Lecture 4: Digital Image Processing

Overview

• Continuing discussion on spatial filtering.
• Brief introduction to Fourier Transform (to be covered in detail later).

Spatial Filtering

Smoothing (Low-Pass Filters)

• Used to reduce sharp transitions in image density and random noise.
• Types of smoothing filters:
  • Linear filters (e.g., Box filters).
  • Non-Linear filters (e.g., Median filters).

Sharpening (High-Pass Filters)

• Highlights intensity transitions in images.
• Enhances edges and discontinuities.

Definitions & Operations

• Neighborhood: Area defined around a pixel for filtering.
• Kernel: Array of weights used in spatial filtering.
  • Example: 3x3 kernel.
  • Calculation involves element-wise multiplication and summation.

Types of Spatial Filters

• Linear Spatial Filters:

  • Produce a sum of products based on the kernel weights.
  • Example Equation:
    • G(x, y) = Σ W(m, n) * I(x+m, y+n)

• Non-Linear Spatial Filters:

  • Different operation than linear (e.g., median filtering).

Convolution vs. Correlation

• Linear Spatial Filtering: Convolution is commonly used.
• Properties of Convolution:
  • Commutative: f * g = g * f.
  • Associative: f * (g * h) = (f * g) * h.
  • Distributive: f * (g + h) = f * g + f * h.

Smoothing Filters

• Box Filters: Simple averaging; uniform weights.
• Gaussian Filters: Circularly symmetric; better for blurring; utilizes standard deviation (σ).
  • Gaussian function is defined as:
    • K(x, y) = (1/(2πσ²)) * e^(-(x²+y²)/2σ²)

Padding Techniques

• Zero Padding: Adds zeros around the image.
• Mirror or Symmetric Padding: Mirrors edge values.
• Replicate Padding: Replicates values adjacent to borders.

Non-Linear Filters

Median Filter

• Orders pixel values in the neighborhood and replaces center pixel with the median.
• Effective for reducing salt and pepper noise.

High-Pass Filters

Sharpening Techniques

• Laplacian Operator: Highlights edges by using second derivatives.
  • Kernel example for Laplacian:
    • [-1, -1, -1]
    • [-1, 8, -1]
    • [-1, -1, -1]
• Gradient Magnitude: Uses first derivatives for edge detection.
• Unsharp Masking: Subtracts a blurred version from the original image to enhance details.

Fourier Transform Overview

• Relates spatial and frequency domains:
  • Convolution in spatial domain = Multiplication in frequency domain.
• Categories of Filters:
  • Low-Pass, High-Pass, Bandpass, Band Reject.
• Fourier Transform computational advantages:
  • Speeds up convolution operations significantly (especially for large kernels).

Future Topics

• Detailed coverage of Fourier Series and Transform in next lectures.
• Convolution in frequency domain and sampling discussed in detail.