Landmark-Based Geometry

• Landmark-based geometry is the study of modeling and analyzing structures via a finite set of key points, enabling precise shape registration and correspondence.
• It leverages rigorous mathematical foundations, including manifold theory and invariant metrics, to support applications in SLAM, medical imaging, and computational anatomy.
• Advanced algorithms and learning methods, from unsupervised mapping to deep neural registration, optimize landmark discovery and matching for efficient geometric inference.

Landmark-based geometry refers to the modeling, analysis, and inference of geometric structure by means of a finite set of distinguished points—“landmarks”—on or in a physical, anatomical, graphical, or conceptual domain. Landmark-based approaches have become fundamental across geometric modeling, statistical shape analysis, registration, SLAM, morphometrics, machine learning, and computational anatomy. Rigorous mathematical definitions and algorithmic strategies have enabled unsupervised, semi-supervised, and supervised discovery, matching, and use of landmarks for both interpretability and efficiency.

1. Mathematical Foundations and Representation

Landmark-based geometry centers on the configuration space of $n$ distinct points $$Q_n = M^n(\mathbb{R}^d) = \{ (x_1,\dots, x_n) \in (\mathbb{R}^d)^n : x_i \neq x_j \ \forall i\neq j \}$$, which forms the open manifold of $n$-tuples of distinct locations in $\mathbb{R}^d$ (Grong et al., 2024). This space underlies the definition of "landmark manifolds" and is equipped with metrics that reflect intrinsic and extrinsic geometry.

Landmark objects in higher-level geometric or semantic domains are often embedded using associated manifolds or algebraic structures. For example, 3D object landmarks for SLAM may be parametrized on the symmetric positive-definite (SPD) matrix manifold $\mathrm{SPD}(3)$, encoding both scale and rotational pose in a single element $P$ representing an ellipsoid (object landmark) (Hu et al., 2022). Array-based approaches such as point distribution models ($\mathbf{s} = [p_1^\top; ...; p_n^\top]$) facilitate PCA and shape variability analysis (Chao et al., 2023).

Landmarks generalize to richer geometric entities: in algebraic settings, lines and planes are landmarks in the affine Grassmannian $\mathrm{Graff}(k,n)$, whose embedding in the classical Grassmannian enables invariant comparisons under rigid-body transformations (Lusk et al., 2022). In functional geometry, boundaries around landmarks are modeled through Dirichlet-Stekloff eigenbases for landmark-adapted functional maps (Panine et al., 2022).

2. Landmark Discovery, Selection, and Learning

Fundamental to landmark-based geometry is the selection or discovery of landmark sets. Strategies fall into two main classes: data-driven optimization and probabilistic selection.

Active landmark selection on manifolds is guided by combined geometric and algebraic criteria. The unified method based on Gershgorin disk deletion minimizes a surrogate bound on semi-supervised error—balancing condition number minimization (for stability) and coverage/maximal affinity (for representativity). The iterative greedy algorithm (GCLS) selects landmarks that optimize the learning error bound $Q(\bar{\mathcal{L}})$ over the alignment matrix (Xu et al., 2017). This approach scales as $\mathcal{O}(NL)$ in the number of samples $N$ and landmarks $L$ and is robust to noise, outperforming classical random, k-means, volume, and kernel-based sampling in regression and classification contexts.

Stochastic discovery may proceed via Gaussian process (GP) priors on shape or surface, using curvature-weighted kernels to drive posterior variance-maximization (active learning). Each landmark is chosen to maximize the predictive uncertainty of the GP, generating designs that cover spatial and semantic features (Gao et al., 2018). Downstream geometric consistency is enforced through multimodal signature filtering and distortion minimization.

In self-supervised contexts, deep neural networks can be trained to extract landmarks by optimizing image registration or deformation reconstruction objectives. By backpropagating through differentiable warping layers (thin-plate splines, diffeomorphisms), the network discovers features that act as effective control points for geometric registration tasks (Bhalodia et al., 2021, Chao et al., 2023). Landmark consistency across instances and population-level correspondence is further driven by anatomical consistency objectives and cross-subject alignment loss.

Contrastive representation learning exposes geometric equivariance and invariance: landmark-sensitive hypercolumn features emerge in deep networks trained on instance- and spatial-discriminative contrastive loss, naturally aligning activations with geometric structures across samples (Cheng et al., 2020).

3. Landmark-Based Embedding, Registration, and Matching

Landmark-based geometry supports a range of embedding and registration algorithms, exploiting both local and global geometric constraints.

Unsupervised neural-geometric mapping, such as Landmark2Vec, casts landmark localization as an embedding optimization: the positions of landmarks are recovered up to a global similarity transform through stochastic-neural factorization of n-gram co-occurrence statistics (Razavi, 2020). The embedding (weight matrix $W$ from input-to-hidden in a bottleneck neural network) represents coordinates in $\mathbb{R}^d$, with cross-entropy loss on neighbor prediction providing the unsupervised signal.

Manifold-wide embedding is further enabled through Laplacian-based spectral algorithms. Neumann eigenmaps (NeuMaps) use the renormalized Neumann Laplacian on the landmark subgraph to provide robust, isometric embeddings that accurately capture the reflecting diffusion distance. The method allows for efficient Nyström-style out-of-sample extension and is stable to the removal of significant landmark points (Sule et al., 10 Feb 2025).

In shape matching, functional maps on landmark-adapted bases (using block-orthogonal Dirichlet–Steklov eigenfunctions) prescribe exact landmark preservation, near-conformal correspondence, and efficient energy-based optimization (Panine et al., 2022). Landmark constraints are realized directly through the harmonic basis decomposition, and the coupled functional map is block-diagonal in the presence of multiple landmark boundaries.

Geodesic registration and diffeomorphic landmark matching use Riemannian geometric flows. Here, Hamilton’s equations are solved on $Q = \Omega^M$ via the momentum representation (encoded in an RKHS via kernel $K_V$). The Bayesian ensemble Kalman filter provides a derivative-free method for inferring initial momentum that achieves effective landmark matching and deformation estimation (Bock et al., 2021).

In high-dimensional anatomical segmentation, hybrid and graph convolutional networks (HybridGNet) decode plausible landmark-based shape sources by spectral Chebyshev convolution on a fixed adjacency graph, efficiently enforcing geometric plausibility and robustness to occlusion (Gaggion et al., 2021).

4. Invariant Landmark Metrics and Optimality

Invariant metric construction on landmark sets is crucial for registration and matching robustness. On the affine Grassmannian, lines and planes are compared using the shifted Grassmannian distance, which becomes invariant to Euclidean motions (SE(3)) when a pre-matching translation is subtracted prior to embedding (Lusk et al., 2022). Graph-based association frameworks such as CLIPPER exploit these invariants to find densest cliques of self-consistent correspondences, greatly improving matching recall and real-time registration.

In object-level SLAM, representations of objects/landmarks as SPD(3) elements eliminate singularities and ambiguities arising from separate rotation/translation/scale parameterizations. Affine-invariant Riemannian metrics and explicit retraction operations maintain global consistency and prevent optimization degeneracies, yielding faster and more robust convergence (Hu et al., 2022).

In the context of state estimation and SLAM, landmark-based optimization problems formulated as QCQP with connection Laplacians and incidence matrices admit efficient global optimality certification. Marginalizing landmarks out of the state space preserves problem size and enables scalable, certificate-based verification of solution optimality in linear time relative to the number of landmarks. Certificate reliability is enhanced by increased landmark-edge density, higher landmark counts, and measurement graph connectivity (Holmes et al., 2022).

5. Applications in Medical Imaging, Robotics, and Geodesy

Landmark-based geometry finds extensive application in computational anatomy, medical image registration, statistical morphometrics, autonomous robotics, and geo-localization.

In anatomical registration, quasi-conformal theory coupled with deep neural networks (LD-Net, CP-Net, DBS-Net) coordinates automatic landmark extraction along sulcal curves, prediction of low-distortion Beltrami coefficients, and ultra-fast quasi-conformal mapping, guaranteeing bijectivity and exact landmark matching at biologically pivotal surfaces (Guo et al., 2022).

Deep learning pipelines for 3D face geometry, such as M$^3$-LRN, simultaneously exploit multi-modal features—image, point, and model—and joint landmark-to-model regression, driving state-of-the-art results in dense reconstruction and face pose even under extreme conditions (Wu et al., 2021).

In geomatics and computer vision, semantic geolocation frameworks unite landmark identity (e.g., building-facade assignment) and geometric signature (facade projection, vanishing point inference) to statistically infer pose probabilities and reduce localization error in complex urban environments (Mousavian et al., 2016).

Functional and statistical morphometrics leverage GP landmarking and active learning to achieve expert-level correspondence and shape-cluster separability. Automatic, curvature-weighted landmarks outperform manual placement for coverage, matching, and population-level taxonomic resolution (Gao et al., 2018).

Self-supervised anatomical shape descriptors founded on landmark discovery via deformation objectives yield robust separation of disease classes and competitive prediction performance with minimal redundancy (Bhalodia et al., 2021, Chao et al., 2023).

6. Theoretical Controllability and Shape Model Simplification

Recent advances in the controllability of landmark manifolds prove that the orbits of $Q_n$ are completely reachable through compositions of flows of just two fixed ambient vector fields—independent of the number of landmarks $n$—by leveraging explicit bracket-generation and the Chow–Rashevsky theorem (Grong et al., 2024). This result implies that shape registration between finite sets of landmarks can be reduced to solving for a non-adaptive, low-dimensional sequence of flow times rather than optimization over the entire diffeomorphism group, thus dramatically reducing computational complexity in control-theoretic shape analysis.

7. Limitations, Extensions, and Open Problems

Landmark-based geometry is subject to several inherent and practical limitations. Disconnected landmark co-occurrence graphs lead to indeterminacy in embedding up to independent transforms (Razavi, 2020). High noise or symmetric arrangements can produce degeneracies, mirror flips, or ambiguity. Under-sampled or poorly placed landmarks risk major geometric distortion.

Extensions include integrating partial ground truth, exploiting deeper architectures for sensor fusion, exploring dynamic—time-varying—landmark embedding, and embedding in higher-dimensional or multimodal sensor spaces. Open problems involve improving computational bounds for combinatorial selection, proving optimal approximation ratios for greedy landmarking, extending frameworks to asymmetric or directed graphs, and synthesizing neural landmark selection with classical geometric criteria.

Landmark-based geometry remains foundational for interpretable, efficient, and rigorous geometric modeling in high-dimensional, data-intensive domains. Its evolution continues to be shaped by advances in manifold theory, kernel methods, probabilistic modeling, neural representation learning, and control theory.