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.

An efficient architecture for discrete wavelet transform and best-basis algorithm for images

The hardware acceleration of the wavelet transform for real-time systems has become an essential research field. In the first part of the thesis, an efficient architecture that performs both forward and inverse lifting-based discrete wavelet transform is proposed. The proposed architecture reduces the hardware requirement by exploiting the redundancy in the arithmetic operation involved in DWT computation. The proposed architecture consists of predict module, update module, address generation module, control unit and a set of registers to establish data communication between predict and update modules. The symmetrical extension of images at the boundary to reduce distorted images has been incorporated in our proposed for both (5,3) wavelet and (9,7) wavelet. Best-basis algorithm that is designed for signal compression and de-noising uses WPT to select the best-basis node for a given additive cost function. In the second part of the thesis, we propose wavelet architecture to perform WPT decomposition. A new algorithm to implement the natural logarithm function using Maclaurin series is proposed to implement the cost function used for best-basis algorithm. These architectures have been described in VHDL at the RTL level and simulated successfully using ModelSim simulation environment. These architectures are implemented in Virex ll Pro FPGA series of Xilinx.

An efficient architecture for discrete wavelet transform and best-basis algorithm for images

The hardware acceleration of the wavelet transform for real-time systems has become an essential research field. In the first part of the thesis, an efficient architecture that performs both forward and inverse lifting-based discrete wavelet transform is proposed. The proposed architecture reduces the hardware requirement by exploiting the redundancy in the arithmetic operation involved in DWT computation. The proposed architecture consists of predict module, update module, address generation module, control unit and a set of registers to establish data communication between predict and update modules. The symmetrical extension of images at the boundary to reduce distorted images has been incorporated in our proposed for both (5,3) wavelet and (9,7) wavelet. Best-basis algorithm that is designed for signal compression and de-noising uses WPT to select the best-basis node for a given additive cost function. In the second part of the thesis, we propose wavelet architecture to perform WPT decomposition. A new algorithm to implement the natural logarithm function using Maclaurin series is proposed to implement the cost function used for best-basis algorithm. These architectures have been described in VHDL at the RTL level and simulated successfully using ModelSim simulation environment. These architectures are implemented in Virex ll Pro FPGA series of Xilinx.

A novel variable filter band discrete wavelet transform: Application

In this study, in order to verify the effectiveness of the variable filter band discrete wavelet transform (VFB-DWT) and construction method of the variable-band filter (VBF), a fetal ECG extraction has been carried out and the main results obtained are as follows. The approach to configuration VBF by selecting the frequency band only where the fetal ECG component is present was effective to configure the optimal base sensible signal. The extraction of the fetal ECG was successful by applying the wavelet shrinkage to VFB-DWT, which used the constructed VBF. The information entropy was selected as an evaluation index, and two kinds of ECG signals are used to evaluate the wavelet transform basis between the wavelet packet transform (WPT) and the VFB-DWT. One is a synthesized signal composed of white noise, the maternal ECG and the fetal ECG. The other signal is the real target signal separated by independent component analysis (ICA) and has the mother's body noise, the maternal ECG and the fetal ECG. The result shows that the basis by VBF of the VFB-DWT is better than the basis of the WPT that was chosen by the best basis algorithm (BBA).