Directional lapped orthogonal transform: theory and design.

Area of Science:

• Signal Processing
• Image Analysis
• Applied Mathematics

Background:

• Paraunitary filter banks are crucial for signal processing, offering properties like orthogonality and perfect reconstruction.
• Existing methods often lack directional adaptability, limiting their performance in analyzing anisotropic data.
• Nonseparable filter banks offer greater flexibility in basis design but pose design challenges.

Purpose of the Study:

• To propose a novel directional design method for 2-D nonseparable linear-phase paraunitary filter banks.
• To introduce and analyze a new vanishing moment (VM) condition, termed trend VM (TVM), for directional constraints.
• To demonstrate the effectiveness of the proposed method in image denoising applications.

Main Methods:

• A lattice structure combining a 2-D separable Discrete Cosine Transform (DCT) block with nonseparable support extension processes.
• Introduction of the trend vanishing moment (TVM) condition to enforce directional properties in wavelet filters.
• Theoretical analysis of TVMs for general 2-D paraunitary systems and application to lattice parameters.

Main Results:

• The proposed method allows for directional bases with critical subsampling, overlapping, orthogonality, symmetry, real-valued, and compact support.
• Numerical confirmation of trend-surface annihilation properties achieved by the TVM condition.
• Image denoising evaluation shows perceptually preferable results compared to existing methods.

Conclusions:

• The directional design method provides enhanced flexibility and performance for 2-D filter banks.
• The trend vanishing moment condition is an effective tool for achieving directional constraints in filter bank design.
• The proposed transforms demonstrate significant potential for improving image denoising quality.