Abstract
Wavelet-based multiscale interpolation operator is often employed to construct the adaptive numerical method for PDEs, in which the computational complexity of the wavelet transform is one of the main factors affecting the algorithm efficiency. As the wavelet transform just acts as the detector of the characteristic points in the interpolation operator, the multiscale wavelet interpolation operator can be viewed as a nonlinear problem. Based on this assumption, we construct an approximate dynamic interpolation operator with the homotopy perturbation method (HPM), which decreases the computational complexity of the wavelet transform appearing in the wavelet interpolation operator fromO((1/3)42J−1)toO(4J), whereJis the amount of the wavelet scales. Then an adaptive algorithm solving the Perona-Malik model on image denoising is constructed with the HPM-based interpolation operator. Last, the quasi-Shannon wavelet is employed to design the experiments on the medical image and some artificial images denoising. The experiment results show that the simplified wavelet interpolation operator based on HPM possesses the adaptability and nonsensitivity to the time step, which is helpful to improve the algorithm efficiency. This illustrates that the HPM-based wavelet interpolation operator is an effective tool to solve the problems in image processing.
Highlights
In recent years, the partial differential equation (PDE) method has been widely used in image denoising, especially in medical images and remote sensing images
Multilevel wavelet numerical method for the nonlinear PDEs has been proposed over ten years, which can take full advantage of the adaptability of the wavelet analysis
The artifacts in image can be eliminated with the wavelet numerical algorithm instead of the infinite difference method, as wavelet basis function possesses many excellent properties such as smoothness and compact support
Summary
The partial differential equation (PDE) method has been widely used in image denoising, especially in medical images and remote sensing images. Perona-Malik model is expressed as a nonlinear 2-dimension partial differential equation, which overcomes the drawback of the scale-space technique introduced by Witkin that involves generating coarser resolution images by convolving the original image with a Gaussian kernel In this approach, a new definition of scale space was suggested, and a class of algorithms was introduced; the accurate locations of the “semantically meaningful” edges at coarse scales using a diffusion process can be obtained; that is, a high quality edge detector which successfully exploits global information was obtained with this new method. The support range of wavelet function is much wider than the basis function in the infinite difference method This leads to a lower computational efficiency of wavelet transform in solving 2D nonlinear PDEs. The multilevel wavelet transform usually appears in the wavelet interpolation operator [7, 8]. The purpose of this research is to construct a dynamic wavelet interpolation operator with HPM and apply it to solve the Perona-Malik model which is a classical image denoising model
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