Stationary wavelet transform

Haar Stationary Wavelet Transform of Lena

The Stationary wavelet transform (SWT)^[1] is a wavelet transform algorithm designed to overcome the lack of translation-invariance of the discrete wavelet transform (DWT). Translation-invariance is achieved by removing the downsamplers and upsamplers in the DWT and upsampling the filter coefficients by a factor of $2^{(j-1)}$ in the $j$th level of the algorithm.^[2][3][4][5] The SWT is an inherently redundant scheme as the output of each level of SWT contains the same number of samples as the input – so for a decomposition of N levels there is a redundancy of N in the wavelet coefficients. This algorithm is more famously known as "algorithme à trous" in French (word trous means holes in English) which refers to inserting zeros in the filters. It was introduced by Holschneider et al.^[6]

Implementation

The following block diagram depicts the digital implementation of SWT.

In the above diagram, filters in each level are up-sampled versions of the previous (see figure below).

Applications

A few applications of SWT are specified below.

• Signal denoising
• Pattern recognition
• Brain image classification ^[7][8]

Synonyms

• Stationary wavelet transform
• Redundant wavelet transform
• Algorithme à trous
• Quasi-continuous wavelet transform
• Translation invariant wavelet transform
• Shift invariant wavelet transform
• Cycle spinning
• Maximal overlap wavelet transform (MODWT)
• Undecimated wavelet transform (UWT)

References

1. James E. Fowler: The Redundant Discrete Wavelet Transform and Additive Noise, contains an overview of different names for this transform.
2. A.N. Akansu and Y. Liu, On Signal Decomposition Techniques, Optical Engineering, pp. 912-920, July 1991.
3. M.J. Shensa, The Discrete Wavelet Transform: Wedding the A Trous and Mallat Algorithms, IEEE Transaction on Signal Processing, Vol 40, No 10, Oct. 1992.
4. M.V. Tazebay and A.N. Akansu, Progressive Optimality in Hierarchical Filter Banks, Proc. IEEE International Conference on Image Processing (ICIP), Vol 1, pp. 825-829, Nov. 1994.
5. M.V. Tazebay and A.N. Akansu, Adaptive Subband Transforms in Time-Frequency Excisers for DSSS Communications Systems , IEEE Transaction on Signal Processing, Vol 43, No 11, pp. 2776-2782, Nov. 1995.
6. M. Holschneider, R. Kronland-Martinet, J. Morlet and P. Tchamitchian. A real-time algorithm for signal analysis with the help of the wavelet transform. In Wavelets, Time-Frequency Methods and Phase Space, pp. 289–297. Springer-Verlag, 1989.
7. Zhang, Y. (2010). "Feature Extraction of Brain MRI by Stationary Wavelet Transform and its Applications". Journal of Biological Systems 18 (s1): 115-132. doi:10.1142/S0218339010003652.
8. Dong, Z. (2015). "MR Brain Image Classification via Stationary Wavelet Transform and Generalized Eigenvalue Proximal SVM". Journal of Medical Imaging and Health Informatics 5 (7): 1–9. doi:10.1166/jmihi.2015.1542.