Two-Dimensional Variational Mode Decomposition (2D-VMD)

Two-Dimensional Variational Mode Decomposition (2D-VMD) is an adaptive image decomposition method that can decompose complex images into multiple mode components with distinct frequency and spatial characteristics. This approach extends the one-dimensional Variational Mode Decomposition (VMD) to handle two-dimensional signal processing, making it particularly suitable for image analysis and feature extraction applications.

The core principle of 2D-VMD involves solving an optimization problem to decompose images, where each mode component possesses specific center frequencies and bandwidths. This ensures that the decomposed components maintain clear physical significance in both frequency and spatial domains. The algorithm employs iterative optimization to adaptively determine the number of modes and their individual characteristics, making it applicable to images with varying complexity levels. Implementation typically involves alternating direction method of multipliers (ADMM) for constraint optimization.

In practical implementation, 2D-VMD utilizes frequency domain transformations, constrained optimization techniques, and iterative updates of mode components. Key computational steps include 2D Fourier transforms for frequency domain processing, Wiener filtering for mode updating, and convergence checks for optimization termination. These operations ensure that the decomposed components preserve critical information from the original image while effectively separating features at different scales. The method finds extensive applications in medical image analysis, remote sensing image processing, and texture classification tasks.

Reference papers and code implementations are available in original research literature, typically containing detailed mathematical derivations and experimental validations. Open-source implementations often include MATLAB or Python code demonstrating the optimization framework, mode extraction procedures, and parameter tuning methods to help understand both theoretical foundations and practical application outcomes.