🧠 Noise Estimation in a Single Image – Tutorial

🧠 Noise Estimation in a Single Image – Tutorial

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🎯 Goal

Estimate the noise variance or noise standard deviation (σ) from an image $g(x, y) = f(x, y) + \eta(x, y)$, where $f$ is the clean image and $\eta$ is additive noise.

We often assume zero-mean white Gaussian noise, i.e., $\eta \sim \mathcal{N}(0, \sigma^2)$.

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📚 Common Methods

🟡 1. High-Pass Filtering Method (Residual-Based)

Assumption: Noise is high-frequency, image content is mostly low-frequency.

📌 Steps:

1. Apply a Gaussian blur to remove low-frequency content.
2. Subtract blurred image from original to isolate residual (mostly noise).
3. Compute the standard deviation of this residual.

🧪 Code (OpenCV/Numpy):

import cv2
import numpy as np

def estimate_noise_highpass(img):
    img = img.astype(np.float32)
    blurred = cv2.GaussianBlur(img, (7, 7), 1.5)
    residual = img - blurred
    return np.std(residual)

img = cv2.imread('image.png', 0)  # grayscale
sigma_est = estimate_noise_highpass(img)
print(f"Estimated noise std: {sigma_est:.2f}")

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🟡 2. MAD on Wavelet Detail Coefficients

MAD: Median Absolute Deviation

📌 Steps:

1. Apply a wavelet decomposition (e.g., using Haar or Daubechies).
2. Use high-frequency subbands (e.g., horizontal, vertical, diagonal).
3. Estimate σ from detail coefficients:

\[\sigma \approx \frac{\text{median}(|c|)}{0.6745}\]

This formula works for Gaussian noise due to properties of MAD.

🧪 Code (PyWavelets):

import pywt
import numpy as np
import cv2

def estimate_noise_wavelet(img):
    coeffs = pywt.dwt2(img, 'db1')
    _, (cH, cV, cD) = coeffs
    detail_coeffs = np.concatenate([cH.ravel(), cV.ravel(), cD.ravel()])
    sigma = np.median(np.abs(detail_coeffs)) / 0.6745
    return sigma

img = cv2.imread('image.png', 0)
sigma_wavelet = estimate_noise_wavelet(img.astype(np.float32))
print(f"Estimated noise σ (wavelet): {sigma_wavelet:.2f}")

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🟡 3. Flat Patch Analysis

Assume flat regions (low gradient) contain mostly noise.

📌 Steps:

1. Divide image into blocks (e.g., 8×8)
2. Compute local variance for each patch
3. Keep patches with low edge activity (low gradient or entropy)
4. Estimate noise as mean or median variance of flat patches

This method is robust to structured content.

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🟡 4. Blind Estimators (No Training)

📘 Method by Immerkaer (1996) – uses Laplacian

\[\sigma \approx \sqrt{\frac{\pi}{2}} \cdot \frac{1}{6(N-2)(M-2)} \sum_{x=2}^{N-1} \sum_{y=2}^{M-1} |(g * \nabla^2)(x, y)|\]

Where $\nabla^2$ is the Laplacian kernel:

\[\begin{bmatrix} 0 & 1 & 0 \\ 1 & -4 & 1 \\ 0 & 1 & 0 \end{bmatrix}\]

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🔬 When to Use What?

Method	Strengths	Weaknesses
High-pass residual	Fast, simple	Sensitive to textures
Wavelet MAD	Robust, well-founded	Needs wavelet lib
Flat patch variance	Good for natural images	Needs threshold tuning
Laplacian estimator	Fully blind, elegant	Assumes Gaussian noise

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✅ Summary

• If you want a quick estimate: use high-pass filtering or wavelet-MAD.
• For robust methods: wavelet-based MAD is commonly used in denoising papers.
• Noise estimation is essential for adaptive filters (Wiener, BM3D, DnCNN).

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