Stroke Expanding

This document details the mathematical framework used to dynamically expand a robotic painting toolpath stroke from a given starting position \((x_0, y_0)\), as implemented in generate_strokes_for_layer.

1. Core Vector Expansion Framework

Once a stroke seed position is established, the path expands iteratively. At each step \(k\), the next position is determined using the current position and a unified, normalized direction vector \(\mathbf{d}_k = (dx, dy)\):

\[\begin{aligned} x_{k+1} &= x_k + dx \cdot \Delta s \\ y_{k+1} &= y_k + dy \cdot \Delta s \end{aligned}\]

Where: * \((x_k, y_k)\) represents the current continuous coordinate. * \(\Delta s\) is the step size constant (stroke_generation_supervisor.gradient_step_size). * \(\mathbf{d}_k\) is a unit vector (\(||\mathbf{d}_k|| = 1\)) computed by blending gradient-orthogonal guidance and an unpainted area attraction force.

2. Gradient-Orthogonal Guidance (Flow Tracking)

To ensure the stroke mimics artistic behavior by flowing along image contours rather than cutting across them, the algorithm computes a vector orthogonal to the image gradient \(\nabla I = (g_x, g_y)\).

Vector Orthogonality

The image gradient points in the direction of the sharpest intensity change. The vector perpendicular to this gradient is obtained via a 90-degree rotation:

\[\mathbf{v}_{\text{grad}} = (-g_y, g_x)\]

Direction Normalization and Consistency

If the vector magnitude \(M = \sqrt{(-g_y)^2 + g_x^2}\) is non-zero, it is converted to a unit vector:

\[\mathbf{\hat{v}}_{\text{grad}} = \left( \frac{-g_y}{M}, \frac{g_x}{M} \right)\]

To prevent the path from making abrupt 180-degree U-turns between steps, the dot product between the new vector and the previous step's direction vector \(\mathbf{d}_{k-1}\) is evaluated. If it opposes the current momentum, the direction is flipped:

\[\mathbf{\hat{v}}_{\text{grad}} = \begin{cases} -\mathbf{\hat{v}}_{\text{grad}}, & \text{if } \mathbf{\hat{v}}_{\text{grad}} \cdot \mathbf{d}_{k-1} < 0 \\ \mathbf{\hat{v}}_{\text{grad}}, & \text{otherwise} \end{cases}\]

3. Unpainted Region Attraction Force

To maximize superpixel area coverage and fill bare spots efficiently, an expansion "pull" vector towards local unpainted regions is calculated within a local neighborhood defined by an attraction radius \(R\) (ATTRACTION_RADIUS).

Center of Mass via Inverse Distance Weighting (IDW)

Let \(U\) be the set of relative offsets \((\Delta x_i, \Delta y_i)\) corresponding to unpainted pixels within the bounding box \([x_k - R, x_k + R] \times [y_k - R, y_k + R]\).

For each unpainted pixel \(i \in U\), its squared Euclidean distance to the brush tip is:

\[d_i^2 = (\Delta x_i)^2 + (\Delta y_i)^2\]

To give closer pixels a stronger pull over distant ones, the algorithm maps an inverse distance weight (safeguarding the brush tip center to avoid division-by-zero):

\[w_i = \frac{1}{\max(1.0, d_i^2)}\]

The total attraction pull vector \(\mathbf{v}_{\text{pull}} = (P_x, P_y)\) acts as a localized center of mass:

\[P_x = \sum_{i \in U} \Delta x_i \cdot w_i, \quad P_y = \sum_{i \in U} \Delta y_i \cdot w_i\]

If \(||\mathbf{v}_{\text{pull}}|| > 0\), it is normalized into a unit vector \(\mathbf{\hat{v}}_{\text{pull}}\).

4. Vector Blending and Final Normalization

The final direction vector \(\mathbf{d}_k\) is a linear combination of the structural stroke path vector and the unpainted coverage attraction vector, modulated by the attraction weight \(w_a\) (ATTRACTION_WEIGHT):

\[\mathbf{v}_{\text{final}} = \mathbf{\hat{v}}_{\text{grad}} + w_a \cdot \mathbf{\hat{v}}_{\text{pull}}\]

Because the addition of two unit vectors does not scale to unity, the blended vector is normalized a final time to produce the directional step components:

\[\mathbf{d}_k = \frac{\mathbf{v}_{\text{final}}}{||\mathbf{v}_{\text{final}}||} = (dx, dy)\]

5. Early Termination Conditions

The loop stops expanding the stroke sequence if any of the following boundaries are crossed:

  1. Length Limit: Loop iterations hit max_stroke_list_length.

  2. Flat Gradient Fallback Failure: If \(M = 0\) and there is no prior historical direction, expansion falls back to a random coordinate shift. If it still fails to exit the current pixel matrix, a flat gradient event is triggered and expansion halts.

  3. Color Deviation Threshold: If the continuous Euclidean color distance between the initial seed color \(C_{\text{seed}}\) and the current canvas pixel canvas color \(C_{k+1}\) surpasses an acceptable threshold while satisfying minimum length rules:

\[\|C_{k+1} - C_{\text{seed}}\|_2 > 120\]