- ized volumes are highly consistent. This indicates that even in cases with significant variability and ambiguous starting configurations, the optimization remains robust to the choice of initialization.
E THE USE OF LARGE LANGUAGE MODELS
We acknowledge GPT-5 for its assistance with grammar correction, sentence shortening, and lan- guage polishing. No part of the research design, analysis, or conclusions was generated by L.MMs.
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Reviews:
[{"review_id": "Dj2YwxVjkv", "reviewer": "Reviewer_xwVJ", "summary": "The paper proposes cryoSplat, a new method for homogeneous reconstruction in cryo-electron microscopy (cryo-EM) using Gaussian mixture models (GMMs). cryoSplat leverages Gaussian splatting, a differentiable rendering technique, to achieve stable and efficient reconstruction directly from raw cryo-EM data. The authors introduce key innovations, including orthogonal projection-aware splatting and FFT-aligned coordinate systems, designed to improve the method's compatibility with cryo-EM's unique imaging physics. Experimental results on real datasets show that cryoSplat outperforms existing methods in terms of resolution, robustness to noise, and speed of convergence, providing a new foundation for GMM-based cryo-EM reconstruction.", "questions": "### Handling Ab Initio Reconstruction:\n\nWhile this paper shows strong results for homogeneous reconstruction with known particle poses, it would be insightful to know if the authors have any plans to extend cryoSplat to handle ab initio reconstruction. Specifically, how would the method perform when the particle orientations are unknown or the dataset is highly heterogeneous?\n\n### Impact of Initialization:\n\nThe paper mentions that cryoSplat can converge from random initialization, but how sensitive is the method to the choice of initialization? Could the authors explore different initialization schemes, particularly in cases where the structure is highly flexible or there are ambiguities in the starting configurations?\n\n### Model Interpretability:\n\nGMMs are known for their interpretability, but in the context of cryo-EM, how do the learned Gaussian parameters (amplitude, mean, covariance) relate to the atomic-level structure? Could the authors provide more insights into how the reconstructed volumes align with the fine structural features of macromolecules?\n\n### Performance in Heterogeneous Systems:\n\nThe current paper focuses on homogeneous reconstruction. How would cryoSplat perform in heterogeneous reconstruction scenarios where different conformations coexist in the same dataset? Could the method be extended or adapted for more complex, heterogeneous systems?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "### Innovative Methodology:\n\nThe integration of Gaussian splatting with cryogenic electron microscopy is novel and promising. This method represents an important step forward in making GMM-based approaches more feasible for cryo-EM.\n\nThe differentiable rendering approach combined with physically accurate projections offers a significant improvement over previous methods that relied on external priors or atomic models for initialization.\n\n### Theoretical Soundness:\n\nThe authors carefully derive their projection model, which aligns well with the cryo-EM imaging process, addressing crucial issues like coordinate system alignment and view-dependent normalization. These innovations are necessary for physically accurate reconstructions and ensure that the method is well-grounded in cryo-EM physics.\n\n### Empirical Validation:\n\nThe paper presents comprehensive experiments on real datasets, including datasets with varying levels of noise and structural complexity. The results show clear improvements in terms of resolution and robustness to heterogeneity.\n\nThe ablation studies and runtime efficiency analysis add further value, demonstrating the method’s ability to scale and deliver faster convergence compared to existing techniques like CryoDRGN.\n\n### Practicality and Efficiency:\n\nThe CUDA-accelerated real-space renderer ensures that the method is computationally efficient, allowing for scalable optimization even with tens of thousands of Gaussians.\n\nThe use of simple but effective mean squared error (MSE) loss simplifies the optimization process, eliminating the need for complex regularizations while ensuring stable training.", "weaknesses": "### Limited Scope of Validation:\n\nWhile the experiments on real datasets demonstrate strong performance, the method's robustness in heterogeneous reconstruction and in the face of ab initio reconstruction (with unknown particle poses) is not fully explored. The current work assumes known poses, which limits the scope of the method's applicability to real-world cryo-EM challenges.\n\nFuture work should explore how cryoSplat performs with pose estimation or heterogeneous reconstruction methods, as this is a critical challenge in cryo-EM.\n\n### Dependency on Known Poses:\n\nThe current version of cryoSplat does not tackle the ab initio reconstruction problem where poses are unknown. It would be valuable to investigate how cryoSplat could be extended to tackle this challenge, which is a key strength of recent methods like CryoDRGN.\n\n### Impact of Initialization:\n\nWhile cryoSplat is robust in its random initialization, it would be useful to further analyze the potential impact of different initialization schemes. In certain cases, random initialization might still pose a challenge, especially for highly flexible or heterogeneous particles. Additional insights into initialization strategies could be helpful.\n\n### Interpretability of Gaussian Mixtures:\n\nThe use of GMMs in cryo-EM reconstruction provides a compact and interpretable model, but more discussion on how this model relates to atomic-level resolution or finer structural details would be beneficial. For instance, how do the learned Gaussian parameters map to the structural features of macromolecules?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Te70PtfjsE", "reviewer": "Reviewer_Mi5f", "summary": "CryoSplat introduces a 3DGS-based homogeneous cryo-EM reconstruction framework that integrates orthographic Gaussian splatting with the physics of electron imaging. The method derives a projection model with view-dependent normalization and FFT-aligned coordinates so that real-space rendering remains consistent with Fourier-based acquisition. A CUDA-accelerated rasterizer supports tens of thousands of anisotropic Gaussians optimized from random initialization using a simple MSE loss. One advantage of cryoSplat comparing to the previous methods such as E2GMM is cryoSplat can converge from random initialization without the need of the traditional consensus reconstruction. Extensive experiments have conducted on EMPIAR-10028, 10049, 10076, and 10180 with provided poses and CTFs, cryoSplat consistently surpasses backprojection and CryoDRGN in FSC resolution, reaching 2.49 Å on EMPIAR-10049.", "questions": "Addressing the following points, which is aligned with the weaknesses above, would help me reassess the paper more favorably.\n\n1. Provide qualitative comparisons or additional reconstructions (e.g., with 50k–100k Gaussians) to check whether sharper details emerge and the FSC gains remain trustworthy.\n2. Extend the Gaussian-count ablation to much smaller mixtures so we can see whether the FSC metric genuinely tracks reconstruction quality when capacity drops.\n3. Include a cryoSPARC baseline that uses the same pose inputs, and discuss about the future direction for adapting cryoSplat to heterogeneous or dynamic reconstruction.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper adapts Gaussian splatting to cryo-EM with a physically grounded orthographic projection, view-dependent normalization, and FFT-aligned coordinates, enabling stable reconstruction from random initial Gaussians without external priors. \n2. The CUDA-based renderer and simple MSE objective deliver practical efficiency, achieving 2–3× higher rendering throughput than CryoDRGN, convergence in five epochs, and support for up to 30k Gaussians.\n3. Experiments on four EMPIAR datasets show consistent FSC gains over backprojection and CryoDRGN—including a 2.49 Å resolution on EMPIAR-10049, and the ablations intelligently cover both fidelity and runtime, showing that larger Gaussian sets improve accuracy while incurring longer training and rendering.\n4. The exposition is clear and well-motivated, the paper is well-written and easy to follow.", "weaknesses": "1. Even with 30k Gaussians, the reconstructions in Fig. 2 remain noticeably smoother than the backprojection or CryoDRGN baselines, raising the concern that the strong Gaussian prior could yield high FSC scores while underfitting finer details. Please explore much larger mixtures—e.g., 50k or 100k components—to see whether the qualitative sharpness catches up and whether the FSC improvements persist or saturate.\n2. The Gaussian-count ablation suggests almost linear gains in FSC as N increases, without clear signs of convergence, so it is unclear how the model behaves in the low-capacity regime. A complementary study reducing N well below 2,048 would clarify whether the FSC metric stays artificially high despite visibly degraded reconstructions, helping disentangle genuine data fit from prior-driven smoothing.\n3. The current evaluation remains limited to homogeneous reconstruction and compares primarily against CryoDRGN, which targets continuous heterogeneity; adding a popular cryoSPARC baseline with the same pose inputs, and outlining how cryoSplat could extend beyond static cases would make the empirical story more compelling.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "dhZQjiPEgA", "reviewer": "Reviewer_qU3z", "summary": "The paper adapts 3D Gaussian Splatting to homogeneous reconstruction in cryo-EM, using only raw picked particle images and assumed known poses. CryoSplat generally exceeds the resolution (measured via Fourier Shell Correlation) of voxel-space backprojection and CryoDRGN, on representative proteins.", "questions": "Please refer to questions embedded in weaknesses. I am open to raising my score if these presentation concerns are addressed.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "The method is a natural combination of 3D Gaussian Splatting, which works well in computer vision, to cryo-EM homogeneous reconstruction. It includes some adaptations necessary for the forward model and high noise of cryo-EM, and quantitative results (FSC) are encouraging.", "weaknesses": "There are some aspects of the presentation that could be improved.\n- The bullet point list of contributions at the end of the introduction slightly overstates the contributions, in my view. In particular, the efficient CUDA implementation would seem to be largely based on that of 3DGS, with fairly minor modifications. This is still a contribution to be sure, but the contribution description should acknowledge if this code is largely based on 3DGS code.\n- Some of the descriptions of voxel-based methods strike me as a bit unfair/unsupported. For example, line 109 describes these methods as “inherently discrete”--this is not the case when voxels use interpolation beyond nearest neighbor. It is ironic that this is described as a limitation of voxels, when (although Gaussians are technically infinitely-supported) 3DGS crops the spatial extent of each Gaussian and is thus more discrete of a representation than a grid with interpolation, since there is no interpolation between distinct Gaussians. Line 124 describes Gaussian mixture models as supporting differentiable optimization; this is only the case when the Gaussians are modeled with their full support, and even so the Gaussian and its gradient are negligible over much of its support, making the gradients of limited value beyond the main ellipsoid of support. Additionally, section 4.3 describes neural representations and Gaussians as more flexible for heterogeneous reconstruction compared to voxel grids. This claim is unsupported (e.g. it is straightforward to imagine combining a canonical 3D model in the form of a voxel grid with a neural or explicit deformation/conformation model, just as one could for a neural or Gaussian representation).\n- The last paragraph on page 3 describes three aspects of the original 3DGS that must be adapted for application to cryo-EM. The first is the switch from perspective to orthographic, which is clearly described. However, the second and third aspects are not clearly described. Aspect (ii) is described as prioritizing photorealism over physical accuracy, and aspect (iii) is described as misaligned coordinates with respect to the Fourier slice theorem. Aspect (ii) is too vague; aspect (iii) makes it sound like the Fourier slice theorem is used in the forward model, but this is not the case since the forward model here is applied in real-space, so it’s not clear why misaligned coordinates would be an issue. Is this because the contrast transfer function is applied in Fourier domain?\n- In Figure 2, the resolution is described as being calculated based on the point where the FSC curve crosses 0.143. However, the graphs of FSC on the right of the figure seem to have arbitrary x axis scales, that do not always show where FSC for CryoSplat crosses 0.143. Moreover, the resolutions reported for CryoSplat do not seem to match these crossing points.\n- In Figure 2, I acknowledge that CryoSplat achieves higher FSC. However, I am confused by this because visually in the ChimeraX figures the reconstructions from CryoSplat appear smoother than those from both backprojection and CryoDRGN, and in some cases (e.g. the second row) there are fine details in which backprojection and CryoDRGN are in agreement with each other, but the same features are not present in CryoSplat. I am not sure how to square this observation with the quantitatively higher FSC for CryoSplat, but I am concerned of a potential bug in computing FSC.\n- Figure 3 caption claims that 10,000 Gaussians are usually sufficient, yet in all four subfigures there is meaningful improvement moving from 10,000 Gaussians to 30,000. Moreover, in Figure 2 CryoSplat uses 30,000 Gaussians. The goal should not be just to outperform baselines, but to maximize resolution and stability of reconstruction.\n- The initialization strategy for the Gaussians is described on line 396, but the rationale behind it is not explained until the appendix. The rationale is more important than the numbers themselves.\n- Section 4.2 describes some reconstructions as producing artifacts, and points out the location of artifacts in Figure 2. How can you know what is an artifact, when you do not have a ground truth volume for comparison?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "98XV6UY6F6", "reviewer": "Reviewer_LabF", "summary": "Single particle cryo electron microscopy (cryo-EM) is a field whose main goal is to reconstruct the 3D shape of molecules from thousands of transmission electron microscope (TEM) images. This field has experienced tremendous progress over the last decade, including several mature software packages for 3D reconstruction such as RELION [1], cryoSPARC [2] and CryoDRGN [3]. The reconstruction pipeline for cryo-EM is comprised of several different steps: particle picking (identifying the location of the molecules in the noisy images), estimation of the point-spread function and ab-initio reconstruction (obtaining an initial low-res reconstruction). This is typically followed by the estimation of viewing angles combined with high-resolution 3D refinement. This paper is focused on the last step: the high-resolution 3D reconstruction of a single (fixed) molecule with known viewing angles. It proposes a simple approach inspired by 3D Gaussian splatting [4] that the authors name CryoSplat. It is a more-or-less direct adaptation of 3D splatting from computer graphics to the cryo-EM domain, making the following changes to match the TEM imaging modality:\n1. The projection is orthogonal, in contrast to the perspective projection (pinhole camera) model typically used in computer graphics.\n2. There are no occlusions. The Gaussians are simply added together. So there's no need for a z-buffer and alpha-blending.\n3. The algorithm does not need to figure out which Gaussians are outside the camera's view since the entire molecule is always in view. The authors still divide the image into tiles, but this is merely an implementation detail for better GPU performance.\n4. The projection image is convolved with a point-spread function.\n\nThe main contribution of this paper is a demonstration that, given input images with matching projection orientations, a high-resolution 3D GMM model can be used to reconstruct the 3D volume using gradient-based optimization.\n\n[1] Scheres, Sjors HW. \"RELION: implementation of a Bayesian approach to cryo-EM structure determination.\" Journal of structural biology 180, no. 3 (2012): 519-530.\n\n[2] Punjani, Ali, John L. Rubinstein, David J. Fleet, and Marcus A. Brubaker. \"cryoSPARC: algorithms for rapid unsupervised cryo-EM structure determination.\" Nature methods 14, no. 3 (2017): 290-296.\n\n[3] Zhong, Ellen D., Tristan Bepler, Bonnie Berger, and Joseph H. Davis. \"CryoDRGN: reconstruction of heterogeneous cryo-EM structures using neural networks.\" Nature methods 18, no. 2 (2021): 176-185.\n\n[4] Kerbl, Bernhard, Georgios Kopanas, Thomas Leimkühler, and George Drettakis. \"3D Gaussian splatting for real-time radiance field rendering.\" ACM Trans. Graph. 42, no. 4 (2023): 139-1.", "questions": "* Is there any \"secret sauce\" in the implementation that I missed? i.e. non-standard choices in the optimization problem definition or implementation that are critical to getting the method to converge or to obtain high-resolution 3D volumes.\n\n* Why did you not include the code in the submission? Are you willing to attach an (anonymized) zip file during the review cycle?\n\nMinor comments (not questions).\n* In Figure 1 the black and orange lines should probably have arrowheads instead of ball heads to clearly show the direction.\n* The last paragraph of Section 3.3.4 seems out of place.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 2, "strengths": "* I like the simplicity of the approach. The authors have demonstrated that one can obtain high-resolution reconstructions simply by running backprop. I am not aware of previous literature that demonstrated this.\n\n* The paper is very clearly written.", "weaknesses": "* The benchmarks do not compare the method to leading packages for homogeneous reconstruction but only to CryoDRGN (and a backprojection baseline). While CryoDRGN is a leading package for heterogeneous reconstruction, it is not the best at homogeneous reconstruction. At a minimum, I do not think the paper can be accepted without a comparison to a leading homogeneous reconstruction method such as RELION or cryoSPARC.\n\n* The novelty of this work is not very high in my opinion. Gaussian mixture models have been used by several authors in cryo-EM reconstruction (e.g. [5],[6]) but mostly to attack the much more difficult problem of heterogeneous reconstruction, where molecules have continuous degrees of freedom. This is probably due to the fact that in cryo-EM the homogeneous reconstruction problem is considered to be \"solved\". The authors acknowledge the works of Chen et al. and others, but claim that they are different from these works because they do not rely on an approximate 3D model of the molecule (\"consensus model\") for initialization. Indeed, CryoSplat does not require such an initial model, but it takes the orientations of the images as input.  The issue is that on experimental data the orientations are typically obtained by rotating a consensus model and comparing the resulting projections to the input images... So a consensus model is still used in the pipeline. As far as I can tell, the optimization is a straightforward application of backpropagation with no \"secret sauce\". Hence, the main contribution of the paper is a demonstration that backprop on a GMM with known orientations can result in high resolution models. The contribution would be much more significant if the method was able to also estimate the viewing angles from scratch.\n\n* This submission seems to have significant overlap to GEM (ICLR submission 6298) even with regards to the choice of datasets. While not a weakness, this is something that the chairs need to take into consideration. In particular, it would not make sense to accept both papers to the same conference.\n\n* The code was not included in the submission so I could not review it.\n\n[5] Chen, Muyuan, Michael F. Schmid, and Wah Chiu. \"Improving resolution and resolvability of single-particle cryoEM structures using Gaussian mixture models.\" Nature methods 21, no. 1 (2024): 37-40.\n\n[6] Shekarforoush, Shayan, David B. Lindell, Marcus A. Brubaker, and David J. Fleet. \"Reconstructing Heterogeneous Biomolecules via Hierarchical Gaussian Mixtures and Part Discovery.\" arXiv preprint arXiv:2506.09063 (2025).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)