Coordinate-wise Projection & Registration

• Coordinate-wise projection and registration is a framework that combines order-preserving convex projections with diverse imaging models to map and align multidimensional data.
• The methodology employs tools like random linear embeddings, closed-form Euclidean projections, and ADMM-based optimizations to enhance computational efficiency and accuracy.
• It underpins multi-modal registration tasks by integrating coordinate-system mapping, differentiable rendering, and projection-based observation models to enable precise and scalable solutions.

Coordinate-wise projection and registration denotes a family of mathematical and computational constructions in which either a projection operator is required to respect a coordinate-wise order, or alignment is inferred after data are mapped through explicit projection models. In convex analysis, the term refers to monotone metric projection under the coordinate-wise partial order on $\mathbb{R}^m$, with isotonic regression cones providing the central example (Németh et al., 2015). In registration, the same vocabulary appears in several distinct but related senses: low-dimensional random projections for information-theoretic matching, projective imaging operators for 2D/3D alignment, slice-wise projections for map merging, and calibrated coordinate-system mappings for histology or helioprojective coordinates (Szabo et al., 2012, Gao et al., 2020, Unlu et al., 2024, Feenstra et al., 2023).

1. Terminological scope and recurring abstractions

Across the cited literature, “projection” does not denote a single operator. It can mean Euclidean metric projection onto a convex set, a random linear embedding, a perspective or cone-beam imaging map, a projection of a 3D object onto a 2D detector, a slice-wise reduction of a 3D map to 2D occupancy images, or the transfer of measurements into a calibrated coordinate system. “Coordinate-wise” is similarly overloaded: in order theory it refers to the partial order induced by the nonnegative orthant, whereas in imaging it may refer to an explicit projection direction or to a target coordinate frame rather than to independent optimization of individual coordinates (Németh et al., 2015, Park et al., 2023).

Sense of projection	Projected object	Representative role
Metric projection	Vector onto a convex set or cone	Order-preserving isotonicity
Random linear projection	High-dimensional feature vectors	Fast entropy-based registration
Projective imaging	3D volumes, surfaces, or points	2D/3D pose estimation
Slice or coordinate mapping	3D maps or tracked locations	Map merging and coordinate transfer

A persistent misconception is that projection-based registration must register coordinates separately. One of the clearest counterexamples is the distributed information-theoretic image registration method based on random projections: it explicitly states that no explicit coordinate-wise registration is performed, because the random projection mixes coordinates linearly and alignment is determined globally through the entropy-based objective (Szabo et al., 2012). By contrast, in the dental CBCT/face-scan method, coordinate-wise projection is literal: the surface is projected along lines parallel to the $y$-axis onto a plane, and the resulting 2D landmarks are triangulated back into 3D (Park et al., 2023). The literature therefore uses a common word for several technically distinct mechanisms.

2. Coordinate-wise order and isotonic metric projection

The order-theoretic foundation is developed for $\mathbb{R}^m$ with the coordinate-wise partial order induced by the nonnegative orthant

$$\mathbb{R}^m_+ = \{x=(x^1,\dots,x^m)\in \mathbb{R}^m : x^i\ge 0,\ i=1,\dots,m\}.$$

The relation

$$x \le_{\mathbb{R}^m_+} y \quad \Longleftrightarrow \quad x^i \le y^i \ \text{for all } i$$

defines the standard coordinate-wise order. For a closed convex set $C\subset \mathbb{R}^m$, the metric projection is

$$P_C x \in C,\qquad \|x-P_Cx\|=\inf_{y\in C}\|x-y\|.$$

The set $C$ is $\mathbb{R}^m_+$-isotonic when

$$x \le_{\mathbb{R}^m_+} y \quad \Longrightarrow \quad P_Cx \le_{\mathbb{R}^m_+} P_Cy.$$

In this sense, increasing the data coordinate-wise cannot decrease the projected point coordinate-wise (Németh et al., 2015).

For cones, the structural constraints are severe. A closed generating cone with coordinate-wise isotonic projection must be polyhedral and representable through halfspaces whose normals are supported on at most two coordinates, with opposite-sign coefficients or one coefficient equal to zero. The resulting cones therefore arise from “two-variable” inequalities only. An immediate corollary is that such a cone has at most $y$0 facets, and a three-dimensional example with six facets shows that the bound is sharp in dimension $y$1 (Németh et al., 2015).

The same work distinguishes coordinate-wise isotonicity from the stronger intrinsic notion of an isotonic projection cone. Its general criterion states that a closed generating cone $y$2 is an isotonic projection cone if and only if its dual cone $y$3 is simplicial in its span and generated by vectors forming pairwise non-acute angles. This criterion becomes decisive for isotonic regression cones. Given positive weights $y$4 and a directed graph $y$5, the isotonic regression cone is

$y$6

and isotonic regression itself is the metric projection onto this cone after rescaling: $y$7

$y$8

Every isotonic regression cone $y$9 is automatically $\mathbb{R}^m$0-isotonic as a projection set. However, $\mathbb{R}^m$1 is an intrinsic isotonic projection cone if and only if the digraph has no two distinct edges with the same tail and no two distinct edges with the same head. In the connected case, the only isotonic regression cone with that stronger property is, up to permutation of coordinates, the weighted monotone cone (Németh et al., 2015). This establishes a sharp separation between ambient coordinate-wise monotonicity and intrinsic cone-induced monotonicity.

3. Computational projection as dimensionality reduction and closed-form optimization

A second line of work uses projection not to enforce order, but to make registration objectives computationally tractable. In distributed high-dimensional information-theoretic image registration, the similarity score is defined through negative joint entropy,

$\mathbb{R}^m$2

and can also be replaced by Rényi entropy, $\mathbb{R}^m$3-mutual information, or $\mathbb{R}^m$4-divergence. The method avoids direct entropy estimation in the original feature space by dividing the $\mathbb{R}^m$5 feature samples into $\mathbb{R}^m$6 disjoint groups, applying an independent random projection to each group, estimating entropy in each low-dimensional embedding, and averaging: $\mathbb{R}^m$7 The support for this strategy is that several entropy estimators are distance-based—kNN, generalized kNN graph, MST, wkNN, and recursive kd-partitioning plug-in estimation—and random projection approximately preserves the pairwise Euclidean distances on which those estimators rely. The reported empirical conclusions include reliable estimates for small random-projection dimensions $\mathbb{R}^m$8, useful group sizes $\mathbb{R}^m$9–$$\mathbb{R}^m_+ = \{x=(x^1,\dots,x^m)\in \mathbb{R}^m : x^i\ge 0,\ i=1,\dots,m\}.$$0, and speed-ups by several orders of magnitude relative to non-dimensionality-reduced estimation (Szabo et al., 2012).

A related but distinct use of projection appears in online rigid-body motion registration via low-rank subspace recovery. There the data matrix of tracked 3D features has rank at most $$\mathbb{R}^m_+ = \{x=(x^1,\dots,x^m)\in \mathbb{R}^m : x^i\ge 0,\ i=1,\dots,m\}.$$1 in the ideal rigid-body case, and initialization is performed by Robust PCA: $$\mathbb{R}^m_+ = \{x=(x^1,\dots,x^m)\in \mathbb{R}^m : x^i\ge 0,\ i=1,\dots,m\}.$$2 After outlier rejection and extraction of a 4-D shape subspace, each new observation is updated by sparse subspace projection,

$$\mathbb{R}^m_+ = \{x=(x^1,\dots,x^m)\in \mathbb{R}^m : x^i\ge 0,\ i=1,\dots,m\}.$$3

through the convex program

$$\mathbb{R}^m_+ = \{x=(x^1,\dots,x^m)\in \mathbb{R}^m : x^i\ge 0,\ i=1,\dots,m\}.$$4

The online stage replaces repeated combinatorial model fitting with a lightweight sparse projection step and yields one to two orders of magnitude speed-up compared to RANSAC solutions while maintaining robustness to noise, gross corruption, outliers, and missing data (Slaughter et al., 2011).

Projection also enters as the primitive operation in global least-squares registration over $$\mathbb{R}^m_+ = \{x=(x^1,\dots,x^m)\in \mathbb{R}^m : x^i\ge 0,\ i=1,\dots,m\}.$$5. After elimination of translations and lifting to a Gram matrix $$\mathbb{R}^m_+ = \{x=(x^1,\dots,x^m)\in \mathbb{R}^m : x^i\ge 0,\ i=1,\dots,m\}.$$6, the problem is split into two sets,

$$\mathbb{R}^m_+ = \{x=(x^1,\dots,x^m)\in \mathbb{R}^m : x^i\ge 0,\ i=1,\dots,m\}.$$7

and solved by ADMM through Euclidean projections onto $$\mathbb{R}^m_+ = \{x=(x^1,\dots,x^m)\in \mathbb{R}^m : x^i\ge 0,\ i=1,\dots,m\}.$$8 and $$\mathbb{R}^m_+ = \{x=(x^1,\dots,x^m)\in \mathbb{R}^m : x^i\ge 0,\ i=1,\dots,m\}.$$9. The dominant cost per iteration is the partial eigendecomposition of an $$x \le_{\mathbb{R}^m_+} y \quad \Longleftrightarrow \quad x^i \le y^i \ \text{for all } i$$0 matrix and $$x \le_{\mathbb{R}^m_+} y \quad \Longleftrightarrow \quad x^i \le y^i \ \text{for all } i$$1 singular value decompositions of $$x \le_{\mathbb{R}^m_+} y \quad \Longleftrightarrow \quad x^i \le y^i \ \text{for all } i$$2 matrices (Ahmed et al., 2019). In this setting, registration is driven by closed-form projection subproblems rather than by explicit correspondence search at every iteration.

4. Projective geometry, differentiable rendering, and 2D/3D registration

In 2D/3D registration, projection is frequently the forward model linking a 3D object to a 2D observation. A probabilistic rigid 3D-to-2D point-registration formulation emphasizes that perspective projection is necessary and orthographic projection is insufficient for aligning 3D model points with 2D image points. The full mapping is written in homogeneous coordinates as

$$x \le_{\mathbb{R}^m_+} y \quad \Longleftrightarrow \quad x^i \le y^i \ \text{for all } i$$3

and registration is posed as a missing-data maximum-likelihood problem with latent correspondences. Under the ECMPR framework, posterior correspondence probabilities $$x \le_{\mathbb{R}^m_+} y \quad \Longleftrightarrow \quad x^i \le y^i \ \text{for all } i$$4 are estimated in the E-step, and rigid parameters are updated either by traversal search or by least-squares estimation based on SVD. In the reported simulated experiments, both algorithms achieved 100% correct matching, with traversal more accurate but much slower and LSE completing in a fraction of a second (Wu, 2018).

Differentiable rendering generalizes this projective viewpoint to end-to-end learning. The Projective Spatial Transformer extends spatial transformers to projective geometry by building a sparse ray grid, transforming its control points by a pose $$x \le_{\mathbb{R}^m_+} y \quad \Longleftrightarrow \quad x^i \le y^i \ \text{for all } i$$5, interpolating voxel values in the volume, and summing along rays to form a projection: $$x \le_{\mathbb{R}^m_+} y \quad \Longleftrightarrow \quad x^i \le y^i \ \text{for all } i$$6 Because the full chain is differentiable with respect to both volume samples and pose parameters, the module supports gradient-based 2D/3D registration and the learning of a similarity function whose gradient approximates that of a geodesic pose loss (Gao et al., 2020).

Subsequent methods combine explicit projection with dense correspondences or learned updates. In single-view rigid 2D/3D registration, projected CT contour points are matched to fluoroscopy through RAFT-based iterative residual optical flow, and the resulting 2D displacements are lifted into a 3D rigid update by a weighted point-to-plane correspondence solver. The reported performance is $$x \le_{\mathbb{R}^m_+} y \quad \Longleftrightarrow \quad x^i \le y^i \ \text{for all } i$$7 mm mean re-projection distance error, 97.0% success ratio, $$x \le_{\mathbb{R}^m_+} y \quad \Longleftrightarrow \quad x^i \le y^i \ \text{for all } i$$8–$$x \le_{\mathbb{R}^m_+} y \quad \Longleftrightarrow \quad x^i \le y^i \ \text{for all } i$$9 mm capture range, and $C\subset \mathbb{R}^m$0 s runtime (Jaganathan et al., 2021). In multi-view registration, differentiable X-ray rendering is combined with cross-view pose and image constraints in $C\subset \mathbb{R}^m$1, followed by test-time pose refinement through the renderer; the reported result is $C\subset \mathbb{R}^m$2 mm mean target registration error on six specimens from the DeepFluoro dataset (Cui et al., 27 Jun 2025).

A complementary strategy avoids direct pose regression and instead estimates dense 2D-to-3D correspondences. In X-ray-to-CT rigid registration using scene coordinate regression, a fully convolutional U-Net predicts a 3D scene coordinate and an uncertainty value for each pixel, the valid correspondences are filtered by predicted variance, and the final rigid pose is recovered by PnP with RANSAC. The reported 50th-percentile performance is 3.79 mm mTRE on simulated test data and 9.65 mm projected mTRE on real fluoroscopic pelvis images (Shrestha et al., 2023). This suggests a recurring pattern: the projection model is fixed and explicit, while learning is used to predict the geometric quantities that make the inverse registration problem well posed.

5. Coordinate-structured reductions of 3D data

Some registration methods use coordinate-structured projections to turn a difficult 3D problem into simpler 2D or constraint-satisfaction subproblems. In automatic registration of dental CBCT and face-scan data, each 3D facial surface is rotated about the $C\subset \mathbb{R}^m$3-axis and projected along lines parallel to the $C\subset \mathbb{R}^m$4-axis onto a plane, producing two 2D projection images at angles $C\subset \mathbb{R}^m$5 and $C\subset \mathbb{R}^m$6. A pre-trained Dlib facial landmark detector yields 68 2D landmarks, 10 stable landmarks are selected on regions with minimal shape change, and the 3D landmark coordinates are reconstructed from the two projected $C\subset \mathbb{R}^m$7-coordinates by closed-form triangulation before least-squares rigid alignment and ICP refinement. For the three CBCT cases, the reported mean surface error is $C\subset \mathbb{R}^m$8 mm and the maximum surface error is $C\subset \mathbb{R}^m$9 mm; in a 10-subject MDCT experiment, the mean surface error is $$P_C x \in C,\qquad \|x-P_Cx\|=\inf_{y\in C}\|x-y\|.$$0 mm (Park et al., 2023).

For gravity-aligned map registration, a tomography-inspired method slices each 3D map into horizontal bands

$$P_C x \in C,\qquad \|x-P_Cx\|=\inf_{y\in C}\|x-y\|.$$1

projects each slice onto the $$P_C x \in C,\qquad \|x-P_Cx\|=\inf_{y\in C}\|x-y\|.$$2-plane as a 2D binary occupancy image, extracts ORB features, and estimates $$P_C x \in C,\qquad \|x-P_Cx\|=\inf_{y\in C}\|x-y\|.$$3 from 2D slice matches under every candidate height difference. The vertical offset $$P_C x \in C,\qquad \|x-P_Cx\|=\inf_{y\in C}\|x-y\|.$$4 is then chosen as the shift producing the largest consensus set of slice-wise pose hypotheses. The method estimates a 4-DoF transform

$$P_C x \in C,\qquad \|x-P_Cx\|=\inf_{y\in C}\|x-y\|.$$5

and is designed to be distributed and parallelizable. Reported memory use is around 100 MB in simulated tests, under 200 MB in noisy cases, and around 500 MB in a large indoor experiment (Unlu et al., 2024).

In sparse texture-less 3D scan alignment, alternating projection appears in a constraint-set sense. Sparse point clouds are re-parameterized into line segments, corners, and edge lines, and registration is formulated as the simultaneous satisfaction of intersection constraints and rigidity constraints. One AP iteration first projects primitive pairs so that line segments intersect or corners lie on edge lines, then projects back onto the rigidity set through rigid alignment. On Kinect data, the method uses 100X downsampled sparse input and still outperforms competing methods operating on full-resolution data; on KITTI, the reported average runtime is 22 ms for the AP solver, 2 ms for inlier counting, and 3 s for line fitting (Ranade et al., 2020). A closely related formulation connects the same line-intersection constraints to generalized relative pose estimation and introduces minimal solvers combining line intersections with plane correspondences (Ranade et al., 2019).

Projection-aware deep networks extend these reductions to large-scale point clouds. RegFormer projects outdoor LiDAR points to cylindrical pseudo-images, uses projection masks to suppress invalid pixels, processes the data with a projection-aware hierarchical transformer, and regresses rigid motion without descriptor matching or RANSAC (Liu et al., 2023). Here projection is neither order-theoretic nor merely visual; it is the structural device that makes large-scale registration computationally manageable.

6. Coordinate-system mapping, multimodal validation, and observation models

A further class of methods treats projection as a calibrated mapping between measurement coordinates and analysis coordinates. The Point Projection Mapping system for optical tissue sensing projects points of interest onto the tissue surface using a calibrated projector-camera setup, records snapshot images with and without projected points, and then transfers the measurement locations into histology space through deformable registration of the specimen image to the histology image. Base-plane calibration is performed by least-squares plane fitting, the camera-to-projector transform is modeled as a rigid 3D transform, and the specimen-to-histology registration uses an unsupervised VoxelMorph-style deformable network trained with mutual information. Reported projection accuracy is approximately 0.59 mm for a custom-built Kinect-projector setup and approximately 0.15 mm for an HP Sprout Pro G2 setup; across 30 breast specimens, Dice improves from median $$P_C x \in C,\qquad \|x-P_Cx\|=\inf_{y\in C}\|x-y\|.$$6 to $$P_C x \in C,\qquad \|x-P_Cx\|=\inf_{y\in C}\|x-y\|.$$7, and mutual information from median $$P_C x \in C,\qquad \|x-P_Cx\|=\inf_{y\in C}\|x-y\|.$$8 to $$P_C x \in C,\qquad \|x-P_Cx\|=\inf_{y\in C}\|x-y\|.$$9 after registration (Feenstra et al., 2023).

In solar imaging, accurate coordinate mapping is the endpoint of registration. APRIL registers ground-based coronal green-line images to SDO/AIA 211 Å images by combining a coarse similarity transform, local statistical correlation in polar coordinates, and RANSAC-based feature-point matching. The method uses 36 overlapping azimuthal subregions on the annulus from $C$0 to $C$1, iterates until $C$2 pixel and $C$3 pixel, and derives the solar center, solar radius, and polar-axis orientation needed to map the data to the Helioprojective Cartesian coordinate system. The reported mapping accuracy is no less than $C$4, and validation spans 100 observing days over an 11-year period (Sha et al., 23 Jul 2025).

Projection may also be the observation model itself rather than a mere preprocessing step. In elastic 3D-2D image registration for microscopy, the 2D image is modeled not by an X-ray integral but by convolution with a point-spread function followed by slicing in the focal plane,

$C$5

and a hyperelastic deformation $C$6 is estimated by minimizing

$C$7

A central claim is that out-of-plane motion can be inferred from the blur pattern induced by the microscope; the blur therefore becomes a source of depth information rather than a nuisance variable (Striewski et al., 2021). This suggests a broader interpretation of projection-based registration: what matters is not only the coordinate transform, but also the observation operator that determines which latent spatial degrees of freedom remain recoverable.

The most explicit methodological separation of these roles appears in the transformation-driven framework for generating comparable projection images from multimodal anatomical scenes. It represents volumes, surface meshes, landmarks, scans, and auxiliary objects as independently transformable scene entities and separates scene representation, projection geometry, acquisition, material interpretation, and presentation. The framework is positioned as useful for registration-related tasks, including validating 2D–3D registration and studying motion observability, but not as a classical registration engine itself (Pojda et al., 15 Jun 2026). That separation clarifies an important distinction running through the entire literature: some methods use projection to define the object of optimization, whereas others use it to define the observation under which registration is meaningful.

Taken together, these works show that coordinate-wise projection and registration is not a single algorithmic doctrine but a set of mathematically precise design choices about monotonicity, dimensionality reduction, projective observation, structured decomposition, and coordinate transfer. The strongest unifying theme is explicit control over how information is mapped from one space into another, whether the goal is monotone metric projection on $C$8, efficient estimation in projected feature spaces, rigid or nonrigid 2D/3D alignment, or the transfer of measurements into anatomically or physically meaningful coordinate systems.