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:

βjk=S2T(n^)ψjk(n^)dΩ\beta_{jk} = \int_{S^2} T(\hat{n}) \, \psi_{jk}(\hat{n}) \, d\Omega

2. Thresholding Rules

Hard Thresholding

β~jk={βjk,if βjk>λj0,otherwise\tilde{\beta}_{jk} = \begin{cases} \beta_{jk}, & \text{if } |\beta_{jk}| > \lambda_j \\ 0, & \text{otherwise} \end{cases}

Soft Thresholding

β~jk={sign(βjk)(βjkλj),if βjk>λj0,otherwise\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.,

λj=σj2logNj\lambda_j = \sigma_j \sqrt{2 \log N_j}

3. Reconstruction

T~(n^)=j,kβ~jkψjk(n^)\tilde{T}(\hat{n}) = \sum_{j,k} \tilde{\beta}_{jk} \, \psi_{jk}(\hat{n})

🔹 Bayesian Shrinkage in Needlet Space

Model Setup

Assume noisy coefficients:

βjkobs=βjktrue+ϵjk,ϵjkN(0,σjk2)\beta_{jk}^{\text{obs}} = \beta_{jk}^{\text{true}} + \epsilon_{jk}, \quad \epsilon_{jk} \sim \mathcal{N}(0, \sigma_{jk}^2)

With prior:

βjktrueN(0,τj2)\beta_{jk}^{\text{true}} \sim \mathcal{N}(0, \tau_j^2)

Posterior Mean (Shrinkage Estimator)

β~jk=τj2τj2+σjk2βjkobs\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.


✅ 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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