Image Sharpening via Sobolev Gradient Flows
Why this work is in the frame
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Bibliographic record
Abstract
Motivated by some recent work in active contour applications, we study the use of Sobolev gradients for PDE-based image diffusion and sharpening. We begin by studying, for the case of isotropic diffusion, the gradient descent/ascent equation obtained by modifying the usual metric on the space of images, which is the $L^2$ metric, to a Sobolev metric. We present existence and uniqueness results for the Sobolev isotropic diffusion, derive a number of maximum principles, and show that the differential equations are stable and well-posed both in the forward and backward directions. This allows us to apply the Sobolev flow in the backward direction for sharpening. Favorable comparisons to the well-known shock filter for sharpening are demonstrated. Finally, we continue to exploit this same well-posed behavior both forward and backward in order to formulate new constrained gradient flows on higher order energy functionals which preserve the first order energy of the original image for interesting combined smoothing and sharpening effects.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.003 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 it