Point Coordinate Mapping (PCM)

• Point Coordinate Mapping (PCM) is a suite of techniques that map, index, and transform point coordinates in geometric, signal, and image domains.
• PCM methods range from classical quantization and affine transforms to modern learned embeddings, enabling robust 3D rendering, mesh generation, and spatial encoding in deep learning.
• Integrating explicit coordinate anchors, affine-invariant schemes, and confidence modeling, PCM enhances accuracy and efficiency in applications like point cloud analysis and vision-language grounding.

Point Coordinate Mapping (PCM) encompasses a class of mathematical and algorithmic techniques for mapping, indexing, or transforming coordinates of points in geometric, signal, or image domains. PCM variants serve as foundational tools for geometric representation, positional encoding in deep learning, quantization, mesh generation, 3D rendering, and robust correspondence estimation. This article presents a comprehensive survey of PCM formalizations: from classical quantization and 3D graphics to contemporary applications in point cloud networks, vision-LLMs, and geometric-invariant matching.

1. Fundamentals and Core Mathematical Formalisms

PCM methods define deterministic or learnable mappings between geometric domains. At their core, these methods specify—by closed-form, learnable embedding, or probabilistic assignment—a rule to map the raw coordinates of a point $p \in \mathbb{R}^d$ to a new representation or another space. Canonical forms of PCM in the literature include:

• Quantization-based PCM: Discretizing (quantizing) signal coefficients, as in pulse-code modulation (PCM) applied to tight frame expansions, where the quantizer $Q_\delta$ maps a real analysis coefficient $c$ to the nearest level in a uniform grid, $A = \delta \mathbb{Z}$, i.e., $$Q_\delta(c) = \delta \left\lfloor \frac{c}{\delta} + \frac{1}{2} \right\rfloor$$ (Wang et al., 2011).
• Affine and polynomial coordinate transforms: Linear (or polynomial) functions mapping between spaces, e.g., 3D to 2D projection $P_{2D} = [x_0, y_0] + P_{3D} M_{3\times2}$ for visualization or graphics, or $x = x(u,v), y = y(u,v)$ employing least-squares polynomial fits for curvilinear mesh generation (Dey et al., 2010, Isshiki et al., 2017).
• Learned spatial mappings: Neural networks trained to map points' coordinates or pixel locations into positional embeddings, as in Point Cloud Mamba—using a per-stage linear layer $\mathrm{Emd}_{\mathrm{pos}} = \mathrm{Linear}(x)$ with $x \in \mathbb{R}^3$ (Zhang et al., 1 Mar 2024).
• Grid or anchor-based discrete mappings: Explicit insertion of coordinate anchors (e.g., "RULER tokens" representing fixed pixel positions) within model token sequences, letting a model reference and interpolate between explicit coordinate anchors rather than relying solely on continuous regression (Wang et al., 3 Oct 2025).
• Affine-invariant probabilistic mappings: Encoding each pixel’s coordinate as a barycentric coordinate relative to three matched anchors (vertices), thereby achieving affine invariance, as in Probabilistic Coordinate Fields (PCFs) (Zhao et al., 2023).

2. PCM in Quantization and Signal Representation

PCM originated in signal quantization as pulse-code modulation, where real-valued analysis coefficients $c_j$ of a signal (when analyzed in a frame or basis $\mathcal{F} = \{e_j\}$) are quantized via $q_j = Q_\delta(c_j)$ for step size $\delta$. In tight frame theory:

• Tight frame reconstruction: For a unit-norm tight frame $\mathcal{F} \subset \mathbb{R}^d$, $$x = \frac{d}{N} \sum_{j=1}^N \langle x, e_j \rangle e_j$$ yields perfect reconstruction. Pulse-code modulation instead reconstructs $\tilde{x} = \frac{d}{N} \sum_{j=1}^N q_j e_j$.
• PCM error bounds: The quantization error for such representations is governed by $\delta$ (quantization granularity) and the frame redundancy $N$, but notably, increasing redundancy does not drive the error to zero, in contrast to Sigma-Delta quantization. Specifically, for analytic uniform frame paths in $\mathbb{R}^d$, $\|x - \tilde{x}\| = O(\delta^{(d+1)/2})$ in the large $N$ regime (Wang et al., 2011).

This formulation positions PCM as both a practical quantization tool and a subject of harmonic analytic paper regarding error rates and optimal frame designs.

3. Coordinate Mapping for 3D Rendering and Numerical Mesh Generation

PCM appears prominently in geometric computing, especially in computer graphics and numerical analysis for PDEs:

• 3D-to-2D oblique projection: For 3D rendering without complex graphics engines, PCM implements a parallel projection mapping $(x, y, z) \mapsto (x', y')$ with user-controlled axes, angles, and optional depth compression. The technique allows for closed-form inverse mapping and for the direct integration of arbitrary 3D affine transforms via precomputed matrices, maintaining $O(N)$ complexity for $N$ points (Dey et al., 2010).
• Algebraic mapping generation for curvilinear coordinates: In mesh generation and computational PDEs, a PCM approach fits polynomials to match a rectangular “computational” domain to an irregular “physical” mesh, solving $x(u, v) = \sum_{m, n} A_{mn} u^m v^n$ and $y(u,v) = \sum_{m, n} B_{mn} u^m v^n$ by least-squares over prescribed data points, forming the basis for efficient mesh generation and PDE discretization (Isshiki et al., 2017).

PCM here provides an explicit, invertible mapping framework, supporting computational geometry and numerical methods beyond the capabilities of pure lookup or black-box approaches.

4. PCM in Point Cloud Deep Learning

Recent advances in point cloud analysis employ PCM to inject geometric structure into neural models:

• Spatial Coordinate Mapping as Positional Embedding: PCM in Point Cloud Mamba (PCM) replaces discrete or periodic positional encodings with a linear transformation of continuous 3D coordinates, $\mathrm{Emd}_{\mathrm{pos}} = \mathrm{Linear}(x)$, where $x$ represents original point coordinates. This approach integrates spatial information while respecting geometric irregularity, contributing to improved modeling capacity and SOTA performance in shape classification and segmentation (Zhang et al., 1 Mar 2024).
• Consistent Traverse Serialization (CTS): PCM in this context serializes unordered 3D points into 1D sequences that preserve local spatial proximity, critical for efficient state-space modeling. Six permutations of axes encoding (“xyz”, “xzy”, etc.) are employed, each augmented with learned prompts, to facilitate global feature propagation in bidirectional state space models (Zhang et al., 1 Mar 2024).
• Ablation and empirical benefit: Multiple PCM components—linear spatial mapping, permutation-based serialization, local KNN context, order prompts—are empirically validated to improve accuracy. For example, on ModelNet40, PCM achieves 93.4% OA, outperforming previous SOTA (Zhang et al., 1 Mar 2024).

5. Position-to-Coordinate Mapping in Vision-Language and GUI Grounding

PCM is a core innovation for robust visual grounding tasks, particularly in aligning fine-grained language references to high-resolution screen elements:

• Explicit coordinate anchors ("RULER tokens"): Rather than regressing pixel coordinates as raw output tokens, PCM injects explicit position markers at regular intervals (e.g., every $s$ patches), allowing models to reference coordinate anchors and predict local offsets, $x=c_j + \delta$, where $c_j$ is the anchor position (Wang et al., 3 Oct 2025).
• Interleaved Rotary Positional Encoding (I-MRoPE): PCM replaces asymmetric 2D frequency allocation with frequency-interleaved encoding, ensuring equal spatial representation for both width and height axes during attention (Wang et al., 3 Oct 2025).
• Generalization and performance: By decoupling position referencing from full-range coordinate regression, PCM methods markedly improve performance and stability, particularly on unseen high-resolution domains. Gains up to +5.3 percentage points in element accuracy were observed over strong baselines on the ScreenSpot-Pro benchmark (Wang et al., 3 Oct 2025).

6. PCM for Geometric-Invariant Matching and Affine-Invariant Fields

Advanced PCM formulations target geometric invariance and robust confidence modeling:

• Barycentric PCM for affine invariance: Probabilistic Coordinate Fields (PCFs) encode each pixel’s position as barycentric coordinates relative to three robustly-matched anchor points, achieving full affine invariance (Zhao et al., 2023).
• Confidence via Gaussian mixture modeling: PCFs use a per-pixel GMM over coordinate reconstruction errors to learn when the barycentric encoding is reliable, yielding a confidence map $M$ and enabling “trusted” coordinate masking.
• Plug-in utility: PCFs seamlessly augment dense and sparse matching pipelines (e.g., SuperGlue, LoFTR) by offering both geometry-invariant embedding and learned confidence masking, resulting in consistent empirical improvements across multiple datasets and tasks (Zhao et al., 2023).

7. Comparison Table of PCM Applications and Methodologies

Domain	PCM Principle	Key Application/Benefit
Signal Processing	Quantization (round-to-nearest)	Tight-frame encoding, error rates (Wang et al., 2011)
Computer Graphics	Oblique/affine projection	Fast 3D-to-2D mapping (Dey et al., 2010)
Numerical Analysis	Polynomial mapping (least-squares)	Curvilinear mesh for PDEs (Isshiki et al., 2017)
Point Cloud ML	Linear coordinate embedding, CTS	Positional encoding, state-space models (Zhang et al., 1 Mar 2024)
Vision-Language Grounding	Discrete anchors, I-MRoPE	Resolution-agnostic GUI localization (Wang et al., 3 Oct 2025)
Geometric Correspondence	Barycentric coordinate fields (PCFs)	Affine-invariant, confidence-masked matching (Zhao et al., 2023)

Each PCM variant addresses characteristic challenges—quantization accuracy, geometric interpretability, spatial encoding, robust generalization, or invariance under transformation—by customizing coordinate mapping formalism and integration strategy for the domain.

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PCM methods unify a diverse set of applications in signal processing, geometry, and modern deep models under the principle of explicit, efficient, and often learnable transformation between coordinate representations. Empirical evidence highlights the importance of tailored PCM formalizations—ranging from analytic quantization to transformer-compatible anchor-based mappings—for achieving accuracy, robustness, and generalization across high-dimensional, irregular, or changing geometric domains.