Gabor Frames and Contact Geometry: From Models of the Primary Visual Cortex to Higher Dimensional Signal Analysis on Manifolds
Bibliographic record
Abstract
This thesis has two objectives: first, to provide a model of the functional architecture of the primary visual cortex $(V_1)$ in terms of both geometry and signal analysis and second to provide a mathematical framework for signal analysis on certain classes of contact manifolds. It is organized in three main parts. \par Firstly, we introduce a model of the primary visual cortex $(V_1)$, which allows the compression and decomposition of a signal by a discrete family of orientation and position dependent receptive profiles. We show in particular that a specific framed sampling set and an associated Gabor system is determined by the Legendrian circle bundle structure of the $3$-manifold of contact elements on a surface (which models the $V_1-$cortex), together with the presence of an almost complex structure on the tangent bundle of the surface (which models the retinal surface). We identify a maximal area of the signal planes, determined by the retinal surface, that provides a finite number of receptive profiles, sufficient for good encoding and decoding. We consider the extension of this model for receptive fields dependent on position, orientation, frequency and phase. \par Moreover, we provide a construction of Gabor Frames that encode local linearizations of a signal detected on a curved smooth manifold of arbitrary dimension. In particular we use Gabor Filters that can detect higher-dimensional boundaries on the manifolds. We describe an application in configuration spaces in robotics with sharp constrains.The construction is a generalization of the geometric framework, developed for the study of the visual cortex. \par Finally, we present a general construction of Gabor analysis on manifolds with coorientable contact distribution, equipped with a Legendrian fibre bundle structure and an almost CR-Structure. This construction is suitable for studying the stability of Gabor frames under contact transformations of the manifold. We prove that Gabor frames with a specific class of window functions are stable under a certain class of contact transformations.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".