Original: Geo-SIC: Learning Deformable Geometric Shapes in Deep Image Classifiers
Revised: Geo-SIC: Learning Deformable Geometric Shapes in Deep Image Classifiers

Original: Anonymous Author(s)

Original: AffiliationAddress email

Original: Abstract
Revised: Abstract

Original: Deformable shapes provide important and complex geometric features of objectspresented in images.
Revised: Deformable shapes provide important and complex geometric features of objects presented in images.

Original: However, such information is oftentimes missing or underutilized as implicit knowledge in many image analysis tasks.
Revised: However, such information is oftentimes missing or underutilized as implicit knowledge in many image analysis tasks.

Original: This paper presents
Revised: This paper presents

Original: Geo-SIC, the first deep learning model to learn deformable shapes in a deformationspace for an improved performance of image classification.
Revised: Geo-SIC, the first deep learning model to learn deformable shapes in a deformation space for an improved performance of image classification.

Original: We introduce a newlydesigned framework that (i) simultaneously derives features from both image andlatent shape spaces with large intra-class variations; and (ii) gains increased modelinterpretability by allowing direct access to the underlying geometric features ofimage data.
Revised: We introduce a newly designed framework that (i) simultaneously derives features from both image and latent shape spaces with large intra-class variations; and (ii) gains increased model interpretability by allowing direct access to the underlying geometric features of image data.

Original: In particular, we develop a boosted classification network, equippedwith an unsupervised learning of geometric shape representations characterized bydiffeomorphic transformations within each class.
Revised: In particular, we develop a boosted classification network, equipped with an unsupervised learning of geometric shape representations characterized by diffeomorphic transformations within each class.

Original: In contrast to previous approachesusing pre-extracted shapes, our model provides a more fundamental approach bynaturally learning the most relevant shape features jointly with an image classifier.
Revised: In contrast to previous approaches using pre-extracted shapes, our model provides a more fundamental approach by naturally learning the most relevant shape features jointly with an image classifier.

Original: We demonstrate the effectiveness of our method on both simulated 2D imagesand real 3D brain magnetic resonance (MR) images.
Revised: We demonstrate the effectiveness of our method on both simulated 2D images and real 3D brain magnetic resonance (MR) images.

Original: Experimental results showthat our model substantially improves the image classification accuracy with anadditional benefit of increased model interpretability.
Revised: Experimental results show that our model substantially improves the image classification accuracy with an additional benefit of increased model interpretability.

Original: All code and data producedin this research will be made publicly available online.
Revised: Our code is publicly available at https://github.com/jw4hv/Geo-SIC

Original:  Introduction
Revised:  Introduction

Original: Deformable shapes have been identified to aid image classification for decades, as they capturegeometric features that describe changes and variability of objects with complex structures fromimages (5; 18; 26).
Revised: Deformable shapes have been identified to aid image classification for decades, as they capture geometric features that describe changes and variability of objects with complex structures from images (5; 18; 26).

Original: Bountiful literature demonstrates that the robustness of shapes to variationsin image intensity and texture (e g., noisy or corrupted data) makes it a reliable cue for imageanalysis tasks (29; 26; 41).
Revised: Bountiful literature demonstrates that the robustness of shapes to variations in image intensity and texture (e.g., noisy or corrupted data) makes it a reliable cue for image analysis tasks (31; 26; 43).

Original: For example, abnormal shape changes of anatomical structures are strongpredictors of serious diseases and health problems, e g., brain shrinkage caused by neurodegenerativedisorders (16; 47), irregular heart motions caused by cardiac arrhythmia (3; 8) and pulmonary edemacaused by infectious lung diseases (40; 42).
Revised: For example, abnormal shape changes of anatomical structures are strong predictors of serious diseases and health problems, e.g., brain shrinkage caused by neurodegenerative disorders (16; 49), irregular heart motions caused by cardiac arrhythmia (3; 8) and pulmonary edema caused by infectious lung diseases (42; 44).

Original: Existing methods have studied various representations ofgeometric shapes, including landmarks (7; 9; 14), point clouds (1), binary segmentations (10; 36),and medial axes (30).
Revised: Existing methods have studied various representations of geometric shapes, including landmarks (7; 9; 14), point clouds (1), binary segmentations (10; 38), and medial axes (32).

Original: A very recent research area in geometric deep learning (11; 31) has investigatedmathematical representations of shapes in the form of analytic graphs or points and then uses themto synthesize shapes.
Revised: A very recent research area in geometric deep learning (11; 33) has investigated mathematical representations of shapes in the form of analytic graphs or points and then uses them to synthesize shapes.

Original: These aforementioned techniques often ignore objects’ interior structures; hence do not capture the intricacies of complex objects in images.
Revised: These aforementioned techniques often ignore objects’ interior structures; hence do not capture the intricacies of complex objects in images.

Original: In contrast, deformation-basedshape representations (based on elastic deformations or fluid flows) focus on highly detailed shapeinformation from images (13; 33).
Revised: In contrast, deformation-based shape representations (based on elastic deformations or fluid flows) focus on highly detailed shape information from images (13; 35).

Original: With the underlying assumption that objects in many genericclasses can be described as deformed versions of an ideal template, descriptors in this class arisenaturally by matching the template to an input image.
Revised: With the underlying assumption that objects in many generic classes can be described as deformed versions of an ideal template, descriptors in this class arise naturally by matching the template to an input image.

Original: This procedure is also known as atlasbuilding (22; 38; 46).
Revised: This procedure is also known as atlas building (22; 40; 48).

Original: The resulting transformation is then consideredas a shape that reflects geometric  changes.
Revised: The resulting transformation is then considered a shape that reflects geometric 36th Conference on Neural Information Processing Systems (NeurIPS 2022). changes.

Original: In this paper, we will feature deformation-based shape representations thatoffer more flexibility in describing shape changes and variability of complex structures.
Revised: In this paper, we will feature deformation-based shape representations that offer more flexibility in describing shape changes and variability of complex structures.

Original: However,our developed framework can be easily adapted to other types of representations, including thosecharacterized by landmarks, binary segmentations, curves, and surfaces.
Revised: However, our developed framework can be easily adapted to other types of representations, including those characterized by landmarks, binary segmentations, curves, and surfaces.

Original: Inspired by the advantages of incorporating shape information in image analysis tasks, current deeplearning-based classification networks have been mostly successful in using pre-extracted shapesfrom images (43; 27; 7).
Revised: Inspired by the advantages of incorporating shape information in image analysis tasks, current deep learning-based classification networks have been mostly successful in using pre-extracted shapes from images (45; 29; 7).

Original: However, these methods require preprocessed shape data and oftentimesachieve a suboptimal solution in identifying shape features that are most representative to differentiatedifferent classes of images.
Revised: However, these methods require preprocessed shape data and oftentimes achieve a suboptimal solution in identifying shape features that are most representative to differentiate different classes of images.

Original: An explicit learning of deformable shapes in deep image classifiershas been missing.
Revised: An explicit learning of deformable shapes in deep image classifiers has been missing.

Original: This limits the power of classification models where quantifying and analyzinggeometric shapes is critical.
Revised: This limits the power of classification models where quantifying and analyzing geometric shapes is critical.

Original: In this paper, we introduce a novel deep learning image classification model, named as Geo-SIC,that jointly learns deformable shapes in a multi-template deformation space.
Revised: In this paper, we introduce a novel deep learning image classification model, named as Geo-SIC, that jointly learns deformable shapes in a multi-template deformation space.

Original: More specifically,
Revised: More specifically,

Original: Geo-SIC provides an unsupervised learning of deformation-based shape representations via a newlydesigned sub-network of atlas building.
Revised: Geo-SIC provides an unsupervised learning of deformation-based shape representations via a newly designed sub-network of atlas building.

Original: Different from previous deep learning based atlas buildingapproaches (15; 20), we employ an efficient parameterization of deformations in a low-dimensional
Revised: Different from previous deep learning based atlas building approaches (15; 20), we employ an efficient parameterization of deformations in a low-dimensional

Original: Fourier space (44) to speed up the training inference.
Revised: Fourier space (46) to speed up the training inference.

Original: The major contribution of Geo-SIC is threefolds:(
Revised: The major contribution of Geo-SIC is three folds:

Original: i)
Revised: (i)

Original: In contrast to previous approaches treating shapes as preprocessed objects from 
Revised: In contrast to previous approaches treating shapes as preprocessed objects from images, Geo-SIC provides a more fundamental approach by merging shape features naturally in the learning process of classification.

Original: images

Original: for classification

Original: tasks, Geo

Original: -SIC provides a more fundamental approach by merging shapefeatures naturally in the learning process
Revised: To the best of our knowledge, Geo-SIC was the first to learn deformation-based shape descriptors within an image classifier.

Original:  of classification To the best
Revised: It provides an image distance function of both intensity and geometric changes that are most relevant to classify different groups.

Original: of

Original: our knowledge, weare

Original: the

Original: first to unify these related tasks

Original: in a joint learning framework. (ii) Geo-SIC performs a simultaneous feature extraction from both image and learned shape spaces.
Revised: (ii) Geo-SIC performs a simultaneous feature extraction from both image and learned shape spaces.

Original: With these integrated features, Geo-SIC achieves an improved accuracy  of image classification.
Revised: With these integrated features, Geo-SIC achieves an improved accuracy and robustness of image classification.

Original: An additional benefit of Geo-SIC has an increased model interpretability because of the direct access to  underlying geometric features of image data.
Revised: An additional benefit of Geo-SIC is increased model interpretability because of its access to the underlying geometric features of image data.

Original: (iii) Geo-SIC provides an efficient sub-network of atlas building in a low dimensional shape space.
Revised: (iii) Geo-SIC provides an efficient geometric learning network via atlas building in a compact and low-dimensional shape space.

Original: This reduces the computational complexity of model training, especiallyfor high-
Revised: This reduces the computational complexity of model training in atlas building, especially

Original: dimensional image data (i.e., 3D brain MR images of large size).
Revised: for high-dimensional image data (i.e., 3D brain MRIs).

Original: We demonstrate the effectiveness of Geo-SIC on both synthetic 2D images and real 3D brain MRimages.
Revised: We demonstrate the effectiveness of Geo-SIC on both synthetic 2D images and real 3D brain MR images.

Original: Experimental results show that our model substantially improves the classification accuracycompared to a wide variety of models without jointly learned geometric features.
Revised: Experimental results show that our model substantially improves the classification accuracy compared to a wide variety of models without jointly learned geometric features.

Original: We then visualizethe class activation maps by using gradient-weighted class activation mapping (Grad-CAM) (34).
Revised: We then visualize the class activation maps by using gradient-weighted class activation mapping (Grad-CAM) (36).

Original: The highlighted regions show that Geo-SIC attracts more attention to geometric shape features thatpositively contribute to the accuracy of classifiers.
Revised: The highlighted regions show that Geo-SIC attracts more attention to geometric shape features that positively contribute to the accuracy of classifiers.

Original:  Background: Deformation-based Shape Representations via Atlas Building
Revised:  Background: Deformation-based Shape Representations via Atlas Building

Original: This section briefly reviews the concept of atlas building (22), which is commonly used to derivedeformation-based shape representations from images.
Revised: This section briefly reviews the concept of atlas building (22), which is commonly used to derive deformation-based shape representations from images.

Original: With the underlying assumption that thegeometric information in the deformations conveys a shape, descriptors in this class arise naturallyby matching a template to an input image with smoothness constraints on the deformation field.
Revised: With the underlying assumption that the geometric information in the deformations conveys a shape, descriptors in this class arise naturally by matching a template to an input image with smoothness constraints on the deformation field.

Original: Given a number of N images { I 1 , · · · , I N } , the problem of atlas building is to find a mean ortemplate image I and deformation fields ϕ 1 , · · · , ϕ N that minimize the energy function
Revised: Given a number of N images { I 1 , · · · , I N } , the problem of atlas building is to find a mean or template image I and deformation fields ϕ 1 , · · · , ϕ N that minimize the energy function

Original: [Equation]
Revised: [Equation]

Original: where σ 2 is a noise variance and ◦ denotes an interpolation operator that deforms image I with anestimated transformation ϕ n .
Revised: where σ 2 is a noise variance and ◦ denotes an interpolation operator that deforms image I with an estimated transformation ϕ n .

Original: The Dist ( · , · ) is a distance function that measures the dissimilaritybetween images, i.e., sum-of-squared differences (6), normalized cross correlation (4), and mutualinformation (39).
Revised: The Dist ( · , · ) is a distance function that measures the dissimilarity between images, i.e., sum-of-squared differences (6), normalized cross correlation (4), and mutual information (41).

Original: The Reg( · ) is a regularization that guarantees the smoothness of transformations.
Revised: The Reg( · ) is a regularization that guarantees the smoothness of transformations

Original: 2.1 Shape Representations In The Tangent Space Of Diffeomorphisms
Revised: 2.1 Shape Representations In The Tangent Space Of Diffeomorphisms

Original: In many applications, it is natural to require the deformation field to be a diffeomorphism, i.e., adifferentiable, bijective mapping with a differentiable inverse.
Revised: In many applications, it is natural to require the deformation field to be a diffeomorphism, i.e., a differentiable, bijective mapping with a differentiable inverse.

Original: Shape representations in the space ofdiffeomorphic transformations highlight a set of desirable features: (i) they capture large geometricvariations within image groups; (ii) the topology of objects in the image remains intact; and (iii) nonon-differentiable artifacts, such as creases or sharp corners, are introduced.
Revised: Shape representations in the space of diffeomorphic transformations highlight a set of desirable features: (i) they capture large geometric variations within image groups; (ii) the topology of objects in the image remains intact; and (iii) no non-differentiable artifacts, such as creases or sharp corners, are introduced.

Original: Moreover, a theoreticalframework of Large Deformation Diffeomorphic Metric Mapping (LDDMM) defines a metric in thespace of diffeomorphic transformations that in turn induces a distance metric on the shape space (6).
Revised: Moreover, a theoretical framework of Large Deformation Diffeomorphic Metric Mapping (LDDMM) defines a metric in the space of diffeomorphic transformations that in turn induces a distance metric on the shape space (6).

Original: Given an open and bounded d -dimensional domain Ω ⊂ R d , we use Diff(Ω) to denote a space ofdiffeomorphisms (i.e., a one-to-one smooth and invertible smooth transformation) and its tangentspace V = T Diff(Ω) .
Revised: Given an open and bounded d -dimensional domain Ω ⊂ R d , we use Diff(Ω) to denote a space of diffeomorphisms (i.e., a one-to-one smooth and invertible smooth transformation) and its tangent space V = T Diff(Ω) .

Original: The LDDMM algorithm (6) provides a distance metric in the deformationbased shape space, which is used as a regularization of atlas building in Eq.
Revised: The LDDMM algorithm (6) provides a distance metric in the deformationbased shape space, which is used as a regularization of atlas building in Eq.

Original: Such a distance metricis formulated as an integral of the Sobolev norm of the time-dependent velocity field v n ( t ) ∈ V ( t ∈[0 , 1]) in the tangent space, i.e.,
Revised: Such a distance metric is formulated as an integral of the Sobolev norm of the time-dependent velocity field v n ( t ) ∈ V ( t ∈ [0 , 1]) in the tangent space, i.e.,

Original: [Equation]
Revised: [Equation]

Original: with
Revised: with

Original: [Equation]
Revised: [Equation]

Original: where L : V → V ∗ is a symmetric, positive-definite differential operator that maps a tangent vectorv ( t ) ∈ V into its dual space as a momentum vector m ( t ) ∈ V ∗ .
Revised: where L : V → V ∗ is a symmetric, positive-definite differential operator that maps a tangent vector v ( t ) ∈ V into its dual space as a momentum vector m ( t ) ∈ V ∗ .

Original: We typically write m ( t ) =
Revised: We typically write m ( t ) =

Original: L v ( t ) ,or v ( t ) =
Revised: L v ( t ) , or v ( t ) =

Original: K m ( t ) , with K being an inverse operator of L .
Revised: K m ( t ) , with K being an inverse operator of L .

Original: In this paper, we adopt a commonly used
Revised: In this paper, we adopt a commonly used

Original: Laplacian operator L = ( − α ∆+ Id ) 3 , where α is a weighting parameter that controls the smoothnessof transformation fields and Id is an identity matrix.
Revised: Laplacian operator L = ( − α ∆+ Id ) 3 , where α is a weighting parameter that controls the smoothness of transformation fields and Id is an identity matrix.

Original: The ( · , · ) is a dual paring, which is similar toan inner product between vectors.
Revised: The ( · , · ) is a dual pairing, which is similar to an inner product between vectors.

Original: The operator D denotes a Jacobian matrix and · represents anelement-wise matrix multiplication.
Revised: The operator D denotes a Jacobian matrix and · represents an element-wise matrix multiplication.

Original: According to a well known geodesic shooting algorithm (37), the minimum of Eq.
Revised: According to a well known geodesic shooting algorithm (39), the minimum of Eq.

Original: (2) is uniquelydetermined by solving a Euler-Poincaré differential equation (EPDiff) (2; 25) with a given initialcondition.
Revised: (2) is uniquely determined by solving a Euler-Poincaré differential equation (EPDiff) (2; 25) with a given initial condition.

Original: This nicely proves that the deformation-based shape descriptor ϕ n can be fully characterized by an initial velocity field v n (0) , which lies in the tangent space of diffeomorphisms.
Revised: This nicely proves that the deformation-based shape descriptor ϕ n can be fully characterized by an initial velocity field v n (0) , which lies in the tangent space of diffeomorphisms.

Original: A recentresearch further identifies an equivalent but more efficient way to reparameterize these initial velocityfields in a low-dimensional bandlimited space (44; 45).
Revised: A recent research further identifies an equivalent but more efficient way to reparameterize these initial velocity fields in a low-dimensional bandlimited space (46; 47).

Original: Let (cid:103) Diff(Ω) and ˜ V denote the bandlimited space of diffeomorphisms and velocity fields respectively.
Revised: Let (cid:103) Diff(Ω) and ˜ V denote the bandlimited space of diffeomorphisms and velocity fields respectively.

Original: The EPDiff is reformulated in a complex-valued Fourier space with much less dimensions, i.e.,
Revised: The EPDiff is reformulated in a complex-valued Fourier space with much lower dimensions, i.e.,

Original: [Equation]
Revised: [Equation]

Original: where ⋆ is the truncated matrix-vector field auto-correlation.
Revised: where ⋆ is the truncated matrix-vector field auto-correlation.

Original: Here ˜ K is an inverse operator of ˜ L ,which is the Fourier transform of a Laplacian operator L .
Revised: Here ˜ K is an inverse operator of ˜ L , which is the Fourier transform of a Laplacian operator L .

Original: The operator ˜ D represents the Fourierfrequencies of a Jacobian matrix with central difference approximation, and ∗ is a circular convolutionwith zero padding to avoid aliasing 1 .
Revised: The operator ˜ D represents the Fourier frequencies of a Jacobian matrix with central difference approximation, and ∗ is a circular convolution with zero padding to avoid aliasing 1 .

Original: The operator ˜ ∇· is the discrete divergence of a vector field, and⊗ represents a tensor product between Fourier frequencies.
Revised: The operator ˜ ∇· is the discrete divergence of a vector field, and ⊗ represents a tensor product between Fourier frequencies.

Original: We are now ready to optimize the problem of atlas building (Eq.
Revised: We are now ready to optimize the problem of atlas building (Eq.

Original: (1)) with reduced computationalcomplexity in a low-dimensional bandlimited space as
Revised: (1)) with reduced computational complexity in a low-dimensional bandlimited space as

Original: [Equation]
Revised: [Equation]

Original: The deformation ϕ − 1 n corresponds to ˜ ϕ − 1 n in Fourier space via the Fourier transform F ( ϕ − 1 n ) =
Revised: The deformation ϕ − 1 n corresponds to ˜ ϕ − 1 n in Fourier space via the Fourier transform F ( ϕ − 1 n ) =

Original: ˜ ϕ − 1 n ,or its inverse ϕ − 1 n = F − 1 (˜ ϕ − 1 n ) .
Revised: ˜ ϕ − 1 n , or its inverse ϕ − 1 n = F − 1 (˜ ϕ − 1 n ) .

Original: Note that we will drop the time index t , i.e., ˜ v n (0) ∆ =
Revised: Note that we will drop the time index t , i.e., ˜ v n (0) ∆ =

Original: ˜ v n , forsimplified notations in next sections.
Revised: ˜ v n , for simplified notations in next sections.

Original:  Our Method: Geo-SIC
Revised:  Our Method: Geo-SIC

Original: In this section, we present a novel deep image classifier (Geo-SIC) that explicitly learns geometricshape representations for an improved performance of accuracy, as well as increased model interpretability.
Revised: In this section, we present a novel deep image classifier (Geo-SIC) that explicitly learns geometric shape representations for an improved performance of accuracy, as well as increased model interpretability.

Original: Geo-SIC consists of two modules: an unsupervised learning of geometric shapes via anatlas building network, and a boosted classification network that integrates features from both imageand learned shape spaces.
Revised: Geo-SIC consists of two modules: an unsupervised learning of geometric shapes via an atlas building network, and a boosted classification network that integrates features from both images and learned shape spaces.

Original: Details of our network architecture are introduced as follows.
Revised: Details of our network architecture are introduced as follows.

Original: Geometric shape learning based on an atlas building network.
Revised: Geometric shape learning based on an atlas building network.

Original: Let ( θ Eg , θ Dg ) be the parametersof an encoder-decoder in our geometric learning network.
Revised: Let ( θ Eg , θ Dg ) be the parameters of an encoder-decoder in our geometric learning network.

Original: Consider a number of J image classes,there exists a number of N j , j ∈ { 1 , . .
Revised: Consider a number of J image classes, there exists a number of N j , j ∈ { 1 , . .

Original: , J } images in each class.
Revised: , J } images in each class.

Original: Our atlas building network willlearn the shape representations, also known as initial velocity fields ˜ v n ( θ Eg , θ Dg ) , n ∈ { 1 , · · · , N j } ,with an updated atlas I j .
Revised: Our atlas building network will learn the shape representations, also known as initial velocity fields ˜ v n ( θ Eg , θ Dg ) , n ∈ { 1 , · · · , N j } , with an updated atlas I j .

Original: We adopt the architecture of UNet (32) in this work, however, other networkstructures such as UNet ++ (48) and TransUNet (12) can be easily applied.
Revised: We adopt the architecture of UNet (34) in this work, however, other network structures such as UNet ++ (50) and TransUNet (12) can be easily applied.

Original: Boosted image classification network.
Revised: Boosted image classification network.

Original: Let θ Ec be the parameters of an encoder that extractsfeatures from image spaces.
Revised: Let θ Ec be the parameters of an encoder that extracts features from image spaces.

Original: We develop a feature fusion module that integrates geometric shape andimage features in a latent space parameterized by θ gc ( θ E g , θ E c ) .
Revised: We develop a feature fusion module that integrates geometric shape and image features in a latent space parameterized by θ gc ( θ E g , θ E c ) .

Original: This boosted classification networkwill predict a class label y nj ( θ gc ) for each input image.
Revised: This boosted classification network will predict a class label y nj ( θ gc ) for each input image.

Original: Network loss.
Revised: Network loss.

Original: The loss function of Geo-SIC includes the loss from both geometric learning andclassification network.
Revised: The loss function of Geo-SIC includes the loss from both geometric learning and classification network.

Original: Given a set of image class labels ˆ y nj , we define Θ =
Revised: Given a set of image class labels ˆ y nj , we define Θ =

Original: ( θ Eg , θ Dg , θ Ec , θ gc ) for allnetwork parameters and formulate the total loss of Geo-SIC as
Revised: ( θ Eg , θ Dg , θ Ec , θ gc ) for all network parameters and formulate the total loss of Geo-SIC as

Original: [Equation]
Revised: [Equation]

Original: where reg ( · ) is a regularity term on the network parameters.
Revised: where reg ( · ) is a regularity term on the network parameters.

Original: An overview of our proposed Geo-SIC network is shown in Fig.
Revised: An overview of our proposed Geo-SIC network is shown in Fig.

Original: [Figure]
Revised: [Figure]

Original: 3.1 Alternating Optimization For Network Training
Revised: 3.1 Alternating Optimization For Network Training

Original: We develop an alternating optimization scheme (28) to minimize the network loss defined in Eq.
Revised: We develop an alternating optimization scheme (30) to minimize the network loss defined in Eq.

Original: All parameters are optimized jointly by alternating between the training of geometric shape learningand image classification networks.
Revised: All parameters are optimized jointly by alternating between the training of geometric shape learning and image classification networks.

Original: Training of geometric shape learning via atlas building.
Revised: Training of geometric shape learning via atlas building.

Original: In contrast to current approaches thatparameterize deformation-based shapes in a high-dimensional image space (20; 15), our modelemploys an efficient reparameterization in a low-dimensional bandlimited space (44).
Revised: In contrast to current approaches that parameterize deformation-based shapes in a high-dimensional image space (20; 15), our model employs an efficient reparameterization in a low-dimensional bandlimited space (46).

Original: This makes thecomputation of geodesic constraints (Eq.
Revised: This makes the computation of geodesic constraints (Eq.

Original: (2) and Eq.
Revised: (2) and Eq.

Original: (3)) required in the loss function substantiallyfaster in each forward/backward propagation during the training process.
Revised: (3)) required in the loss function substantially faster in each forward/backward propagation during the training process.

Original: The loss of training thesub-module of atlas building network is
Revised: The loss of training in the sub-module of atlas building network is

Original: [Equation]
Revised: [Equation]

Original: Similar to (20; 15), we treat atlas I j as a network parameter and update it accordingly.
Revised: Similar to (20; 15), we treat atlas I j as a network parameter and update it accordingly.

Original: To guaranteethe network optimization stays in the tangent space of diffeomorphisms, we pull back the networkgradient with regard to initial velocity fields by backward integrating adjoint jacobi fields (44) eachtime after the forward pass.
Revised: To guarantee the network optimization stays in the tangent space of diffeomorphisms, we pull back the network gradient with regard to initial velocity fields by backward integrating adjoint jacobi fields (46) each time after the forward pass.

Original: More details are included in the supplementary materials.
Revised: More details are included in the supplementary materials.

Original: Training of boosted image classifier.
Revised: Training of boosted image classifier.

Original: As highlighted in  red box in Fig.
Revised: As highlighted in the red box in Fig.

Original: 1, Geo-SIC extracts bothimage and geometric features and then integrate them into a feature fusion block.
Revised: 1, Geo-SIC extracts both images and geometric features and then integrates them into a feature fusion block.

Original: This boostedclassifier is optimized over the loss defined as
Revised: This boosted classifier is optimized over the loss defined as

Original: [Equation]
Revised: [Equation]

Original: A summary of our joint learning of Geo-SIC through an alternating optimization is in Alg.1.
Revised: A summary of our joint learning of Geo-SIC through an alternating optimization is in Alg.

Original:  Experimental Evaluation 
Revised:  Experimental Evaluation 4.1 Data

Original: 4.1 Data

Original: 2D data set.
Revised: 2D data set.

Original: We choose 50000 images (including five classes, circle, cloud, envelope, square, andtriangle are shown in Fig 3) of Google Quickdraw dataset (21), a collection of categorized drawingscontributed by online players in a drawing game.
Revised: We choose 50000 images (including five classes, circle, cloud, envelope, square, and triangle are shown in Fig 3) of Google Quickdraw dataset (21), a collection of categorized drawings contributed by online players in a drawing game.

Original: We run affine transformation within each class asprepossessing and upsample each image from 28 × 28 to 100 × 100 .3D
Revised: We run affine transformation within each class as prepossessing and upsample each image from 28 × 28 to 100 × 100 .

Original:  brain MRI.
Revised: 3D brain MRI.

Original: For brain data, we include 373 public T1-weighted brain MRI scans from Open
Revised: For brain data, we include 373 public T1-weighted brain MRI scans from Open

Original: Access Series of Imaging Studies (OASIS) (17).
Revised: Access Series of Imaging Studies (OASIS) (17).

Original: All 150 subjects are aged from 60 to 96 with
Revised: All 150 subjects are aged from 60 to 96 with

Original: Alzheimer’s disease diagnosis (79 cases for dementia and 71 cases for non-demented).
Revised: Alzheimer’s disease diagnosis (79 cases for dementia and 71 cases for non-demented).

Original: All MRIswere all pre-processed as 256 × 256 × 256 , 1 . 25 mm 3 isotropic voxels, and underwent skull-stripped,intensity normalized, bias field corrected, and pre-aligned with affine transformation.
Revised: All MRIs were all pre-processed as 256 × 256 × 256 , 1 . 25 mm 3 isotropic voxels, and underwent skull-stripped, intensity normalized, bias field corrected, and pre-aligned with affine transformation.

Original: 4.2 Experiments
Revised: 4.2 Experiments

Original: Classification evaluation.
Revised: Classification evaluation.

Original: We demonstrate the effectiveness of our model on both 2D synthetic dataand 3D brain MRI scans.
Revised: We demonstrate the effectiveness of our model on both 2D synthetic data and 3D brain MRI scans.

Original: We select four classification backbones (AlexNet (23), a five-layer CNN,
Revised: We select four classification backbones (AlexNet (23), a five-block 3D CNN,

Original: ResNet18 (19), and VGG19 (35)) as baseline methods.
Revised: ResNet18 (19), and VGG19 (37)) as baseline methods.

Revised: For CNN, we use a 3D convolutional layer with a 5 × 5 × 5 convolutional kernel size, a batch normalization (BN) layer with activation functions (PReLU or ReLU), and a 2 × 2 × 2 3D max-pooling in each CNN block.

Original: For a fair comparison, we show the resultsof Geo-SIC by replacing the backbone in our model with all baselines (named asGeo-SIC:Alex,
Revised: For a fair comparison, we show the results of Geo-SIC by replacing the backbone in our model with all baselines (named as

Original:  Geo-SIC:CNN, Geo-SIC:Res, and Geo-SIC:VGG).
Revised: Geo-SIC:Alex, Geo-SIC:CNN, Geo-SIC:Res, and Geo-SIC:VGG).

Original: To further investigate the advantages of our joint learning, we compare with two-step approaches(Two-step Alex, Two-step CNN, Two-step Res, and Two-step VGG), where the geometric learningnetwork is treated as a preprocessing step for geometric feature extraction.
Revised: To further investigate the advantages of our joint learning, we compare with two-step approaches (Two-step Alex, Two-step CNN, Two-step Res, and Two-step VGG), where the geometric learning network is treated as a preprocessing step for geometric feature extraction.

Original: We report the averageaccuracy, F1-score, AUC, sensitivity, and specificity for all methods.
Revised: We report the average accuracy, F1-score, AUC, sensitivity, and specificity for all methods.

Original: We also show the receiveroperating characteristic (ROC) curves by plotting the true positive rate (TPR) against the false positiverate (FPR) at various threshold settings.
Revised: We also show the receiver operating characteristic (ROC) curves by plotting the true positive rate (TPR) against the false positive rate (FPR) at various threshold settings.

Revised: Robustness and interpretability.

Revised: We demonstrate the robustness of Geo-SIC to variations in image intensity by adding different scales of universal adversarial noises (28; 27) in both 2D and 3D images.

Revised: We adopt an iterative algorithm (27) that computes the universal perturbations to send perturbed images outside of the decision boundary of the classifier while fooling most images without visibly changing image data.

Revised: We then compare the image classification accuracy for the baseline (selected from the best performance in backbones) and our model.

Original: To better understand the model interpretability in terms of network attention of all models, wevisualize the Grad-CAM (34) of the last neural network layer.
Revised: To better understand the model interpretability in terms of network attention of all models, we visualize the Grad-CAM (36) of the last neural network layer.

Original: Atlas evaluation.
Revised: Atlas evaluation.

Original: We also evaluate the performance of our newly designed atlas building network,which serves as a core part of Geo-SIC.
Revised: We also evaluate the performance of our newly designed atlas building network, which serves as a core part of Geo-SIC.

Original: We compare the estimated atlas of Geo-SIC that achieves thehighest AUC with three atlas-building methods: a diffeomorphic autoencoder (20) (LagoMorph), adeep learning based conditional template estimation method (15) (Con-Temp) and a Bayesian atlasbuilding framework with hyper priors (38) (Hier-Baye).
Revised: We compare the estimated atlas of Geo-SIC that achieves the highest AUC with three atlas-building methods: a diffeomorphic autoencoder (20) (LagoMorph), a deep learning based conditional template estimation method (15) (Con-Temp) and a Bayesian atlas building framework with hyper priors (40) (Hier-Baye).

Original: We report the total computational time and memory consumption on 3D real brain images.
Revised: We report the total computational time and memory consumption on 3D real brain images.

Original: To measurethe sharpness of estimated atlas I , we adopt a metric of normalized standard deviation computedfrom randomly selected 4000 image patches (24).
Revised: To measure the sharpness of estimated atlas I , we adopt a metric of normalized standard deviation computed from randomly selected 4000 image patches (24).

Original: Given M ( i ) , a patch around a voxel i of an atlas I ,the local measure of the sharpness at voxel i is defined as sharpness ( I ( i ))
Revised: Given M ( i ) , a patch around a voxel i of an atlas I , the local measure of the sharpness at voxel i is defined as sharpness ( I ( i ))

Original: =
Revised: =

Original: sd M ( i ) ( I ) / avg M ( i ) ( I ) ,where sd and avg denote the standard deviation and the mean of M i .
Revised: sd M ( i ) ( I ) / avg M ( i ) ( I ) , where sd and avg denote the standard deviation and the mean of M i .

Original: Parameter Setting.
Revised: Parameter Setting.

Original: We set an optimal dimension of the low-dimensional shape representation as16 2 for 2D dataset and 32 3 for 3D dataset.
Revised: We set an optimal dimension of the low-dimensional shape representation as 16 2 for 2D dataset and 32 3 for 3D dataset.

Original: We set parameter α = 3 for the operator ˜ L , the number oftime steps for Euler integration in EPDiff (Eq.
Revised: We set parameter α = 3 for the operator ˜ L , the number of time steps for Euler integration in EPDiff (Eq.

Original: as 10 .
Revised: as 10 .

Original: We set the noise variance σ = 0 .
Revised: We set the noise variance σ = 0 .

Original: We setthe batch size as 16 and use the cosine annealing learning rate schedule that starts from a learning rateη = 1 e − 3 for network training.
Revised: We set the batch size as 16 and use the cosine annealing learning rate schedule that starts from a learning rate η = 1 e − 3 for network training.

Original: We run 1000 epochs with the Adam optimizer and save networkswith the best validation performance for all models.
Revised: We run 1000 epochs with the Adam optimizer and save networks with the best validation performance for all models.

Original: All networks are trained with an i7, 9700K CPUwith 32 GB internal memory.
Revised: All networks are trained with an i7, 9700K CPU with 32 GB internal memory.

Original: The training and prediction procedure of all learning-based methodsare performed on four Nvidia GTX 2080Ti GPUs.
Revised: The training and prediction procedure of all learning-based methods are performed on four Nvidia GTX 2080Ti GPUs.

Original: For both 2D and 3D datasets, we split the imagesby using 70% as training images, 15% as validation images, and 15% as testing images.
Revised: For both 2D and 3D datasets, we split the images by using 70% as training images, 15% as validation images, and 15% as testing images.

Original: 4.3 Results 
Revised: 4.3 Results 4.3.1 Classification on 2D synthetic images

Original: 4.3.1 Classification on 2D synthetic images

Original: Table.
Revised: Table.

Original: 1 reports the classification performance across all methods over model accuracy and five microaveraged evaluation metrics.
Revised: 1 reports the classification performance across all methods over model accuracy and five microaveraged evaluation metrics.

Original: Our method achieves the highest model accuracy with comparablemicro-averaged AUC, F-1 score, precision, sensitivity, and specificity.
Revised: Our method achieves the highest model accuracy with comparable micro-averaged AUC, F-1 score, precision, sensitivity, and specificity.

Original: Compare to all baselinesincluding two-step approaches, Geo-SIC achieves the best classification performance.
Revised: Compare to all baselines including two-step approaches, Geo-SIC achieves the best classification performance.

Original: Fig.

Original: 2 visualizes the micro-averages ROC curve for baselines and our proposed Geo-SIC.
Revised: 2 visualizes the micro-averages ROC curve for baselines and our proposed Geo-SIC.

Original: For foursets of comparisons, all curves produced by our classifiers are with a larger AUC and are closer to thetop-left corner, which indicates a better classification performance.Fig.
Revised: For four sets of comparisons, all curves produced by our classifiers are with a larger AUC and are closer to the top-left corner, which indicates a better classification performance.

Original: 3 (left) displays the heat maps overlaid with 2D QuickDraw data for all methods.

Original: Geo-SICproduces more explainable heat maps that are geometrically aligned with the original shapes.

Original: Itindicates that the latent shape space attract more attention to geometric features that positivelycontribute to the performance of classifiers.

Original: Fig.

Original: 3 (right) visualizes a comparison of atlases generatedby baselines and Geo-SIC.

Original: It shows that Geo-SIC produces image atlases with the best visual quality,e.g., clearer circle/cloud edges, sharper triangle/square corners.

Original: [Table]
Revised: [Table]

Original: [Figure]

Original:  P o s i t i v e
Revised: ue P o s i t i v e

Original: R a t e
Revised: R a t e

Original: T r ue P o s i
Revised: T r ue P o s i

Original: t i v e
Revised: t i v e

Original: R a t e
Revised: R a t e

Original: T r ue P o s i
Revised: T r ue P o s i

Original: t i v e
Revised: t i v e

Original: R a t e
Revised: R a t e

Revised: Fig.

Revised: 3 (left) displays the heat maps overlaid with 2D QuickDraw data for all methods.

Revised: Geo-SIC produces more explainable heat maps that are geometrically aligned with the original shapes.

Revised: It indicates that the latent shape space attracts more attention to geometric features that positively contribute to the performance of classifiers.

Original:  [Figure]
Revised: Fig. 3 (right) visualizes a comparison of atlases generated by baselines and Geo-SIC. It shows that Geo-SIC produces image atlases with the best visual quality, e.g., clearer circle/cloud edges, and sharper triangle/square corners. [Figure]

Original: 4.3.2 Classification on 3D real brain MRIs
Revised: 4.3.2 Classification on 3D real brain MRIs

Original: Table.
Revised: Table.

Original: 2 reports the model performance for brain images.
Revised: 2 reports the model performance for brain images.

Original: Our method Geo-SIC outperforms allbaselines with comparable accuracy, AUC, F1-score, precision, sensitivity, and specificity. It showsthat Geo-SIC substantially improves the classification performance with  reduced misclassificationrate.
Revised: Our method Geo-SIC outperforms all baselines with comparable accuracy, AUC, F1-score, precision, sensitivity, and specificity that Geo-SIC substantially improves the classification performance with a reduced misclassification rate. [Table]

Original: Fig.
Revised: Fig.

Original: 4 visualizes the ROC curves for all algorithms.
Revised: 4 visualizes the ROC curves for all algorithms.

Original: For four sets of comparisons, our boostedclassifiers offer curves closer to the top-left corner with higher AUC values.
Revised: For four sets of comparisons, our boosted classifiers offer curves closer to the top-left corner with higher AUC values.

Original: It shows that Geo-SIChas better performance in distinguishing between healthy control and Alzheimer’s diseases groups. 
Revised: It shows that Geo-SIC has better performance in distinguishing between healthy control and Alzheimer’s disease groups. T r ue P o s i t i v e R a t e

Original: [Table]
Revised: False Positive Rate False Positive Rate False Positive Rate False Positive Rate

Original:  P o s i t i v e R a t e
Revised: T r ue P o s i t i v e R a t e

Original: T r ue
Revised: T r ue

Original: P o s i t i v e
Revised: P o s i t i v e

Original: R a t e
Revised: R a t e

Original: T r ue
Revised: T r ue

Original: P o s i t i v e
Revised: P o s i t i v e

Original: R a t e
Revised: R a t e

Revised: 4.3.3 Robustness and interpretability

Original: Fig.
Revised: Fig.

Revised: 5 shows that Geo-SIC consistently achieves better classification accuracy ( ∼ 10% higher in average ) than baseline algorithms (i.e., VGG19) across different levels of adversarial attacks (i.e., ϵ = 5 e − 3 , 5 e − 2 , 5 e − 1 ) on image intensity.

Revised: This indicates that our model is able to improve the classification model robustness by providing jointly learned geometric features.

Original: 5 displays the heat maps (produced by Grad-CAM) overlaid with brain MRIs.
Revised: [Figure] Fig. 6 displays the heat maps (produced by Grad-CAM) overlaid with brain MRIs.

Original: Our generatedheat maps are fairly aligned with brain structures (e.g., ventricle), which are critical to Alzheimer’sdiseases diagnosis.
Revised: Our generated heat maps are fairly aligned with brain structures (e.g., ventricle), which are critical to Alzheimer’s disease diagnosis.

Original: It profiles the fact that the latent geometric features guide the neural networks tocapture the most anatomically meaningful brain regions for distinguishing healthy vs. disease groups.
Revised: It profiles the fact that the latent geometric features guide the neural networks to capture the most anatomically meaningful brain regions for distinguishing healthy vs. disease groups.

Original: For both dementia and non-dementia cases, the heat maps of Geo-SIC are more explainable.
Revised: For both dementia and non-dementia cases, the heat maps of Geo-SIC are more explainable.

Original: Thisshows that our model has great clinical potential for better analysis of Alzheimer’s diseases.
Revised: This shows that our model has great clinical potential for better analysis of Alzheimer’s disease. N on - de m en t ed c a s e s D e m en t i a c a s e s [Figure]

Revised: 4.3.4 Atlas of 3D images

Original: Fig.
Revised: Fig.

Original: 6 (top) visualizes a comparison of the atlas on real brain MRI scans.
Revised: 7 (top) visualizes a comparison of the atlas on real brain MRI scans.

Original: Our model significantlyimproves the  quality of the atlas with sharper details.
Revised: With the benefits of reparameterizing the deformation fields in a low-dimensional bandlimited space (46), our model obtains better quality of the atlas with sharper details.

Original: More specifically, Geo-SIC offers  better brainatlas with clearer anatomical structures, e.g., ventricle, grey and white matter.
Revised: More specifically, Geo-SIC offers a better brain atlas with clearer anatomical structures, e.g., ventricle, grey and white matter.

Original: Fig.
Revised: Fig.

Original: 6 (bottom left)quantitatively reports the sharpness metric of all methods.
Revised: 7 (bottom left) quantitatively reports the sharpness metric of all methods.

Original: Fig.
Revised: Fig.

Original: 6 (bottom right) shows the comparisonof computational time and memory consumption across all methods.
Revised: 7 (bottom right) shows the comparison of computational time and memory consumption across all methods.

Original: Although Con-Temp is slightlyfaster than Geo-SCI due to a very different parameterization of velocity fields (stationary velocityrather than the time-dependent velocity in other methods), it achieves a less sharper atlas and stillrequires larger memory consumption than our model.
Revised: Although Con-Temp is slightly faster than Geo-SCI due to a very different parameterization of velocity fields (stationary velocity rather than the time-dependent velocity in other methods), it achieves a less sharp atlas and still requires larger memory consumption than our model.

Original: Comparing with the other methods (Lagomorphand Hier-Baye) that employs time-dependent velocity fields, Geo-SIC substantially reduces the timeconsumption and memory consumption.
Revised: Compared with the other methods (Lagomorph and Hier-Baye) that employ time-dependent velocity fields, Geo-SIC substantially reduces time consumption and memory consumption.

Original:  Conclusions & Discussion
Revised:  Conclusions & Discussion

Original: This paper presents a novel deep learning model, named as Geo-SIC, that for the first time incorporatesdeformable geometric shape learning into deep image classifiers.
Revised: This paper presents a novel deep learning model, named as Geo-SIC, that for the first time incorporates deformable geometric shape learning into deep image classifiers.

Original: We jointly learn a boosted classifierwith an unsupervised shape learning network via atlas building.
Revised: We jointly learn a boosted classifier with an unsupervised shape learning network via atlas building.

Original: To achieve this goal, we define a newjoint loss function with an alternating optimization scheme.
Revised: To achieve this goal, we define a new joint loss function with an alternating optimization scheme.

Original: An additional benefit is that Geo-SICprovides an efficient shape representations in a low-dimensional bandlimited space.
Revised: An additional benefit is that Geo-SIC provides efficient shape representations in a low-dimensional bandlimited space.

Original: Experimentalresults on both 2D synthetic data and 3D brain MRI scans show that our model gains an improvedclassification performance while producing  sharper atlas with better visual quality.
Revised: Experimental results on both 2D synthetic data and 3D brain MRI scans show that our model gains an improved classification performance while producing a sharper atlas with better visual quality.

Original: In addition,compared with the state-of-the-arts, our model is more explainable in terms of interpreting thenetwork attention on geometric features.
Revised: In addition, [Figure] compared with the state-of-the-arts, our model is more explainable in terms of interpreting the network attention on geometric features.

Original: The theoretical tools developed in this paper are generic to awide variety of combinations of shape representations and classification backbones.
Revised: The theoretical tools developed in this paper are generic to a wide variety of combinations of shape representations and classification backbones.

Original: Geo-SIC not onlyhas a great potential to impact clinically diagnostic routine, such as Alzheimer’s disease detection, or N on - d e m en t ed c a s e s D e m en t i a c a s e s AlexNet
Revised: Geo-SIC not only has a great potential to impact clinically diagnostic routines, such as Alzheimer’s disease detection, or post-treatment for patient care, but also bridges the gap between the developed deformable shape learning theories and classification-based applications.

Original: [Figure]

Original: Res

Original: [Figure] post-treatment for patient care, but also bridges the gap between the developed deformable shapelearning theories and classification-based applications. A promising future workis to incorporateimages with missing data values that are caused by corrupted signals [Figure]
Revised: Future work to extend our Geo-SIC can be (i) modeling multiple templates within each class to capture multimodal image distributions, and (ii) incorporating images with missing data values that are caused by occlusions, or appearance changes such as tumor growth.

Original: or appearance changes such asbrain tumor resection, into our framework.

Original: Checklist
Revised: Checklist

Original: For all authors... (a) Do the main claims made in the abstract and introduction accurately reflect the paper’scontributions and scope?
Revised: For all authors... (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope?

Original: [Yes] .
Revised: [Yes] .

Original: (b) Did you describe the limitations of your work?
Revised: (b) Did you describe the limitations of your work?

Original: [No] (c) Did you discuss any potential negative societal impacts of your work?[No]
Revised: [Yes] (c) Did you discuss any potential negative societal impacts of your work?

Original:  (d) Have you read the ethics review guidelines and ensured that your paper conforms tothem?
Revised: [No] (d) Have you read the ethics review guidelines and ensured that your paper conforms to them?

Original: [Yes] 
Revised: [Yes] 2.

Original: If you are including theoretical results... (a) Did you state the full set of assumptions of all theoretical results?
Revised: If you are including theoretical results... (a) Did you state the full set of assumptions of all theoretical results?

Original: [Yes] (b) Did you include complete proofs of all theoretical results?
Revised: [Yes] (b) Did you include complete proofs of all theoretical results?

Original: [Yes] 
Revised: [Yes] 3.

Original: If you ran experiments...
Revised: If you ran experiments...

Original: (a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?
Revised: (a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?

Original: [Yes]

Original:  (b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?
Revised: [Yes] (b) Did you specify all the training details (e g., data splits, hyperparameters, how they were chosen)?

Original: [Yes] (c) Did you report error bars (e
Revised: [Yes] (c) Did you report error bars (e

Original: g., with respect to the random seed after running experiments multiple times)?
Revised: g., with respect to the random seed after running experiments multiple times)?

Original: [Yes]
Revised: [Yes]

Original: (d) Did you include the total amount of compute and the type of resources used (e g., typeof GPUs, internal cluster, or cloud provider)?
Revised: (d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?

Original: [Yes] 
Revised: [Yes] 4.

Original: If you are using existing assets (e
Revised: If you are using existing assets (e

Original: g., code, data, models) or curating/releasing new assets... (a) If your work uses existing assets, did you cite the creators?
Revised: g., code, data, models) or curating/releasing new assets... (a) If your work uses existing assets, did you cite the creators?

Original: [N/A] (b) Did you mention the license of the assets?
Revised: [N/A] (b) Did you mention the license of the assets?

Original: [N/A] (c) Did you include any new assets either in the supplemental material or as a URL?
Revised: [N/A] (c) Did you include any new assets either in the supplemental material or as a URL?

Original: [N/A] (d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating?
Revised: [N/A] (d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating?

Original: [N/A] (e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content?
Revised: [N/A] (e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content?

Original: [N/A] 
Revised: [N/A] 5.

Original: If you used crowdsourcing or conducted research with human subjects... (a ) Did you include the full text of instructions given to participants and screenshots, if 
Revised: If you used crowdsourcing or conducted research with human subjects... (a) Did you include the full text of instructions given to participants and screenshots, if applicable?

Original: applicable?

Original: [N/A] b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?
Revised: [N/A] (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?

Original: [N/A] c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?
Revised: [N/A] (c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?

Original: [N/A]
Revised: [N/A]