The Projection Method for Multidimensional Framelet and Wavelet Analysis
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Bibliographic record
Abstract
The projection method is a simple way of constructing functions and filters by integrating multidimensional functions and filters along parallel superplanes in the space domain. Equivalently expressed in the frequency domain, the projection method constructs a new function by simply taking a cross-section of the Fourier transform of a multidimensional function. The projection method is linked to several areas such as box splines in approximation theory and the projection-slice theorem in image processing. In this paper, we shall systematically study and discuss the projection method in the area of multidimensional framelet and wavelet analysis. We shall see that the projection method not only provides a painless way for constructing new wavelets and framelets but also is a useful analysis tool for studying various optimal properties of multidimensional refinable functions and filters. Using the projection method, we shall explicitly and easily construct a tight framelet filter bank from every box spline filter having at least order one sum rule. As we shall see in this paper, the projection method is particularly suitable to be applied to frequency-based nonhomogeneous framelets and wavelets in any dimensions, and the periodization technique is a special case of the projection method for obtaining periodic wavelets and framelets from wavelets and framelets on Euclidean spaces.
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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 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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