Mathematical Foundations of PASDE

0. Patent & IP Notice

Important: The formulas on this page are illustrative and describe general mathematical concepts commonly used in physics-based vision and signal processing. They are not a disclosure of the internal algorithms, parameter choices, or proprietary logic used in Phocoustic’s PASDE, PADR, SOEC, PEQM, PQRC, or related patented systems.

The actual production pipeline includes additional filters, admissibility gates, quantization logic, and cross-domain constraints that are protected by the Phocoustic / VisualAcoustic.ai patent family.

1. Frames as Physical Fields

PASDE starts from the simple idea that each captured frame is not just an image, but a measurement of a physical field. For a given modality (e.g., visible light), we can write the intensity at pixel location \((x, y)\) and time \(t\) as:

\( I(x, y, t) \)

For multispectral or acoustic-derived channels, \(I\) may be vector-valued:

\( \mathbf{I}(x, y, t) = \big(I_1(x,y,t), I_2(x,y,t), \dots, I_M(x,y,t)\big) \)

where each component can represent a wavelength band, polarization state, structured-light return, or acoustic intensity mapped into the image plane.

2. Spatial Structure: Gradients and Geometry

To understand local surface geometry and edges, PASDE relies on spatial gradients. A standard way to express this is:

\( \nabla I(x,y,t) = \left( \frac{\partial I}{\partial x}(x,y,t), \frac{\partial I}{\partial y}(x,y,t) \right) \)

The gradient captures how intensity changes across the surface. Regions with large gradient magnitudes often correspond to edges, corners, curvature changes, or structured-light distortions — all of which are important indicators of physical shape.

A simple scalar measure of local spatial change is:

\( \left\lVert \nabla I(x,y,t) \right\rVert = \sqrt{ \left(\frac{\partial I}{\partial x}\right)^2 + \left(\frac{\partial I}{\partial y}\right)^2 } \)

In the PASDE framework, such quantities are used as ingredients for detecting physically meaningful structures, not as the full algorithm.

3. Temporal Drift: Measuring Change Over Time

To quantify how a scene changes between frames, we consider temporal differences. An illustrative finite-difference drift field between frame \(t-1\) and frame \(t\) can be written as:

\( D_t(x,y) = I(x,y,t) - I(x,y,t-1) \)

In a multi-channel setting:

\( \mathbf{D}_t(x,y) = \mathbf{I}(x,y,t) - \mathbf{I}(x,y,t-1) \)

This raw difference captures any change between frames: motion, deformation, illumination variation, sensor noise, or glare. PASDE’s role is to separate true physical drift from spurious non-physical change.

4. Drift Magnitude and Direction (Illustrative)

Drift at a point can be decomposed into a magnitude and a direction. A simple illustrative magnitude is:

\( \left\lVert D_t(x,y) \right\rVert = \big| I(x,y,t) - I(x,y,t-1) \big| \)

When additional motion or optical-flow estimates are available, a local drift vector \(\mathbf{v}(x,y,t)\) can be introduced and its direction characterized via normalized vectors, dot products, or angles.

For example, the cosine of the angle between two consecutive drift vectors might be written as:

\( \cos \theta(x,y,t) = \dfrac{ \mathbf{v}(x,y,t) \cdot \mathbf{v}(x,y,t-1) }{ \lVert \mathbf{v}(x,y,t) \rVert \,\lVert \mathbf{v}(x,y,t-1) \rVert } \)

Stable physical motion tends to preserve direction over short horizons, whereas flicker, noise, and many glare artifacts do not.

5. Persistence: Multi-Frame Stability

A key idea in PASDE is that real physical change persists and remains coherent across multiple frames, while noise and brief disturbances do not. One illustrative persistence score over a temporal window of length \(N\) is:

\( P(x,y) = \frac{1}{N} \sum_{k=0}^{N-1} \left\lVert D_{t-k}(x,y) \right\rVert \)

In practice, PASDE applies more sophisticated multi-frame logic, but the core intuition is the same: only persistent, structured drift should be promoted into higher-level representations.

6. Physical Continuity and Admissibility

Not every pattern of change is physically plausible. For example, isolated pixels that flip back and forth with no spatial support are more likely to be sensor noise than a real crack, warp, or deformation.

One illustrative way to encode a continuity constraint is to look at the spatial variation of the drift field itself:

\( C(x,y) = \left\lVert \nabla D_t(x,y) \right\rVert \)

Drift patterns with excessively large \(C(x,y)\) may be considered non-physical and rejected. A conceptual admissibility condition might be written as:

\( C(x,y) < \beta \)

where \(\beta\) is a continuity threshold chosen based on domain, resolution, and sensor characteristics. The true PASDE/PEQM system uses a richer combination of thresholds and rules protected by patent.

7. Drift Admissibility Envelope (Conceptual)

Combining persistence, directional stability, and continuity leads to the idea of a drift admissibility envelope. A point or region may be considered physically admissible if it passes multiple checks at once. An illustrative envelope can be written as:

\[ E(x,y) = \Big\{ D_t(x,y) \; \big|\; P(x,y) > \alpha,\; C(x,y) < \beta,\; \cos \theta(x,y,t) > \gamma \Big\} \]

where:

\(\alpha\) encodes minimum persistence,
\(\beta\) encodes maximum allowed discontinuity, and
\(\gamma\) encodes minimum directional consistency.

In the full PASDE architecture, admissibility involves additional cross-frame, cross-channel, and physics-aware checks (including dual-domain mappings and structured-light geometry) that remain proprietary.

8. From Drift Fields to Structured Representations

Once admissible drift has been isolated, it can be encoded into compact, structured forms for downstream processing. In the VisualAcoustic.ai platform, this includes quantized patterns and ranking codes that summarize:

where drift is occurring,
how strong it is,
which directions it prefers, and
how it evolves over time.

Conceptually, one can think of these codes as mapping continuous drift values into discrete bins or glyphs that preserve essential physical information while remaining compact and explainable. The exact quantization strategies, encoding layouts, and ranking logic are part of Phocoustic’s patented PQRC/SPQRC and related structures and are not specified here.

9. Educational Summary

The mathematics behind PASDE follows straightforward physical intuition:

Treat frames as measurements of a real physical field.
Measure how that field changes in space and time.
Reject changes that are too inconsistent, too isolated, or too unstable to be physical.
Promote only persistent, coherent drift into higher-level representations.

By grounding every step in measurable physical behavior, PASDE provides a stable foundation for anomaly detection, inspection, and physics-anchored semantic interpretation without relying on traditional training data or statistical models.

For full technical details, including implementation-specific logic, admissibility gates, and quantization schemes, please refer to the Phocoustic / VisualAcoustic.ai patent family and associated technical documentation provided under appropriate NDA.