| # AIFlow Math Ink 0.5: geometric gridding method |
|
|
| ## Abstract |
|
|
| AIFlow Math Ink 0.5 is a deterministic preprocessing method for partitioning |
| online handwritten mathematics into image cells suitable for downstream |
| image-to-LaTeX recognition. It uses stroke trajectories and pen widths rather |
| than predicted symbols. The method constructs a sparse semantic approximation |
| through pairwise geometric predicates, augments it with fraction-bar and tall |
| delimiter bridges, and extracts connected components with disjoint-set union. |
| It is not a learned model and makes no claim of semantic parsing. |
|
|
| ## 1. Input representation |
|
|
| Let a canvas contain ordered strokes |
|
|
| \[ |
| \mathcal{S} = \{s_i\}_{i=1}^{n}, |
| \qquad |
| s_i = (\mathrm{id}_i, \{(x_{ik},y_{ik})\}_{k=1}^{m_i}, w_i). |
| \] |
| |
| Each stroke has a stable identifier, an ordered point sequence, and pen width |
| \(w_i\). Its axis-aligned bounding box expands the point extrema by |
| \(w_i/2\). This preserves the visible support of thick strokes. |
| |
| The parser applies finite caps to stroke count, points per stroke, total points, |
| coordinate magnitude, and pen width. These caps are part of the runtime safety |
| contract rather than the mathematical grouping rule. |
| |
| ## 2. Scale estimation |
| |
| Let \(h_i\) be the height of stroke box \(B_i\). The characteristic scale is |
| |
| \[ |
| \tilde h = \operatorname{median}_{i}(h_i). |
| \] |
| |
| Median height is resistant to a small number of long fraction bars or tall |
| delimiters. Four distances are then defined: |
| |
| \[ |
| \tau_b = \max(24, 0.50\tilde h), \quad |
| \tau_x = \max(72, 1.25\tilde h), |
| \] |
| |
| \[ |
| \tau_s = \max(52, 0.80\tilde h), \quad |
| \tau_a = \max(18, 0.18\tilde h). |
| \] |
| |
| The constants are defaults exposed by `GridConfig`; they are not learned |
| parameters. |
| |
| ## 3. Geometric relation graph |
| |
| For boxes \(B_i,B_j\), let \(g_x(i,j)\) and \(g_y(i,j)\) be non-negative |
| axis gaps, and let \(c_y(i)\) be vertical center. |
|
|
| A same-baseline edge is created when |
|
|
| \[ |
| |c_y(i)-c_y(j)| \le \tau_b |
| \quad\land\quad |
| g_x(i,j) \le \tau_x. |
| \] |
| |
| A local-attachment edge handles detached pieces and scripts: |
| |
| \[ |
| g_x(i,j)\le\tau_a |
| \quad\land\quad |
| g_y(i,j)\le\tau_a |
| \] |
| |
| \[ |
| \quad\land\quad |
| |c_y(i)-c_y(j)| |
| \le 1.35\max(h_i,h_j)+\tau_a. |
| \] |
|
|
| These predicates intentionally favor recall of one coherent expression. They |
| can over-merge dense neighboring work, which is a documented limitation. |
|
|
| ## 4. Structural bridges |
|
|
| A box is treated as a horizontal bridge candidate when |
|
|
| \[ |
| \operatorname{width}(B_i)\ge30 |
| \quad\land\quad |
| \operatorname{width}(B_i)\ge4\operatorname{height}(B_i). |
| \] |
| |
| It connects to a nearby box when horizontal overlap is at least 20% of the |
| smaller width and vertical axis gap is at most \(\tau_s\). This approximates the |
| role of a fraction bar without recognizing its symbol. |
|
|
| A box is treated as a vertical bridge candidate when |
|
|
| \[ |
| \operatorname{height}(B_i)\ge45 |
| \quad\land\quad |
| \operatorname{height}(B_i)\ge2.5\operatorname{width}(B_i). |
| \] |
| |
| It connects nearby vertically aligned strokes, approximating tall parentheses, |
| brackets, absolute-value marks, and matrix delimiters. |
| |
| ## 5. Connected components |
| |
| All accepted relations are union operations over a disjoint-set forest with |
| path compression and union by size. Each connected component becomes one |
| formula cell. The cell boundary is the union of member boxes plus configurable |
| padding. |
| |
| Cells are deterministically ordered by a quantized vertical band followed by |
| their left coordinate. This is a stable geometric order, not a full |
| two-dimensional mathematical parse. |
| |
| ## 6. Rendering contract |
| |
| For every cell, member strokes are redrawn on a white RGB canvas and cropped to |
| the padded component boundary. Redrawing suppresses unrelated marks that happen |
| to occupy the rectangular crop. The returned object is a list of ordinary |
| `PIL.Image.Image` values and contains no recognizer-specific tensor transform. |
| |
| This image-level contract is compatible with downstream recognizers that accept |
| formula image batches, including TexTeller-style pipelines. No recognizer is |
| included. |
| |
| ## 7. Complexity |
| |
| The pairwise relation phase is \(O(n^2)\). Disjoint-set operations take |
| \(O(\alpha(n))\) amortized time, where \(\alpha\) is the inverse Ackermann |
| function. Rendering cost is proportional to retained stroke points and output |
| pixels. Bounded input size is therefore required for untrusted server traffic. |
| |
| ## 8. Scientific status and limitations |
| |
| AIFlow Math Ink 0.5 is an engineering heuristic with explicit assumptions: |
| |
| - axis-aligned boxes are sufficient local geometry; |
| - median stroke height approximates device scale; |
| - long horizontal and vertical strokes are useful structural bridges; |
| - connected components are adequate formula-cell hypotheses. |
| |
| The release does not include a learned grouping head, probabilistic confidence, |
| ground-truth dataset, or benchmark claim. Appropriate evaluation would report |
| partition exact match, over-merge and under-merge rates, writer/device-disjoint |
| slices, and end-to-end expression accuracy separately. |
| |
| |
