Histogram of Oriented Displacements (HoD)

• Histogram of Oriented Displacements (HoD) is a technique that converts 3D displacement vectors into a compact, amplitude-weighted angular histogram.
• It employs spherical coordinate transformation and discrete binning to robustly capture directional and magnitude information from vector fields.
• The method supports dimensionality reduction, clustering, and machine learning integration, proving valuable for radiotherapy and other medical imaging applications.

The Histogram of Oriented Displacements (HoD) is a dimensionality reduction and pattern extraction technique for vector fields, specifically designed to summarize the directional structure and amplitude of multivariate displacement data within a compact, histogram-based representation. The HoD framework has achieved particular relevance in the analysis of deformable registration vector fields in medical imaging (notably, four-dimensional CT data in radiotherapy) and offers general applicability in any context where high-dimensional vector fields must be efficiently encoded for comparison, clustering, or downstream analysis (Madesta et al., 25 Nov 2024).

1. Mathematical Formulation and Core Principles

The HoD methodology encodes the essential statistics of a 3D displacement vector field by mapping each displacement vector $v = [x, y, z]^T$ to spherical coordinates: magnitude $r$ and angular components $\theta$ and $\phi$, where

$$r = \sqrt{x^2 + y^2 + z^2},\qquad \theta = \arccos\left(\frac{z}{r}\right),\qquad \phi = \operatorname{arctan2}(y, x)$$

This transformation enables the separation of displacement amplitude from its directional orientation, which is particularly suited to capturing patterns in systems (such as respiratory-induced lung motion) where movement has predominant spatial directions and varying magnitude.

Each vector is then binned into a fixed-resolution 2D histogram $H(\phi_\text{bin}, \theta_\text{bin})$ indexed by discrete $(\phi, \theta)$ bins. Contribution to each histogram bin is weighted by the vector’s magnitude, i.e.,

$$H(\phi_\text{bin}, \theta_\text{bin}) \leftarrow H(\phi_\text{bin}, \theta_\text{bin}) + r$$

After iterating across the entire domain (typically restricted by a meaningful anatomical mask, such as a lung mask), the histogram is normalized with respect to the number of voxels, yielding a compact, probability-like distribution summarizing the orientation and amplitude statistics of the 3D vector field.

2. Algorithmic Implementation and Workflow

The construction of the HoD representation unfolds in sequential stages:

1. Spherical Coordinate Transformation: Each voxel’s displacement vector from the registration-derived vector field is mapped to $(r, \theta, \phi)$ as above. This step may require appropriate axis alignment, especially in organ-specific data.
2. Histogram Binning: The $(\phi, \theta)$ angular domain is partitioned into discrete bins (e.g., 8 bins in $\phi$, 16 in $\theta$; choice tunable for resolution requirements). For each voxel, the assigned bin is identified, and the magnitude $r$ is accumulated.
3. Magnitude Thresholding (optional): Small-magnitude displacement vectors (potentially arising from noise) may be omitted or downweighted, further increasing the robustness of the summary.
4. Histogram Normalization: After accumulation, the histogram is normalized either per total number of included voxels or total included magnitude sum, resulting in a per-voxel or per-field scaled descriptor.

The end result is an oriented histogram of displacements—a fixed-sized array (e.g., $8 \times 16$)—that provides a concise, amplitude-weighted angular distribution of the vector field.

3. Applications in Medical Image Analysis

The HoD approach is strongly motivated by clinical needs in radiotherapy, especially for patient stratification and treatment optimization under complex organ motion (such as respiratory-driven lung deformation) (Madesta et al., 25 Nov 2024). By encoding patient motion signatures into HoD representations, vector field comparisons become computationally efficient, facilitating:

• Dimensionality Reduction: 3D vector fields comprising millions of voxel vectors are reduced to a manageable histogram, enabling high-throughput analysis.
• Patient Clustering: Sequences of HoDs (across breathing phases) can be further condensed using autoencoders and then embedded with techniques such as Uniform Manifold Approximation and Projection (UMAP) for unsupervised clustering. With local neighborhood preservation (e.g., using cosine distance), this yields clusters which stratify patients by motion similarity.
• Knowledge Transfer Between Patients: Clustering structures enable treatment protocols or deformable registration hyperparameters to be transferred between patients with similar HoD-based motion characteristics.
• Retrospective and Prospective Analysis: The method supports both analyzing historical patient datasets and providing real-time support for prospective patient management using only pre-treatment information.

4. Generalizability and Extensions of the Oriented Histogram Approach

Although initially developed for lung radiotherapy, the HoD framework is inherently general. Any application involving the summarization and comparison of high-dimensional vector fields can benefit, including:

• Organ motion analysis outside the thorax (e.g., cardiac or abdominal imaging).
• Tracking changes in registration vector fields in longitudinal or inter-modality studies.
• Non-medical vector fields, provided a physical directional interpretation is meaningful.

The strategy of amplitude-weighted angular histogramming retains the essential structural information while substantially reducing the computational and storage overhead required for downstream tasks such as learning, clustering, or anomaly detection.

5. Integration with Nonlinear Embedding and Machine Learning

The compactness and interpretability of HoD representations permit their seamless integration into machine learning workflows. The pipeline reported in (Madesta et al., 25 Nov 2024) encompasses:

1. Autoencoder Compression: HoD histograms (potentially in sequence for multiple respiratory phases) are further compressed into low-dimensional latent vectors via an unsupervised autoencoder, optimized for $L_2$ reconstruction loss.
2. UMAP Embedding: The autoencoder latents are projected into 2D manifolds by UMAP, with local structure preservation parameters (e.g., neighborhood size of 5, cosine metric).
3. Patient Stratification and Protocol Decision Support: Clusters in the UMAP embedding correspond to populations with shared motion traits, guiding protocol adaptation and potentially enabling individualized or groupwise optimization of treatment strategies.

A plausible implication is that similar approaches can be adopted in other domains where complex vector fields arise, facilitating meta-analyses or cohort-based decision pipelines.

6. Summary Table: HoD Workflow Components

Stage	Input	Output/Role
Spherical Transform	3D displacement vectors	$(r, \theta, \phi)$ for each voxel
Histogram Binning	Spherical coordinates per voxel	Amplitude-weighted $(\phi, \theta)$ histogram
Normalization	Raw histogram	Scaled/probability-like histogram
Autoencoder	Seq. of histograms	Dense latent vector per patient/field
UMAP Embedding	Latent vectors (patients)	2D manifold for clustering/stratification

The sequence of steps realizes a robust transformation of complex motion fields into actionable features amenable to unsupervised analysis, groupwise protocol development, or direct inclusion in predictive modeling pipelines.

7. Theoretical and Practical Considerations

• Resolution and Parameter Choice: The angular binning resolution and magnitude thresholding must be tailored to the particular application and anatomical context for optimal pattern preservation.
• Noise Sensitivity: Amplitude weighting, as opposed to simple tallying, mitigates noise effects by de-emphasizing low-magnitude (and often unreliable) motion vectors.
• Interpretability: The orientation histogram encodes interpretable statistics, mapping directly to known directional behaviors (e.g., predominant cranio-caudal lung motion).
• Extensibility: The method admits extension to higher-dimensional histograms (e.g., including time, additional anatomical axes), although with increased data sparsity and potential trade-offs in interpretability.

In summary, the Histogram of Oriented Displacements provides a mathematically rigorous, computationally tractable, and generalizable framework for encoding and comparing high-dimensional vector field information, with particular demonstrated efficacy in medical imaging applications such as 4D CT-based radiotherapy planning (Madesta et al., 25 Nov 2024).