Implementation of Wavelet Transform Using the À Trous Algorithm (Undecimated Wavelet Transform)

We can implement wavelet transform using the à trous algorithm (undecimated wavelet transform). This algorithm is particularly suitable for one-dimensional signal processing and enables better analysis of signal frequency domain characteristics. The implementation typically involves iterative convolution operations with progressively dilated filters while maintaining the original signal length at each decomposition level. By applying this algorithm, we obtain more detailed spectral information, allowing for more accurate understanding of signal properties and variations. The key advantage lies in its translation-invariant property achieved through the undecimated approach, which preserves all temporal information. This method finds widespread applications in various fields such as image processing, audio analysis, and video compression. The algorithm's implementation commonly involves scaling and wavelet filter banks with zero-padding between coefficients (the "trous" or holes) to achieve dilation. Therefore, employing this algorithm helps us better understand and process one-dimensional signals through comprehensive multi-resolution analysis.