Gary P. T. Choi

Gary P. T. Choi, Chris H. Rycroft

In this paper, we are concerned with the problem of creating flattening maps of simply-connected open surfaces in $\mathbb{R}^3$. Using a natural principle of density diffusion in physics, we propose an effective algorithm for computing density-equalizing flattening maps with any prescribed density distribution. By varying the initial density distribution, a large variety of mappings with different properties can be achieved. For instance, area-preserving parameterizations of simply-connected open surfaces can be easily computed. Experimental results are presented to demonstrate the effectiveness of our proposed method. Applications to data visualization and surface remeshing are explored.

Yuchen Guo, Qiguang Chen, Gary P. T. Choi et al.

In medical imaging, surface registration is extensively used for performing systematic comparisons between anatomical structures, with a prime example being the highly convoluted brain cortical surfaces. To obtain a meaningful registration, a common approach is to identify prominent features on the surfaces and establish a low-distortion mapping between them with the feature correspondence encoded as landmark constraints. Prior registration works have primarily focused on using manually labeled landmarks and solving highly nonlinear optimization problems, which are time-consuming and hence hinder practical applications. In this work, we propose a novel framework for the automatic landmark detection and registration of brain cortical surfaces using quasi-conformal geometry and convolutional neural networks. We first develop a landmark detection network (LD-Net) that allows for the automatic extraction of landmark curves given two prescribed starting and ending points based on the surface geometry. We then utilize the detected landmarks and quasi-conformal theory for achieving the surface registration. Specifically, we develop a coefficient prediction network (CP-Net) for predicting the Beltrami coefficients associated with the desired landmark-based registration and a mapping network called the disk Beltrami solver network (DBS-Net) for generating quasi-conformal mappings from the predicted Beltrami coefficients, with the bijectivity guaranteed by quasi-conformal theory. Experimental results are presented to demonstrate the effectiveness of our proposed framework. Altogether, our work paves a new way for surface-based morphometry and medical shape analysis.

Yanwen Huang, Lok Ming Lui, Gary P. T. Choi

Many imaging problems require computing spatial transformations induced by spatially varying intensity, feature, or density fields. Canonical examples include distortion correction, deformable image registration, atlas-based segmentation, and deformation-driven image analysis. These tasks can be formulated as geometric mapping problems in which the transformation is constrained to preserve local structure, control boundary behavior, or regulate angular distortion. Such formulations typically lead to variational models, diffusion processes, or elliptic partial differential equations. However, repeatedly solving high-resolution systems becomes computationally expensive when the underlying parameter fields vary across instances. In this work, we propose a resolution-free neural surrogate for geometric parameterization and mapping problems. Given a spatially varying parameter field $p:Ω\to\mathbb{R}^m$ and query locations $\{x_i\}_{i=1}^N\subsetΩ$, the model predicts mapped locations $\{u(x_i)\}_{i=1}^N$ on arbitrary structured or unstructured point sets. To avoid dependence on a fixed grid, we use a multi-resolution geometric encoding strategy that conditions the network on coordinate-augmented samples of the parameter field. The model is trained without labeled solution data by enforcing geometry-aware constraints derived from variational energies, diffusion-based density equalization, and quasi-conformal theory. Experimental results on quasi-conformal mapping and density-equalizing mapping problems are presented to demonstrate the effectiveness of our proposed method.

Trenton Lau, Gary P. T. Choi

The Normalized Maximum Likelihood (NML) codelength, or stochastic complexity, represents a principled criterion for universal coding. While recent coarea-based formulations provided a calculation method for smooth models, this framework collapses for the non-smooth estimators ubiquitous in modern machine learning (e.g., Lasso, Sparse SVMs). In this work, we provide a rigorous framework for computing the NML for regular path-differentiable Lipschitz (PDL) estimators. By applying classical geometric measure theory and bridging the coarea formula with conservative Jacobians, we prove that the stochastic complexity for non-smooth models is well-posed and theoretically consistent with the outputs of modern Automatic Differentiation. To compute this quantity exactly, we introduce the Propose-and-Project Metropolis-Hastings (PDL-PPMH) sampler, a geometric MCMC algorithm capable of traversing the non-differentiable level sets of the maximum likelihood estimator. We theoretically justify its components, including a stochastic tangent space proposal and a provably convergent non-smooth projection solver. We demonstrate the method's robustness by sampling from a high-dimensional Lasso posterior ($P=2000$), while simultaneously quantifying the computational scaling that governs the trade-off between exactness and mixing time. Crucially, we empirically demonstrate that our exact NML criterion provides a highly data-efficient alternative to cross-validation, achieving statistically indistinguishable predictive optima without requiring data splitting. Altogether, our work paves the way for the theoretical analysis of the NML codelength for regular non-smooth models.

Hangyu Li, Gary P. T. Choi

The study of shapes is one of the most fundamental problems in life sciences. Although numerous methods have been developed for the morphometry of planar biological shapes over the past several decades, most of them focus solely on either the outer silhouettes or the interior features of the shapes without capturing the coupling between them. Moreover, many existing shape mapping techniques are limited to establishing correspondence between planar structures without further allowing for the quantitative analysis or modelling of shape changes. In this work, we introduce FDA-QC, a novel planar morphometry method that combines functional shape data analysis (FDA) techniques and quasi-conformal (QC) mappings, taking both the boundary and interior of the planar shapes into consideration. Specifically, closed planar curves are represented by their square-root velocity functions and registered by elastic matching in the function space. The induced boundary correspondence is then extended to the entire planar domains by a quasi-conformal map, optionally with landmark constraints. Moreover, the proposed FDA-QC method can naturally lead to a unified framework for shape morphing and shape variation quantification. We apply the FDA-QC method to various leaf and insect wing datasets, and the experimental results show that the proposed combined approach captures morphological variation more effectively than purely boundary-based or interior-based descriptions. Altogether, our work paves a new way for understanding the growth and form of planar biological shapes.

Ka Ho Lai, Hei Tung Tsang, Gary P. T. Choi et al.

Origami structures, particularly Miura-ori patterns, offer unique capabilities for surface approximation and deployable designs. In this study, a constrained mapping optimization algorithm is designed for designing surface-aligned Miura-ori via a narrow band approximation of the input surface. The Miura-fold, embedded in the narrow band, is parameterized to a planar domain, and a mapping is computed on the parameter pattern by optimizing certain energy terms and constraints. Extensive experiments are conducted, showing the significance and flexibility of our methods.

Kaikwan Lau, Gary P. T. Choi

Generating novel, biologically plausible three-dimensional morphological structures is a fundamental challenge in computational evolutionary biology, hampered by extreme data scarcity and the requirement that generated shapes respect phylogenetic relationships among species. In this work, we present PhyloSDF, a phylogenetically-conditioned neural generative model for 3D biological morphology that integrates two innovations: (1) a DeepSDF auto-decoder regularized by a novel Phylogenetic Consistency Loss that structures the latent space to correlate with evolutionary distances (Pearson $r=0.993$); (2) a Residual Conditional Flow Matching (Residual CFM) architecture that factorizes generation into analytic species-centroid lookup and learned residual prediction, enabling generation from as few as ~4 specimens per species. We evaluate PhyloSDF on 100 micro-CT-scanned skulls of Darwin's Finches and their relatives across 24 species. The model generates novel meshes achieving 88-129% of real intra-species variation at the code level, with all 180 generated meshes verified as non-memorized. Residual CFM surpasses denoising diffusion (which fails entirely at this scale), standard flow matching (which mode-collapses to 3-6% variation), and a Gaussian mixture baseline in both fidelity (Chamfer Distance 0.00181 vs. 0.00190) and morphometric Fréchet distance (10,641 vs. 13,322). Leave-one-species-out experiments across 18 species demonstrate phylogenetic extrapolation capability, and smooth latent interpolations produce biologically plausible ancestral skull reconstructions.

Zhiyuan Lyu, Lok Ming Lui, Gary P. T. Choi

Surface parameterization is a fundamental task in geometry processing and plays an important role in many science and engineering applications. In recent years, the density-equalizing map, a shape deformation technique based on the physical principle of density diffusion, has been utilized for the parameterization of simply connected and multiply connected open surfaces. More recently, a spherical density-equalizing mapping method has been developed for the parameterization of genus-0 closed surfaces. However, for genus-0 closed surfaces with extreme geometry, using a spherical domain for the parameterization may induce large geometric distortion. In this work, we develop a novel method for computing density-equalizing maps of genus-0 closed surfaces onto an ellipsoidal domain. This allows us to achieve ellipsoidal area-preserving parameterizations and ellipsoidal parameterizations with controlled area change. We further propose an energy minimization approach that combines density-equalizing maps and quasi-conformal maps, which allows us to produce ellipsoidal density-equalizing quasi-conformal maps for achieving a balance between density-equalization and quasi-conformality. Using our proposed methods, we can significantly improve the performance of surface remeshing for genus-0 closed surfaces. Experimental results on a large variety of genus-0 closed surfaces are presented to demonstrate the effectiveness of our proposed methods.

Jiani Fang, Xiaoxiao Ding, Gary P. T. Choi

Knitted fabrics exemplify a broad class of architected materials capable of large deformations, enabling shape morphing, mechanical biocompatibility, and embedded multifunctionality without material damage. Although geometric nonlinearity has been intuitively utilized in their design, a quantitative description of stitch-resolved deformation and its temporal evolution remains lacking. Here, we introduce a geometric quantification framework that reconstructs smooth yarn centerlines and fabric surfaces from sparse yarn-level representations and extracts interpretable descriptors across dimensions. Applied to representative knitted structures, this framework resolves how global deformation is distributed among stitch reorientation, loop bending, surface bending, and dilation. Moreover, it reveals how regions of large geometric variation emerge, persist, and redistribute over time. Rather than directly measuring stress, these geometric descriptors define a unified geometric state space for comparing knitted structures and identifying candidate regions of mechanical localization. The framework provides a quantitative language for nonlinear deformation in knits and establishes a geometry-based representation that can be coupled to constitutive models, experimental measurements, and graph-based inverse-design workflows.

Zhiyuan Lyu, Gary P. T. Choi

Over the past several decades, geometric mapping methods have been extensively developed and utilized for many practical problems in science and engineering. To assess the quality of geometric mappings, one common consideration is their conformality. In particular, it is well-known that conformal mappings preserve angles and hence the local geometry, which is beneficial in many applications. Therefore, many existing works have focused on the angular distortion as a measure of the conformality of mappings. More recently, quasi-conformal theory has attracted increasing attention in the development of geometric mapping methods, in which the Beltrami coefficient has also been considered as a representation of the conformal distortion. However, the precise connection between these two concepts has not been analyzed. In this work, we study the connection between the two concepts and establish a series of theoretical results. In particular, we discover a simple relationship between the norm of the Beltrami coefficient of a mapping and the absolute angular distortion of triangle elements under the mapping. We can further estimate the maximal angular distortion using a simple formula in terms of the Beltrami coefficient. We verify the developed theoretical results and estimates using numerical experiments on multiple geometric mapping methods, covering conformal mapping, quasi-conformal mapping, and area-preserving mapping algorithms, for a variety of surface meshes in biology and engineering. Altogether, by establishing the theoretical foundation for the relationship between the angular distortion and Beltrami coefficient, our work opens up new avenues for the quantification and analysis of surface mapping algorithms.

Shunyu Yao, Gary P. T. Choi

Tube-like surfaces are widely encountered in geometry processing, engineering structures, and medical anatomy, yet their intrinsic longitudinal and circumferential topology is not well preserved by conventional planar annular or rectangular parameterization domains. In this work, we propose a conformal parameterization framework for open tube-like surfaces with two boundary components. The proposed method first constructs a fixed-boundary tubular parameterization by cutting the input mesh, computing a disk-to-rectangle conformal map, and lifting the result to a three-dimensional tubular domain. To reduce residual distortion introduced near the cut seam, we further introduce a localized quasi-conformal correction scheme formulated on an annular domain, which improves conformality while leaving regions away from the seam unchanged. To handle noisy or irregular input boundaries, we also develop a free-boundary variant based on boundary extension and cycle-Laplacian smoothing, allowing the prescribed boundary constraints to be imposed on artificial outer rings rather than directly on the original surface. Finally, we derive two conformal toroidal bending maps that transform the tubular parameterization into toroidal geometries while preserving the underlying tube topology. Experiments on synthetic tube meshes and real vascular surfaces demonstrate that the proposed framework produces low-distortion parameterizations, effectively mitigates seam-induced artifacts, improves robustness for boundary-noisy inputs, and provides flexible tubular and toroidal target domains for downstream surface processing tasks.

Hugo Hiu Chak Cheng, Gary P. T. Choi

Kirigami, the art of paper cutting, has been widely used in the modern design of mechanical metamaterials. In recent years, many kirigami-based metamaterials have been designed based on different planar tiling patterns and applied to different science and engineering problems. However, it is natural to ask whether one can create deployable kirigami structures based on the simplest forms of tilings, namely the monotile patterns. In this work, we answer this question by proving the existence of periodic and aperiodic monotile kirigami structures via explicit constructions. In particular, we present a comprehensive collection of periodic monotile kirigami structures covering all 17 wallpaper groups and aperiodic monotile kirigami structures covering various quasicrystal patterns as well as polykite tilings. We further perform theoretical and computational analyses of monotile kirigami patterns in terms of their shape and size changes under deployment. Altogether, our work paves a new way for the design and analysis of a wider range of shape-morphing metamaterials.

Yanwen Huang, Lok Ming Lui, Gary P. T. Choi

Density-equalizing map (DEM) serves as a powerful technique for creating shape deformations with the area changes reflecting an underlying density function. In recent decades, DEM has found widespread applications in fields such as data visualization, geometry processing, and medical imaging. Traditional approaches to DEM primarily rely on iterative numerical solvers for diffusion equations or optimization-based methods that minimize handcrafted energy functionals. However, these conventional techniques often face several challenges: they may suffer from limited accuracy, produce overlapping artifacts in extreme cases, and require substantial algorithmic redesign when extended from 2D to 3D, due to the derivative-dependent nature of their energy formulations. In this work, we propose a novel learning-based density-equalizing mapping framework (LDEM) using deep neural networks. Specifically, we introduce a loss function that enforces density uniformity and geometric regularity, and utilize a hierarchical approach to predict the transformations at both the coarse and dense levels. Our method demonstrates superior density-equalizing and bijectivity properties compared to prior methods for a wide range of simple and complex density distributions, and can be easily applied to surface remeshing with different effects. Also, it generalizes seamlessly from 2D to 3D domains without structural changes to the model architecture or loss formulation. Altogether, our work opens up new possibilities for scalable and robust computation of density-equalizing maps for practical applications.

Siheng Chen, Fabio Giardina, Gary P. T. Choi et al.

Geometric graph models of systems as diverse as proteins, robots, and mechanical structures from DNA assemblies to architected materials point towards a unified way to represent and control them in space and time. While much work has been done in the context of characterizing the behavior of these networks close to critical points associated with bond and rigidity percolation, isostaticity, etc., much less is known about floppy, under-constrained networks that are far more common in nature and technology. Here we combine geometric rigidity and algebraic sparsity to provide a framework for identifying the zero-energy floppy modes via a representation that illuminates the underlying hierarchy and modularity of the network, and thence the control of its nestedness and locality. Our framework allows us to demonstrate a range of applications of this approach that include robotic reaching tasks with motion primitives, and predicting the linear and nonlinear response of elastic networks based solely on infinitesimal rigidity and sparsity, which we test using physical experiments. Our approach is thus likely to be of use broadly in dissecting the geometrical properties of floppy networks using algebraic sparsity to optimize their function and performance.

Gary P. T. Choi, Amita Giri, Lalan Kumar

The parameterization of open and closed anatomical surfaces is of fundamental importance in many biomedical applications. Spherical harmonics, a set of basis functions defined on the unit sphere, are widely used for anatomical shape description. However, establishing a one-to-one correspondence between the object surface and the entire unit sphere may induce a large geometric distortion in case the shape of the surface is too different from a perfect sphere. In this work, we propose adaptive area-preserving parameterization methods for simply-connected open and closed surfaces with the target of the parameterization being a spherical cap. Our methods optimize the shape of the parameter domain along with the mapping from the object surface to the parameter domain. The object surface will be globally mapped to an optimal spherical cap region of the unit sphere in an area-preserving manner while also exhibiting low conformal distortion. We further develop a set of spherical harmonics-like basis functions defined over the adaptive spherical cap domain, which we call the adaptive harmonics. Experimental results show that the proposed parameterization methods outperform the existing methods for both open and closed anatomical surfaces in terms of area and angle distortion. Surface description of the object surfaces can be effectively achieved using a novel combination of the adaptive parameterization and the adaptive harmonics. Our work provides a novel way of mapping anatomical surfaces with improved accuracy and greater flexibility. More broadly, the idea of using an adaptive parameter domain allows easy handling of a wide range of biomedical shapes.

Daoping Zhang, Gary P. T. Choi, Jianping Zhang et al.

With the advancement of computer technology, there is a surge of interest in effective mapping methods for objects in higher-dimensional spaces. To establish a one-to-one correspondence between objects, higher-dimensional quasi-conformal theory can be utilized for ensuring the bijectivity of the mappings. In addition, it is often desirable for the mappings to satisfy certain prescribed geometric constraints and possess low distortion in conformality or volume. In this work, we develop a unifying framework for computing $n$-dimensional quasi-conformal mappings. More specifically, we propose a variational model that integrates quasi-conformal distortion, volumetric distortion, landmark correspondence, intensity mismatch and volume prior information to handle a large variety of deformation problems. We further prove the existence of a minimizer for the proposed model and devise efficient numerical methods to solve the optimization problem. We demonstrate the effectiveness of the proposed framework using various experiments in two- and three-dimensions, with applications to medical image registration, adaptive remeshing and shape modeling.

Ho Law, Gary P. T. Choi, Ka Chun Lam et al.

Image registration has been widely studied over the past several decades, with numerous applications in science, engineering and medicine. Most of the conventional mathematical models for large deformation image registration rely on prescribed landmarks, which usually require tedious manual labeling and are prone to error. In recent years, there has been a surge of interest in the use of machine learning for image registration. In this paper, we develop a novel method for large deformation image registration by a fusion of quasiconformal theory and convolutional neural network (CNN). More specifically, we propose a quasiconformal energy model with a novel fidelity term that incorporates the features extracted using a pre-trained CNN, thereby allowing us to obtain meaningful registration results without any guidance of prescribed landmarks. Moreover, unlike many prior image registration methods, the bijectivity of our method is guaranteed by quasiconformal theory. Experimental results are presented to demonstrate the effectiveness of the proposed method. More broadly, our work sheds light on how rigorous mathematical theories and practical machine learning approaches can be integrated for developing computational methods with improved performance.

Gary P. T. Choi, Di Qiu, Lok Ming Lui

In this work, we develop a framework for shape analysis using inconsistent surface mapping. Traditional landmark-based geometric morphometrics methods suffer from the limited degrees of freedom, while most of the more advanced non-rigid surface mapping methods rely on a strong assumption of the global consistency of two surfaces. From a practical point of view, given two anatomical surfaces with prominent feature landmarks, it is more desirable to have a method that automatically detects the most relevant parts of the two surfaces and finds the optimal landmark-matching alignment between those parts, without assuming any global 1-1 correspondence between the two surfaces. Our method is capable of solving this problem using inconsistent surface registration based on quasi-conformal theory. It further enables us to quantify the dissimilarity of two shapes using quasi-conformal distortion and differences in mean and Gaussian curvatures, thereby providing a natural way for shape classification. Experiments on Platyrrhine molars demonstrate the effectiveness of our method and shed light on the interplay between function and shape in nature.

Albert Pumarola, Jordi Sanchez, Gary P. T. Choi et al.

Recent advances in 3D human shape estimation build upon parametric representations that model very well the shape of the naked body, but are not appropriate to represent the clothing geometry. In this paper, we present an approach to model dressed humans and predict their geometry from single images. We contribute in three fundamental aspects of the problem, namely, a new dataset, a novel shape parameterization algorithm and an end-to-end deep generative network for predicting shape. First, we present 3DPeople, a large-scale synthetic dataset with 2.5 Million photo-realistic images of 80 subjects performing 70 activities and wearing diverse outfits. Besides providing textured 3D meshes for clothes and body, we annotate the dataset with segmentation masks, skeletons, depth, normal maps and optical flow. All this together makes 3DPeople suitable for a plethora of tasks. We then represent the 3D shapes using 2D geometry images. To build these images we propose a novel spherical area-preserving parameterization algorithm based on the optimal mass transportation method. We show this approach to improve existing spherical maps which tend to shrink the elongated parts of the full body models such as the arms and legs, making the geometry images incomplete. Finally, we design a multi-resolution deep generative network that, given an input image of a dressed human, predicts his/her geometry image (and thus the clothed body shape) in an end-to-end manner. We obtain very promising results in jointly capturing body pose and clothing shape, both for synthetic validation and on the wild images.

Gary P. T. Choi, Hei Long Chan, Robin Yong et al.

Shape analysis is important in anthropology, bioarchaeology and forensic science for interpreting useful information from human remains. In particular, teeth are morphologically stable and hence well-suited for shape analysis. In this work, we propose a framework for tooth morphometry using quasi-conformal theory. Landmark-matching Teichmüller maps are used for establishing a 1-1 correspondence between tooth surfaces with prescribed anatomical landmarks. Then, a quasi-conformal statistical shape analysis model based on the Teichmüller mapping results is proposed for building a tooth classification scheme. We deploy our framework on a dataset of human premolars to analyze the tooth shape variation among genders and ancestries. Experimental results show that our method achieves much higher classification accuracy with respect to both gender and ancestry when compared to the existing methods. Furthermore, our model reveals the underlying tooth shape difference between different genders and ancestries in terms of the local geometric distortion and curvatures.

Gary P. T. Choi, Bernard Chiu, Chris H. Rycroft

Carotid atherosclerosis is a focal disease at the bifurcations of the carotid artery. To quantitatively monitor the local changes in the vessel-wall-plus-plaque thickness (VWT) and compare the VWT distributions for different patients or for the same patients at different ultrasound scanning sessions, a mapping technique is required to adjust for the geometric variability of different carotid artery models. In this work, we propose a novel method called density-equalizing reference map (DERM) for mapping 3D carotid surfaces to a standardized 2D carotid template, with an emphasis on preserving the local geometry of the carotid surface by minimizing the local area distortion. The initial map was generated by a previously described arc-length scaling (ALS) mapping method, which projects a 3D carotid surface onto a 2D non-convex L-shaped domain. A smooth and area-preserving flattened map was subsequently constructed by deforming the ALS map using the proposed algorithm that combines the density-equalizing map and the reference map techniques. This combination allows, for the first time, one-to-one mapping from a 3D surface to a standardized non-convex planar domain in an area-preserving manner. Evaluations using 20 carotid surface models show that the proposed method reduced the area distortion of the flattening maps by over 80% as compared to the ALS mapping method.

Chun Pang Yung, Gary P. T. Choi, Ke Chen et al.

With the advancement in the digital camera technology, the use of high resolution images and videos has been widespread in the modern society. In particular, image and video frame registration is frequently applied in computer graphics and film production. However, conventional registration approaches usually require long computational time for high resolution images and video frames. This hinders the application of the registration approaches in the modern industries. In this work, we first propose a new image representation method to accelerate the registration process by triangulating the images effectively. For each high resolution image or video frame, we compute an optimal coarse triangulation which captures the important features of the image. Then, we apply a surface registration algorithm to obtain a registration map which is used to compute the registration of the high resolution image. Experimental results suggest that our overall algorithm is efficient and capable to achieve a high compression rate while the accuracy of the registration is well retained when compared with the conventional grid-based approach. Also, the computational time of the registration is significantly reduced using our triangulation-based approach.