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.

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🔹 Hard/Soft Thresholding in Needlet Space

1. Needlet Transform

Decompose the map $T(\hat{n})$ into needlet coefficients:

$$\beta_{jk} = \int_{S^2} T(\hat{n}) \, \psi_{jk}(\hat{n}) \, d\Omega$$

2. Thresholding Rules

Hard Thresholding

$$\tilde{\beta}_{jk} = \begin{cases} \beta_{jk}, & \text{if } |\beta_{jk}| > \lambda_j \\ 0, & \text{otherwise} \end{cases}$$

Soft Thresholding

$$\tilde{\beta}_{jk} = \begin{cases} \text{sign}(\beta_{jk}) (|\beta_{jk}| - \lambda_j), & \text{if } |\beta_{jk}| > \lambda_j \\ 0, & \text{otherwise} \end{cases}$$

Threshold $\lambda_j$ is often chosen based on the noise level at scale $j$, e.g.,

$$\lambda_j = \sigma_j \sqrt{2 \log N_j}$$

3. Reconstruction

$$\tilde{T}(\hat{n}) = \sum_{j,k} \tilde{\beta}_{jk} \, \psi_{jk}(\hat{n})$$

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🔹 Bayesian Shrinkage in Needlet Space

Model Setup

Assume noisy coefficients:

$$\beta_{jk}^{\text{obs}} = \beta_{jk}^{\text{true}} + \epsilon_{jk}, \quad \epsilon_{jk} \sim \mathcal{N}(0, \sigma_{jk}^2)$$

With prior:

$$\beta_{jk}^{\text{true}} \sim \mathcal{N}(0, \tau_j^2)$$

Posterior Mean (Shrinkage Estimator)

$$\tilde{\beta}_{jk} = \frac{\tau_j^2}{\tau_j^2 + \sigma_{jk}^2} \, \beta_{jk}^{\text{obs}}$$

This is equivalent to Wiener filtering in needlet space.

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✅ 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