Local Phase-Based Modeling

• Local-phase-based approaches are defined by explicit modeling of local phase information paired with amplitude to effectively capture image textures and edge orientations.
• They utilize energy-based probabilistic models with 2D subspace representations, enabling the extraction of quadrature-pair features that mimic biological vision.
• This method enhances image analysis tasks such as denoising, segmentation, and feature extraction through explicit phase coupling and hierarchical statistical modeling.

A local-phase-based approach is a class of modeling and inference strategies across physics, signal processing, materials science, and machine learning that seeks to describe, infer, or manipulate the structure and behavior of systems by explicitly capturing, modeling, or utilizing local phase information—typically in conjunction with, but distinct from, global amplitude or magnitude features. In the context of imaging, signal analysis, and materials characterization, this contrasts with methods that treat phase only globally or ignore it altogether. The following sections provide a comprehensive treatment of the main theoretical frameworks, model constructions, algorithmic methodologies, and applications of local-phase-based approaches as articulated in foundational research (Cadieu et al., 2010).

1. Mathematical Foundation: Factorized Energy Models and Subspace Decomposition

The core mathematical infrastructure is based on energy-based probabilistic models, particularly extensions of the factorized third-order Boltzmann machine. In the prototype model (mcRBM), the covariance energy for input vector $v \in \mathbb{R}^D$ and binary latent vector $h$ is given by: $$E^c(v, h^c) = -\frac{1}{2} \sum_{n=1}^N h_n^{(c)} \sum_{f=1}^F P_{fn} \left( \sum_{i=1}^D C_{if} \frac{v_i}{\|v\|} \right)^2 - \sum_{n=1}^N b_n^{(c)} h_n^{(c)}$$ where $C$, $P$, and $b$ are learned filters, weights, and biases respectively.

The local-phase-based extension introduces $L_p$-spherically symmetric subspaces by grouping factors—each now indexed by $l=1,...,L$—via a new tensor of filters $C_{ifl}$. The energy function becomes: $$E^p(v, h^p) = -\frac{1}{2} \sum_{n=1}^N h_n^{(p)} \sum_{f=1}^F P_{fn} \left\{ \sum_{l=1}^L \left[ \left( \sum_{i=1}^D C_{ifl} \frac{v_i}{\|v\|} \right)^\alpha \right] \right\}^{1/\alpha} - \sum_{n=1}^N b_n h_n$$ For $L=2$ and $\alpha=2$, this yields two-dimensional subspaces (interpretable as complex pairs), enabling explicit modeling of amplitude (via $L_2$ norm) and phase (arctangent of filter pair responses).

2. Local Amplitude–Phase Representation and Subspace Structure

Within each 2D subspace (i.e., a pair of quadrature filters), the response to input $v$ is decomposed into an amplitude

$A_f = \sqrt{ (C_{if,1} v_i)^2 + (C_{if,2} v_i)^2 }$

and a phase

$$\theta_f = \arg\left[ \sum_{i=1}^D C_{if,1} v_i + j \sum_{i=1}^D C_{if,2} v_i \right].$$

These subspaces, when learned from natural images, evolve into pairs of Gabor-like filters—a configuration that mirrors the structure of complex cells in early visual cortex and is efficient for encoding local oriented structure and edges.

This decomposition is critical: amplitude captures local energy/contrast, while phase aligns with spatial alignment, contour structure, and texture periodicities in images. The separation is a key feature distinguishing local-phase-based methods from global phase recovery or pure-magnitude models.

3. Training and Emergence of Gabor-like Representations

Training proceeds on preprocessed (mean-subtracted, whitened, L2-normalized) image patches. Parameters are updated to maximize the log-likelihood using Contrastive Divergence and hybrid Monte Carlo sampling. The grouping weights $P$ are adapted to aggregate responses from subspaces with similar position and frequency selectivity, further enhancing the model’s capacity to encode structured image features.

Empirically, the filters in each subspace converge toward quadrature-pair Gabor functions. This structure reflects the statistical regularities of natural images—specifically, the prevalence of local edge and texture arrangements that are phase-coupled.

4. Explicit Phase Coupling: Higher-Order Dependencies

To model the joint structure of phase variables across subspaces, additional hidden units $h^k$ are introduced, and the responses are represented via their sine and cosine components: $$x_{f1} = \cos(\theta_f),\quad x_{f2} = \sin(\theta_f)$$ The phase coupling energy term is defined as: $$E^k(v, h^k) = -\frac{1}{2} \sum_{t=1}^T h_t^{(k)} \sum_{g=1}^G R_{gt} \left( \sum_{l=1}^L \sum_{f=1}^F Q_{flg} x_{fl} \right)^2 - \sum_{t=1}^T b_t^{(k)} h_t^{(k)}$$ This construction enables the model to capture combinatorial mixtures of phase-alignment distributions (concentrated in the sum and difference of phase pairs). The resulting form is analogous to a mixture of von Mises–type couplings, permitting explicit modeling of dependencies such as co-linearity and alignment relevant to extended edge structures.

The conditional distribution over phase vectors given the hidden state $h^k$ is: $$p(\theta | h^k) \propto \exp\left(-\frac{1}{2} x^T K x \right),\quad K = Q\,\mathrm{diag}(R h^k) Q^T$$ With $h^k$ binary, this allows a large expressive class of phase-coupling distributions to emerge.

5. Applications: Image Analysis and Beyond

This framework provides foundational mechanisms for several downstream image analysis tasks:

• Denoising: Joint modeling of amplitude and phase improves the ability to reconstruct local texture and edge information under noise by leveraging phase coherence.
• Segmentation and Edge Detection: The representation is particularly well suited to discriminating regions and orientation structure by exploiting spatial phase alignment statistics.
• Feature Extraction: The learned representation maps tightly onto biological vision models, providing a rationale for the efficiency of quadrature-pair coding in cortex.
• Extension to Hierarchies and Invariances: The explicit modeling of phase offers a path toward constructing deeper generative models capable of learning invariances (e.g., to local phase shifts) and richer hierarchical structure—all while providing an interpretable intermediate description.

6. Implications, Limitations, and Future Perspectives

The explicit factorization of amplitude and phase, combined with higher-order phase coupling, advances beyond classical sparse coding and earlier Boltzmann machine models by enabling a more precise statistical account of local image structure. The design encourages future research along multiple axes:

• Learning optimal subspace dimensionality (potentially with higher $L_p$ symmetry).
• Exploring nested and hierarchical models to capture even more sophisticated dependencies.
• Extending the phase-coupling framework to video, volumetric, or multi-modal settings.

A notable limitation is computational complexity in learning and inference, especially as the number of subspaces or phase-coupling hidden units grows. Additionally, while the model captures many key statistical properties of natural scenes, its utility in domains beyond imaging (e.g., audio, remote sensing) requires tailored analysis of local phase relevance.

7. Summary Table: Structural Components of the Local-Phase-Based Boltzmann Machine

Model Component	Mathematical Structure	Role in Model
Subspace Filters	$C_{ifl}$ (tensor for $L$-dim. subspaces)	Defines 2D orientation-selective responses
Amplitude/Phase	$A_f$, $\theta_f$	Encodes local energy and alignment
Covariance Units	$h^p$	Model amplitude (contrast) dependencies
Phase-Coupling Units	$h^k$	Model higher-order phase dependencies across subspaces
Coupling Matrices	$Q$, $R$	Weight and combine phase responses for joint modeling

This architecture achieves robust extraction and modeling of local image phase, including the explicit coupling required for realistic image generation and advanced computer vision inference. The methodology exemplifies how local-phase-based statistical modeling forms a bridge between rigorous mathematical energy models and the biologically inspired understanding of visual coding (Cadieu et al., 2010).