Original: ACE-EM: B OOSTED ab initio C RYO -EM 3D R ECONSTRUCTION WITH A SYMMETRIC C OMPLEMENTARY A UTOENCODER
Revised: ACE-EM: B OOSTED ab initio C RYO -EM 3D R ECONSTRUCTION WITH A SYMMETRIC C OMPLEMENTARY A UTOENCODER

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: Cryo-electron microscopy (cryo-EM) is an imaging technique for obtaining highresolution biomolecular structures.
Revised: Cryo-electron microscopy (cryo-EM) is an imaging technique for obtaining highresolution biomolecular structures.

Original: The central problem in cryo-EM is to recover the underlying 3-dimensional (3D) objects from 2-dimensional (2D) projection images.
Revised: The central problem in cryo-EM is to recover the underlying 3-dimensional (3D) objects from 2-dimensional (2D) projection images.

Original: Aside from signal corruptions and extremely low signal-to-noise ratio, a major challenge in cryo-EM 3D reconstruction is to estimate the poses of 3D objects during the low-quality projection image formation, which are missing from experimental measurements.
Revised: Aside from signal corruptions and extremely low signal-to-noise ratio, a major challenge in cryo-EM 3D reconstruction is to estimate the poses of 3D objects during the low-quality projection image formation, which are missing from experimental measurements.

Original: Recent methods attempted to solve the pose estimation problem with the autoencoder architecture.
Revised: Recent methods attempted to solve the pose estimation problem with the autoencoder architecture.

Original: However, a key issue with this approach is that the latent vector is only indirectly updated through the decoder.
Revised: However, a key issue with this approach is that the latent vector is only indirectly updated through the decoder.

Original: Thus, pose estimation is not explicitly constrained, and the pose can be easily trapped in a local subspace, resulting in suboptimal pose inferences and inferior 3D reconstruction quality.
Revised: Thus, pose estimation is not explicitly constrained, and the pose can be easily trapped in a local subspace, resulting in suboptimal pose inferences and inferior 3D reconstruction quality.

Original: Here we present an improved autoencoder architecture called ACE (Asymmetric Complementary autoEncoder) and designed the ACEEM method to solve this issue, which consists of two learning tasks.
Revised: Here we present an improved autoencoder architecture called ACE (Asymmetric Complementary autoEncoder) and designed the ACEEM method to solve this issue, which consists of two learning tasks.

Original: The first task takes projection images and outputs predicted images using an image-to-pose encoder followed by a pose-to-image decoder.
Revised: The first task takes projection images and outputs predicted images using an image-to-pose encoder followed by a pose-to-image decoder.

Original: The second task explicitly learns the pose estimation by reversing the order of encoder and decoder, which takes randomly sampled poses and outputs predicted poses.
Revised: The second task explicitly learns the pose estimation by reversing the order of encoder and decoder, which takes randomly sampled poses and outputs predicted poses.

Original: The two tasks complement each other and achieve a more balanced training of the autoencoder parameters.
Revised: The two tasks complement each other and achieve a more balanced training of the autoencoder parameters.

Original: Compared to previous methods, our proposed ACE-EM reaches higher pose space coverage within the same training time and has achieved the Nyquist resolution of 3D reconstruction for several cryo-EM benchmark datasets.
Revised: Compared to previous methods, our proposed ACE-EM reaches higher pose space coverage within the same training time and has achieved the Nyquist resolution of 3D reconstruction for several cryo-EM benchmark datasets.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Single-particle cryo-electron microscopy (cryo-EM) is one of the most important structural biology techniques (Frank, 2017).
Revised: Single-particle cryo-electron microscopy (cryo-EM) is one of the most important structural biology techniques (Frank, 2017).

Original: This technique can be divided into four stages: biological sample preparation, electron microscopy image collection, 3-dimensional (3D) reconstruction, and atomic structural model building.
Revised: This technique can be divided into four stages: biological sample preparation, electron microscopy image collection, 3-dimensional (3D) reconstruction, and atomic structural model building.

Original: 3D reconstruction of molecular volumes from 2-dimensional (2D) electron microscopy images is the most challenging and time-consuming step in cryo-EM data analysis.
Revised: 3D reconstruction of molecular volumes from 2-dimensional (2D) electron microscopy images is the most challenging and time-consuming step in cryo-EM data analysis.

Original: There are two major challenges in cryo-EM 3D reconstruction: unknown projection poses (orientations and positions) and low signal-to-noise ratio (SNR).
Revised: There are two major challenges in cryo-EM 3D reconstruction: unknown projection poses (orientations and positions) and low signal-to-noise ratio (SNR).

Original: During electron microscopy imaging, the 3D poses of the biological molecules in the sample can not be directly measured.
Revised: During electron microscopy imaging, the 3D poses of the biological molecules in the sample can not be directly measured.

Original: The SNR of a typical cryo-EM image is very low, which can vary from -10 dB to -20 dB (average around -10 dB)
Revised: The SNR of a typical cryo-EM image is very low, which can vary from -10 dB to -20 dB (average around -10 dB)

Original: (Bepler et al., 2020), making it extremely challenging to accurately estimate poses and perform the 3D reconstruction.
Revised: (Bepler et al., 2020), making it extremely challenging to accurately estimate poses and perform the 3D reconstruction.

Original: Currently, many machine-learning (ML)-based methods have been proposed for solving cryo-EM 3D reconstruction (Donnat et al., 2022), utilizing architectures like GAN (Goodfellow et al., 2014) and auto-encoders (Kramer, 1991).
Revised: Currently, many machine-learning (ML)-based methods have been proposed for solving cryo-EM 3D reconstruction (Donnat et al., 2022), utilizing architectures like GAN (Goodfellow et al., 2014) and auto-encoders (Kramer, 1991).

Original: Nevertheless, ML-based methods of cryo-EM 3D reconstruction are still at an early stage.
Revised: Nevertheless, ML-based methods of cryo-EM 3D reconstruction are still at an early stage.

Original: The highest possible cryo-EM reconstruction resolution (2 pixels) at FSC threshold 0.5 has not been achieved by ML-based methods, even in simulated datasets without noise.
Revised: The highest possible cryo-EM reconstruction resolution (2 pixels) at FSC threshold 0.5 has not been achieved by ML-based methods, even in simulated datasets without noise.

Original: For experimental datasets like the 80S, some methods with amortized inference methods failed to reconstruct the object of certain size, or found the highest possible resolution (2 pixels) of half-map
Revised: For experimental datasets like the 80S, some methods with amortized inference methods failed to reconstruct the object of certain size, or found the highest possible resolution (2 pixels) of half-map

Original: FSC (Rosenthal & Rubinstein, 2015) at 0.143 not reachable (Levy et al., 2022).
Revised: FSC (Rosenthal & Rubinstein, 2015) at 0.143 not reachable (Levy et al., 2022).

Original: For widely utilized architecture auto-encoders, an image-to-pose encoder extracts the image-projection poses from 2D input cryo-EM images, while a pose-to-image decoder generates the images to match the inputs.
Revised: For widely utilized architecture auto-encoders, an image-to-pose encoder extracts the image-projection poses from 2D input cryo-EM images, while a pose-to-image decoder generates the images to match the inputs.

Original: However, as the poses are intermediate latent variables without supervised loss, the estimated poses can be inaccurate and easily trapped in local minima of the orientation space.
Revised: However, as the poses are intermediate latent variables without supervised loss, the estimated poses can be inaccurate and easily trapped in local minima of the orientation space.

Original: These pose estimation errors lead to an inferior resolution in the 3D reconstruction output.
Revised: These pose estimation errors lead to an inferior resolution in the 3D reconstruction output.

Original: To improve the pose estimation and 3D reconstruction quality, here we propose a new framework called ACE-EM (ACE for Asymmetric Complementary autoEncoder).
Revised: To improve the pose estimation and 3D reconstruction quality, here we propose a new framework called ACE-EM (ACE for Asymmetric Complementary autoEncoder).

Original: In particular, ACE-EM consists of two training tasks: (1) Image-pose-image (IPI).
Revised: In particular, ACE-EM consists of two training tasks: (1) Image-pose-image (IPI).

Original: The task is the same as previous work, which takes projection images as inputs, and outputs predicted images, by an image-to-pose encoder followed by a pose-to-image decoder.
Revised: The task is the same as previous work, which takes projection images as inputs, and outputs predicted images, by an image-to-pose encoder followed by a pose-to-image decoder.

Original: (2) Pose-image-pose (PIP).
Revised: (2) Pose-image-pose (PIP).

Original: The task explicitly learns the pose estimation in a self-supervised fashion, which takes randomly sampled poses as inputs, and outputs predicted poses, using the same encoder and decoder as in IPI but reversing their order.
Revised: The task explicitly learns the pose estimation in a self-supervised fashion, which takes randomly sampled poses as inputs, and outputs predicted poses, using the same encoder and decoder as in IPI but reversing their order.

Original: The two tasks complement each other and achieve a more balanced training of the autoencoder parameters.
Revised: The two tasks complement each other and achieve a more balanced training of the autoencoder parameters.

Original: Our main contributions are listed below.
Revised: Our main contributions are listed below.

Original: • As far as we know, ACE-EM is the first deep-learning model in cryo-EM reconstruction that enhances the pose estimation by the self-supervised PIP task.
Revised: • As far as we know, ACE-EM is the first deep-learning model in cryo-EM reconstruction that enhances the pose estimation by the self-supervised PIP task.

Original: With better pose estimation, ACEEM can converge much faster than previous methods, efficiently cover more pose spaces, and achieve better cryo-EM 3D reconstruction quality.
Revised: With better pose estimation, ACEEM can converge much faster than previous methods, efficiently cover more pose spaces, and achieve better cryo-EM 3D reconstruction quality.

Original: • ACE-EM can boost performance regardless of decoder types.
Revised: • ACE-EM can boost performance regardless of decoder types.

Original: For example, some decoders, that failed in previous autoencoder architectures, can be successfully used in ACE-EM.
Revised: For example, some decoders, that failed in previous autoencoder architectures, can be successfully used in ACE-EM.

Original: • Experimental results demonstrate that ACE-EM can perform well in both simulated and realworld experimental datasets.
Revised: • Experimental results demonstrate that ACE-EM can perform well in both simulated and realworld experimental datasets.

Original: In particular, ACE-EM outperformed all the baseline methods and reached 2 pixels, the Nyquist resolution (the highest possible resolution) at FSC threshold 0.5, in spliceosome and spike simulated datasets with SNR 200 dB and -10 dB. And ACE-EM reached 7 . 54 ˚ A , the Nyquist resolution of half-map FSC at 0.143 in the 80S
Revised: In particular, ACE-EM outperformed all the baseline methods and reached 2 pixels, the Nyquist resolution (the highest possible resolution) at FSC threshold 0.5, in spliceosome and spike simulated datasets with SNR 200 dB and -10 dB. And ACE-EM reached 7 . 54 ˚ A , the Nyquist resolution of half-map FSC at 0.143 in the 80S

Original: experimental dataset, which is the only architecture that reached the Nyquist resolution with amortized inference methods.
Revised: experimental dataset, which is the only architecture that reached the Nyquist resolution with amortized inference methods.

Original:  R ELATED WORK
Revised:  R ELATED WORK

Original: Overview of cryo-EM methods 3D reconstruction and view synthesis is a popular field in computer vision.
Revised: Overview of cryo-EM methods 3D reconstruction and view synthesis is a popular field in computer vision.

Original: Many new methods have been emerged in recent years, such as NeRF (Mildenhall et al., 2020), Plenoxels (Yu et al., 2021), DirectVoxGo (Sun et al., 2022), BARF (Lin et al., 2021),
Revised: Many new methods have been emerged in recent years, such as NeRF (Mildenhall et al., 2020), Plenoxels (Yu et al., 2021), DirectVoxGo (Sun et al., 2022), BARF (Lin et al., 2021),

Original: SCNeRF (Jeong et al., 2021), and GNeRF (Jeong et al., 2021).
Revised: SCNeRF (Jeong et al., 2021), and GNeRF (Jeong et al., 2021).

Original: These methods can reconstruct 3D scenes from natural images with high signal-to-noise ratios (SNR).
Revised: These methods can reconstruct 3D scenes from natural images with high signal-to-noise ratios (SNR).

Original: In ab initio cryo-EM 3D reconstruction, the reconstruction problem is much harder due to the low-SNR nature of cryo-EM images and missing image projection parameters.
Revised: In ab initio cryo-EM 3D reconstruction, the reconstruction problem is much harder due to the low-SNR nature of cryo-EM images and missing image projection parameters.

Original: In addition to traditional cryo-EM reconstruction methods like RELION (Scheres, 2012) and cryoSPARC (Punjani et al., 2017), many ML-based cryo-EM reconstruction methods have been developed in the past few years.
Revised: In addition to traditional cryo-EM reconstruction methods like RELION (Scheres, 2012) and cryoSPARC (Punjani et al., 2017), many ML-based cryo-EM reconstruction methods have been developed in the past few years.

Original: Given only the 2D cryo-EM projection images without any labels, cryo-EM 3D reconstruction can be formulated as a self-supervised learning problem.
Revised: Given only the 2D cryo-EM projection images without any labels, cryo-EM 3D reconstruction can be formulated as a self-supervised learning problem.

Original: Two ML frameworks have been explored:
Revised: Two ML frameworks have been explored:

Original: GAN (Goodfellow et al., 2014) and autoencoder (Kramer, 1991).
Revised: GAN (Goodfellow et al., 2014) and autoencoder (Kramer, 1991).

Original: CryoGAN (Gupta et al., 2021) and Multi-CryoGAN (Gupta et al., 2020) adopted the GAN framework.
Revised: CryoGAN (Gupta et al., 2021) and Multi-CryoGAN (Gupta et al., 2020) adopted the GAN framework.

Original: In these methods, the generator contains the predicted volume information and generates projection images, while the discriminator adopts the generated and authentic images and discriminates the source of the images.
Revised: In these methods, the generator contains the predicted volume information and generates projection images, while the discriminator adopts the generated and authentic images and discriminates the source of the images.

Original: However, due to the limitations of GANs, these methods lack pose estimation and 3D reconstruction quality is inferior to other methods.
Revised: However, due to the limitations of GANs, these methods lack pose estimation and 3D reconstruction quality is inferior to other methods.

Original: For cryo-EM, the images are fed to the encoder to generate a latent variable (usually representing the pose estimation), and the decoder produces the reconstructed images based on this variable.
Revised: For cryo-EM, the images are fed to the encoder to generate a latent variable (usually representing the pose estimation), and the decoder produces the reconstructed images based on this variable.

Original: The model is trained by reducing the loss of the original and reconstructed ones.
Revised: The model is trained by reducing the loss of the original and reconstructed ones.

Original: After training, the 3D reconstruction density map can be obtained from the decoder.
Revised: After training, the 3D reconstruction density map can be obtained from the decoder.

Original: Cryo-EM 3D reconstruction with variational autoencoder (VAE)
Revised: Cryo-EM 3D reconstruction with variational autoencoder (VAE)

Original: (Ullrich et al., 2020), CryoDRGN (Zhong et al., 2019; 2021a;b),
Revised: (Ullrich et al., 2020), CryoDRGN (Zhong et al., 2019; 2021a;b),

Original: AtomVAE (Rosenbaum et al., 2021), CryoPoseNet (Nashed et al., 2021), and cryoAI (Levy et al., 2022) adopt the autoencoder architecture.
Revised: AtomVAE (Rosenbaum et al., 2021), CryoPoseNet (Nashed et al., 2021), and cryoAI (Levy et al., 2022) adopt the autoencoder architecture.

Original: They improve the performance of autoencoders in cryoEM 3D reconstructions by using more powerful networks for encoders and decoders, adding or modifying loss functions, modifying the latent variable designs and other strategies.
Revised: They improve the performance of autoencoders in cryoEM 3D reconstructions by using more powerful networks for encoders and decoders, adding or modifying loss functions, modifying the latent variable designs and other strategies.

Original: Pose estimation Pose estimation is concerned with estimating the missing orientation and position (together known as pose) parameters of the images based on the underlying 3D object.
Revised: Pose estimation Pose estimation is concerned with estimating the missing orientation and position (together known as pose) parameters of the images based on the underlying 3D object.

Original: Existing pose estimation methods can be divided into two classes: Per-image pose search methods and amortized inference methods.
Revised: Existing pose estimation methods can be divided into two classes: Per-image pose search methods and amortized inference methods.

Original: The first class of methods performs a global pose search for each projection image in the input dataset.
Revised: The first class of methods performs a global pose search for each projection image in the input dataset.

Original: Traditional software, like RELION (Scheres, 2012) and cryoSPARC (Punjani et al., 2017), and some ML-based methods, inluding cryoDRGN-BNB Zhong et al.
Revised: Traditional software, like RELION (Scheres, 2012) and cryoSPARC (Punjani et al., 2017), and some ML-based methods, inluding cryoDRGN-BNB Zhong et al.

Original: (2019) and CryoDRGN2(Zhong et al., 2021b), fall into this category.
Revised: (2019) and CryoDRGN2(Zhong et al., 2021b), fall into this category.

Original: The per-image pose search method is not an ML-based strategy, To control the Pose estimation as an ML process, the second class method, amortized inference, is first proposed for cryo-EM 3D reconstruction by Ullrich et.
Revised: The per-image pose search method is not an ML-based strategy, To control the Pose estimation as an ML process, the second class method, amortized inference, is first proposed for cryo-EM 3D reconstruction by Ullrich et.

Original: al.
Revised: al.

Original: (Ullrich et al., 2020).
Revised: (Ullrich et al., 2020).

Original: It focuses on learning a parameterized function for mapping image
Revised: It focuses on learning a parameterized function for mapping image

Original: Y i to its pose ϕ
Revised: Y i to its pose ϕ

Original: i , then it is widely accepted by autoencoder solutions as the encoder.
Revised: i , then it is widely accepted by autoencoder solutions as the encoder.

Original: For autoencoders with amortized inference method, they suffer from too many local minima in their probabilistic pose estimation framework.
Revised: For autoencoders with amortized inference method, they suffer from too many local minima in their probabilistic pose estimation framework.

Original: Rosenbaum et.
Revised: Rosenbaum et.

Original: al. attempted to overcome local-minima traps using variational autoencoder (VAE) (Kingma & Welling, 2014).
Revised: al. attempted to overcome local-minima traps using variational autoencoder (VAE) (Kingma & Welling, 2014).

Original: CryoAI (Levy et al., 2022) extended CryoPoseNet (Nashed et al., 2021) work by introducing “symmetry loss” to facilitate pose learning by adding a 180 ◦ -rotated input image as input.
Revised: CryoAI (Levy et al., 2022) extended CryoPoseNet (Nashed et al., 2021) work by introducing “symmetry loss” to facilitate pose learning by adding a 180 ◦ -rotated input image as input.

Original: In practice, we found these strategies are insufficient for avoiding poses being trapped in local minima.
Revised: In practice, we found these strategies are insufficient for avoiding poses being trapped in local minima.

Original: The proposed ACE-EM method also falls into this category of amortized inference.
Revised: The proposed ACE-EM method also falls into this category of amortized inference.

Original: Decoder or generator choices
Revised: Decoder or generator choices

Original: In the view of cryo-EM 3D reconstruction, a decoder (in autoencoder) or a generator (in GAN) is a structure that contains the volume representation, which adopts the pose or other variables and outputs the predicted 2D projection images.
Revised: In the view of cryo-EM 3D reconstruction, a decoder (in autoencoder) or a generator (in GAN) is a structure that contains the volume representation, which adopts the pose or other variables and outputs the predicted 2D projection images.

Original: The underlying target 3D object can have either real-space or frequency-space representations in the space domain, and either neural network type or voxel grid type for volume parameter representations (Donnat et al., 2022).
Revised: The underlying target 3D object can have either real-space or frequency-space representations in the space domain, and either neural network type or voxel grid type for volume parameter representations (Donnat et al., 2022).

Original: Traditional softwares represent the reconstruction target volume as a 3D voxel grid of frequency space, like Relion (Scheres, 2012) and cryoSPARC (Punjani et al., 2017).
Revised: Traditional softwares represent the reconstruction target volume as a 3D voxel grid of frequency space, like Relion (Scheres, 2012) and cryoSPARC (Punjani et al., 2017).

Original: Many recently developed deep-learning-based reconstruction methods mainly use two types of decoders (generators): real-space voxel grid type and frequency-space neural network type.
Revised: Many recently developed deep-learning-based reconstruction methods mainly use two types of decoders (generators): real-space voxel grid type and frequency-space neural network type.

Original: CryoDRGN (Zhong et al., 2019; 2021a;b) and CryoAI (Levy et al., 2022) uses the frequency-space neural network as their decoders.
Revised: CryoDRGN (Zhong et al., 2019; 2021a;b) and CryoAI (Levy et al., 2022) uses the frequency-space neural network as their decoders.

Original: Real-space voxel grid methods are also used by many algorithms, such as CryoGAN (Gupta et al., 2021), Multi-CryoGAN (Gupta et al., 2020), 3DFlex (Punjani & Fleet, 2021), and
Revised: Real-space voxel grid methods are also used by many algorithms, such as CryoGAN (Gupta et al., 2021), Multi-CryoGAN (Gupta et al., 2020), 3DFlex (Punjani & Fleet, 2021), and

Original: AtomVAE (Rosenbaum et al., 2021).
Revised: AtomVAE (Rosenbaum et al., 2021).

Original: CryoPoseNet (Nashed et al., 2021) stores a real space voxel grid representation of the reconstruction target, but the projection was made in Fourier space.
Revised: CryoPoseNet (Nashed et al., 2021) stores a real space voxel grid representation of the reconstruction target, but the projection was made in Fourier space.

Original: In this work, we show that the ACE-EM method is effective with both real-space voxel grid type and frequency-space neural network type of decoders.
Revised: In this work, we show that the ACE-EM method is effective with both real-space voxel grid type and frequency-space neural network type of decoders.

Original: Cyclic training
Revised: Cyclic training

Original: Our method uses encoder-decoder(E-D) and decoder-encoder(D-E) two orders for training in autoencoders, and this cyclic training is widely used in style transfer, like MUNIT (Huang et al., 2018), DRIT (Lee et al., 2019) and CDD (Gonzalez-Garcia et al., 2018).
Revised: Our method uses encoder-decoder(E-D) and decoder-encoder(D-E) two orders for training in autoencoders, and this cyclic training is widely used in style transfer, like MUNIT (Huang et al., 2018), DRIT (Lee et al., 2019) and CDD (Gonzalez-Garcia et al., 2018).

Original: These methods differ from ours in 4 main aspects, architecture setting, data format, aims, and challenges.
Revised: These methods differ from ours in 4 main aspects, architecture setting, data format, aims, and challenges.

Original: For the architecture setting, MUNIT uses E-D and (E-)D-E two orders (Huang et al., 2018), DRIT uses one task E-D-E-D to finish the cycle, while more complex tasks are used in CDD.
Revised: For the architecture setting, MUNIT uses E-D and (E-)D-E two orders (Huang et al., 2018), DRIT uses one task E-D-E-D to finish the cycle, while more complex tasks are used in CDD.

Original: MUNIT and
Revised: MUNIT and

Original: DRIT need two random images inputted together as a group for cross-domain translation (Lee et al., 2019; Gonzalez-Garcia et al., 2018), CDD uses paired data for supervised learning (Gonzalez-Garcia et al., 2018), but paired or grouped data is not required by ACE-EM.
Revised: DRIT need two random images inputted together as a group for cross-domain translation (Lee et al., 2019; Gonzalez-Garcia et al., 2018), CDD uses paired data for supervised learning (Gonzalez-Garcia et al., 2018), but paired or grouped data is not required by ACE-EM.

Original: Style transfer aims to decouple images’ style and content, and our aim is to settle the poses in an accurate distribution.
Revised: Style transfer aims to decouple images’ style and content, and our aim is to settle the poses in an accurate distribution.

Original: Also, our methods face specific reconstruction challenges, like noise problems.
Revised: Also, our methods face specific reconstruction challenges, like noise problems.

Original:  A PPROACH 3. O VERVIEW OF ACE-EM
Revised:  A PPROACH 3. O VERVIEW OF ACE-EM

Original: ACE-EM is an autoencoder-based model and is trained with two unsupervised learning tasks.
Revised: ACE-EM is an autoencoder-based model and is trained with two unsupervised learning tasks.

Original: The autoencoder contains an image-to-pose encoder ( E IP ) and a pose-to-image decoder ( D PI ).
Revised: The autoencoder contains an image-to-pose encoder ( E IP ) and a pose-to-image decoder ( D PI ).

Original: The
Revised: The

Original: E IP takes projection images and outputs projection pose parameters.
Revised: E IP takes projection images and outputs projection pose parameters.

Original: The D PI can be viewed as a cryo-EM image projection physics simulator.
Revised: The D PI can be viewed as a cryo-EM image projection physics simulator.

Original: It takes the pose parameters and outputs projection images corresponding to the given poses, which are post-processed by applying CTF (Konstantinidis, 2014).
Revised: It takes the pose parameters and outputs projection images corresponding to the given poses, which are post-processed by applying CTF (Konstantinidis, 2014).

Original: The first task of ACE-EM is image-pose-image(IPI) task which follows the standard unsupervised autoencoder architecture.
Revised: The first task of ACE-EM is image-pose-image(IPI) task which follows the standard unsupervised autoencoder architecture.

Original: With the pipeline of E IP and D PI , reconstructed images are generated corresponding to the given projection images.
Revised: With the pipeline of E IP and D PI , reconstructed images are generated corresponding to the given projection images.

Original: The second task is the pose-image-pose (PIP) task, a self-supervised learning, that uses the same encoder-decoder in IPI but in the reversed order.
Revised: The second task is the pose-image-pose (PIP) task, a self-supervised learning, that uses the same encoder-decoder in IPI but in the reversed order.

Original: With the reverse pipeline, a corresponding pose is predicted from a random-selected pose, and the gap between the two poses is minimized in training.
Revised: With the reverse pipeline, a corresponding pose is predicted from a random-selected pose, and the gap between the two poses is minimized in training.

Original: The IPI task and PIP task can be trained simultaneously or in alternating steps.
Revised: The IPI task and PIP task can be trained simultaneously or in alternating steps.

Original: With either training strategy, the E IP and D PI parameters are shared between the two tasks.
Revised: With either training strategy, the E IP and D PI parameters are shared between the two tasks.

Original: As a result, the two tasks of ACE-EM complement each other and form a more balanced training of the E IP and D PI parameters.
Revised: As a result, the two tasks of ACE-EM complement each other and form a more balanced training of the E IP and D PI parameters.

Original: 3. E NCODER E IP AND D ECODER D PI
Revised: 3. E NCODER E IP AND D ECODER D PI

Original: The E IP represents a function that maps an input image Y i to its corresponding projection pose parameters ( R i , t i ) .
Revised: The E IP represents a function that maps an input image Y i to its corresponding projection pose parameters ( R i , t i ) .

Original: R i ∈
Revised: R i ∈

Original: SO (3) ⊂
Revised: SO (3) ⊂

Original: R 3 × 3 is a rotation matrix for mapping a reference orientation to the projection orientation.
Revised: R 3 × 3 is a rotation matrix for mapping a reference orientation to the projection orientation.

Original: t i ∈
Revised: t i ∈

Original: R 2 is the 2D translation vector to account for the 2D image shift in the input image
Revised: R 2 is the 2D translation vector to account for the 2D image shift in the input image

Original: Y i .
Revised: Y i .

Original: The D PI takes pose ( R i , t i ) and predicts the projection image Y predi .The
Revised: The D PI takes pose ( R i , t i ) and predicts the projection image Y predi .The

Original: trainable parameters of D PI contain the cryo-EM volume in an explicit or implicit way, and the volume can be obtained from the D PI after training.
Revised: trainable parameters of D PI contain the cryo-EM volume in an explicit or implicit way, and the volume can be obtained from the D PI after training.

Original: CTF was applied to the output images to create a more realistic projection image Y predi .
Revised: CTF was applied to the output images to create a more realistic projection image Y predi .

Original: The details of the encoder and decoder structure can be found in appendix A.
Revised: The details of the encoder and decoder structure can be found in appendix A.

Original: [Equation]
Revised: [Equation]

Original: Choices of decoders ACE-EM provides an architecture that can work with different encoders and decoders.
Revised: Choices of decoders ACE-EM provides an architecture that can work with different encoders and decoders.

Original: Cryo-EM 3D reconstruction can have either real-space or frequency-space representations in the space domain, and either neural network type or voxel grid type for volume parameter representations.
Revised: Cryo-EM 3D reconstruction can have either real-space or frequency-space representations in the space domain, and either neural network type or voxel grid type for volume parameter representations.

Original: To prove ACE-EM is universal and can boost the performances regardless of the decoder types, we tested a real-space voxel grid decoder VoxelGrid R which was used in cryoGAN (Gupta et al., 2021), and partially used in the CryoPoseNet (Nashed et al., 2021), and a frequency-space neural network decoder FourierNet which was shown to outperform other similar methods (Levy et al., 2022).
Revised: To prove ACE-EM is universal and can boost the performances regardless of the decoder types, we tested a real-space voxel grid decoder VoxelGrid R which was used in cryoGAN (Gupta et al., 2021), and partially used in the CryoPoseNet (Nashed et al., 2021), and a frequency-space neural network decoder FourierNet which was shown to outperform other similar methods (Levy et al., 2022).

Original: 3. IPI(I MAGE -P OSE -I MAGE ) TASK
Revised: 3. IPI(I MAGE -P OSE -I MAGE ) TASK

Original: The IPI task follows the standard autoencoder architecture as shown in Figure 1.
Revised: The IPI task follows the standard autoencoder architecture as shown in Figure 1.

Original: Using the notations defined earlier, the IPI task can be formally defined as follows.
Revised: Using the notations defined earlier, the IPI task can be formally defined as follows.

Original: [Equation]
Revised: [Equation]

Original: IPI loss function Since both the input and output are images, the objective of the IPI task is to minimize their differences by mean squared error (MSE) loss function for each training batch of size
Revised: IPI loss function Since both the input and output are images, the objective of the IPI task is to minimize their differences by mean squared error (MSE) loss function for each training batch of size

Original: B and with an image edge length of L .
Revised: B and with an image edge length of L .

Original: [Equation]
Revised: [Equation]

Original: Cryo-EM 3D reconstruction is prone to spurious 2-fold planar mirror symmetry (Levy et al., 2022).
Revised: Cryo-EM 3D reconstruction is prone to spurious 2-fold planar mirror symmetry (Levy et al., 2022).

Original: A tentative solution is to use the “symmetric loss” employed in cryoAI (Levy et al., 2022), where Γ cryoAI is the cryo-AI autoencoder pipeline and R π represents an in-plane rotation of angle π operation applied to the input image
Revised: A tentative solution is to use the “symmetric loss” employed in cryoAI (Levy et al., 2022), where Γ cryoAI is the cryo-AI autoencoder pipeline and R π represents an in-plane rotation of angle π operation applied to the input image

Original: Y i .
Revised: Y i .

Original: [Equation]
Revised: [Equation]

Original: In practice, we found that this loss function cannot prevent the formation of the spurious cyclic symmetry artifact with VoxelGrid R when the target 3D object has an approximate cyclic symmetry.
Revised: In practice, we found that this loss function cannot prevent the formation of the spurious cyclic symmetry artifact with VoxelGrid R when the target 3D object has an approximate cyclic symmetry.

Original: We have generalized the above symmetric loss to include in-plane image rotation by arbitrary angles and also included mirror transformation.
Revised: We have generalized the above symmetric loss to include in-plane image rotation by arbitrary angles and also included mirror transformation.

Original: “generalized symmetric loss” as below.
Revised: “generalized symmetric loss” as below.

Original: A 1 [ Y i ] and A 

Revised: A 1 [ Y i ] and A

Original: [ Y i ] are two different affine transformations with random image rotations and/or mirror flipping.
Revised: [ Y i ] are two different affine transformations with random image rotations and/or mirror flipping.

Original: The generalized symmetry loss is only applied when using the voxel-grid-based decoder VoxelGrid R .
Revised: The generalized symmetry loss is only applied when using the voxel-grid-based decoder VoxelGrid R .

Original: [Equation]
Revised: [Equation]

Original: In addition to image loss, we also added an L1-regularization term for image shift to avoid unrealistic large shift predictions and keep the reconstructed 3D object near the origin of the coordinate system.
Revised: In addition to image loss, we also added an L1-regularization term for image shift to avoid unrealistic large shift predictions and keep the reconstructed 3D object near the origin of the coordinate system.

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: IPI warm-up labeling High-frequency features are difficult to learn in at an early training stage, especially for projection images with low SNR.
Revised: IPI warm-up labeling High-frequency features are difficult to learn in at an early training stage, especially for projection images with low SNR.

Original: We found that the training results can be improved by using the low-pass filtered input images as training labels instead of using original image labels, as a training warm-up.
Revised: We found that the training results can be improved by using the low-pass filtered input images as training labels instead of using original image labels, as a training warm-up.

Original: The training image label ˜
Revised: The training image label ˜

Original: Y i is defined below.
Revised: Y i is defined below.

Original: The details of f filter-k can be found in appendix A. f filter-1 is the first Gaussian filter with the lowest Gaussian convolution kernel variance, while f filter-k is the k-th Gaussian filter with higher convolution kernel variance.
Revised: The details of f filter-k can be found in appendix A. f filter-1 is the first Gaussian filter with the lowest Gaussian convolution kernel variance, while f filter-k is the k-th Gaussian filter with higher convolution kernel variance.

Original: N IPI warm-up is the iteration threshold for switching to the original input images.
Revised: N IPI warm-up is the iteration threshold for switching to the original input images.

Original: In practice, using k = 5 or k =is sufficient. .
Revised: In practice, using k = 5 or k =is sufficient. .

Original: [Equation]
Revised: [Equation]

Original: 3. PIP(P OSE -I MAGE -P OSE ) TASK
Revised: 3. PIP(P OSE -I MAGE -P OSE ) TASK

Original: The PIP task is similar to the IPI task but with E IP and D PI placed in reverse order.
Revised: The PIP task is similar to the IPI task but with E IP and D PI placed in reverse order.

Original: Besides, an additive Gaussian noise ϵ ∼ N ( µ, σ 2 I ) is added to the output image from D PI to create more realistic inputs for the E IP .
Revised: Besides, an additive Gaussian noise ϵ ∼ N ( µ, σ 2 I ) is added to the output image from D PI to create more realistic inputs for the E IP .

Original: The µ and σ of the Gaussian noise is sampled based on the background of the input images.
Revised: The µ and σ of the Gaussian noise is sampled based on the background of the input images.

Original: Compared to the IPI task, the inputs of PIP have changed from images to synthesized projection parameters ( R i , t i ) drawn from certain distributions.
Revised: Compared to the IPI task, the inputs of PIP have changed from images to synthesized projection parameters ( R i , t i ) drawn from certain distributions.

Original: The loss function is defined as the MSE loss of rotation matrices { R i } i ∈B and translation vectors { t i } i ∈B for each batch
Revised: The loss function is defined as the MSE loss of rotation matrices { R i } i ∈B and translation vectors { t i } i ∈B for each batch

Original: B of size B as shown below, where ∥·∥ F is the Frobenius matrix norm.
Revised: B of size B as shown below, where ∥·∥ F is the Frobenius matrix norm.

Original: [Equation]
Revised: [Equation]

Original: With only IPI, the pose estimation of E IP is not distributed in the whole pose space, and some part of the pose space is never reached.
Revised: With only IPI, the pose estimation of E IP is not distributed in the whole pose space, and some part of the pose space is never reached.

Original: By applying PIP, in which the input pose is randomly sampled in the whole pose space, E IP is forced to match the image-pose pairs over all possible pose spaces.
Revised: By applying PIP, in which the input pose is randomly sampled in the whole pose space, E IP is forced to match the image-pose pairs over all possible pose spaces.

Original: Therefore, PIP gives E IP a proper output distribution and lets the model measure infinity of pose and images instead of only images in datasets.
Revised: Therefore, PIP gives E IP a proper output distribution and lets the model measure infinity of pose and images instead of only images in datasets.

Original: With a substantially correct volume representation of D PI , PIP even turns the unsupervised task into a supervised image-pose matching task.
Revised: With a substantially correct volume representation of D PI , PIP even turns the unsupervised task into a supervised image-pose matching task.

Original: When training the PIP task, one has the choice of freezing the D PI parameters and only training the E IP , or allowing both the E IP and D PI parameters to be updated.
Revised: When training the PIP task, one has the choice of freezing the D PI parameters and only training the E IP , or allowing both the E IP and D PI parameters to be updated.

Original: 3. T RAINING OF THE IPI AND PIP TASKS
Revised: 3. T RAINING OF THE IPI AND PIP TASKS

Original: The IPI and PIP tasks can be trained together or in succession.
Revised: The IPI and PIP tasks can be trained together or in succession.

Original: When trained together, we designed the following training schedule with a warm-up period of N trainwarm-up iterations, which allows the IPI task to learn an approximate description of the underlying 3D object before adding the PIP task.
Revised: When trained together, we designed the following training schedule with a warm-up period of N trainwarm-up iterations, which allows the IPI task to learn an approximate description of the underlying 3D object before adding the PIP task.

Original: Although we found the following simple schedule was sufficient in our benchmark tests, other schedules of β are also possible.
Revised: Although we found the following simple schedule was sufficient in our benchmark tests, other schedules of β are also possible.

Original: [Equation]
Revised: [Equation]

Original:  R ESULTS
Revised:  R ESULTS

Original: To evaluate the ab initio 3D reconstruction quality of ACE-EM, we benchmarked this method against both the traditional cryo-EM 3D reconstruction software cryoSPARC (Punjani et al., 2017) and recent methods, including cryoPoseNet (Nashed et al., 2021), cryoDRGN

Revised: To evaluate the ab initio 3D reconstruction quality of ACE-EM, we benchmarked this method against both the traditional cryo-EM 3D reconstruction software cryoSPARC (Punjani et al., 2017) and recent methods, including cryoPoseNet (Nashed et al., 2021), cryoDRGN

Original: (Zhong et al., 2021b), and cryoAI (Levy et al., 2022), on both simulated and experimental cryo-EM datasets.
Revised: (Zhong et al., 2021b), and cryoAI (Levy et al., 2022), on both simulated and experimental cryo-EM datasets.

Original: 4. D ATASET P REPARATION AND T RAINING S ETUP
Revised: 4. D ATASET P REPARATION AND T RAINING S ETUP

Original: Datasets Both simulated and experimental cryo-EM datasets are prepared following the setup of cryoAI (Levy et al., 2022).
Revised: Datasets Both simulated and experimental cryo-EM datasets are prepared following the setup of cryoAI (Levy et al., 2022).

Original: For simulated datasets, two protein molecules, the pre-catalytic spliceosome and the SARS-CoV-2 spike ectodomain structure, are selected as the reconstructed targets.
Revised: For simulated datasets, two protein molecules, the pre-catalytic spliceosome and the SARS-CoV-2 spike ectodomain structure, are selected as the reconstructed targets.

Original: The simulated dataset generation method and the detailed dataset setting can be seen in the appendix B.
Revised: The simulated dataset generation method and the detailed dataset setting can be seen in the appendix B.

Original: The shape of each projection image is 128 × 128 .
Revised: The shape of each projection image is 128 × 128 .

Original: Each reconstructed volume is a voxel with 128 × 128 × 128 shape.
Revised: Each reconstructed volume is a voxel with 128 × 128 × 128 shape.

Original: The experimental cryo-EM dataset 80S ribosome (EMPIAR-10028) consists of 105,247 images with a shape of 360 × 360 , which are downsampled to 128 × 128 for training with ACE-EM.
Revised: The experimental cryo-EM dataset 80S ribosome (EMPIAR-10028) consists of 105,247 images with a shape of 360 × 360 , which are downsampled to 128 × 128 for training with ACE-EM.

Original: Note that as discussed in the original publication (Levy et al., 2022), the cryoAI algorithm could not handle 128 × 128 -downsampled images due to convergence issues and an input image size of 256 × 256 was used during training.
Revised: Note that as discussed in the original publication (Levy et al., 2022), the cryoAI algorithm could not handle 128 × 128 -downsampled images due to convergence issues and an input image size of 256 × 256 was used during training.

Original: The output shape of the reconstructed volume from cryoAI was set to 128 × 128 × 128 voxels, the same as ACE-EM and other methods.
Revised: The output shape of the reconstructed volume from cryoAI was set to 128 × 128 × 128 voxels, the same as ACE-EM and other methods.

Original: Accuracy assessment The benchmark metrics follow cryoAI (Levy et al., 2022).
Revised: Accuracy assessment The benchmark metrics follow cryoAI (Levy et al., 2022).

Original: 3D reconstruction quality was assessed by the Fourier Shell Correlation (FSC).
Revised: 3D reconstruction quality was assessed by the Fourier Shell Correlation (FSC).

Original: It measures the correlation at every corresponding frequency or resolution(reciprocal of the frequency) between two volumes.
Revised: It measures the correlation at every corresponding frequency or resolution(reciprocal of the frequency) between two volumes.

Original: The FSC resolution at value k is defined as the specific resolution when the correlation is dropped to k .
Revised: The FSC resolution at value k is defined as the specific resolution when the correlation is dropped to k .

Original: Rotation matrix error (Rot.) was calculated using the mean/median square Frobenius norm relative to ground-truth R i .
Revised: Rotation matrix error (Rot.) was calculated using the mean/median square Frobenius norm relative to ground-truth R i .

Original: Translation matrix error (Trans.) was calculated by the mean square L2-norm of the predicted and ground-truth translation matrix.
Revised: Translation matrix error (Trans.) was calculated by the mean square L2-norm of the predicted and ground-truth translation matrix.

Original: Training setup We ran our benchmark tests on a server with 8 Nvidia V100 GPUs and 84 CPU cores.
Revised: Training setup We ran our benchmark tests on a server with 8 Nvidia V100 GPUs and 84 CPU cores.

Original: When training with the FourierNet decoder, we used the batch size of 384 and the learning rate of 1e-4 for both the encoder and decoder.
Revised: When training with the FourierNet decoder, we used the batch size of 384 and the learning rate of 1e-4 for both the encoder and decoder.

Original: FourierNet was trained for 40,000 iterations.
Revised: FourierNet was trained for 40,000 iterations.

Original: When training with the VoxelGrid R decoder, we used the batch-size of 1,024 and training iterations of 30,000.
Revised: When training with the VoxelGrid R decoder, we used the batch-size of 1,024 and training iterations of 30,000.

Original: The learning rate for the encoder was 1e-3 and 1.0 for the decoder.
Revised: The learning rate for the encoder was 1e-3 and 1.0 for the decoder.

Original: The duration of IPI warm-up labeling N IPI warm-up usually set to 3000-6000 iterations for VoxelGrid R .
Revised: The duration of IPI warm-up labeling N IPI warm-up usually set to 3000-6000 iterations for VoxelGrid R .

Original: The“WarmupMultiStepLR” warm-up schedule and AdamW(Loshchilov & Hutter, 2017) optimizer were used in both cases.
Revised: The“WarmupMultiStepLR” warm-up schedule and AdamW(Loshchilov & Hutter, 2017) optimizer were used in both cases.

Original: N trainwarm-up is usually set to 0 or 2000-3000 iterations, PIP parameter β is usually set to 0.9 for 200 dB dataset and 0.05-0.09 for -10db dataset in both cases.
Revised: N trainwarm-up is usually set to 0 or 2000-3000 iterations, PIP parameter β is usually set to 0.9 for 200 dB dataset and 0.05-0.09 for -10db dataset in both cases.

Original: The decoder is frozen during the PIP process.
Revised: The decoder is frozen during the PIP process.

Original: 4. 3D RECONSTRUCTION ON SIMULATED DATASETS
Revised: 4. 3D RECONSTRUCTION ON SIMULATED DATASETS

Original: Dataset Two protein molecules are selected as the targets: the pre-catalytic spliceosome (PDB ID: 5NRL) and the SARS-CoV-2 spike ectodomain structure (PDB ID: 6VYB).
Revised: Dataset Two protein molecules are selected as the targets: the pre-catalytic spliceosome (PDB ID: 5NRL) and the SARS-CoV-2 spike ectodomain structure (PDB ID: 6VYB).

Original: For comparing the performances of methods with different noise levels, we synthesized each dataset with Gaussian noises of different SNRs, 200 dB, -10 dB or even -

Revised: For comparing the performances of methods with different noise levels, we synthesized each dataset with Gaussian noises of different SNRs, 200 dB, -10 dB or even -

Original: dB(Figure 7).
Revised: dB(Figure 7).

Original: With high SNR at 200 dB, images can be regarded as noise-free.
Revised: With high SNR at 200 dB, images can be regarded as noise-free.

Original: -10 dB is the most common SNR in experimental projection image datasets, thus, the model performance of this condition is a focus.
Revised: -10 dB is the most common SNR in experimental projection image datasets, thus, the model performance of this condition is a focus.

Original: With -20 dB, the images become indistinguishable from human eyes and reconstruction becomes more challenging.
Revised: With -20 dB, the images become indistinguishable from human eyes and reconstruction becomes more challenging.

Original: An additional dataset spliceosome at the SNR of -20 dB is applied to test the noise tolerance of our method.
Revised: An additional dataset spliceosome at the SNR of -20 dB is applied to test the noise tolerance of our method.

Original: Baselines We benchmarked against CryoPoseNet (Nashed et al., 2021), CryoSPARC (Punjani et al., 2017) (traditional method), cryoDRGN2 (Zhong et al., 2021b), and cryoAI (Levy et al., 2022) in terms of pose estimation and 3D reconstruction accuracy.
Revised: Baselines We benchmarked against CryoPoseNet (Nashed et al., 2021), CryoSPARC (Punjani et al., 2017) (traditional method), cryoDRGN2 (Zhong et al., 2021b), and cryoAI (Levy et al., 2022) in terms of pose estimation and 3D reconstruction accuracy.

Original: For simulated datasets, FSC resolution at a threshold of 0.5 between the ground-truth and predicted volumes is the evaluation criterion.
Revised: For simulated datasets, FSC resolution at a threshold of 0.5 between the ground-truth and predicted volumes is the evaluation criterion.

Original: Results
Revised: Results

Original: The result of the simulated datasets can be seen in Table 1.
Revised: The result of the simulated datasets can be seen in Table 1.

Original: Results for CryoPoseNet were taken from previously reported data (Levy et al., 2022), cryoDRGN2’s results were from (Zhong et al., 2021b), while the cryoSPARC and cryoAI’s results were obtained by re-running these methods.
Revised: Results for CryoPoseNet were taken from previously reported data (Levy et al., 2022), cryoDRGN2’s results were from (Zhong et al., 2021b), while the cryoSPARC and cryoAI’s results were obtained by re-running these methods.

Original: We found that cryoAI failed frequently due to the spurious planar symmetries.
Revised: We found that cryoAI failed frequently due to the spurious planar symmetries.

Original: CryoSPARC were very consistent except on the spliceosome (-20 dB) dataset where three out of six runs failed.
Revised: CryoSPARC were very consistent except on the spliceosome (-20 dB) dataset where three out of six runs failed.

Original: In comparison, all ACE-EM reconstructions succeeded.
Revised: In comparison, all ACE-EM reconstructions succeeded.

Original: The pose estimation accuracy and 3D reconstruction quality are comparable or better than these baseline methods using either VoxelGrid R or FourierNet decoder.
Revised: The pose estimation accuracy and 3D reconstruction quality are comparable or better than these baseline methods using either VoxelGrid R or FourierNet decoder.

Original: In terms of reconstruction resolution, ACE-EM with FourierNet outperformed all the baseline methods.
Revised: In terms of reconstruction resolution, ACE-EM with FourierNet outperformed all the baseline methods.

Original: It reached 2 pixels, the Nyquist resolution, using an FSC threshold of 0.5 in all datasets with SNR 200 dB and -10 dB. Visualization of 3D reconstruction result can be seen in Figure 5 and Figure 6.
Revised: It reached 2 pixels, the Nyquist resolution, using an FSC threshold of 0.5 in all datasets with SNR 200 dB and -10 dB. Visualization of 3D reconstruction result can be seen in Figure 5 and Figure 6.

Original: Our reconstructed volumes have more details compared to the result of cryoAI.
Revised: Our reconstructed volumes have more details compared to the result of cryoAI.

Original: Based on the Nyquist–Shannon sampling theorem (Shannon, 1949), the highest measurable resolution of the dispersed volume representation is 2 pixels.
Revised: Based on the Nyquist–Shannon sampling theorem (Shannon, 1949), the highest measurable resolution of the dispersed volume representation is 2 pixels.

Original: The resolution of ACE-EM with
Revised: The resolution of ACE-EM with

Original: FourierNet reached the physical limitation, which is not reachable by other methods.
Revised: FourierNet reached the physical limitation, which is not reachable by other methods.

Original: To be specific, the  correlation of resolution 2.03 pixels is still very high (0.87) for the spliceosome (200dB) dataset, well above the standard threshold (0.5).
Revised: To be specific, the FSC correlation coefficient at 2.03 pixels is still very high (0.87) for the spliceosome (200 dB) dataset, well above the standard threshold (0.5).

Original: We also benchmarked ACE-EM on an even more challenging dataset (SNR -20 dB; without image shifts).
Revised: We also benchmarked ACE-EM on an even more challenging dataset (SNR -20 dB; without image shifts).

Original: The FSC resolution of the output 3D object reached 2.1 pixels, which is even higher than the results of -10 dB provided by other ML-based baselines.
Revised: The FSC resolution of the output 3D object reached 2.1 pixels, which is even higher than the results of -10 dB provided by other ML-based baselines.

Original: Also, the performance of -20 dB SNR proves the potential of ACE-EM with FourierNet for working in really low SNR situations.
Revised: Also, the performance of -20 dB SNR proves the potential of ACE-EM with FourierNet for working in really low SNR situations.

Original: When using VoxelGrid R as the decoder, ACE-EM reached a similar or better FSC resolution among other baselines.
Revised: When using VoxelGrid R as the decoder, ACE-EM reached a similar or better FSC resolution among other baselines.

Original: In detail, on the similar SNR -10dB with the experimental environment, it provided higher resolutions on the spliceosome dataset compared to other baselines (except ACE-EM with FourierNet ).
Revised: In detail, on the similar SNR -10dB with the experimental environment, it provided higher resolutions on the spliceosome dataset compared to other baselines (except ACE-EM with FourierNet ).

Original: With VoxelGrid R , CryoPoseNet failed in the spike datasets, indicating that our method gave VoxelGrid R back the competitiveness for cryo-EM 3D reconstruction.
Revised: With VoxelGrid R , CryoPoseNet failed in the spike datasets, indicating that our method gave VoxelGrid R back the competitiveness for cryo-EM 3D reconstruction.

Original: For rotation errors and translation errors, our methods get comparable or better results with other baselines.
Revised: For rotation errors and translation errors, our methods get comparable or better results with other baselines.

Original: Our methods achieve qualified pose estimations on noise-free datasets as most baselines.
Revised: Our methods achieve qualified pose estimations on noise-free datasets as most baselines.

Original: Moreover, our methods acquire better performances of noisy datasets among ML-based methods, especially on the evaluation of mean rotation error.
Revised: Moreover, our methods acquire better performances of noisy datasets among ML-based methods, especially on the evaluation of mean rotation error.

Original: 4. 3D RECONSTRUCTION ON EXPERIMENTAL DATASETS
Revised: 4. 3D RECONSTRUCTION ON EXPERIMENTAL DATASETS

Original: Dataset 80S EMPIAR-10028 is an experimental cryo-EM reconstruction dataset used for comparing baselines.
Revised: Dataset 80S EMPIAR-10028 is an experimental cryo-EM reconstruction dataset used for comparing baselines.

Original: The size is downsampled to 128 for cryoDRGN2 and our methods, and it is downsampled to 256 for cryoSPARC and cryoAI due to the failure of reconstruction with cryoAI (Levy et al., 2022) in size 128.
Revised: The size is downsampled to 128 for cryoDRGN2 and our methods, and it is downsampled to 256 for cryoSPARC and cryoAI due to the failure of reconstruction with cryoAI (Levy et al., 2022) in size 128.

Original: Baselines For experimental datasets, evaluation criterion is half-map FSC (Rosenthal & Rubinstein, 2015) resolution at 0.143, which splits the dataset evenly into two half datasets, then trains the two datasets separately to get two volumes for comparison.
Revised: Baselines For experimental datasets, evaluation criterion is half-map FSC (Rosenthal & Rubinstein, 2015) resolution at 0.143, which splits the dataset evenly into two half datasets, then trains the two datasets separately to get two volumes for comparison.

Original: Results
Revised: Results

Original: The result of the experimental datasets can be see in Table 2.
Revised: The result of the experimental datasets can be see in Table 2.

Original: Results for cryoSPARC, cryoDRGN2 and cryoAI were taken from previously reported data (Levy et al., 2022), The input image size and output volume size setting for different methods can also be seen in Table 2.
Revised: Results for cryoSPARC, cryoDRGN2 and cryoAI were taken from previously reported data (Levy et al., 2022), The input image size and output volume size setting for different methods can also be seen in Table 2.

Original: CryoAI claimed it is the first amortized inference method to demonstrate proper volume reconstruction on an experimental dataset.
Revised: CryoAI claimed it is the first amortized inference method to demonstrate proper volume reconstruction on an experimental dataset.

Original: However, CryoAI did not properly converge when fed with input images of size 128.
Revised: However, CryoAI did not properly converge when fed with input images of size 128.

Original: A compromise is reached by training with input images with a 256-pixel side but producing a 128-pixel side volume in cryoAI.
Revised: A compromise is reached by training with input images with a 256-pixel side but producing a 128-pixel side volume in cryoAI.

Original: CryoAI is a standard auto-encoder architecture with
Revised: CryoAI is a standard auto-encoder architecture with

Original: FourierNet as its decoder.
Revised: FourierNet as its decoder.

Original: Therefore, using ACE-EM with FourierNet is almost like adding the PIP task to the CryoAI structure.
Revised: Therefore, using ACE-EM with FourierNet is almost like adding the PIP task to the CryoAI structure.

Original: The convergence problem is preliminarily resolved by applying our method.
Revised: The convergence problem is preliminarily resolved by applying our method.

Original: After adding the PIP task, the model can adequately converge with input size 128 × 128 .
Revised: After adding the PIP task, the model can adequately converge with input size 128 × 128 .

Original: Also, the resolution of ACE-EM with FourierNet on a 128-pixel input size is better than CryoAI on a 256-pixel input size, which shows the robustness and effectiveness of our method.
Revised: Also, the resolution of ACE-EM with FourierNet on a 128-pixel input size is better than CryoAI on a 256-pixel input size, which shows the robustness and effectiveness of our method.

Original: Meanwhile, the result of ACE-EM with VoxelGrid R is on the Nyquist resolution, representing its capacity for the experimental situation.
Revised: Meanwhile, the result of ACE-EM with VoxelGrid R is on the Nyquist resolution, representing its capacity for the experimental situation.

Original: Visualization of 3D reconstruction result can be seen in Figure 5.
Revised: Visualization of 3D reconstruction result can be seen in Figure 5.

Original: [Table] [Table]
Revised: [Table] [Table]

Original: 4. A BLATION STUDIES
Revised: 4. A BLATION STUDIES

Original: Without PIP, ACE-EM is degenerate into a traditional autoencoder structure.
Revised: Without PIP, ACE-EM is degenerate into a traditional autoencoder structure.

Original: Comparisons are made between training with and without PIP tasks in two different decoders, to prove that our work ACEEM can boost the performance of an autoencoder by applying PIP regardless of decoder types.
Revised: Comparisons are made between training with and without PIP tasks in two different decoders, to prove that our work ACEEM can boost the performance of an autoencoder by applying PIP regardless of decoder types.

Original: 4.4. PIP ABLATION WITH FourierNet
Revised: 4.4. PIP ABLATION WITH FourierNet

Original: To test the effect of PIP with FourierNet , we ran an ablation study on the spliceosome dataset (- dB) using a batch size of 384 for 300 epochs.
Revised: To test the effect of PIP with FourierNet , we ran an ablation study on the spliceosome dataset (- dB) using a batch size of 384 for 300 epochs.

Original: As shown in Figure 2, all the performances, including
Revised: As shown in Figure 2, all the performances, including

Original: FSC-resolution, mean and median rotation matrix prediction errors, are much worse without the
Revised: FSC-resolution, mean and median rotation matrix prediction errors, are much worse without the

Original: PIP task.
Revised: PIP task.

Original: Though the median rotation error is similar at the end, the gap in the mean rotation error is noticeable.
Revised: Though the median rotation error is similar at the end, the gap in the mean rotation error is noticeable.

Original: Without PIP task, the mean rotation error is 1.98 at the 300th epoch, which is three orders of magnitude larger than 0.002, the mean error of ACE-EM.
Revised: Without PIP task, the mean rotation error is 1.98 at the 300th epoch, which is three orders of magnitude larger than 0.002, the mean error of ACE-EM.

Original: IPI reduces the existence of exaggerated rotation errors(see in Figure 8), then gets a better mean rotation error and further improves the resolution.
Revised: IPI reduces the existence of exaggerated rotation errors(see in Figure 8), then gets a better mean rotation error and further improves the resolution.

Original: We also compare the coverage of the projection orientation space during training in terms of the distribution of the oriented z-axis (projection direction), which is equivalent to the last column of the predicted rotation matrix.
Revised: We also compare the coverage of the projection orientation space during training in terms of the distribution of the oriented z-axis (projection direction), which is equivalent to the last column of the predicted rotation matrix.

Original: When creating the simulated dataset, the projection orientations were sampled from a uniform distribution.
Revised: When creating the simulated dataset, the projection orientations were sampled from a uniform distribution.

Original: Therefore the directions of the oriented z-axis should also be uniformly distributed.
Revised: Therefore the directions of the oriented z-axis should also be uniformly distributed.

Original: As shown in Figure 3, the predicted projection direction vectors covered the entire orientation space with the PIP task starting from epoch 50.
Revised: As shown in Figure 3, the predicted projection direction vectors covered the entire orientation space with the PIP task starting from epoch 50.

Original: In contrast, without the PIP task, the coverage of the orientation space is still incomplete at around 20% in 300 epochs.
Revised: In contrast, without the PIP task, the coverage of the orientation space is still incomplete at around 20% in 300 epochs.

Original: And the insufficient coverage performance reflected in the rotation error is the low median value along with the high mean value.
Revised: And the insufficient coverage performance reflected in the rotation error is the low median value along with the high mean value.

Original: After adding PIP task, the training time is increased to 1.25-1.50 times of the original duration for one epoch.
Revised: After adding PIP task, the training time is increased to 1.25-1.50 times of the original duration for one epoch.

Original: But the time consumption is worthy of faster convergence and better performances.
Revised: But the time consumption is worthy of faster convergence and better performances.

Original: 4.4. PIP ABLATION WITH VoxelGrid R
Revised: 4.4. PIP ABLATION WITH VoxelGrid R

Original: We also perform an ablation study for the PIP task using VoxelGrid R as the decoder on the same spliceosome dataset (-10 dB) using a batch size of 1,024 for 300 epochs.
Revised: We also perform an ablation study for the PIP task using VoxelGrid R as the decoder on the same spliceosome dataset (-10 dB) using a batch size of 1,024 for 300 epochs.

Original: As shown in Table 4 and
Revised: As shown in Table 4 and

Original: Figure 4, 3D reconstruction failed without either the PIP task or warm-up labeling (Section 3.3).
Revised: Figure 4, 3D reconstruction failed without either the PIP task or warm-up labeling (Section 3.3).

Original: The reconstruction can be completed when using warm-up labeling for loss calculations at the beginning of training.
Revised: The reconstruction can be completed when using warm-up labeling for loss calculations at the beginning of training.

Original: When PIP task was included, the mean rotation matrix prediction error was reduced significantly, and the FSC resolution was also improved.
Revised: When PIP task was included, the mean rotation matrix prediction error was reduced significantly, and the FSC resolution was also improved.

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: In this work, we have developed a new unsupervised learning framework ACE and applied it to the ab initio cryo-EM 3D reconstruction problem.
Revised: In this work, we have developed a new unsupervised learning framework ACE and applied it to the ab initio cryo-EM 3D reconstruction problem.

Original: Compared to existing methods, the most significant advantage of ACE-EM is its ability to learn the pose space much more effectively and accurately.
Revised: Compared to existing methods, the most significant advantage of ACE-EM is its ability to learn the pose space much more effectively and accurately.

Original: The ACE-EM method can work with different types of decoders, and boost the FSC-resolution even to Nyquist resolution in some simulated and experimental datasets.
Revised: The ACE-EM method can work with different types of decoders, and boost the FSC-resolution even to Nyquist resolution in some simulated and experimental datasets.

Original: In the future work on ACE-EM, we plan to extend the ACE framework to work with heterogeneous cryo-EM 3D reconstruction and novel view synthesis for natural images.
Revised: In the future work on ACE-EM, we plan to extend the ACE framework to work with heterogeneous cryo-EM 3D reconstruction and novel view synthesis for natural images.

Original:  R EPRODUCIBILITY S TATEMENT
Revised:  R EPRODUCIBILITY S TATEMENT

Original: The model structure can be seen in Section 3 and appendix A. The datasets and training setup can be seen in Section 4.1.
Revised: The model structure can be seen in Section 3 and appendix A. The datasets and training setup can be seen in Section 4.1.

Original: The simulated dataset generation method and the detailed dataset setting can be seen in the appendix B. The code to reproduce our experiments will be open-sourced upon publication.
Revised: The simulated dataset generation method and the detailed dataset setting can be seen in the appendix B. The code to reproduce our experiments will be open-sourced upon publication.

Original: A E NCODER AND DECODER A. E NCODER E IP
Revised: A E NCODER AND DECODER A. E NCODER E IP

Original: The E IP represents a function that maps an input image Y i to its corresponding projection pose parameters ( R i , t i ) .
Revised: The E IP represents a function that maps an input image Y i to its corresponding projection pose parameters ( R i , t i ) .

Original: R i ∈
Revised: R i ∈

Original: SO (3) ⊂
Revised: SO (3) ⊂

Original: R 3 × 3 is a rotation matrix for mapping a reference orientation to the projection orientation.
Revised: R 3 × 3 is a rotation matrix for mapping a reference orientation to the projection orientation.

Original: t i ∈
Revised: t i ∈

Original: R 2 is the 2D translation vector to account for the 2D image shift in the input image
Revised: R 2 is the 2D translation vector to account for the 2D image shift in the input image

Original: Y i .
Revised: Y i .

Original: [Equation]
Revised: [Equation]

Original: Cryo-EM projection images are extremely noisy.
Revised: Cryo-EM projection images are extremely noisy.

Original: To improve the E IP performance, we adopted the same image preprocessing strategy as in cryoAI (Levy et al., 2022).
Revised: To improve the E IP performance, we adopted the same image preprocessing strategy as in cryoAI (Levy et al., 2022).

Original: Each image Y
Revised: Each image Y

Original: i is passed on to a set of five Gaussian filters, which are implemented as 2D convolutions with a kernel size ofpixels and output a 5-channel image Y 5 C i .
Revised: i is passed on to a set of five Gaussian filters, which are implemented as 2D convolutions with a kernel size ofpixels and output a 5-channel image Y 5 C i .

Original: The Gaussian kernel variances follow a geometric series, i.e., 10 − 2 , 10 − 1 , 1 , 10 , 10 2 in pixel 2 for output image channels 1 to 5.
Revised: The Gaussian kernel variances follow a geometric series, i.e., 10 − 2 , 10 − 1 , 1 , 10 , 10 2 in pixel 2 for output image channels 1 to 5.

Original: The filtered images are given to a ResNet-

Revised: The filtered images are given to a ResNet-

Original: (He et al., 2016) network with two MLP networks for the inference of R i and t i respectively.
Revised: (He et al., 2016) network with two MLP networks for the inference of R i and t i respectively.

Original: The raw prediction of t i is passed through a Sigmoid function to restrict the range of values to (0 , 1) , because t i is expressed in unit of fraction of the image edge length.
Revised: The raw prediction of t i is passed through a Sigmoid function to restrict the range of values to (0 , 1) , because t i is expressed in unit of fraction of the image edge length.

Original: Other feature extractors (FE) can also be used in place of ResNet-18.
Revised: Other feature extractors (FE) can also be used in place of ResNet-18.

Original: [Equation]
Revised: [Equation]

Original: The complete E IP can be defined as follows, where “ ◦ ” means function composition.
Revised: The complete E IP can be defined as follows, where “ ◦ ” means function composition.

Original: the trainable parameters are the network work parameters in f FE and f MLP .
Revised: the trainable parameters are the network work parameters in f FE and f MLP .

Original: [Equation]
Revised: [Equation]

Original: There are many representations of R i such as Euler angles and quaternions.
Revised: There are many representations of R i such as Euler angles and quaternions.

Original: It has been shown that all 3D rotation representations in the real Euclidean spaces with no more than four dimensions are discontinuous and difficult to learn using neural networks (Zhou et al., 2019).
Revised: It has been shown that all 3D rotation representations in the real Euclidean spaces with no more than four dimensions are discontinuous and difficult to learn using neural networks (Zhou et al., 2019).

Original: We chose the 6dimensional vector representation in R 

Revised: We chose the 6dimensional vector representation in R

Original: (Zhou et al., 2019), which can be converted into a rotation matrix in R 3 × 3 using PyTorch3D (Johnson et al., 2020).
Revised: (Zhou et al., 2019), which can be converted into a rotation matrix in R 3 × 3 using PyTorch3D (Johnson et al., 2020).

Original: A. D ECODER D PI
Revised: A. D ECODER D PI

Original: The D PI takes pose ( R i , t i ) and predicts the projection image Y predi .
Revised: The D PI takes pose ( R i , t i ) and predicts the projection image Y predi .

Original: [Equation]
Revised: [Equation]

Original: There are many possible choices for D PI .
Revised: There are many possible choices for D PI .

Original: we tested a real-space voxel grid decoder VoxelGrid R which was used in cryoGAN (Gupta et al., 2021) and partially used in CryoPoseNet (Nashed et al., 2021), and a frequency-space neural network decoder FourierNet from cryoAI (Levy et al., 2022).
Revised: we tested a real-space voxel grid decoder VoxelGrid R which was used in cryoGAN (Gupta et al., 2021) and partially used in CryoPoseNet (Nashed et al., 2021), and a frequency-space neural network decoder FourierNet from cryoAI (Levy et al., 2022).

Original: The trainable parameters of D PI depend on the implementation.
Revised: The trainable parameters of D PI depend on the implementation.

Original: In VoxelGrid R , the trainable parameters are stored in
Revised: In VoxelGrid R , the trainable parameters are stored in

Original: a 3D tensor V ∈
Revised: a 3D tensor V ∈

Original: R L × L × L , which is a discrete representation of 3D reconstruction target in real space.
Revised: R L × L × L , which is a discrete representation of 3D reconstruction target in real space.

Original: In FourierNet , the trainable parameters are the network parameters for the two Sinusoidal Representation Networks (SIRENs) (Sitzmann et al., 2020) as detailed in cryoAI (Levy et al., 2022).
Revised: In FourierNet , the trainable parameters are the network parameters for the two Sinusoidal Representation Networks (SIRENs) (Sitzmann et al., 2020) as detailed in cryoAI (Levy et al., 2022).

Original: B B ENCHMARK DATASET
Revised: B B ENCHMARK DATASET

Original: To create the ground-truth volumes, we first generate simulated cryo-EM density maps based on published atomic structure coordinate (pdb) files using the molmap command from ChimeraX (Pettersen et al., 2021).
Revised: To create the ground-truth volumes, we first generate simulated cryo-EM density maps based on published atomic structure coordinate (pdb) files using the molmap command from ChimeraX (Pettersen et al., 2021).

Original: In each density map, each atom is described by a 3D Gaussian distribution with a width proportional to a chosen “resolution” parameter (6 ˚A for both proteins).
Revised: In each density map, each atom is described by a 3D Gaussian distribution with a width proportional to a chosen “resolution” parameter (6 ˚A for both proteins).

Original: Then, the groundtruth volumes were generated by re-sampling the simulated density maps onto cubic voxel grids with 128 pixels on each side.
Revised: Then, the groundtruth volumes were generated by re-sampling the simulated density maps onto cubic voxel grids with 128 pixels on each side.

Original: A set of projection images were generated using cryoSPARC (Punjani et al., 2017) with uniformly sampled projection orientations over the ground-truth volume.
Revised: A set of projection images were generated using cryoSPARC (Punjani et al., 2017) with uniformly sampled projection orientations over the ground-truth volume.

Original: To reduce the difference between the simulated projection images and the experimental ones, random translations and CTF were applied to each projection image.
Revised: To reduce the difference between the simulated projection images and the experimental ones, random translations and CTF were applied to each projection image.

Original: Random translations were sampled from a Gaussian distribution ( µ = 0 , σ = 20 ˚A).
Revised: Random translations were sampled from a Gaussian distribution ( µ = 0 , σ = 20 ˚A).

Original: [Table]
Revised: [Table]

Original: C V ISUALIZATION OF 2D IMAGES AND 3D RECONSTRUCTION
Revised: C V ISUALIZATION OF 2D IMAGES AND 3D RECONSTRUCTION

Original: D A BLATION STUDY OF THE PIP TASK
Revised: D A BLATION STUDY OF THE PIP TASK

Original: [Table]
Revised: [Table]

Original: E B ACKGROUND
Revised: E B ACKGROUND

Original: Cryo-EM is concerned with the inverse problem of inferring the 3D object associated with a set of projection images { Y i } i ∈D where D is index set for the image dataset.
Revised: Cryo-EM is concerned with the inverse problem of inferring the 3D object associated with a set of projection images { Y i } i ∈D where D is index set for the image dataset.

Original: Here we briefly introduce the image formation model in cryo-EM and relevant terminologies.
Revised: Here we briefly introduce the image formation model in cryo-EM and relevant terminologies.

Original: E. C RYO -EM IMAGE FORMATION MODEL
Revised: E. C RYO -EM IMAGE FORMATION MODEL

Original: Cryo-EM projection images are formed by collecting the electrons scattered by the atoms from macromolecules (e.g., protein molecules) in the sample embedded in thin ice.
Revised: Cryo-EM projection images are formed by collecting the electrons scattered by the atoms from macromolecules (e.g., protein molecules) in the sample embedded in thin ice.

Original: The raw cryo-EM data are usually referred to as micrographs, which consist of the 2D projections of hundreds of 3D objects.
Revised: The raw cryo-EM data are usually referred to as micrographs, which consist of the 2D projections of hundreds of 3D objects.

Original: In a so-called “particle picking” step, the projection images of individual 3D objects are cropped out of each micrograph and collected into a stack of projection images.
Revised: In a so-called “particle picking” step, the projection images of individual 3D objects are cropped out of each micrograph and collected into a stack of projection images.

Original: The 3D reconstruction target object can be defined as a function V for mapping a 3D coordinate to a real value.
Revised: The 3D reconstruction target object can be defined as a function V for mapping a 3D coordinate to a real value.

Original: [Equation]
Revised: [Equation]

Original: Each projection image is associated with a 3D object oriented in a certain direction.
Revised: Each projection image is associated with a 3D object oriented in a certain direction.

Original: Here we focus on the homogenous 3D reconstruction problem where the 3D objects corresponding to all the projection images are identical except for their orientations which can be modeled as a rotation matrix
Revised: Here we focus on the homogenous 3D reconstruction problem where the 3D objects corresponding to all the projection images are identical except for their orientations which can be modeled as a rotation matrix

Original: R i ∈ SO (3) ⊂ R 3 × 3 .
Revised: R i ∈ SO (3) ⊂ R 3 × 3 .

Original: Each 3D objection can be defined as follows, where the 3D coordinate ⃗r =
Revised: Each 3D objection can be defined as follows, where the 3D coordinate ⃗r =

Original: [ x, y, z ] T .
Revised: [ x, y, z ] T .

Original: [Equation]
Revised: [Equation]

Original: The projection image X i of object V i is described as integration along the projection direction, usually along the z-axis.
Revised: The projection image X i of object V i is described as integration along the projection direction, usually along the z-axis.

Original: [Equation]
Revised: [Equation]

Original: Due to the wavelike nature of electrons and the magnetic lens systems in the electron microscopes, the projection image signals X i are corrupted by optical interference effects which are modeled by the Point Spread Function (PSF)
Revised: Due to the wavelike nature of electrons and the magnetic lens systems in the electron microscopes, the projection image signals X i are corrupted by optical interference effects which are modeled by the Point Spread Function (PSF)

Original: f PSF in real space or the Contrast Transfer Function (CTF) in Fourier space (see the next section for details).
Revised: f PSF in real space or the Contrast Transfer Function (CTF) in Fourier space (see the next section for details).

Original: Furthermore, the cropping process of projection images for individual 3D objects from large micrographs is imperfect which can result in small translational image shifts.
Revised: Furthermore, the cropping process of projection images for individual 3D objects from large micrographs is imperfect which can result in small translational image shifts.

Original: Lastly, cryo-EM images are affected by noise from various sources (Baxter et al., 2009) which is assumed to be zero mean, uncorrelated, independently distributed (Penczek, 2010).
Revised: Lastly, cryo-EM images are affected by noise from various sources (Baxter et al., 2009) which is assumed to be zero mean, uncorrelated, independently distributed (Penczek, 2010).

Original: A common practice is to model it as a Gaussian noise
Revised: A common practice is to model it as a Gaussian noise

Original: ϵ i ∼ N (0 , σ 2 I ) .
Revised: ϵ i ∼ N (0 , σ 2 I ) .

Original: Taken together, the entire cryo-EM image formation model can be described by the following equation (Levy et al., 2022; Bendory et al., 2020) where Y i is the final projection image considering all factors discussed above, 2D coordinate ⃗p =
Revised: Taken together, the entire cryo-EM image formation model can be described by the following equation (Levy et al., 2022; Bendory et al., 2020) where Y i is the final projection image considering all factors discussed above, 2D coordinate ⃗p =

Original: [ x, y ] T , and 2D translation ⃗τ =
Revised: [ x, y ] T , and 2D translation ⃗τ =

Original: [∆ x, ∆ y ] T .
Revised: [∆ x, ∆ y ] T .

Original: [Equation]
Revised: [Equation]

Original: Since translation of a function by τ is equivalent to convolution with a shifted delta function:
Revised: Since translation of a function by τ is equivalent to convolution with a shifted delta function:

Original: [Equation]
Revised: [Equation]

Original: x ) , where δ τ := δ ( x + τ )
Revised: x ) , where δ τ := δ ( x + τ )

Original: We can rewrite the above equation as below.
Revised: We can rewrite the above equation as below.

Original: [Equation]
Revised: [Equation]

Original: The real-space VoxelGrid decoder used in this work was designed based on this image formation model.
Revised: The real-space VoxelGrid decoder used in this work was designed based on this image formation model.

Original: E. F OURIER SLICE THEOREM
Revised: E. F OURIER SLICE THEOREM

Original: An alternative and more computationally efficient method of calculating image projection is based on the Fourier slice theorem.
Revised: An alternative and more computationally efficient method of calculating image projection is based on the Fourier slice theorem.

Original: It states that the Fourier transformation of a projection image is the same as a slice of the Fourier transform of the corresponding 3D object.
Revised: It states that the Fourier transformation of a projection image is the same as a slice of the Fourier transform of the corresponding 3D object.

Original: The orientation of this slice is the same as the projection plane.
Revised: The orientation of this slice is the same as the projection plane.

Original: [Equation]
Revised: [Equation]

Original: Here ˆ Y i and ˆ V
Revised: Here ˆ Y i and ˆ V

Original: i are the Fourier transform of the projection image Y i and ˆ V i , respectively.
Revised: i are the Fourier transform of the projection image Y i and ˆ V i , respectively.

Original: S i is the volume slicing operator which is defined as a mapping from frequency coordinates ⃗k =
Revised: S i is the volume slicing operator which is defined as a mapping from frequency coordinates ⃗k =

Original: [ k x , k y , k z ] T on a rotated 2D Fourier plane (passing through the origin) to the value of ˆ V
Revised: [ k x , k y , k z ] T on a rotated 2D Fourier plane (passing through the origin) to the value of ˆ V

Original: i at that point in Fourier space.
Revised: i at that point in Fourier space.

Original: [Equation]
Revised: [Equation]

Original: Using the Fourier slice theorem, the cryo-EM image formation model in Fourier space can be simplified as follows, where 2D frequency coordinate ⃗q =
Revised: Using the Fourier slice theorem, the cryo-EM image formation model in Fourier space can be simplified as follows, where 2D frequency coordinate ⃗q =

Original: [ k x , k y ] T .
Revised: [ k x , k y ] T .

Original: [Equation]
Revised: [Equation]

Original: F CTF is the contrast transfer function which is the Fourier transform of the Point Spread Function (Penczek, 2010).
Revised: F CTF is the contrast transfer function which is the Fourier transform of the Point Spread Function (Penczek, 2010).

Original: ˆ ϵ is the Gaussian white noise in Fourier space.
Revised: ˆ ϵ is the Gaussian white noise in Fourier space.

Original: F ⃗τ is the phase-shift function defined below, which is equivalent to translation by ⃗τ in real space: F ⃗τ ( ⃗q ) :=
Revised: F ⃗τ is the phase-shift function defined below, which is equivalent to translation by ⃗τ in real space: F ⃗τ ( ⃗q ) :=

Original: e − 2 πi⃗τ · ⃗q .
Revised: e − 2 πi⃗τ · ⃗q .

Original: A majority of the cryo-EM 3D reconstruction algorithms are based on this theorem, such as RELION (Scheres, 2012), cryoSPARC (Punjani et al., 2017), cryoDRGN (Zhong et al., 2019; 2021a;b), and cryoAI (Levy et al., 2022).
Revised: A majority of the cryo-EM 3D reconstruction algorithms are based on this theorem, such as RELION (Scheres, 2012), cryoSPARC (Punjani et al., 2017), cryoDRGN (Zhong et al., 2019; 2021a;b), and cryoAI (Levy et al., 2022).