Image Pyramid

1. What is an Image Pyramid?

An image pyramid is a multi-scale representation of an image. It’s a hierarchical structure consisting of multiple versions of the same image at different resolutions, typically arranged in a pyramid-like shape.

• Levels: Each level of the pyramid represents the image at a specific scale.
• Base (Bottom Level): The original, full-resolution image.
• Higher Levels: Each subsequent level is a reduced-resolution (smaller) version of the image at the previous level. The resolution typically decreases by a factor of 2 in each dimension as you go up the pyramid.
• Apex (Top Level): The smallest, lowest-resolution version of the image.

2. Types of Image Pyramids

• Gaussian Pyramid:
  • The most common type.
  • Each level is created by low-pass filtering (usually with a Gaussian filter) and then downsampling the previous level.
  • This process effectively removes high-frequency details, creating a blurred and smaller version of the image.
  • Used in various applications, including feature detection, object detection, and image blending.
• Laplacian Pyramid:
  • Constructed from a Gaussian pyramid.
  • Each level stores the difference between that level in the Gaussian pyramid and an upsampled version of the next higher level in the Gaussian pyramid.
  • Essentially, each level captures the details (high-frequency information) that are lost when going to a lower resolution in the Gaussian pyramid.
  • Useful for image compression, image blending, and image enhancement.

3. Key Operations: Subsampling and Upsampling

• Subsampling (Downsampling):
  • Purpose: To reduce the resolution of an image.
  • Steps:
    1. Low-pass Filtering: Apply a low-pass filter (typically Gaussian) to the image. This is crucial to prevent aliasing, which occurs when high-frequency information is misrepresented as lower frequencies after downsampling, leading to artifacts.
    2. Downsampling: Select a subset of the filtered pixels. For a downsampling factor of 2, every other pixel is kept in both the horizontal and vertical directions.
• Upsampling:
  • Purpose: To increase the resolution of an image.
  • Steps:
    1. Interpolation: Insert new pixels between the existing ones and estimate their values using interpolation methods. Common methods include:
       • Nearest Neighbor: Assigns the value of the closest existing pixel (fast but can produce blocky results).
       • Bilinear: Calculates a weighted average of the four nearest existing pixels (smoother than nearest neighbor).
       • Bicubic: Uses a weighted average of the 16 nearest pixels (produces even smoother results but is more computationally expensive).
    2. Filtering (Optional): A low-pass filter can be applied after upsampling to smooth out any artifacts introduced by interpolation.

4. Why are Image Pyramids Useful?

• Multi-scale Analysis: They allow algorithms to process images at different scales, making them robust to variations in object size.
• Computational Efficiency: Operations performed on smaller images are faster. Coarse-to-fine strategies leverage this by first finding approximate solutions at lower resolutions and then refining them.
• Feature Detection: Many feature detectors (e.g., SIFT, SURF) use image pyramids to find scale-invariant features.
• Object Detection: Detecting objects of different sizes becomes easier by searching at each level of the pyramid.
• Image Blending: Laplacian pyramids enable seamless blending by combining different frequency bands separately.
• Image Compression: Laplacian pyramids can be more compressible than the original image due to their sparse nature (many values near zero).
• Other Applications: Optical flow, image registration, image segmentation, etc.

5. Laplacian Pyramid: Decomposing into Frequency Bands

• Construction:

  1. Build a Gaussian pyramid.
  2. For each level (except the top), subtract an upsampled version of the next higher level from the current level in the Gaussian pyramid. This difference image is the corresponding level in the Laplacian pyramid.

• Properties:

  • Each level represents a band-pass filtered version of the original image, capturing details within a specific frequency range.
  • The top level of the Laplacian pyramid is the same as the top level of the Gaussian pyramid (the lowest resolution image).
  • The original image can be reconstructed by reversing the process: upsample the top level of the Laplacian pyramid and add it to the next lower level, repeat until you reach the base.