250409-peaks finder
connected objects (or "blobs")
1. peak finder
tips: maybe find peak after denoise is better?
todo: implement spherical peak finder maybe better?
2. denoise
🌀 Needlet-Based Denoising for CMB Maps
📌 Overview
Needlets are spherical wavelets localized in both harmonic and pixel space. Denoising in needlet space leverages their multiscale nature to suppress noise without overly smoothing the signal.
🔹 Hard/Soft Thresholding in Needlet Space
1. Needlet Transform
Decompose the map $T(\hat{n})$ into needlet coefficients:
2. Thresholding Rules
Hard Thresholding
Soft Thresholding
Threshold $\lambda_j$ is often chosen based on the noise level at scale $j$, e.g.,
3. Reconstruction
🔹 Bayesian Shrinkage in Needlet Space
Model Setup
Assume noisy coefficients:
With prior:
Posterior Mean (Shrinkage Estimator)
This is equivalent to Wiener filtering in needlet space.
✅ Comparison Summary
Method
Thresholding
Local Adaptivity
Bayesian
Continuous Output
Hard Thresholding
Yes
Yes
No
No
Soft Thresholding
Yes
Yes
No
Yes
Bayesian Shrinkage
No (implicit)
Yes
Yes
Yes
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