Image Enhancement

Image Enhancement 

Image enhancement improves the visual appearance of an image or makes it more suitable for analysis and interpretation.

Spatial Domain Image Enhancement

Spatial domain methods operate directly on image pixels.

$$g(x,y)=T[f(x,y)]$$

Where:

• f(x,y) → input image
• g(x,y) → enhanced image
• T → transformation function

Gray Level Transformations

Gray level transformations modify pixel intensity values.

Image Negative

Used to enhance white or gray details in dark regions.

$$s=L-1-r$$

Input (r)	Output (s)
Dark	Bright
Bright	Dark

Application: Medical imaging, X-ray images

Logarithmic Transformation

Expands dark pixel values and compresses bright values.

$$s=c\log (1+r)$$

Use: Fourier spectrum display

Power-Law (Gamma) Transformation

$$$$

γ Value	Effect
γ < 1	Brightens image
γ > 1	Darkens image

Application: Gamma correction in monitors

Piecewise Linear Transformation

Used for contrast stretching.

Technique	Purpose
Contrast stretching	Improves contrast
Gray level slicing	Highlights ranges
Bit-plane slicing	Binary image analysis

Histogram Processing

A histogram represents the distribution of gray levels.

Image Histogram

Feature	Description
x-axis	Gray levels
y-axis	Frequency

Histogram Equalization

Redistributes gray levels to improve contrast.

Advantages:

• Automatic contrast enhancement
• No parameter tuning

Limitation:

• May over-enhance noise

Histogram Specification (Matching)

Maps histogram to a desired shape.

Application: Medical & satellite images

Histogram Equalization vs Specification

Feature	Equalization	Specification
Output	Uniform	User-defined
Control	Low	High
Complexity	Simple	Moderate

Basics of Spatial Filtering

Spatial filtering modifies pixel values using a filter mask (kernel).

Spatial Filtering Equation

Where:

• w(i,j) → filter mask
• f(x,y) → input image

Types of Spatial Filters

Filter Type	Purpose
Smoothing	Noise removal
Sharpening	Edge enhancement

Smoothing Spatial Filtering

Used to reduce noise and detail.

Mean (Averaging) Filter

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Effect: Blurs image

Weighted Mean Filter

Gives higher weight to center pixel.

Median Filter

Replaces pixel with median value.

Best for: Salt-and-pepper noise

Comparison of Smoothing Filters

Filter	Noise Removal	Edge Preservation
Mean	Good	Poor
Weighted	Better	Moderate
Median	Excellent	Good

Sharpening Spatial Filtering

Used to highlight edges and fine details.

First-Order Derivative (Gradient)

Operator	Mask
Roberts	2×2
Prewitt	3×3
Sobel	3×3

Sobel Mask (Horizontal)

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Second-Order Derivative (Laplacian)

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Property: Highlights regions of rapid intensity change

Unsharp Masking & High-Boost Filtering

k	Effect
1	Unsharp masking
>1	High-boost

Smoothing vs Sharpening

Feature	Smoothing	Sharpening
Purpose	Noise reduction	Edge enhancement
Effect	Blur	Highlight details
Filters	Mean, Median	Sobel, Laplacian

Applications

Technique	Application
Histogram equalization	Medical images
Median filter	Noise removal
Sobel operator	Edge detection
Laplacian	Image sharpening

Introduction to Frequency Domain Processing

In frequency domain processing, an image is transformed from the spatial domain to the frequency domain, processed, and then converted back.

Basic Idea

Where:

• f(x,y) → input image
• F(u,v) → frequency representation
• g(x,y) → enhanced image

Why Frequency Domain?

Reason	Explanation
Low frequencies	Represent smooth areas
High frequencies	Represent edges & noise
Filtering	Easy separation of components

Fourier Transform (FT)

2D Discrete Fourier Transform

$$F(u,v)=\sum _{x=0}^{M-1}\sum _{y=0}^{N-1}f(x,y)e^{-j2\pi (\frac{ux}{M}+\frac{vy}{N})}$$

Inverse DFT

$$f(x,y)=\frac{1}{MN}\sum _{u=0}^{M-1}\sum _{v=0}^{N-1}F(u,v)e^{j2\pi (\frac{ux}{M}+\frac{vy}{N})}$$

Properties of Fourier Transform

Property	Meaning
Linearity	Sum of transforms
Periodicity	Repeating spectrum
Symmetry	Complex conjugate
Convolution	Multiplication in frequency domain

Frequency Domain Filtering

Filtering Equation

$$G(u,v)=H(u,v)F(u,v)$$

Where:

• H(u,v) → frequency filter

Smoothing (Low-Pass) Frequency Domain Filters

Low-pass filters suppress high-frequency noise.

Ideal Low-Pass Filter (ILPF)

$$H(u,v) = \begin{cases} 1 & D(u,v) \leq D_0 \\ 0 & D(u,v) > D_0 \end{cases}$$

Feature	Description
Cutoff	Sharp
Ringing	High
Practical use	Limited

Butterworth Low-Pass Filter (BLPF)

$$H(u,v)=\frac{1}{1+(\frac{D(u,v)}{D_0}{)}^{2\mathrm{n}}}$$

Parameter	Meaning
D₀	Cutoff frequency
n	Filter order

Gaussian Low-Pass Filter (GLPF)

$$H(u,v)=e^{-\frac{D^2(u,v)}{2D_0^{\mathrm{2}}}}$$

Advantage	Explanation
No ringing	Smooth transition
Best visual result	Preferred

Sharpening (High-Pass) Frequency Domain Filters

High-pass filters enhance edges and fine details.

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Types of High-Pass Filters

Filter	Description
Ideal HPF	Sharp cutoff
Butterworth HPF	Smooth cutoff
Gaussian HPF	No ringing

High-Pass Filter Relation

$$H_{HP}(u,v)=1-H_{LP}(u,v)$$

Comparison of Frequency Domain Filters

Feature	Ideal	Butterworth	Gaussian
Transition	Abrupt	Gradual	Very smooth
Ringing	Severe	Moderate	None
Practical use	Low	Medium	High

Homomorphic Filtering

Purpose: Separates illumination and reflectance components of an image.

Image Model

$$f(x,y)=i(x,y)\cdot r(x,y)$$

Taking log:

$$\ln f=\ln i+\ln r$$

Steps in Homomorphic Filtering

• Log transformation
• Fourier transform
• High-pass filtering
• Inverse FT
• Exponential operation

Advantages

Benefit	Use
Illumination correction	Uneven lighting
Contrast enhancement	Medical images

Color Image Enhancement

Color image enhancement improves visual quality of color images.

RGB-Based Enhancement

• Enhance each channel separately
• Risk of color distortion

HSI-Based Enhancement (Preferred)

Step	Description
Convert RGB → HSI	Separate intensity
Enhance Intensity	Histogram or filtering
Convert back	HSI → RGB

Advantage: Preserves original color information

Frequency Domain Color Enhancement

• Apply frequency filtering on Intensity component
• Used in satellite & medical images

Applications

Technique	Application
Gaussian LPF	Noise reduction
Butterworth HPF	Edge sharpening
Homomorphic filtering	Illumination normalization
Color enhancement	Remote sensing

Exam-Oriented Key Points

• Draw frequency filter shapes
• Write ILPF, BLPF, GLPF formulas
• Explain homomorphic filtering steps
• Compare Ideal vs Butterworth vs Gaussian