Image Restoration: Complete Guide with Examples

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1. What is Image Restoration?

Image Restoration is the process of recovering the original image from a degraded or noisy image.
Degradation can occur due to noise, blurring, or transmission errors. The goal is to improve image quality.

Noise Model:

g(x,y) = f(x,y) + η(x,y)

• g(x,y): observed (noisy) image
• f(x,y): original image
• η(x,y): noise

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2. Types of Noise with Examples

2.1 Gaussian Noise

Noise follows a normal distribution (bell curve). Common in cameras and sensors.

PDF: P(z) = (1 / (√(2π)σ)) * exp(-(z-μ)² / 2σ²)

Original 3×3 patch:

52	55	61
62	59	55
63	65	66

Gaussian noise values (assumed, μ=0, σ=5):

2	-5	2
3	1	-3
-3	3	-2

Noisy patch:

54	50	63
65	60	52
60	68	64

Restoration using Arithmetic Mean Filter (3×3)

• Neighborhood for center pixel (60): [54,50,63,65,60,52,60,68,64]
• Sum = 536
• Average = 536 / 9 ≈ 59.56 → replace center pixel 60

Resulting patch after filter:

54	50	63
65	59.56	52
60	68	64

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2.2 Salt & Pepper Noise

Random pixels become black (0) or white (255). Common in transmission errors.

Original 3×3 patch:

52	55	61
62	59	55
63	65	66

Noisy patch:

52	255	61
0	59	55
63	65	255

Restoration using Median Filter (3×3)

• Neighborhood for center pixel (59): [52,255,61,0,59,55,63,65,255]
• Sorted: [0,52,55,59,61,63,65,255,255]
• Median = 61 → replace center pixel 59

Resulting patch after filter:

52	255	61
0	61	55
63	65	255

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2.3 Uniform Noise

Noise is evenly distributed between two values. Adds small random variations.

Original 3×3 patch:

50	52	55
53	54	52
51	53	54

Noisy patch (±2):

52	50	55
54	56	50
49	55	53

Restoration using Midpoint Filter (3×3)

• Neighborhood for center pixel (56): [52,50,55,54,56,50,49,55,53]
• Max = 56, Min = 49 → Midpoint = (56+49)/2 = 52.5

Resulting patch after filter:

52	50	55
54	52.5	50
49	55	53

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2.4 Rayleigh Noise

Common in radar images. Example 3×3 patch:

50	52	51
53	54	52
51	53	54

Noisy patch (Rayleigh noise, assumed values):

53	55	52
55	57	53
52	56	55

Restoration using Arithmetic Mean Filter:

• Neighborhood sum for center pixel = 53+55+52+55+57+53+52+56+55=488
• Average = 488/9 ≈ 54.22 → replace center pixel

Resulting patch:

53	55	52
55	54.22	53
52	56	55

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2.5 Erlang / Gamma Noise

Occurs in multiplicative processes like sonar.

Original patch:

50	51	53
52	54	52
51	53	55

Noisy patch (assumed Erlang noise):

52	52	54
53	56	53
52	54	57

Restoration using Arithmetic Mean Filter:

• Neighborhood sum for center pixel = 52+52+54+53+56+53+52+54+57=483
• Average = 483/9 ≈ 53.67 → replace center pixel

Resulting patch:

52	52	54
53	53.67	53
52	54	57

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2.6 Exponential Noise

Common in low-light or photon-limited sensors.

Original patch:

50	51	52
53	54	52
51	53	55

Noisy patch (assumed exponential noise):

51	52	53
54	56	53
52	54	56

Restoration using Arithmetic Mean Filter:

• Neighborhood sum for center pixel = 51+52+53+54+56+53+52+54+56=481
• Average = 481/9 ≈ 53.44 → replace center pixel

Resulting patch:

51	52	53
54	53.44	53
52	54	56

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3. Summary Table of Filters

Noise Type	Best Filter	Example Computation
Gaussian	Arithmetic / Geometric Mean	Average of neighborhood: (54+50+63+65+60+52+60+68+64)/9 ≈ 59.56
Salt & Pepper	Median Filter	Median of [52,255,61,0,59,55,63,65,255] = 61
Uniform	Midpoint Filter	Max=56, Min=49 → Midpoint = 52.5
Rayleigh	Arithmetic Mean	Average of neighborhood: sum/9 ≈ 54.22
Erlang / Gamma	Arithmetic Mean	Average of neighborhood: sum/9 ≈ 53.67
Exponential	Arithmetic Mean	Average of neighborhood: sum/9 ≈ 53.44