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COMPUTER VISION

Image Noise: Types, Models and Denoising

Noise corrupts images during capture, transmission, and processing. Gaussian noise follows a normal distribution; salt-and-pepper is impulse noise. Each type has a matched denoising strategy.

1Sources of Noise in Images

Image noise is an unwanted variation in pixel values that degrades the visual quality of an image. It is an inherent property of photography — there will always be some noise in any captured image. The key question is not how to eliminate it, but how to model and reduce it.

Three main sources produce noise: insufficient light (low-light photography forces high ISO, amplifying the sensor's electronic noise), transmission interference (impulse noise corrupts pixel values as data travels over noisy channels), and sensor heat (thermal noise adds random electron motion, especially in long exposures). Each source produces a different noise distribution.

Sensor noise
Caused by electronic noise in the imaging sensor. Increases with ISO and temperature. Approximately Gaussian in distribution at the pixel level.

Transmission noise
Bits flipped during data transfer. Manifests as isolated pixels set to 0 (black) or 255 (white) — the salt-and-pepper pattern.

Quantisation noise
Rounding error when converting continuous sensor values to discrete integers. Small and uniform — usually negligible compared to sensor noise.

2Gaussian Noise

Gaussian noise adds a random value drawn from a normal distribution N(0, σ²) to each pixel: noisy(x,y) = f(x,y) + ε, where ε ~ N(0,σ²). The result is a smooth, continuous corruption — every pixel is slightly off, but extreme deviations are rare. σ controls the severity: σ=10 is barely visible, σ=50 is very noisy.

Signal-to-Noise Ratio (SNR) quantifies noise severity: SNR = mean signal / σ_noise. Higher SNR means a cleaner image. In practice, Gaussian noise is well-matched by linear filters (Gaussian blur, mean filter) because the averaging operation reduces variance by 1/k² for a k×k kernel.

Gaussian noise is statistically independent at each pixel. Averaging N independent noisy observations of the same scene reduces noise power by a factor of N — this is why burst photography (taking multiple frames and averaging them) is so effective for low-light imaging.

3Salt-and-Pepper Noise

Salt-and-pepper noise (impulse noise) sets a small fraction of pixels to extreme values — 0 (pepper, black) or 255 (salt, white). Unlike Gaussian noise, it is sparse and extreme: most pixels are clean, but corrupted ones are maximally wrong. It commonly arises from transmission errors or faulty sensors.

Best filter: Median

The median filter sorts the neighbourhood window values and takes the middle value. Because 0 and 255 are extreme outliers, they rarely win the median — so the corrupted pixel is replaced by the true local value.

Why mean fails

The mean filter averages all neighbourhood values including the 0/255 outliers. A single salt pixel in a 3×3 window contributes 255/9 ≈ 28 to the average — enough to visibly corrupt the output.

4Denoising Strategies

The choice of denoising filter must match the noise model. Gaussian/mean filter: optimal for Gaussian noise — blurs the image slightly but reduces variance. Median filter: optimal for salt-and-pepper — preserves edges better than mean but is slower. Bilateral filter: edge-preserving Gaussian — weights nearby pixels by both spatial distance and intensity similarity, smoothing uniform regions while keeping edges sharp.

Mean filter
Simple average of k×k neighbourhood. Reduces Gaussian noise variance by 1/k². Fast but blurs edges. O(k²) per pixel.

Median filter
Sorts neighbourhood values, takes middle. Robust to extreme outliers. Non-linear — cannot be expressed as a convolution. Best for impulse noise.

Bilateral filter
Gaussian in both space and intensity. Preserves edges by down-weighting pixels that differ strongly in value. Much slower than mean/median.

Blur: Gaussian, Median and Bilateral →Point Operators →Image Sharpening →

Add Gaussian and salt-and-pepper noise; apply mean and median filters

import numpy as np

np.random.seed(7)

# 6x6 synthetic image (clean)
clean = np.array([
    [100, 100, 100, 100, 100, 100],
    [100, 200, 200, 200, 200, 100],
    [100, 200, 255, 255, 200, 100],
    [100, 200, 255, 255, 200, 100],
    [100, 200, 200, 200, 200, 100],
    [100, 100, 100, 100, 100, 100],
], dtype=float)

def clamp(arr):
    return np.clip(arr, 0, 255)

# Gaussian noise (sigma=20)
gauss_noise = np.random.normal(0, 20, clean.shape)
noisy_gauss = clamp(clean + gauss_noise).astype(int)

# Salt-and-pepper noise (10% pixels)
noisy_sp = clean.copy()
mask = np.random.random(clean.shape)
noisy_sp[mask < 0.05] = 0    # pepper
noisy_sp[mask > 0.95] = 255  # salt
noisy_sp = noisy_sp.astype(int)

def mean_filter(img, k=3):
    pad = k // 2
    out = np.zeros_like(img, dtype=float)
    p = np.pad(img, pad, mode='edge')
    for i in range(img.shape[0]):
        for j in range(img.shape[1]):
            out[i,j] = p[i:i+k, j:j+k].mean()
    return out.astype(int)

def median_filter(img, k=3):
    pad = k // 2
    out = np.zeros_like(img, dtype=float)
    p = np.pad(img, pad, mode='edge')
    for i in range(img.shape[0]):
        for j in range(img.shape[1]):
            out[i,j] = np.median(p[i:i+k, j:j+k])
    return out.astype(int)

print("Centre pixel (3,3):")
print("  Clean:          ", int(clean[3,3]))
print("  Gaussian noisy: ", noisy_gauss[3,3])
print("  S&P noisy:      ", noisy_sp[3,3])
print()
print("After 3x3 mean filter on Gaussian noise:")
denoised_gauss = mean_filter(noisy_gauss)
print("  Pixel (3,3):    ", denoised_gauss[3,3], " (clean was 255)")
print()
print("After 3x3 median filter on S&P noise:")
denoised_sp = median_filter(noisy_sp)
print("  Pixel (3,3):    ", denoised_sp[3,3], " (clean was 255)")
print()
print("Mean filter MSE on Gauss:  ", round(float(np.mean((denoised_gauss - clean)**2)), 2))
print("Median filter MSE on S&P:  ", round(float(np.mean((denoised_sp - clean)**2)), 2))