Title: Weakly-Supervised Cortical Surfaces Reconstruction from Brain Ribbon Segmentations
Venue: NeurIPS 2024
Accepted: no
Keywords: Cortical surface reconstruction

diffeomorphic deformation

ODE

Brain MRI
Abstract: Deep learning-based cortical surface reconstruction (CSR) approaches typically rely on supervision information provided by pseudo ground truth generated by conventional CSR methods, subject to errors associated with the supervision information and also increasing computational cost of training data preparation. We propose a new method to jointly reconstruct multiple cortical surfaces using weak supervision from brain MRI ribbon segmentation results. Our approach initializes a midthickness surface, which is then deformed inward and outward to form the inner (white matter) and outer (pial) cortical surfaces, respectively, by jointly learning diffeomorphic flows by minimizing loss functions to optimize the surfaces towards the boundaries of the cortical ribbon segmentation maps. Specifically, a boundary surface loss drives the initialization surface to the inner and outer boundaries, while an inter-surface normal consistency loss regularizes the pial surface in challenging deep cortical sulci regions. Additional regularization terms are utilized to enforce edge length uniformity and smoothness of the reconstructed surfaces. Our method has been evaluated on two large-scale adult brain MRI datasets and one infant brain MRI dataset, demonstrating comparable or superior performance in CSR in terms of accuracy and surface regularity compared to alternative supervised deep learning methods.

Abstract

Deep learning-based cortical surface reconstruction (CSR) approaches typically rely on supervision information provided by pseudo ground truth generated by conventional CSR methods, subject to errors associated with the supervision information and also increasing computational cost of training data preparation.We propose a new method to jointly reconstruct multiple cortical surfaces using weak supervision from brain MRI ribbon segmentation results. Our approach initializes a midthickness surface, which is then deformed inward and outward to form the inner (white matter) and outer (pial) cortical surfaces, respectively, by jointly learning diffeomorphic flows by minimizing loss functions to optimize the surfaces towards the boundaries of the cortical ribbon segmentation maps. Specifically, a boundary surface loss drives the initialization surface to the inner and outer boundaries, while an inter-surface normal consistency loss regularizes the pial surface in challenging deep cortical sulci regions. Additional regularization terms are utilized to enforce edge length uniformity and smoothness of the reconstructed surfaces. Our method has been evaluated on two large-scale adult brain MRI datasets and one infant brain MRI dataset, demonstrating comparable or superior performance in CSR in terms of accuracy and surface regularity compared to alternative supervised deep learning methods.

1 Introduction

Cortical surface reconstruction (CSR) is a crucial step for both qualitative visualization and quantitative characterization of cortical surfaces in imaging studies of brain morphology [15, 51], neurodegenerative diseases [6, 12, 43], and psychological disorders [42]. Well-established cortical analysis pipelines, such as BrainSuite [48], FreeSurfer [17], Connectome Workbench [18], and iBEAT V2.0 [52], have achieved significant success in reconstructing cortical surfaces from brain MRI data. However, these pipelines typically involve multiple processing steps, including iterative surface deformation and topology check and correction, resulting in lengthy processing time (e.g., ∼ 6h/subject). Moreover, each pipeline requires meticulously tuned parameters, posing challenges for generalization across diverse data domains, age groups, or acquisition protocols.

Deep learning (DL) approaches have significantly accelerated CSR, demonstrating orders of magni29 tude faster inference speeds while maintaining high accuracy and topology correctness [8, 11, 13, 30 22, 26, 30-32, 41, 47, 54]. One line of research predicts implicit surface representations, such as 31 signed distance functions [13, 21] or level sets [41], from which 3D meshes are extracted using the 32 Marching Cube (MC) algorithm [27] and refined with topology correction algorithms [4] to detect 33 and rectify topology errors, ensuring that the reconstructed surface conforms to a sphere-like topology. 34 Another line of research focuses on learning explicit surface deformations, using methods such as 35 flow-based [8, 11, 22, 26, 47, 54] or NODE-based techniques [30, 31]), to deform an initial mesh 36 towards target cortical surfaces. However, all these methods heavily rely on supervision information 37

provided by pseudo ground truth (pGT) of cortical surfaces generated by conventional CSR methods , 38 regardless of whether they use implicit or explicit surface representations. The prolonged processing 39 time for generating pGT surfaces limits the collection of sufficiently large datasets for training, 40 and a general pipeline capable of extracting pGT surfaces across various data domains (e.g., age, 41 modality) is currently lacking. Conversely, segmentation of brain structures is comparatively simpler, 42 inspiring us to explore avenues to eliminate the need for supervised learning in CSR and to generalize 43 DL-based CSR approaches to scenarios where ribbon segmentation results are readily available. 44

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The key challenges for achieving accurate weakly supervised reconstruction of cortical surfaces span three primary aspects. First , devising sub-voxel supervision signals presents a formidable hurdle. While existing approaches can produce precise segmentations [7, 20, 29, 45, 52], voxel-level representations may struggle to capture the intricate morphology of the cerebral cortex, especially its thin and highly-folded structure, due to the partial volume effect (PVE) inherent in brain MRI scans. This problem becomes particularly prominent in deep cortical sulci [17], where the two banks of grooves nearly converge, or in low-resolution images [52], such as under-sampled or infant MRIs. Second , effectively modeling the interdependence between multiple surfaces is crucial. Incorporating this prior knowledge into the design of models and training algorithms can alleviate the complexity of reconstructing both the inner (white matter) and outer (pial) surfaces, ensuring the spherical topology of the reconstructed surfaces [8, 54]. However, in the absence of pGT, it becomes more challenging to forcibly deform surfaces and less stable to optimize multiple surfaces concurrently. Third , maintaining optimal surface topology is paramount. Mesh uniformity, smoothness, and topology are susceptible to distortion during large deformations if networks are optimized based on randomly sampled vertices in 3D space for dense volumetric fields.

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In this paper, we introduce SegCSR , a novel weakly supervised DL framework aimed at reconstructing multiple cortical surfaces using ribbon segmentations derived from brain MRIs. We address the diffeomorphic deformation problem in a continues coordinate space, deforming the initialization midthickness surface towards the target inner and outer surfaces via innovative loss functions. Specifically, the boundary surface loss function based on the ribbon segmentations and the intensity gradient loss function based on the raw image facilitate sub-voxel-level surface movement. The inter-surface normal consistency loss function explicitly integrates the normal directions of the WM, midthickness, and pial surfaces, thereby regularizing the pial surface in challenging deep cortical sulci regions. Furthermore, we devise a customized edge length loss, in conjunction with the known normal consistency loss, to ensure surface uniformity and smoothness. Our main contributions can be summarized as follows:

We propose a new weakly supervised paradigm for reconstructing multiple cortical surfaces, reducing the dependence on pGT cortical surfaces in training, unlike existing DL methods.

We design two loss functions to optimize the surfaces towards the boundary of the cortical ribbon segmentation maps, along with regularization terms to enforce regularity of surfaces.

We conduct extensive experiments on two large-scale adult brain MRI datasets and one infant brain MRI dataset. Our new method achieves comparable or superior performance compared to existing supervised DL-based CSR alternatives.

2 Related Works

Cortical Surface Reconstruction (CSR). (I) Traditional CSR methods typically rely on empirically defined automatic image/surface processing techniques to accomplish tissue segmentation (e.g., WM, GM, cerebrospinal fluid (CSF)), hemisphere separation, subcortical filling, topology correction, WM surface reconstruction, and pial surface reconstruction sequentially. Established pipelines such as FreeSurfer [17], BrainSuite [48], and HCP [18] are tailored for processing adult brain images, while dHCP [34] and iBEAT V2.0 [52] are designed for neonatal brain images, which exhibit distinct differences in intensity values, size, and shape compared to adult brains. Despite achieving sub-voxel accuracy and maintaining spherical topology, the iterative surface deformation and topology check and correction procedures lead to lengthy processing times. (II) DL-based CSR methods have significantly enhanced reconstruction speed while preserving high accuracy. Approaches like SegRecon [19] and DeepCSR [13] predict a signed distance map for implicit surface representation, embedding the target surface as the zero level-set and extracting it using MC algorithms. However, these methods require topology correction to eliminate artifacts and ensure spherical topology. Alternatively, PialNN [32], TopoFit [22], Vox2cortex [8], the CorticalFlow series [26, 47], SurfFlow [11], CortexODE [31],

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and CoCSR [54] leverage explicit representation to maintain good topology and overcome PVE by learning volumetric or vertex-wise diffeomorphic deformations and progressively deforming genus-0 template meshes. However, both implicit and explicit methods heavily rely on the supervision of pGT of cortical surfaces generated by traditional pipelines. Our proposed method is based on the explicit representation but differs significantly from them by utilizing ribbon segmentation maps for weakly supervising the model training process.

Weakly-/Un-supervised Mesh Reconstruction. Although geometric DL methods for general computer vision tasks have been extensively studied, research on mesh reconstruction from 3D images under weakly-/un-supervised settings is relatively underexplored. One approach involves constructing mesh-to-image rasterizer loss functions, as demonstrated in [36], where 2D projection views are extracted from predicted 3D meshes and compared with ground truth segmentations. Another line of research, exemplified by [39], focuses on learning the correspondence between a template image and a target image, which is then utilized to deform the template mesh to the target location. However, these methods have primarily been applied to biomedical tasks involving organs with relatively simple shapes, such as the liver and heart. But the cerebral cortex presents a highly-folded thin structure with a significantly complex shape, necessitating more advanced methods.

Diffeomorphic Deformation. Diffeomorphic deformation is a spatial transformation that guarantees both smoothness and invertibility in the mapping process [46]. It has been widely used in the modeling and analysis of brain morphometry, including image registration and surface reconstruction tasks. LDDMM [5] computes diffeomorphic deformation based on a time-dependent velocity vector field, while Arsigny et al. [2] employ a stationary velocity field (SVF) in conjunction with the scaling and squaring method to reduce computation complexity. Learning-based methods [3, 28, 38] improve the computation efficiency, with regularizations such as smoothness [3] and inverseconsistency [38] enchancing the diffeomorphic property of the deformation. In the CSR task, diffeomorphic deformation strategies have been adopted to solve an ordinary differential equation (ODE) modeling the trajectories of each vertex of a surface. For instance, CoticalFlow methods [26, 47] propose solving the ODE vertex-wise and derive a numerical condition to ensure homeomorphism of integration by training a chain of diffeomorphic deformation models in sequential stages. Recently, with the advances in neural ODE solver [10], CortexODE [31] parameterizes the trajectories of vertices on the surface as ODEs and proposes a pipeline to reconstruct WM and pial surfaces sequentially . Our method builds upon these works [31, 47, 54] and integrates multiple CSR tasks into a single framework, leveraging the efficiency and diffeomorphic properties of these strategies.

3 Methodology

Our proposed framework, depicted in Fig. 1, is designed to reconstruct multiple cortical surfaces simultaneously, eliminating the dependency on pGT generated by conventional and time-consuming CSR pipelines. We leverage as weak supervision the brain ribbon segmentation maps that are less accurate than pGT surfaces but more accessible. Section3.1 outlines the network structure that couples multiple cortical surfaces to reduce the learning difficulty. Section 3.2 describes the loss functions devised to supervise the network optimization, facilitating sub-voxel reconstruction accuracy and preserving optimal surface topology.

3.1 Coupled Cortical Surface Reconstruction

Existing supervised methods require pGT obtained from traditional CSR pipelines to provide precise 134 sub-voxel supervision. They can effectively learn the deformation field, even from distant initial 135 locations, to accurately align the initialization surface with the target surfaces [11, 26, 47]. However, 136 brain ribbon segmentation maps are inherently discrete voxel grids, offering much coarser supervision. 137 Consequently, the selection of the initialization surface becomes more critical. Moreover, given the 138 intricate folded patterns of the cerebral cortex, the proximity of the two banks of grooves in deep 139 cortical sulci often poses a considerable risk of generating topology errors (e.g., handles, holes) in the 140 reconstructed surfaces. Conversely, voxels closer to the WM surface exhibit clearer contrast, enabling 141 a distinct separation between sulci (Fig. 2 (b)). Thus, following [54], we opt for the midthickness 142 layer, positioned midway between the WM and pial surfaces, to serve as a connection for coupling 143 the reconstructions of both surfaces and achieve a balanced performance for both surfaces. 144

As illustrated in Fig. 1, SegCSR employs a neural network to jointly model three diffeomorphic flows: 145 F θ ( I, S 0 ) = ( v m , v o , v i ) . Here, I represents a multi-channel input consisting of brain MRI, cortical 146

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ribbon masks, and signed distance functions (SDFs); S 0 denotes the initialization midthickness 147 surface; and v m , v o , v i correspond to the velocity fields that drive S 0 towards the true midthickness 148 surface S M , outward to the pial surface S G , and inward to the WM surface S W , respectively. The 149 SegCSR establishes an explicit one-to-one mapping between multiple surfaces and is trained by 150 minimizing weakly supervised losses between the predicted mesh and the ribbon segmentations. 151

The diffeomorphic deformation between the initialization surface and the target surface can be 152 computed as the integration of an ODE [1] based on the velocity field v : 153

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where Φ( x , t ) defines a trajectory from the source position x (0) = Φ( x , 0) to the target position x (1) = Φ( x , 1) . According to the Cauchy-Lipschitz theorem [50], if the velocity field is Lipschitz continuous, the resulting mapping Φ is bijective with continuous inverse (i.e., a diffeomorphism). To solve this initial value problem, we perform the integration on the predicted velocity fields using standard numerical integration techniques, such as the Euler method and the Runge-Kutta method [9]. Specifically, for each integration step t ∈ [0 , 1] , each vertex's coordinates can be updated by x ( t +1) = x ( t ) + h v (Φ( x , t ) , t ) , where h = 1 T is the step size and T is the total time steps, and the velocity vector v for a vertex is trilinearly interpolated from its neighboring velocity vectors [54].

3.2 Weak Supervision Loss Functions

Mesh Loss . Weak supervision for SegCSR is derived from cortical ribbon segmentation maps of 163 WMand GM (see Fig. 1, the filled interior area of WM and pial surfaces), which can be obtained 164 from existing segmentation approaches [7, 20, 29, 45, 52]. Although these ribbon segmentation 165 maps do not perfectly represent the intricate pial surface, the WM surface is relatively easier to 166 recognize due to its clear local intensity contrast, providing a better-separable boundary (see Fig. 2 167 (a-b)). Therefore, we use the boundary of the pGT WM segmentation to supervise the WM surface 168 reconstruction. Inspired by [31, 54], we generate an SDF for the WM surface by using a distance 169 transform algorithm, where voxels with values of zero represent the surface boundaries and voxels 170 with negative or positive values encode their distances to the surface boundaries inward or outward, 171

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respectively. We then apply a fast topology check and correction algorithm [4] to the SDF to ensure 172 the surface maintains spherical topology. The WM surface S W ∗ is extracted using the Marching 173 Cubes algorithm [27]. The distance of the vertices between the predicted surface S W and the pGT 174 surface S W ∗ is minimized using the bi-directional Chamfer distance [26]: 175

where p and p ∗ are the coordinates of vertices on meshes. See Fig. 2 (c-1) for illustration. 176

For the pial surface, GM segmentation may fail to delineate the boundary in deep cortical sulci. 177 As shown in Fig. 2 (c-2), using a similar pGT surface generation protocol as the WM surface to 178 generate the pial surface S G ∗ fail to capture cortical folding accurately. Directly fitting to S G ∗ with 179 bi-directional Chamfer loss causes the model to predict similarly inaccurate cortical sulci. To address 180 this issue, we propose the boundary surface loss, which uses a uni-directional Chamfer distance to 181 compute the shortest distance from the pGT pial surface S G ∗ to the predicted pial surface S G : 182

In this way, the deformed surface is not influenced by the inaccuracies of S G ∗ and does not move 183 outward from the deep sulci. The overall mesh loss is computed as L mesh = L chW + L chG . 184

Inter-Mesh Normal Consistency Loss . To further alleviate the difficulty of constraining the pial 185 surface using the WM and midthickness surfaces, we propose leveraging the prior knowledge that 186 the cerebral cortex has a sheet-like topology (i.e., the inner, middle, and outer surfaces are locally 187 parallel to each other). As shown in Fig. 2 (d), this loss is defined to ensure that the deformation 188 of the midthickness surface aligns with its normal direction, thereby maintaining similar normal 189 directions on the target surfaces: 190

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where n p G and n p W are the normal vectors of the deformed vertex p on S M and S G respectively.

Intensity Gradient Loss . In addition to ribbon segmentaions, inspired by the fact that traditional methods utilize raw image intensity contrast to define and optimize the target surfaces, we propose to adjust the nuance between GT target surface and the pGT segmentation boundaries. By definition [17, 52], the WM (or pial) surface lies at the WM/GM (or GM/CSF) interface where image intensity change most drastically. We sample K points along the extended lines on each side of the normal direction at vertex p , and compute the gradients of neighboring points: L grad = 1 |S W | ∑ p ∈S M ∑ K i =1 grad i ( p ) + 1 |S G | ∑ p ∈S G ∑ K i =1 grad i ( p ) .

Cycle Consistency Loss. We utilize the midthickness layer to establish a correspondence between the inner and outer surfaces, thereby reducing the difficulty of learning large deformations. However, there is no true midthickness surface available for supervision, nor a definitive criterion for choosing between bi-directional or uni-directional approaches for different regions on the midthickness surface. Additionally, the learned velocity fields v o and v i could potentially cause non-inverse transformations at the midthickness surface. To address these issues, we propose a loss function that enforces the midthickness surface resides halfway between the WM and pial surfaces and maintains consistency along the entire trajectory:

where p Φ b ◦ Φ a represents deforming a vertex p ∈ S M with velocity field v a and v b sequentially, and L Mid → GM ( p ) is the accumulated trajectory length over T steps of deformation. For example, as shown in Fig. 2 (f), the deformations move a vertex p Mid outward to p GM using v o and then inward to p ′ Mid using v i , in which the two trajectories are aligned by minimizing the distance between p Mid and p ′ Mid . Similarly, we enforce the consistency between p Φ G ◦ Φ W and p . Furthermore, starting from the midthickness layer, the trajectory lengths of the vertex moving to the WM and pial surfaces should be equal, which is regularized by the third term in the equation above.

Mesh Quality Loss. First, the reconstructed surface should be composed of uniformally distributed triangles. To accommodate various sizes of brain volume and image resolution, we devise a customized edge length loss to constrain the size of triangles in the predicted meshes for each subject. Specifically, we assume an ideal prediction where the faces are equilateral and of the same area A and drive the edge length to the target edge length µ el = 2 √ A √ 3 (see Fig. 2 (g)). Second, we employ a normal consistency loss to promote the surfaces' smoothness. The mesh quality loss is defined as:

where S denotes the predicted mesh, N ( p ) are the neighbors of vertex p , e is an edge, f 0 and f 1 are e 's two neighboring faces with their unit normals n f 0 and n f 1 .

In summary, we combine all the losses to jointly optimize our SegCSR model: L = λ 1 L mesh + λ 2 L imnc + λ 3 L grad + λ 4 L cyc + λ 5 L qua , where { λ i } i =1 , ··· , 5 are weights to balance the loss terms.

4 Experiments

4.1 Experimental Setups

Datasets. We evaluate our method on two large-scale adult datasets and one infant dataset of low resolution. The ADNI-1 [24] dataset consists of 817 subjects aged 55 to 90. We randomly split it into subsets of 654, 50, and 113 subjects for training, validation, and testing, respectively. The OASIS1 [35] dataset consists of 413 subjects aged 18 to 96. We randomly split it into subsets of 330, 25, and 58 subjects for training, validation, and testing, respectively. We followed a pre-processing protocol used in previous works [8, 13, 26, 31] for fair comparison. The T1-weighted MRI scans were aligned to the MNI152 template and clipped to the size of 192 × 224 × 192 at 1 mm 3 isotropic resolution. The pseudo ground-truth (pGT) of ribbon segmentation and cortical surfaces were generated using FreeSurfer v7.2.0 [17]. The BCP [23] dataset consists of 121 subjects ranging in age from 2 weeks to 12 months. We randomly allocate 90, 12, and 19 subjects for training, validation, and testing,

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respectively. Rigid registration was applied to the T1w and T2w image pairs. The pGT of ribbon segmentation and cortical surfaces were generated by the iBEAT v2.0 [52]. The intensity values of MRI scans, ribbon segmentation maps, and SDFs were normalized to [0 , 1] and the coordinates of the vertices were normalized to [ -1 , 1] . All the models were trained on the training set until they reached a loss plateau on the validation set and evaluated on the test set.

Implementation Details Our framework was implemented in PyTorch [40] and trained on a workstation with 12 GB NVIDIA P100 GPU. The 3D U-Net [44] for segmentation of ribbons was trained for 200 epochs using Adam [25] optimization and achieved an average Dice index of 0.96 on the testing set. The SegCSR model utilized T = 5 steps (i.e., step size is 0 . 2 ) in Euler solver. We trained our SegCSR model using Adam optimizer ( β 1 = 0 . 9 , β 2 = 0 . 999 , ϵ = 1 e -10 , learning rate 1 e -4 ) for 400 epochs to reconstruct both WM, midthickness, and pial surfaces of both brain hemispheres. We set λ 1 = λ 4 = 1 and λ 2 = λ 3 = λ 5 = 0 . 1 . The surface meshes had ∼ 130 k vertices. More details can be found in the Supplementary Materials.

Evaluation Metrics We utilized three distance-based metrics to measure the CSR accuracy: Chamfer distance (CD), average symmetric surface distance (ASSD), and 90th-percentile Hausdorff distance (HD). CD [16, 53] measures the mean distance between two sets of vertices. ASSD [13] and HD [13, 49] measure the average and maximum distance between two surfaces. They were computed bidirectionally over ∼ 130 k points uniformly sampled from the predicted and target surfaces. A lower distance means a better result. Since topology is also important in CSR, we utilized the ratio of self-intersection faces (SIF) [13, 14, 31, 54] to measure reconstructed surface quality.

4.2 Comparison with Related Works

We compare SegCSR with both implicit and explicit learning-based cortical surface reconstruction approaches described in Section 1 and summarize the experimental results in Table 5.

On Adult Datasets. (I) Comparison with Implicit Approaches . We compare SegCSR with two representative implicit representation approaches on the ADNI and OASIS datasets. As shown in Table 5, SegCSR achieves superior geometric accuracy. Note that both DeepCSR [13] and 3D U-Net [44] require post-processing to correct topology and extract a mesh, resulting in SIFs of 0. Without post-processing, the SIFs for 3D U-Net's WM and pial surfaces range from 3% to 15% . SegCSR produces a negligible number of self-intersecting faces, ∼ 0 . 3% on average for both white and pial surfaces. Fig. 3 shows that SegCSR effectively deforms the pial surface into deep sulci, while the baseline approaches exhibit large geometric errors due to the PVE problem of brain MRI.

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(a) DeepCSR

(b) CortexODE

(c) SegCSR (S0 setting)

(d) SegCSR (S0* setting)

(e) pGT from FreeSurfer

Additionally, SegCSR requires only 0.37s of runtime per brain hemisphere, orders of magnitude faster than traditional FreeSurfer pipelines. (II) Comparison with Explicit Approaches . We compare SegCSR with explicit learning-based approaches, including CorticalFlow++ [47], Vox2Cortex [8], CortexODE [31], and CoCSR [54]. These methods are trained with pGT surfaces generated by conventional pipelines, providing more accurate supervision than pGT ribbon segmentations. For a fair comparison, we employ the same network structure for the current best CoCSR [54] and our SegCSR, with CoCSR serving as an upper-bound performance benchmark for our weakly supervised SegCSR. As shown in Table 5, SegCSR surprisingly surpasses some supervised baselines in terms of both geometric and morphological accuracy, demonstrating its potential to replace existing methods when accurate surface supervision is not available.

On Infant Dataset. Infant brain MRIs present additional challenges due to the smaller size of fetal brains, limited image resolution, and lower image contrast, which together make the reconstruction task more difficult. Consequently, overall performance is inferior compared to adult datasets. We compare SegCSR with both implicit and explicit representation approaches. The results in Table 5 show that SegCSR achieves superior performance than the implicit DeepCSR and 3D U-Net methods, and comparable performance to explicit methods like CorticalFlow++, CortexODE, and CoCSR.

4.3 Ablation Studies

Loss Functions. We evaluated the contribution of different losses of our method to the surface reconstruction performance in terms of both accuracy (CD, ASSD, HD) and topological correctness (SIF). The results are summarized in Table 2 (Top). The setting S4 represents using our proposed Chamfer loss (i.e., uni-directional for the pial surface) alone, while S4 ⋆ referes to using existing bi-directional Chamfer loss for both WM and pial surfaces. The results of S4 and S4 ⋆ indicated that the model using bi-directional Chamfer loss overfitted to the pGT segmentation boundary and failed to fit the deep cortical sulci. Another pair of comparison, S0 and S0 ⋆ , showed a similar phenomenon. Enforcing the inter-mesh normal consistency of the WM and pial surfaces (S3, L imnc ) improved geometric accuracy by explicitly constraining the nromal direction of two surfaces but slightly worsened the topology, which might be caused by the discrepancy between the midthickness and the WM (and pial) surface. The proposed intensity gradient loss (S2, L grad ) helped adjust the deformed surfaces locally, leading to slightly improved geometric accuracy and reduced topology error. Enforcing equality of the trajectories from the midthickness surface to the WM and pial surfaces

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and symmetric cycle consistency of two trajectories (S1, L cyc ) helped optimize the midthickness surface and promoted the invertibility of deformations. Moreover, the inclusion of regularization terms on the uniformity and smoothness of the reconstructed surfaces (S0, L qua ) enhanced the surface quality and significantly reduce the self-intersection face ratio. Overall, our proposed method struck a balance between geometric accuracy and topology quality, with each component playing a complementary role.

Initialization Surface Location. Table 2 (Bottom) shows the impact of the initialization surface location. Starting from either the WM or midthickness surfaces leads to satisfactory results. Conversely, initializing from the GM surface introduced more difficulty in learning large deformations into deep sulci due to the severe partial volume effect, resulting in worse average geometric accuracy for both surfaces. The results also indicated that the closer the initial surface was to its target surface, the higher the reconstruction accuracy achieved. Therefore, starting from the midthickness surface strikes a balance between WM and pial surface reconstruction outcomes.

4.4 Reproducibility

We conducted an experiment on the Test-Retest dataset [33], which comprises 40 MRIs collected within a short period for each of the 3 subjects. The cortical surfaces of the same subject should be nearly identical. Following the experimental setup outlined in [8, 13, 31, 54], we utilized the iterative closest-point algorithm to align image pairs and computed the geometric distance between surfaces. The results for the left hemisphere are presented in Table 3, showing that SegCSR obtained superior reproducibility compared with DeepCSR (implicit representation; weakly supervised) and was comparable to the conventional FreeSurfer pipeline and supervised DL-based CSR

methods. This implied that the results generated by SegCSR can be reliably used for downstream analyses, such as investigating cortical thickness changes in patients.

5 Conclusions

We introduce SegCSR, a novel approach to jointly reconstruct multiple cortical surfaces using weak supervision from ribbon segmentations derived from brain MRIs. Our method initializes a midthickness surface and then deforms it inward and outward to the inner and outer cortical surfaces by jointly learning diffeomorphic flows. The new boundary loss function optimizes the surfaces toward the boundaries of the cortical ribbon segmentation maps while the inter-surface normal consistency loss regularizes the pial surface in complex and challenging cortical sulci regions. Additional regularization terms are incorporated to enforce reconstructed surfaces' uniformity, smoothness, and topology. Extensive experiments conducted on large-scale adult and infant brain MRI datasets demonstrate superior performance in terms of accuracy and surface regularity compared to existing supervised DL-based alternatives.

Limitations and Future Directions. The efficacy of SegCSR is influenced by the quality of pGT segmentations. Also, We can utilize brain tissue segmentation as auxiliary functions to supervise the model training. SegCSR constrains the inter-mesh consistency of the deformation on the midthickness surface, potentially affecting anatomical fidelity of pial surfaces. The method should be tested on more diverse cohorts of subjects to demonstrate its efficacy on real world neuroimage analysis tasks.

Societal Impact. Our proposed method has been rigorously evaluated on four real-world brain MRI datasets, showcasing its capacity to assist doctors and scientists in both quantitative and qualitative analyses of the cerebral cortex. Nonetheless it is imperative to conduct more thorough evaluation on a larger cohort of subjects and across various imaging qualities. And the deployment of the model in clinical settings should be approached with caution and under human supervision.

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A Model Details

A.1 Cortical Ribbon Segmentation Network Architecture

Fig. 4 (Left) shows the detailed network architecture of our cortical ribbon segmentation network, which is a 5-level hierarchical encoder-decoder with skip connections. The network processes a 3D brain MRI to produce a cortical ribbon segmentation map. The white matter (WM) segmentation includes the interior of the WM surface, encompassing cortical WM, deep gray matter, ventricles, hippocampus, and other tissues within the surface. Similarly, the gray matter (GM) segmentation includes the interior of the pial surface. The output map has five classes: left hemisphere WM and GM, right hemisphere WM and GM, and background. In the encoder, each level uses a 3 × 3 × 3 convolutional layer with a stride of 2 to downsample the features. In the decoder, features are upsampled by 2 × at each scale, concatenated with the corresponding features from the encoder via skip connections, and then fused using a 3 × 3 × 3 convolutional layer with a stride of 1. For feature extraction at the input, a 3 × 3 × 3 convolutional layer with a stride of 1 is used. Before the final prediction, three consecutive convolutional layers are applied. Each convolutional layer is followed by a leaky ReLU activation function, except for the last one, which uses a Softmax function before computing the cross-entropy loss with the ground truth.

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As shown in Fig.4 (Right), our cortical surface reconstruction (CSR) network operates at five scales. 513

To conserve memory, we downsample the input image using a 3 × 3 × 3 convolution with a stride of 514

2 and skip complex feature fusion via skip connections in the decoding path at this scale. To improve 515

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B Experimental Settings

B.1 Dataset Preprocessing

We preprocessed all the MRIs of the ADNI-1 [24] and OASIS-1 [35] datasets with the same protocols as following: Based on the standard processing protocol in FreeSurfer V7.2.0 [17], the original images were conformed and normalized (saved as orig.mgz ), affinely registered to the MNI152 template [8] using the NiftyReg toolbox [37]. The respective ribbon segmentation maps, SDFs, and pseudo-ground-truth surfaces were also transformed using the computed transformation. Similarly, we utilize iBEAT V2.0 [52] to process the BCP [23] dataset and merge the brain tissue segmentation results as the ribbon segmentation maps.

B.2 Baselines

We compared our SegCSR with representatives from the two categories of existing DL-based CSR methods and evaluated their performance for both WM and pial surface reconstruction. DeepCSR [13] and 3D U-Net [44] represent implicit surface reconstruction methods, while others fall into the category of explicit methods. Note that we modify the 3D U-Net method to first generate SDFs based on the ribbon segmentation results, then perform topology correction, and finally utilize the Marching Cubes algorithm to extract the mesh. Since it does not require pGT surfaces from FreeSurfer for training supervision, it can be treated as a weakly supervised learning-based baseline. CorticalFlow++[47] utilizes smoothed convex hulls as the initialization template, trains a chain of deformation fields, and employs a fourth-order Runge-Kutta (RK4) solver to compute the integration for the initial value problem. CortexODE[31] uses WM segmentation for surface initialization and Neural ODE for deformation computation. Vox2cortex [8] deforms averaged surface templates with a GNN-based network to reconstruct multiple surfaces. CoCSR [54] integrates multiple cortical surface reconstructions into a single network. A summary of the state-of-the-art CSR methods is provided in Table 4.

C More Experimental Results

C.1 Quantitative comparison of our methods with Related Works

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe submission presents a deep learning-based approach for cortical surface reconstruction (CSR) from brain MRI data using weak supervision derived from cortical brain segmentation maps. The claimed contributions are: \n\n1. Weak Supervision Paradigm: The authors introduce a new weakly supervised paradigm for reconstructing multiple cortical surfaces, significantly reducing the reliance on pseudo ground truth (pGT) surfaces generated by conventional CSR methods.\n2. New Loss Functions: Two novel loss functions are designed to optimize the surfaces towards the boundaries of the cortical ribbon segmentation maps. Regularization terms are also introduced to enforce surface uniformity and smoothness.\n3. Evaluation and Performance: The proposed method is extensively evaluated on two large-scale adult brain MRI datasets and one infant brain MRI dataset, demonstrating comparable or superior performance to existing supervised DL-based CSR methods.\n\nSTRENGTHS:\n1. The paper presents an approach to leverage weak supervision from segmentation maps instead of relying on pGT surfaces, which is a significant departure from traditional methods.\n2. The methodology is explained and the experimental setup is described. The authors conduct evaluations on multiple datasets, evaluating the efficacy and efficiency.\n3. The paper is well-structured, with clear descriptions of the problem, methodology, and results. The figures and tables effectively illustrate the performance and comparisons.\n4. The approach addresses a critical bottleneck in CSR by reducing the dependency on time-consuming and error-prone pGT surfaces, potentially broadening the applicability of CSR methods to more diverse datasets and clinical scenarios.\n\nWEAKNESSES:\nMethod\n1. It seems that this work combines [1] and [2], and thus has limited technical novelty. The architecture in Figure 1 and the circle consistency loss (Eq. 5) are almost identical to CoCSR [1]. The boundary surface loss and inter-mesh normal consistency loss (Eq. 3-4 and Figure 2) are very similar to the loss functions proposed by [2].\n\n2. Additionally, the customized edge length loss (Eq. 6) has also been proposed by [3]. Considering the large individual differences across human brains, how did the authors choose the area A without knowing the pGT cortical surfaces?\n\n3. It is confusing that the ribbon segmentations are used as both input and pGT. The authors claimed that the ribbon segmentations are inaccurate weak supervision, but still generated the initial surface based on ribbon segmentations according to Figure 1.\n\n4. The velocity field defined in Eq. 1 is time dependent. How did the authors learn non-stationary velocity fields through a 3D U-Net?\n\n5. In line 156, a bijective mapping with continuous inverse is called homeomorphism. A diffeomorphism is defined as a smooth/differentiable bijection with smooth/differentiable inverse.\n\n6. As shown in Figure 2 (b), it is clear to observe that the WM and pial surfaces do not have the same normal directions in some regions. The inter-mesh normal consistency loss could cause inaccurate surface reconstruction. Could the authors provide more insights to solve this problem?\n\n\nResults\n1. The experimental results are unreliable and unconvincing. After careful comparison, it seems that the baseline results (CorticalFlow++, CortexODE, Vox2Cortex, DeepCSR) on the ADNI and OASIS datasets in Table 1 were directly copied and pasted from Table 2 in [1]. This leads to unfair comparisons.\n\n2. Furthermore, as reported in Table 1, SegCSR produced no more than 0.061% of self-intersecting faces (SIF), whereas the authors claimed in line 264 that there are ∼0.3% on average for both white and pial surfaces. This is confusing. Which result is correct?\n\n3. In line 263, the authors claimed that DeepCSR and U-Net produced a large number of SIFs without post-processing. However, the Marching Cubes algorithm only produces topological errors such as holes no SIFs.\n\n4. The BCP dataset only includes 19 test subjects. Cross-validation should be conducted to ensure fair evaluation of the performance.\n\n5. The flow ODE was integrated using the forward Euler method with T=5 steps. Such a large step size could cause unstable ODE solutions and failure in preventing self-intersections. The value of the Lipschitz constant should be reported to examine the numerical stability of the ODE solver.\n\n6. The authors reported that SegCSR requires only 0.37s of runtime per brain hemisphere. However, SegCSR adopted a topology correction algorithm, which may take several seconds to a few minutes, to create an initial midthickness surface for each subject. This should be included in the total runtime. A breakdown of runtime should be reported and compared to SOTA baseline approaches. \n\n\n[1] Zheng, H., Li, H. and Fan, Y. Coupled reconstruction of cortical surfaces by diffeomorphic mesh deformation. Advances in Neural Information Processing Systems, 2023.\n\n[2] Ma, Q., Li, L., Robinson, E.C., Kainz, B. and Rueckert, D. Weakly Supervised Learning of Cortical Surface Reconstruction from Segmentations. arXiv preprint arXiv:2406.12650\n\n[3] Chen, X., Zhao, J., Liu, S., Ahmad, S. and Yap, P.T. SurfFlow: A Flow-Based Approach for Rapid and Accurate Cortical Surface Reconstruction from Infant Brain MRI. MICCAI, 2023.\n\nQUESTIONS:\n1. Can the authors elaborate on the key differences between their approach and [1,2,3], particularly in terms of methodology and experimental setup?\n2. How does the proposed boundary surface loss function improve upon the traditional bi-directional Chamfer loss used in existing methods?\n3. Can the authors provide more details on the computational efficiency and runtime comparisons with existing CSR pipelines?\n\nLIMITATIONS:\nThe authors have addressed some limitations, but further clarity on the following aspects would be beneficial:\n\n1. The efficacy of SegCSR is influenced by the quality of pGT segmentations. More discussion on how to handle low-quality segmentations would be helpful.\n2. The constraint on inter-mesh consistency of deformation might affect the anatomical fidelity of pial surfaces. Further exploration of this trade-off is necessary.\n3. The method could be tested on more diverse cohorts to demonstrate its efficacy across various imaging qualities and subject demographics.", "TITLE": "Official Review", "RECOMMENDATION": 2, "REVIEWER_CONFIDENCE": 5, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-06-27"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors proposed a novel new method to jointly reconstruct multiple cortical surfaces using weak supervision from brain MRI ribbon segmentation results, which deforms midthickness surface deformed inward and outward to form the inner (white matter) and outer (pial) cortical surfaces. The proposed method is evaluated on two large-scale adult brain MRI datasets and one infant brain MRI dataset, demonstrating comparable or superior performance in CSR in terms of accuracy and surface regularity.\n\nSTRENGTHS:\n1.\tPropose a new weakly supervised paradigm for reconstructing multiple cortical surfaces, reducing the dependence on pGT cortical surfaces in training, unlike existing DL methods.\n2.\tDesign two loss functions to optimize the surfaces towards the boundary of the cortical ribbon segmentation maps, along with regularization terms to enforce the regularity of surfaces.\n3.\tConduct extensive experiments on two large-scale adult brain MRI datasets and one infant brain MRI dataset.\n\nWEAKNESSES:\n1.\tIt seems overclaim in the manuscript. The ‘pseudo’ ground-truth surface mentioned in the manuscript is actually the ground-truth mesh in other approaches, obtained by Marching cube/Free surfer. Since the chamfer distance is used to guide the network training, why do the authors claim the proposed method is weakly supervised?\n2.\tIt is not clear how the original images are overlaid with the predicted mesh. Is any registration used? Details are missing.\n3.\tIt seems the main contribution of the proposed SegCSR is the boundary loss function?\n\nQUESTIONS:\n1.\tWhy not use the total ADNI datasets for network training as what is used in previous research like DeepCSR and voxel2cortex?\n2.\tHow the predicted meshes are overlaid on the original images?  Details should be given.\n3.\tWhat does the ‘L-Pial Surface’ and ‘L-WM Surface’ in the tables mean? The Pial and WM surface of the left hemisphere. Why not also present the results for the right hemisphere?\n\nLIMITATIONS:\nThe limitations are discussed in the msnuscript.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-09"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper presents a deep learning approach to jointly reconstruct multiple cortical surfaces using weak supervision from brain ribbon segmentations derived from brain MRIs. The method leverages the midthickness surface and deforms it inward and outward to fit the inner and outer cortical surfaces by jointly learning diffeomorphic flows. Regularization terms are included to promote uniformity, smoothness, and topology preservation across the surfaces. Experiments are conducted on large-scale adult and infant brain MRI datasets.\n\nSTRENGTHS:\n- The approach is novel in its use of weak supervision from readily available segmentation datasets, which reduces the burden of preparing pseudo-ground truth surfaces.\n- The paper is well-written and structured, with a clear motivation for the method.\n- The methodology is explained in detail, and the experiments are comprehensive.\n- The approach has the potential to democratize the use of deep learning in cortical surface reconstruction by leveraging existing segmentation datasets.\n\nWEAKNESSES:\n- The paper's central contribution of weak supervision is undermined by the fact that the model is trained on pseudo ground truth surfaces for white matter and pial surfaces.\n- The experimentation is limited to brain cortical surfaces and MRI images. Broader experiments involving different anatomies (e.g., bone cortical surfaces, heart walls) and imaging modalities would enhance the paper's impact.\n- Results lack statistical significance analysis to validate sub-millimeter reconstruction errors.\n- There is no evidence showing that improvements in mesh reconstructions correlate with enhanced performance in downstream analysis tasks.\n- The robustness of the method regarding input noise/perturbation and images from multiple centers is not evaluated.\n- There is no analysis of the computational complexity, including the resources and time savings provided by the proposed weak supervision.\n- There is no sensitivity analysis on the choice of weights used to weigh the different components of the overall loss.\n- The impact of ribbon segmentations quality (e.g., voxel spacing) as weak supervision is not investigated.\n\nQUESTIONS:\n1. Can you provide evidence or analysis showing that improvements in mesh reconstructions lead to enhanced performance in downstream analysis tasks?\n2. How does the method perform with input noise or perturbations? What is the expected performance under domain shifts?\n3. What are the computational resources and time requirements saved by using weak supervision compared to traditional methods?\n4. How does the quality of ribbon segmentations (e.g., voxel spacing) impact the reconstruction accuracy?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-12"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper presents a novel deep learning method for the reconstruction of cortical surfaces from 3D MRI. The proposed method follows an approach learning explicit surface deformations, in which a CNN is used to predict three velocity fields, corresponding to the pial, white matter and midthickness surfaces. Unlike previous techniques which use cortical surface pseudo ground truth (e.g., generated using FreeSurfer), the proposed method trains the network with faster-to-obtain segmentation pseudo ground truth. In addition to the standard surface prediction losses (based on Chamfer distance), the method uses 1) an Inter-Mesh Normal Consistency loss that encourages the pial and WM surface to be locally parallel, 2) an Intensity Gradient loss that place the surfaces at regions of high intensity gradients, 3) a Cycle Consistency loss enforcing inverse consistency between the midthickness-to-pial deformation and the midthickness-to-WM one, and 4) a Mesh Quality loss that helps having regular surface meshes (uniform sized triangles and smoothly varying normals). The method is evaluated on the ADNI, OASIS and BCP datasets, where its performance is compared to that of implicit and explicit approaches. Results show that the method obtains a better reconstruction accuracy compared to other techniques trained in a weakly supervised setting (pGT segmentation mask), but a lower performance than those trained with pGT cortical surfaces.\n\nSTRENGTHS:\n* The proposed method differs from previous approaches that explicit surface deformations by predicting a midthickness surface and incorporating additional loss terms that compensate for the weak supervision of pGT segmentation.\n\n* Experiments, involving three different datasets and comparing against several recent baselines, as well as including various ablation variants, are well designed. Results indicate superior performance in the weakly supervised setting.\n\nWEAKNESSES:\n* The main motivation of the proposed method is doubtful. Authors motivate the need for their weakly-supervised cortical reconstruction method by the \"prolonged processing time for generating pGT surfaces\". However, as the pGT cortical surfaces can be generated automatically in an offline step, I believe the argument is weak. Moreover, recent pipelines for brain image processing, such as FastSurfer, can extract surfaces with comparable accuracy in a fraction of the time.\n\n* The accuracy of the proposed method is considerably lower than approaches which train on cortical surfaces. Furthermore, while it produces fewer topological artifacts like self-intersecting faces, those can be removed via post-proicessing in implicit methods like DeepCSR. Combined with my previous comment, the advantages of the method are unclear.\n\n* The ablation study in Table 2 indicates that most of the proposed loss terms have limited impact on the overall performance. For example, adding the Mesh quality loss seems to actually degrade performance in terms of CD, ASSD and HD.\n\nQUESTIONS:\n* How does your method compare to other approaches in terms of training and inference time ? \n\n* The proposed method has several hyper-parameters (lambda1-5) that need to be tuned. How were the values selected for these hyper-parameters, and how sensitive is the method to the chosen values?\n\n* In Figure, why is the pial surface represented with two different colors (orange and purple) ?\n\n* In Eq (4), how do you compute the pial and WM surface normals if the point is on the midthickness surface?\n\n* p6: \"where npG and npW are the normal vectors of the deformed vertex p on SM and SG respectively\": Do you mean on S_G and S_W ?\n\n* p6: \"segmentaions\" \n\n* Section 4.2: Do you mean Table 1 ? \n\n* p8: \"nromal\"\n\n* p9: \"Also, We can\"\n\nSee weaknesses for main comments to answer.\n\nLIMITATIONS:\nLimitations are reasonably identified in the Conclusions section of the paper.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-13"}