Population-level analysis is an essential task in brain research. Recent studies have investigated brain development and aging patterns using data from large age ranges (Habes et al., 2016, Mills et al., 2021), and some have created comprehensive brain charts spanning the entire human lifespan (Bethlehem et al., 2022, Sun et al., 2024). In such studies, standard brain templates are needed to register these cross-individual and cross-age data into a common reference space to ensure data consistency and comparability. To generate these templates, researchers typically use group-wise registration to establish anatomical correspondences across all images in a group, creating a standard structural brain template that serves as an important reference for population-specific image analysis. Although conventional and deep network-based algorithms have been proposed (Liu and Wang, 2014, Dey et al., 2021), group-wise registration is faced with some challenges in a large number of brain images with large anatomical differences, such as computationally expensive and the registration accuracy is difficult to guarantee. Therefore, accurate and rapid group-wise registration is in great demand which can be applied for template construction and population imaging analysis.

Single reference-based group-wise registration is considered an intuitive extension of pair-wise registration in terms of spatially mapping all the images to the common space of a selected or generated reference image. Initially, the reference image was mainly either chosen from the image group (Collins et al., 1994, Park et al., 2005, Uhlich et al., 2018) or by computing the median of the image manifold (Yang et al., 2009) followed by minimizing the sum of the geodesic distances from all images to the template. However, in these methods, the heavy reliance on a selected reference image poses unavoidable bias to the final registration (Ding and Niethammer, 2022) and consequently may lead to inaccurate demographic analysis. To relieve the bias, the group-mean-based methods (Joshi et al., 2004, Wu et al., 2011) computed the mean of all images as a reference image in each iteration, then registered all group images in accordance to the latest estimated reference image, thus maximizing the similarity of images within the entire group. Avants et al. (2010) introduced a more robust approach to group-wise registration by leveraging symmetric normalization (SyN) and optimizing both forward and backward transformations to achieve diffeomorphic mappings. Although these group-mean-based methods can alleviate the dependence on a particular image, the estimation of template image initializing with a fuzzy mean image affects the overall accuracy of the group-wise registration.

The group-wise registration free from the constraints of a single reference offers advantages for the images with large inter-subject anatomical variations by avoiding the bias and blur due to the selection and estimation of a reference image. Clustering-based methods (Wang et al., 2009, Wang et al., 2010, Jia et al., 2010) were proposed to cluster images into small-scale subgroups and then the intra-subgroup registration was performed to generate its center reference image that was the mean or median of all aligned images. Finally, all the center reference images or sub-templates were further registered to achieve group-wise registration. Since the group-wise registration at higher hierarchies was guided by subgroup reference images, the registration errors would be inevitably accumulated and escalated. Instead, reference-free graph-based methods (Ying et al., 2014, Fu et al., 2020) improved registration performance by taking account of the global and local data distribution of the registration population. Specifically, in these methods, the image manifold was represented with a graph in which nodes represented images and the edges defined the geodesic pathway between the two nodes, and then, the graph shrank gradually in the meanwhile preserving its topology. While the optimization was based on the global data distribution of the population, the registration between images was still iterative optimization process that was computationally expensive and hence limited their application to large-scale populations.

Due to their efficiency in inference cases, convolutional neural networks (CNNs) have been trained to establish anatomical correspondences within all images in the group. A common deep group-wise registration mechanism was to apply the existing deep pair-wise registration framework (Che et al., 2023, Fan et al., 2019, Zhu et al., 2019, van der Ouderaa et al., 2020, He and Chung, 2020) to enable registration of multiple images to their geodesic average, in which all the images were in turn defined as reference images. In these deep group-wise registration models, the input images were fed into multiple channels of the deep learning network, which posed strict limitations on the registration capacity. Besides, in view of the representation learning ability of the deforming autoencoder (Shu et al., 2018), researchers (Siebert and Heinrich, 2020, Gupta et al., 2021, He and Chung, 2022) adapted the concept of deforming autoencoders to obtain the latent representation of an individual image. Then, the template was reconstructed from the latent vectors through the common or average features of a set of images. However, it is challenging for the deforming autoencoders to learn the specific representations of the images with large anatomical deformation and properties, and hence limits their adaptability to the large population.

More recently, based on the generative models, deformable template estimation methods were proposed (Dalca et al., 2019, Dey et al., 2021, Yu et al., 2020, Pei et al., 2021) to generate either universal or conditional templates. Then, these templates were deformed to register with each individual image based on a similarity loss or a discriminator and improved both registration accuracy and template sharpness. Subsequently, to further improve the registration performance, Ding and Niethammer (2022) extended the previous framework (Dalca et al., 2019) by incorporating pair-wise image losses and by computing their alignments in template space as well as in image space. These methods can generate deformable templates under specific conditions with higher efficiency compared to the state-of-the-art (SOTA) methods. However, these template construction models were performed in one step by simply registering each image to a reference image (e.g., the mean of all training images) directly, and hence lacked adaptability when anatomical variations were large across images within the group. Besides, the template generation for a group required training models for specific groups, which reduced the efficiency of the model.

For 3D medical images of a large population with diverse variations, the current group-wise registration is facing three major challenges including (1) the model’s adaptability to large variations in the images, (2) registration efficiency, and (3) the model’s capacity to large-scale population. To address these challenges, we propose a novel hierarchical group-wise registration framework to construct age-related template images for brain magnetic resonance imaging (MRI) populations (see Fig. 1). Firstly, to enhance the model’s adaptability to large variation populations, we propose a progressive subgrouping strategy to obtain the hierarchical sequence of partitions in an integrated manner, which can provide variation reduction prior to group-wise registration. Secondly, to improve registration efficiency and model capacity, we design a nested hierarchical group-wise registration module, including hierarchical templates generated from local subgroups to the global population, hierarchical local manifold shrink registration within subgroups, and the deformation fields are estimated by a hierarchical deep registration network based on our previous work (Che et al., 2023).

Our major contributions can be summarized as below:

Different from typical hierarchical group-wise registration, the proposed NHGR embraces nested hierarchical strategy from population level, subgroup level, and image level to enhance the efficiency and capacity of the model for large-scale populations.

A novel subgrouping strategy is proposed by progressively clustering images with similar features at expanding scale factors, reducing anatomical variations prior to group-wise registration.

Further, the age-related sub-templates are constructed from subgroups by forwarding sub-templates to involve registration at higher hierarchies to reduce the registration burden for larger subgroup size, and backward passing deformation fields to further refine the registration of subject images at lower hierarchies.