Describe Wavelets. How are Two-Dimensional Wavelets Constructed?
Wavelets are a type of multi-resolution image representation, similar to image pyramids but with several important enhancements. Unlike Gaussian or Laplacian pyramids that simply create low-pass and band-pass versions of images, wavelets decompose images into frequency subbands that isolate spatial details in horizontal, vertical, and diagonal directions.
A key advantage of wavelets is that they offer both spatial and frequency localization, making them particularly effective for tasks such as image compression, denoising, texture analysis, and feature detection.
Concept of Wavelet Transform
The wavelet transform is a signal processing technique that breaks down a signal (or image) into a set of basis functions called wavelets, which are localized in both space and frequency. This is unlike the Fourier transform, which provides only frequency information and loses spatial localization.
Wavelets can be viewed as filters:
- A low-pass filter captures the coarse approximation of the image (the average).
- High-pass filters capture details at different orientations: horizontal, vertical, and diagonal.
These filters are applied recursively on the low-pass (approximation) image to create a hierarchy of detail and approximation coefficients.
1D to 2D Wavelet Extension
To extend wavelets from 1D to 2D images:
- The transform is first applied along rows (horizontal direction).
- Then applied along columns (vertical direction).
This results in four sub-band images at each level of decomposition:
- LL (Low-Low): approximation image
- LH (Low-High): vertical details
- HL (High-Low): horizontal details
- HH (High-High): diagonal details
The LL sub-band (coarse approximation) is then further decomposed at the next level.
Construction of Two-Dimensional Wavelets
The construction process of 2D wavelets follows these steps:
Step 1: Filtering
- Convolve each row of the image with:
- A low-pass filter to get approximate values.
- A high-pass filter to get detail values.
- Downsample the result by 2 (keep every second pixel).
Step 2: Column Filtering
- Apply the same process to each column of the filtered rows.
- This produces four sub-bands (LL, LH, HL, HH) for each level.
Step 3: Recursion
- Repeat the process on the LL image for multiple levels of decomposition.
This process results in a pyramid-like structure, but with more directional detail. Each wavelet level stores three-quarters of the original pixels (details) and one-quarter (approximation) is passed to the next level.
Properties of Wavelets
- Perfect reconstruction: The original image can be reconstructed exactly using inverse wavelet transform.
- Compact representation: Sparse coefficients make it suitable for image compression (used in JPEG 2000).
- Directional sensitivity: Captures details along different orientations.
- Multi-scale: Useful for analyzing images at multiple resolutions.
Application Example
In image compression, wavelet coefficients can be quantized and stored efficiently. Due to the hierarchical structure, most of the image energy is in the LL bands, while finer details (HH, HL, LH) may be aggressively compressed without significant perceptual loss.
