Multi-band wavelets have applied in many areas such as signal processing and numerical analysis due to their richer parameter space to have a more flexible time-frequency tiling,to give better energy compaction than 2-band wavelets.The sub-band operators are studied,an optimization model is built,and the wavelets with the smallest norm of the sub-band operator can be selected from the symmetric multi-band biorthogonal wavelets with free parameters,which can be applied in digital image processing based on wavelet theory and its fast algorithms.Firstly,the sub-band operator which is an infinite-dimensional matrix,is introduced,circular matrix theory is developed,and the norm of a sub-band operator associated to four-band wavelets with fast calculation structure is obtained.It is easy to obtain the norm of the sub-band operator with some structure by construction a function and computing the maximum.Secondly,a model to minimize the norm is built and four-band biorthogonal wavelets filter bands with fast calculation structure are designed.Lastly,an example is provided to illustrate the proposed results.
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