2D Wavelet à trous Algorithm Image Decomposition Based on FFT and Dyadic Wavelets

We have developed a 2D wavelet à trous algorithm for image decomposition based on Fast Fourier Transform (FFT) and dyadic wavelets. This method employs efficient FFT computations to handle convolution operations in the wavelet transform, while the dyadic wavelet scheme provides optimal multi-resolution analysis. The algorithm achieves excellent decomposition results, effectively separating image details from overall features through multi-scale analysis. Key implementation aspects include: using FFT for rapid convolution calculations in frequency domain, implementing dyadic wavelet filters with progressively increasing dilation factors, and applying the à trous algorithm to maintain translation invariance without subsampling. The decomposition process preserves image size across scales and enables precise extraction of high-frequency details and low-frequency approximations. This approach has broad applications in image processing, pattern recognition, and computer vision fields. Our research contributes to enhanced image analysis capabilities, particularly benefiting applications requiring detailed multi-scale feature extraction and texture analysis. The method's translation invariance makes it suitable for texture classification and edge detection tasks where spatial relationships are critical. The algorithm implementation typically involves: constructing dyadic wavelet filters in frequency domain, performing FFT-based convolution for each decomposition level, and reconstructing subbands using inverse transforms. This framework provides researchers and practitioners with an effective tool for advanced image decomposition and multi-scale analysis.