Stone Spectrum System

Stone Spectrum Analysis 

1. Keep your LiDAR math as the sensor front-end

This may have the LiDAR side cleanly defined:

1. 3D point from angles + range
   [
   \vec{P}_j
   = R_j
   \begin{bmatrix}
   \cos(\omega_j)\cos(\alpha_j)\
   \cos(\omega_j)\sin(\alpha_j)\
   \sin(\omega_j)
   \end{bmatrix}
   ]

2. Spectrum sweep per laser ( i )
   [
   \lambda_i(t) = \lambda_{\min} + (\lambda_{\max} - \lambda_{\min}) f_{\text{spec}}(i,t)
   ]

3. Angle modulation
   [
   \alpha_i(t) = \alpha_0 + f_\alpha(i,t), \qquad
   \omega_i(t) = \omega_0 + f_\omega(i,t)
   ]

4. Full data tuple for each hit
   [
   D_j = (t_j, i_j, \lambda_j, \alpha_j, \omega_j, R_j, A_j, \phi_j)
   ]

5. Full scan as a high-D cloud
   [
   \mathcal{S} = \bigcup_{j=1}^M D_j
   ]

So: LiDAR gives one a stream of high-dimensional “points.” Now we turn that into a mapping system your recursive/QCAD engine can work on.

2. Define the mapping domain (the “world grid”)

Let’s define a continuous world space (can later be discretized):

• Spatial coordinates: ((x,y,z))

• Spectral coordinate: (\lambda)

• Modulation/encoding coordinate: (\phi)

Define the environment field you want to reconstruct:

[
F(x,y,z,\lambda,\phi, t)
]

Interpretation:

• At time (t), (F) encodes what the environment “looks like” at position ((x,y,z)), spectral band (\lambda), and modulation channel (\phi) (e.g., reflectivity, material class, confidence, etc.).

3. Map raw LiDAR hits into the field

For each LiDAR hit (D_j), we know:

• 3D position (\vec{P}_j = (x_j,y_j,z_j))

• Spectrum (\lambda_j)

• Modulation (\phi_j)

• Amplitude (A_j)

We can deposit these into the field with kernel smoothing:

[
F(x,y,z,\lambda,\phi,t)
= \sum_{j} A_j ,
K_{\text{sp}}(x - x_j, y - y_j, z - z_j),
K_{\lambda}(\lambda - \lambda_j),
K_{\phi}(\phi - \phi_j),
K_{t}(t - t_j)
]

Where:

• (K_{\text{sp}}) is a spatial kernel (e.g. Gaussian, top-hat, voxel indicator)

• (K_{\lambda}) a spectral kernel

• (K_{\phi}) a modulation/phase kernel

• (K_{t}) a temporal kernel / window

In practice: this becomes a multi-dimensional voxel grid or StoneCube lattice where each cell accumulates contributions from nearby hits.

4. Compress to per-cell “state” for recursion

For each cell (voxel) (v) with center ((x_v,y_v,z_v,\lambda_v,\phi_v)), define a cell state:

[
s_v(t) = G\big(F(x_v,y_v,z_v,\lambda_v,\phi_v,t)\big)
]

Where (G(\cdot)) is any function that compresses / normalizes:

• E.g. log amplitude, normalized intensity, probability of occupancy, material classification score, etc.

So now one may have:

• A set of cells (v)

• Each with a scalar or small vector state (s_v(t))

This is exactly the kind of “scalar state” the recursive amplifier and QCAD-style logic can operate on.

5. Apply recursive amplifier per cell

For each cell (v) and time step (t_n), define:

1. Pre-processed state (perturbed state from environment & control):

[
\tilde{s}_v(t_n) = s_v(t_n) + \alpha u_v(t_n) + \delta_v(t_n)
]

• (u_v(t_n)): local control or attention weight (e.g., how much users are actively scanning that region)

• (\delta_v(t_n)): environment noise / model mismatch

1. Recursive amplifier (Stone power-tower-style map):

Start from:

[
x_{v,0}(t_n) = \sigma\big(\tilde{s}_v(t_n)\big)
]

Then iterate:

[
x_{v,k+1}(t_n) = f\big(x_{v,k}(t_n)\big)
]

Example:

• Bounded power recursion:
  [
  f(x) = \text{clip}\big(x^x, x_{\min}, x_{\max}\big)
  ]

• Or simpler exponential:
  [
  f(x) = x^\lambda,; \lambda > 1
  ]

Stop when:

[
\left|x_{v,k+1}(t_n) - x_{v,k}(t_n)\right| < \varepsilon
]

Then define:

[
z_v(t_n) = x_{v,K_v}(t_n)
]

This (z_v(t_n)) is now your amplified mapping value for that cell, where subtle differences in (s_v) or trajectory are blown up near extremes.

6. Use recursion to classify each map cell (bifurcation layer)

For each voxel (v), compute:

[
v_v(t_n) = z_v(t_n) - z_v(t_{n-1}) \quad\text{(discrete derivative / “velocity”)}
]

Now one can classify each cell:

• Stable / background region
  [
  |v_v(t_n)| < \theta_{\text{stable}}
  \Rightarrow \text{background / steady geometry}
  ]

• Edge / change region (moving object, boundary, or new feature)
  [
  |v_v(t_n)| \geq \theta_{\text{edge}}
  \Rightarrow \text{dynamic or boundary}
  ]

• Bifurcation / anomaly region
  If recursion fails to converge or oscillates →
  [
  \text{cell classified as anomalous / critical bifurcation}
  ]

This is where your QCAD language plugs in:

• Convergence → static / stable geometry

• Divergence / oscillation → moving objects, transitions, anomalies

The LiDAR map becomes a QCAD-classified spatial-spectral-modulation field.

7. Close the loop: LiDAR scanning strategy as a control problem

One can now make the LiDAR scanning itself adaptive using the same math:

For each laser (i), choose its next angles and wavelength based on the map:

[
\alpha_i(t_{n+1}) = \alpha_0 + f_\alpha\big(i, t_n, {z_v(t_n)}{v}\big)
]
[
\omega_i(t{n+1}) = \omega_0 + f_\omega\big(i, t_n, {z_v(t_n)}{v}\big)
]
[
\lambda_i(t{n+1}) = \lambda_{\min} + (\lambda_{\max}-\lambda_{\min}) f_{\text{spec}}\big(i, t_n, {z_v(t_n)}_{v}\big)
]

Examples:

• Focus more scans (larger f values) on regions with large (|v_v|) (fast-changing)

• Reduce scans in fully converged, stable regions

• Change wavelength where anomalies are detected to probe material composition

So:

This LiDAR isn’t just mapping the world; the world map (through the recursive amplifier) is driving how LiDAR chooses what to see next.

That is a true mapping system: sensing ↔ mapping ↔ control, all wrapped in the recursive/QCAD logic.