Author: Gilles Colling
Date: 2026-05-20

# Advanced Topics

#### Gilles Colling

#### 2026-05-20

## Overview

This vignette serves three purposes:

1. Build intuition: Use pixel morphing as a visual analogy for assignment problems

2. Scale up: Demonstrate approximation strategies when exact LAP becomes infeasible

3. Scientific applications: Show how matching applies to ecology, physics, and chemistry

This vignette is different: Unlike the other couplr documentation, it emphasizes understanding over doing. If you’re looking to solve a matching problem today, start with `vignette("getting-started")` or `vignette("matching-workflows")`. Come here when you want to understand why these algorithms work and when approximations are appropriate.

### Who This Vignette Is For

Audience: Advanced users, researchers, algorithm developers, curious minds

Prerequisites:

- Familiarity with `lap_solve()` (`vignette("getting-started")`)

- Basic complexity analysis (Big-O notation)

- Interest in algorithm design or scientific computing

What You’ll Learn:

- Why exact LAP becomes infeasible for large n

- Three approximation strategies and their trade-offs

- How matching problems appear in ecology, physics, and chemistry

- Mathematical connections to optimal transport theory

Time to complete: 45-60 minutes (conceptual reading)

### Documentation Roadmap

Vignette | Focus | Difficulty
--- | --- | ---
Getting Started | Basic LAP solving, API introduction | Beginner
Algorithms | Mathematical foundations, solver selection | Intermediate
Matching Workflows | Production matching pipelines | Intermediate
Pixel Morphing | Scientific applications, approximations | Advanced

You are here: Pixel Morphing (Advanced)

### Why Pixels?

Pixels provide an ideal testbed for understanding assignment problems:

- Each pixel is an entity with measurable properties

- Color = feature (what it looks like)

- Position = spatial location (where it is)

- The matching is visually verifiable: you can see if it worked

The same algorithms that morph images smoothly also track particles in physics, align molecules in chemistry, and match vegetation plots in ecology.

### Scientific Applications at a Glance

Domain | Entities | Features | Spatial
--- | --- | --- | ---
Ecology | Vegetation plots | Species composition | Geographic location
Physics | Particles | Intensity, size | Predicted position
Chemistry | Atoms | Element type | 3D coordinates
Images | Pixels | RGB color | (x, y) position

## The General Matching Problem

### Problem Formulation

Given two sets of entities \(A = \{a_1, \ldots, a_n\}\) and \(B = \{b_1, \ldots, b_n\}\), find the optimal one-to-one correspondence by minimizing

\[ \min_{\pi} \sum_{i=1}^{n} c_{i,\pi(i)} \]

where the cost combines feature similarity and spatial proximity:

\[ c_{ij} = \alpha \, d_{\text{feature}}(a_i, b_j) + \beta \, d_{\text{spatial}}(\mathbf{x}_i, \mathbf{x}_j). \]

Feature distance \(d_{\text{feature}}\): domain-specific similarity

- Ecology: Bray-Curtis dissimilarity between species vectors

- Physics: difference in particle intensity or size

- Chemistry: penalty for mismatched atom types

- Images: Euclidean distance in RGB color space

Spatial distance \(d_{\text{spatial}}\): physical proximity

- Ecology: geographic distance between plot centers

- Physics: Euclidean distance accounting for predicted motion

- Chemistry: 3D distance between atomic coordinates

- Images: 2D pixel position distance

Weights \(\alpha, \beta \ge 0\) balance feature matching vs. spatial coherence.

### Computational Challenge

Exact solution: solve the full \(n \times n\) LAP.

- Complexity: \(O(n^3)\) using Jonker-Volgenant

- Feasible: up to \(n \approx 1000\) (about \(30 \times 30\) images, or \(1000\) plots/particles/atoms)

- Prohibitive: for \(n = 10\,000\) (\(100 \times 100\) images), runtime and memory become expensive

Real applications often involve

- High-resolution images: \(200 \times 200 = 40\,000\) pixels

- Large ecological surveys: \(5000+\) plots

- Particle tracking: \(10\,000+\) particles per frame

- Molecular dynamics: \(100\,000+\) atoms

We therefore need approximations that are much faster but still produce high-quality matchings.

## Visual Illustration: Pixel Morphing

To make the abstract ideas concrete, we visualize them using image morphing where

- entities = pixels

- features = RGB color values

- spatial position = \((x, y)\) coordinates

We first show the static input images (all at \(80 \times 80\) for display), then the animated morphs produced by different matching strategies.

The first pair are real photographs, the second pair are simple geometric shapes. Internally, all matching is computed on logical \(40 \times 40\) grids; we then upscale to \(80 \times 80\) purely for clearer display.

### Exact Pixel Matching

The exact pixel morph uses a full LAP solution on a \(1600 \times 1600\) cost matrix. For each pair of pixels \((i, j)\) we compute

\[ c_{ij} = \alpha \,\lVert \text{RGB}_i^A - \text{RGB}_j^B \rVert_2 + \beta \,\lVert (x_i, y_i) - (x_j, y_j) \rVert_2, \]

where color distances are normalized to \([0, \sqrt{3}]\) (RGB in \([0,1]\)) and spatial distances to \([0,1]\) using the image diagonal.

This yields an optimal one-to-one assignment of pixels. The resulting animations are smooth and artifact-free but require solving the full LAP.

### Feature Quantization Morph (Strategy 1)

In the feature quantization morph, similar colors are grouped, and groups are matched rather than individual pixels. Colors move as coherent “bands,” preserving global color structure but losing fine-grained per-pixel detail.

### Hierarchical Morph (Strategy 2)

The hierarchical morph first matches large patches, then refines within patches. The motion is locally coherent and scales well to large problems, at the price of potentially missing some globally optimal cross-patch matches.

## Three Approximation Strategies

We now describe the three approximation strategies in detail. The animations above correspond directly to these methods.

### Strategy 1: Feature Quantization

Core idea: reduce problem size by grouping entities with similar features, then match groups.

#### Mathematical Formulation

1. Quantize features

Map continuous feature space to a finite palette

\[ \text{quantize}: \mathbb{R}^d \to \{1, \ldots, k\}, \]

where \(k \ll n\) (for example \(k \approx 64\) for \(n = 1600\)).

2. Group by palette

Form groups \[ G_A^{(c)} = \{ i : \text{quantize}(f_i) = c \} \] and similarly for \(B\).

3. Match groups

Solve a \(k \times k\) LAP between palette entries with costs

\[ c'_{ij} = \alpha \, d(p_i, p_j) + \beta \, d(\bar{\mathbf{x}}_i, \bar{\mathbf{x}}_j), \]

where \(p_i\) is the palette color and \(\bar{\mathbf{x}}_i\) the centroid position of group \(i\).

4. Assign entities

Every entity in \(G_A^{(c)}\) is assigned according to the group-to-group match.

#### Complexity Reduction

- Original: \(O(n^3)\) for an \(n \times n\) LAP

- Quantized: \(O(k^3 + n k)\) for the \(k \times k\) LAP plus group assignment

- Speedup: approximately \((n/k)^3\)

For example, with \(n = 1600\) (a \(40 \times 40\) image) and \(k = 64\) you get

\[ \left(\frac{1600}{64}\right)^3 = 25^3 \approx 15\,000 \]

times fewer LAP operations.

#### Quality Trade-offs

Advantages

- Very large speedups for big \(n\)

- Preserves global structure (similar features stay together)

- Produces smooth, band-like motion without large jumps

Disadvantages

- Loses detail within each palette group

- Quantization artifacts when \(k\) is too small

- May miss optimal local pairings between similar but distinct feature values

The corresponding GIFs are the color walk morphs shown earlier.

### Strategy 2: Hierarchical Decomposition

Core idea: split the domain into smaller subproblems by spatial partitioning, solve subproblems, and combine.

#### Mathematical Formulation

1. Spatial partitioning

Divide the domain into \(m \times m\) patches (for example \(m = 4\) so you get \(16\) patches). Denote the subset of entities of \(A\) in patch \(k\) by

\[ P_A^{(k)} = \{ a_i : \mathbf{x}_i \in \text{Patch}_k \}. \]

2. Patch-level matching

Form patch representatives: centroid position and mean features per patch. Solve an \(m^2 \times m^2\) LAP between patches, with costs defined using the same feature and spatial distances but now at patch level.

3. Recursive refinement

Within each matched patch pair \((P_A^{(k)}, P_B^{(l)})\):

- If \(\lvert P_A^{(k)} \rvert \le \tau\) (a threshold, e.g. \(\tau = 50\)) solve the subproblem exactly.

- Otherwise, partition that patch pair again and repeat.

4. Combine solutions

Concatenate assignments from all leaf subproblems to obtain the global matching.

#### Complexity (Sketch)

With \(d\) levels of decomposition (each level splitting into four patches), the work can be made close to \(O(n \log n)\) in practice, compared to \(O(n^3)\) for a single full LAP. Intuitively, the LAPs near the leaves are very small, and the costly large LAP is replaced by a series of much smaller ones.

#### Quality Trade-offs

Advantages

- Scales to very large \(n\) (tens of thousands of entities)

- Preserves local structure: nearby entities tend to be matched within the same spatial patch

- No feature discretization, so feature precision is retained

Disadvantages

- May miss globally optimal cross-patch matches

- Quality depends on partitioning scheme and threshold \(\tau\)

- Possible boundary artifacts if important structure crosses patch boundaries

#### High-Level Algorithm

    // Pseudocode for hierarchical LAP matching

    FUNCTION match_hierarchical(region_A, region_B, threshold, level):

      // Base case: region small enough for exact LAP
      IF size(region_A) <= threshold THEN
        cost <- compute_cost_matrix(region_A, region_B, α, β)
        RETURN lap_solve(cost)
      END IF

      // Divide into 2x2 spatial grid (4 patches)
      patches_A <- spatial_partition(region_A, grid = 2x2)
      patches_B <- spatial_partition(region_B, grid = 2x2)

      // Compute patch representatives
      FOR each patch p DO
        centroid[p]      <- mean(positions in p)
        mean_feature[p]  <- mean(features in p)
      END FOR

      // Match patches using 4x4 LAP
      patch_cost <- matrix(4, 4)
      FOR i = 1 TO 4 DO
        FOR j = 1 TO 4 DO
          patch_cost[i, j] <- α·distance(mean_feature_A[i], mean_feature_B[j]) +
                             β·distance(centroid_A[i], centroid_B[j])
        END FOR
      END FOR

      patch_assignment <- lap_solve(patch_cost)

      // Recursively solve within matched patches
      assignments <- []
      FOR i = 1 TO 4 DO
        j <- patch_assignment[i]
        sub_assignment <- match_hierarchical(
          patches_A[i],
          patches_B[j],
          threshold,
          level + 1
        )
        assignments <- append(assignments, sub_assignment)
      END FOR

      RETURN concatenate(assignments)
    END FUNCTION

The `couplr` implementation adds pragmatic details such as normalization of color and spatial distances, conversion between \((x, y)\) coordinates and raster indexing, and handling remainder patches when the grid does not divide evenly.

### Strategy 3: Resolution Reduction

Core idea: solve the LAP on a coarse grid, then lift/upscale the assignment to the full-resolution grid.

#### Mathematical Formulation

1. Downscale

Reduce spatial resolution by a factor \(s\) (for example \(s = 2\)):

\[ A' = \text{downsample}(A, s), \qquad B' = \text{downsample}(B, s). \]

Now \(A'\) and \(B'\) each have \(n' = n / s^2\) entities.

2. Solve at low resolution

Compute an exact LAP solution on the \(n' \times n'\) problem:

\[ \pi' = \arg\min_{\pi'} \sum_{i=1}^{n'} c'_{i,\pi'(i)}. \]

3. Upscale assignment

Map the low-resolution assignment back to full resolution:

\[ \pi(i) = \text{upscale}\!\bigl(\pi'(\text{coarse\_index}(i)), s\bigr), \]

where each full-resolution entity inherits the assignment of its coarse cell.

#### Complexity

- Original: \(O(n^3)\)

- Downscaled: \(O\bigl((n/s^2)^3\bigr) = O(n^3 / s^6)\)

- Speedup: \(s^6\)

For \(s = 2\) this gives a \(64\times\) reduction in LAP work.

#### Quality Trade-offs

Advantages

- Very simple to implement

- Exact LAP at the coarse level

- Large speedups for moderate \(s\)

Disadvantages

- Loss of fine detail and blocky artifacts

- Assignment is no longer a true permutation at pixel level (multiple fine pixels can map to the same coarse target)

- Quality deteriorates quickly for larger \(s\)

In practice, resolution reduction is most useful as a crude initialization step for very large problems (\(n > 100\,000\)).

### Strategy Comparison

Approach | Speedup (vs. exact) | Quality | Best for
--- | --- | --- | ---
Exact LAP | \(1\times\) | Optimal | \(n \le 1000\)
Feature quantization | \((n/k)^3\) | Good global structure | Distinct feature groups
Hierarchical | \(\approx n^{3/2}\) | Good local structure | Large \(n\), strong spatial structure
Resolution reduction | \(s^6\) | Moderate | Very large \(n\), rough initialization

Practical rules of thumb

- \(n < 1000\): use the exact LAP.

- \(1000 < n < 5000\): feature quantization or a shallow hierarchy.

- \(n > 5000\): hierarchical decomposition with 2-3 levels.

- \(n > 50\,000\): combine \(s = 2\) resolution reduction with a hierarchical method.

## Implementation Details of Exact Pixel Matching

We now spell out the exact LAP-based morph more concretely.

We again use the cost

\[ c_{ij} = \alpha \,\lVert \text{RGB}_i^A - \text{RGB}_j^B \rVert_2 + \beta \,\lVert (x_i, y_i) - (x_j, y_j) \rVert_2. \]

The algorithm:

    // Pseudocode for exact pixel matching

    // Step 1: Compute full cost matrix (normalized)
    n_pixels <- height x width
    cost <- matrix(0, n_pixels, n_pixels)

    FOR i = 1 TO n_pixels DO
      FOR j = 1 TO n_pixels DO
        // RGB color distance (normalized to [0, sqrt(3)])
        color_dist <- sqrt((R_A[i] - R_B[j])^2 +
                          (G_A[i] - G_B[j])^2 +
                          (B_A[i] - B_B[j])^2) / (255 · sqrt(3))

        // Spatial distance (normalized to [0, 1] by diagonal)
        spatial_dist <- sqrt((x_A[i] - x_B[j])^2 +
                            (y_A[i] - y_B[j])^2) / diagonal_length

        // Combined cost
        cost[i, j] <- α · color_dist + β · spatial_dist
      END FOR
    END FOR

    // Step 2: Solve with Jonker-Volgenant
    assignment <- lap_solve(cost, method = "jv")

    // Step 3: Generate morph frames by linear interpolation
    FOR frame_idx = 1 TO n_frames DO
      t <- frame_idx / n_frames  // Time parameter in [0, 1]

      FOR pixel_i = 1 TO n_pixels DO
        j <- assignment[pixel_i]  // Matched target pixel

        // Interpolate position
        x_new[pixel_i] <- (1 - t) · x_A[pixel_i] + t · x_B[j]
        y_new[pixel_i] <- (1 - t) · y_A[pixel_i] + t · y_B[j]

        // Keep source color (transport-only, no blending)
        RGB_new[pixel_i] <- RGB_A[pixel_i]
      END FOR

      frames[frame_idx] <- render(x_new, y_new, RGB_new)
    END FOR

The `couplr` implementation handles indexing, raster layout, and shows or saves the resulting GIFs.

Approximate performance: up to about \(100 \times 100\) (10 000 pixels) on typical hardware is fine with the exact LAP.

## Application to Scientific Domains

We now return from pixel morphs to the scientific settings that motivated them.

### Ecology: Vegetation Plot Matching

Problem: match \(n\) vegetation plots surveyed at time \(t\) to \(n\) plots at time \(t + \Delta t\) to track community dynamics.

Feature distance: Bray-Curtis dissimilarity between species abundance vectors

\[ d_{\text{BC}}(a, b) = \frac{\sum_s \lvert a_s - b_s \rvert} {\sum_s (a_s + b_s)}, \]

where \(a_s, b_s\) are abundances of species \(s\) in plots \(a\) and \(b\).

Spatial distance: geographic distance (e.g. in kilometers) between plot centers.

Exact solution for small studies (\(n < 100\)):

    // Pseudocode for ecological plot matching
    FOR i = 1 TO n_plots_t DO
      FOR j = 1 TO n_plots_tplus DO

        // Bray-Curtis dissimilarity for species composition
        numerator   <- sum over species s of |abundance_t[i, s] - abundance_tplus[j, s]|
        denominator <- sum over species s of (abundance_t[i, s] + abundance_tplus[j, s])
        bc_distance <- numerator / denominator

        // Geographic distance (kilometers)
        geo_distance <- sqrt((x_t[i] - x_tplus[j])^2 +
                            (y_t[i] - y_tplus[j])^2)

        // Combined cost (α = 0.7 emphasizes species composition)
        cost[i, j] <- 0.7 · bc_distance + 0.3 · (geo_distance / max_distance)

      END FOR
    END FOR

    plot_correspondence <- lap_solve(cost)

For large studies (\(n > 1000\)) a hierarchical approach by region is more practical:

    // Hierarchical decomposition by geographic region

    // 1. Divide landscape into spatial grid (e.g. 10 km x 10 km cells)
    regions_t     <- spatial_partition(plots_t,     grid_size = 10 km)
    regions_tplus <- spatial_partition(plots_tplus, grid_size = 10 km)

    // 2. Compute region representatives
    FOR each region r DO
      mean_composition[r] <- average species vector across plots in r
      centroid[r]         <- geographic center of r
    END FOR

    // 3. Match regions (small LAP: ~100 regions)
    region_cost       <- compute_cost(mean_composition, centroids, α = 0.7, β = 0.3)
    region_assignment <- lap_solve(region_cost)

    // 4. Within matched regions, solve plot-level LAP
    full_assignment <- []
    FOR r = 1 TO n_regions DO
      r_matched <- region_assignment[r]
      plots_A   <- plots in regions_t[r]
      plots_B   <- plots in regions_tplus[r_matched]

      // Local LAP (smaller problem, e.g. 50 x 50)
      cost_local       <- compute_plot_cost(plots_A, plots_B, α = 0.7, β = 0.3)
      local_assignment <- lap_solve(cost_local)

      full_assignment <- append(full_assignment, local_assignment)
    END FOR

    RETURN full_assignment

This allows tracking individual plot trajectories across time, distinguishing stable communities, successional trends, and invasion fronts.

### Physics: Particle Tracking

Problem: track \(n\) particles between frame \(t\) and \(t + \Delta t\) in experimental video.

Feature distance: differences in intensity, size, or shape.

Spatial distance: displacement relative to predicted motion:

\[ d_{\text{spatial}}(i, j) = \bigl\| \mathbf{x}_i + \mathbf{v}_i \Delta t - \mathbf{x}_j \bigr\|_2, \]

where \(\mathbf{v}_i\) is the estimated velocity from previous frames.

We also impose a maximum displacement \(d_{\max}\) beyond which matches are physically implausible.

Exact solution (moderate \(n\)):

    // Pseudocode for particle tracking with velocity prediction

    // Initialize cost matrix as forbidden everywhere
    cost <- matrix(Inf, n_particles_t, n_particles_tplus)

    FOR i = 1 TO n_particles_t DO
      // Predict position using previous velocity
      x_predicted <- x_t[i] + v_x_t[i] · Δt
      y_predicted <- y_t[i] + v_y_t[i] · Δt

      FOR j = 1 TO n_particles_tplus DO
        // Distance from predicted position
        dx <- x_predicted - x_tplus[j]
        dy <- y_predicted - y_tplus[j]
        spatial_distance <- sqrt(dx^2 + dy^2)

        // Only consider physically plausible matches
        IF spatial_distance <= max_displacement THEN
          // Feature similarity (intensity, size, etc.)
          feature_distance <- |intensity_t[i] - intensity_tplus[j]|

          // Combined cost
          cost[i, j] <- α · feature_distance + β · spatial_distance
        END IF
      END FOR
    END FOR

    // Solve assignment (Inf entries are forbidden)
    particle_tracks <- lap_solve(cost)

    // Update velocities from assignments
    FOR i = 1 TO n_particles_t DO
      j <- particle_tracks[i]
      velocity_new[i] <- (position_tplus[j] - position_t[i]) / Δt
    END FOR

For dense tracking (\(n > 5000\)), we can first cluster particles:

    // Two-stage: clustering then local matching

    // Stage 1: spatial clustering
    clusters_t     <- spatial_cluster(particles_t,     radius = 2 · pixel_size)
    clusters_tplus <- spatial_cluster(particles_tplus, radius = 2 · pixel_size)

    // Compute cluster representatives
    FOR each cluster c DO
      centroid[c]       <- mean position of particles in c
      mean_intensity[c] <- mean intensity
      mean_velocity[c]  <- mean velocity (if available)
    END FOR

    // Match clusters
    cluster_cost   <- compute_cluster_similarity(clusters_t, clusters_tplus)
    cluster_tracks <- lap_solve(cluster_cost)

    // Stage 2: within matched clusters, track individual particles
    full_tracks <- []
    FOR c = 1 TO n_clusters DO
      c_matched   <- cluster_tracks[c]
      particles_A <- particles in clusters_t[c]
      particles_B <- particles in clusters_tplus[c_matched]

      cost_local <- compute_particle_distance(
        particles_A, particles_B,
        max_displacement = 5,
        α = 0.3,
        β = 0.7
      )

      local_tracks <- lap_solve(cost_local)
      full_tracks  <- append(full_tracks, local_tracks)
    END FOR

    RETURN full_tracks

This yields efficient and robust trajectories even for very dense particle fields.

### Chemistry: Molecular Conformation Alignment

Problem: align two conformations of the same molecule (e.g. a protein) with \(n\) atoms to compute RMSD and analyze structural change.

Feature distance: strict element matching

\[ d_{\text{element}}(i, j) = \begin{cases} 0, & \text{if } \text{element}_i = \text{element}_j, \\ \infty, & \text{otherwise.} \end{cases} \]

Spatial distance: 3D Euclidean distance between atomic coordinates.

Exact LAP for small molecules:

    // Pseudocode for molecular conformation alignment

    n_atoms <- number of atoms in molecule
    cost    <- matrix(0, n_atoms, n_atoms)

    FOR i = 1 TO n_atoms DO
      FOR j = 1 TO n_atoms DO

        // Enforce strict element type matching
        IF element_type_A[i] ≠ element_type_B[j] THEN
          cost[i, j] <- Inf
        ELSE
          dx <- x_A[i] - x_B[j]
          dy <- y_A[i] - y_B[j]
          dz <- z_A[i] - z_B[j]
          cost[i, j] <- sqrt(dx^2 + dy^2 + dz^2)
        END IF

      END FOR
    END FOR

    // Solve alignment
    alignment <- lap_solve(cost)

    // Compute RMSD
    sum_sq_dist <- 0
    FOR i = 1 TO n_atoms DO
      j            <- alignment[i]
      sum_sq_dist <- sum_sq_dist + cost[i, j]^2
    END FOR

    rmsd <- sqrt(sum_sq_dist / n_atoms)

For large biomolecules, we again use a hierarchical strategy, this time by secondary structure elements (helices, sheets, loops, etc.), aligning segments first and then atoms within matched segments.

## Implementation Notes

### Customizing Morph Duration

The morphing examples use default settings, but you can customize the number of frames and speed:

    # From inst/scripts/generate_examples.R
    generate_morph <- function(assignment, pixels_A, pixels_B,
                               n_frames    = 30,   # number of frames
                               frame_delay = 0.1)  # delay between frames (seconds)
    {
      frames <- lapply(seq(0, 1, length.out = n_frames), function(t) {
        interpolate_frame(t, assignment, pixels_A, pixels_B)
      })

      save_gif(frames, delay = frame_delay)
    }

Total animation duration is `n_frames * frame_delay` seconds.

### Using the Example Code

The morphing implementation is provided in `inst/scripts/generate_examples.R`:

    # View the source
    example_script <- system.file("scripts", "generate_examples.R", package = "couplr")
    file.show(example_script)

    # Or source to use its helpers
    source(example_script)

    # Apply to your own data
    my_cost       <- build_cost_matrix(my_data_A, my_data_B)
    my_assignment <- lap_solve(my_cost)

### Regenerating Examples

To regenerate all demo GIFs and PNGs:

    source("inst/scripts/generate_examples.R")

This will write the assets under `inst/extdata`.

## Mathematical Foundation: Optimal Transport

The matching problems discussed here are discrete instances of optimal transport.

### Monge Problem

The original Monge formulation (1781) seeks a transport map \(T: A \to B\) minimizing

\[ \int_A c(\mathbf{x}, T(\mathbf{x})) \,\mathrm{d}\mu(\mathbf{x}). \]

### Kantorovich Relaxation

Kantorovich (1942) relaxed this to a transport plan \(\gamma\) on \(A \times B\):

\[ \min_{\gamma} \int_{A \times B} c(\mathbf{x}, \mathbf{y}) \,\mathrm{d}\gamma(\mathbf{x}, \mathbf{y}) \]

subject to marginal constraints on \(\gamma\).

### Discrete Linear Assignment

For discrete uniform distributions with \(n\) points in \(A\) and \(B\) we obtain exactly the linear assignment problem:

\[ \min_{\pi \in S_n} \sum_{i=1}^n c_{i,\pi(i)}, \]

which is what `couplr` solves efficiently.

### Wasserstein Distance

With \(c_{ij} = d(\mathbf{x}_i, \mathbf{x}_j)\) (often Euclidean distance), the optimal cost defines the \(1\)‑Wasserstein distance:

\[ W_1(\mu, \nu) = \min_{\pi \in S_n} \sum_{i=1}^n c_{i,\pi(i)}. \]

This appears in

- Earth mover’s distance for image retrieval

- Distributional similarity in statistics

- Generative modeling (e.g. Wasserstein GANs)

## Further Reading

### Optimal Transport Theory

- Peyré, G., & Cuturi, M. (2019). Computational Optimal Transport. Foundations and Trends in Machine Learning.

- Villani, C. (2008). Optimal Transport: Old and New. Springer.

### Scientific Applications

Ecology

- Anderson, M. J. et al. (2011). Navigating the multiple meanings of beta diversity. Ecology Letters.

- Legendre, P., & Legendre, L. (2012). Numerical Ecology. Elsevier.

Physics

- Adrian, R. J., & Westerweel, J. (2011). Particle Image Velocimetry. Cambridge University Press.

- Crocker, J. C., & Grier, D. G. (1996). Methods of digital video microscopy. Journal of Colloid and Interface Science.

Chemistry

- Kabsch, W. (1976). A solution for the best rotation to relate two sets of vectors. Acta Crystallographica.

- Coutsias, E. A. et al. (2004). Using quaternions to calculate RMSD. Journal of Computational Chemistry.

### Assignment Algorithms

- Burkard, R., Dell’Amico, M., & Martello, S. (2009). Assignment Problems. SIAM.

- For implementation details in this package see `vignette("algorithms")`.

## Connection to couplr Workflows

The approximation strategies in this vignette become relevant when working with couplr’s practical matching functions.

### When Do These Strategies Apply?

Strategy | couplr Implementation | When to Use
--- | --- | ---
Exact LAP | `match_couples(method = "jv")` | n < 3,000
Feature quantization | Implicit via `scale = "robust"` | Reduces effective feature space
Hierarchical | `matchmaker(block_type = "cluster")` | n > 3,000, use blocking
Resolution reduction | Custom code | Very large n

### Practical Recommendations

For n < 3,000: Use `match_couples()` with exact algorithms:

    result <- match_couples(left, right, vars = c("x", "y", "z"), auto_scale = TRUE)

For 3,000 < n < 10,000: Use blocking to create smaller subproblems:

    blocks <- matchmaker(left, right, block_type = "cluster", n_blocks = 10)
    result <- match_couples(blocks$left, blocks$right, vars = vars, block_id = "block_id")

For n > 10,000: Use greedy matching:

    result <- greedy_couples(left, right, vars = vars, strategy = "sorted")

For n > 50,000: Combine strategies (blocking + greedy within blocks), or implement custom approximations using the techniques in this vignette.

### Limitations of Approximation Strategies

Each approximation trades accuracy for speed. Know the failure modes:

Strategy | Works Well When | Fails When
--- | --- | ---
Feature quantization | Features cluster naturally | Continuous features, fine distinctions matter
Hierarchical | Spatial locality is meaningful | Optimal matches cross boundaries
Resolution reduction | Coarse structure suffices | Fine detail matters

## Summary

This vignette explored optimal matching through the lens of pixel morphing and scientific applications.

Key Ideas:

1. Assignment = matching: LAP finds optimal correspondences between two sets

2. Scalability matters: \(O(n^3)\) becomes prohibitive for \(n > 3{,}000\)

3. Three approximations: Feature quantization, hierarchical decomposition, resolution reduction

4. Same math, different domains: Pixels, particles, plots, and atoms all use the same algorithms

The same algorithms that morph images smoothly also track particles in physics, align molecules in chemistry, and match vegetation plots in ecology. Together, the methods in `couplr` let you move between exact optimal matchings and principled approximations, depending on problem size and accuracy requirements.

## See Also

- `vignette("getting-started")` - Basic LAP solving

- `vignette("algorithms")` - Mathematical foundations

- `vignette("matching-workflows")` - Production matching pipelines

- `?lap_solve`, `?match_couples`, `?greedy_couples`