eGHWT: The Extended Generalized Haar–Walsh Transform

Extending computational harmonic analysis tools from the classical setting of regular lattices to the more general setting of graphs and networks is very important, and much research has been done recently. The generalized Haar–Walsh transform (GHWT) developed by Irion and Saito (2014) is a multiscale transform for signals on graphs, which is a generalization of the classical Haar and Walsh–Hadamard transforms. We propose the extended generalized Haar–Walsh transform (eGHWT), which is a generalization of the adapted time–frequency tilings of Thiele and Villemoes (1996). The eGHWT examines not only the efficiency of graph-domain partitions but also that of “sequency-domain” partitions simultaneously. Consequently, the eGHWT and its associated best-basis selection algorithm for graph signals significantly improve the performance of the previous GHWT with the similar computational cost, $$O(N \log N)$$ O ( N log N ) , where N is the number of nodes of an input graph. While the GHWT best-basis algorithm seeks the most suitable orthonormal basis for a given task among more than $$(1.5)^N$$ ( 1.5 ) N possible orthonormal bases in $$\mathbb {R}^N$$ R N , the eGHWT best-basis algorithm can find a better one by searching through more than $$0.618\cdot (1.84)^N$$ 0.618 · ( 1.84 ) N possible orthonormal bases in $$\mathbb {R}^N$$ R N . This article describes the details of the eGHWT best-basis algorithm and demonstrates its superiority using several examples including genuine graph signals as well as conventional digital images viewed as graph signals. Furthermore, we also show how the eGHWT can be extended to 2D signals and matrix-form data by viewing them as a tensor product of graphs generated from their columns and rows and demonstrate its effectiveness on applications such as image approximation.

Improved Denoising Method for Ultrasonic Echo with Mother Wavelet Optimization and Best-Basis Selection

<p>Weak features of ultrasonicnondestructive test signals are usually immersed in noisy signals. So, in this paper, we proposed an improved scheme for noise reduction and feature extraction based on discrete wavelet transform. The basis of the mother wavelet was selected to be matched to a given signal. Three different constraints were presented to minimize the error between the denoised and the given signal. It should be mentioned that such an optimum wavelet can represent the signal more compactly with a few large coefficients which can be considered as the signal features. Standard signals and simulated ultrasonic echo were used to evaluate the performance of the presented algorithms. Signal to error ratio was used to compare the designed wavelet performance with that of standard wavelets. Simulation results revealed that the proposed method outperformed the other presented methods and even standard wavelets. The results also has shown that the signal-based noise reduction algorithms make the feature extraction more reliable. Finally, the performance of the proposed algorithm was compared with other methods from different literatures.</p>

Improved Denoising Method for Ultrasonic Echo with Mother Wavelet Optimization and Best-Basis Selection

<p>Weak features of ultrasonicnondestructive test signals are usually immersed in noisy signals. So, in this paper, we proposed an improved scheme for noise reduction and feature extraction based on discrete wavelet transform. The basis of the mother wavelet was selected to be matched to a given signal. Three different constraints were presented to minimize the error between the denoised and the given signal. It should be mentioned that such an optimum wavelet can represent the signal more compactly with a few large coefficients which can be considered as the signal features. Standard signals and simulated ultrasonic echo were used to evaluate the performance of the presented algorithms. Signal to error ratio was used to compare the designed wavelet performance with that of standard wavelets. Simulation results revealed that the proposed method outperformed the other presented methods and even standard wavelets. The results also has shown that the signal-based noise reduction algorithms make the feature extraction more reliable. Finally, the performance of the proposed algorithm was compared with other methods from different literatures.</p>