International Journal of Wavelets, Multiresolution and Information Processing

Time-frequency analysis methods such as wavelet analysis are applied to investigate characteristic from non-stationary signals. In this study, we proposed redundant morphological wavelet analysis that was a kind of nonlinear discrete wavelet and redundant wavelet. This method analyzes a transition of shape information from signals in detail since this method keeps property of shift invariance though information of decomposition includes redundancy. Local pattern spectrum which corresponds to nonlinear short time Fourier transform is derived from this nonlinear wavelet. The characteristics of these methods were confirmed by applying to simulation data and actual data.

We present a new family of triangular biorthogonal wavelets that is defined on a triangular lattice by introducing a new operation to generalize two-dimensional lifting, which we call twist. The resulting filters inherit several remarkable features of the early triangular biorthogonal wavelet filters such as the hexagonal symmetry of low-pass filters, symmetrical arrangement of three high-pass filters on the lattice, and that the wavelet decomposition produces uniform energy distributions over three detail components, preserving the isotropy of decomposed images. Additionally, these filters are a biorthogonal set of truly nonseparable two-dimensional wavelet filters that have much larger support, which provides much larger portions of the total energy to three detail components of decomposed images. We show that this plays a crucial role when extracting the edge structure of an image.

The theorems, giving the condition of perfect translation invariance for discrete wavelet transforms, have already been proven. Based on these theorems, the dual-tree complex discrete wavelet transform, the 2-dimensional discrete wavelet transform, the complex wavelet packet transform, the variable-density complex discrete wavelet transform and the real-valued discrete wavelet transform, having perfect translation invariance, were proposed. However, their customizability of wavelets in the frequency domain is limited. In this paper, also based on these theorems, a new type of complex discrete wavelet transform is proposed, which achieves perfect translation invariance with high degree of customizability of wavelets in the frequency domain.

In correcting a real linear code y = Bx + w by ℓ₁ linear programming, where the encoding matrix B ∈ ℝ^{m × n} has full rank with m ≥ n and the noise w ∈ ℝ^m is a sparse random vector, it is numerically observed that the breakdown points of 50% successes in recovering the input vector x ∈ ℝⁿ from the corrupted oversampled measurement y lie on the Donoho–Tanner curves when reflected in their midpoint. The curves of 50% successes in solving underdetermined systems, z = Aw, by ℓ₁ linear programming with uniformly distributed compressed sensing matrices A ∈ ℝ^{d × m}, where d < m and w is a sparse vector, have been numerically observed and recently shown to coincide with the Donoho–Tanner curves for normally-distributed compressed sensing matrices A derived from geometric combinatorics. When n ≤ m/2, correcting a linear code is faster if done directly by ℓ₁ linear programming. However, when n > m/2, to save computing time, this problem can be transformed into an underdetermined compressed sensing problem, Aw = z := Ay, for the syndrome z by a full rank matrix A ∈ ℝ^{d × m}, d = m – n, such that AB = 0. For this purpose, to have equivalently high mean breakdown points by ℓ₁ linear programming, one can use uniformly distributed random matrices A ∈ ℝ^{(m-n) × m} and matrices B ∈ ℝ^{m × n} with orthonormal columns spanning the null space of A. Two exceptional cases have been found. Numerical results are collected in figures and tables.

We propose a new digital image watermarking method for authentication based on the dual-tree complex discrete wavelet transform (DT-CDWT) and interval arithmetic (IA). This method is a kind of content-based watermarking, because it generates a watermark from the host image and authenticates the test image by comparing two types of watermarks constructed using the test image. We describe our watermarking and authentication procedure in detail and present experimental results demonstrating that our method produces high-quality watermarked images and can locate the areas that have been tampered with.

Method for data hiding based on LeGall 5/3 of Cohen-Daubechies-Feauveau: CDF 5/3 wavelet with data compression and random scanning of secret imagery data together with steganography is proposed. Invisibility of secret imagery data is evaluated based on Peak Signal to Noise Ratio: PSNR with SIDBA standard image database. The experimental results show that PSNR of LeGall based wavelet utilized data hiding ranges from 43.82 to 46.9 while that of Daubechies based method ranges from 44.33 to 44.75 when the coded secret imagery data is inserted in the first 3 digits from Least Significant Bit: LSB of the original image. Data compression ratio for the secret imagery data ranges from 1.3 to 19.4 which depends on the complexity of the secret imagery data. Meanwhile, PSNR of data hidden image ranges from 46.83 to 47.41. Consequently, the proposed data hiding method is permissive because PSNR is over 40 dB results in satisfaction on invisibility of the secret imagery data in the data hidden image.

The three-dimensional orthogonal wavelet multi-resolution technique was applied to analyze flow structures of various scales around an externally mounted vehicle mirror. Firstly, the three-dimensional flow of mirror wake was numerically analyzed at a Reynolds number of 10⁵ by using the large-eddy simulation (LES). Then the instantaneous velocity and vorticity were decomposed into the large-, intermediate- and relatively small-scale components by the wavelet multi-resolution technique. It was found that a three-dimensional large-scale vertical vortex dominates the mirror wake flow and makes a main contribution to vorticity concentration. Some intermediate- and relatively small-scale vortices were extracted from the LES and were clearly identifiable.

Quantum chemistry program based on Multiresolution Multiwavelet (MRMW) basis set is much simpler and can be more efficient than the conventional one based on Gaussian basis set in molecular geometry optimization because the Hellmann Fynman Theorem (HFT) holds for the complete space which MRMW provides. It is numerically shown that large Gaussian basis sets can provide the chemical information with chemical accuracy for very small molecular systems despite the incompleteness of their basis set. However, even for the relatively small size molecule of CH₄, the error from the basis set incompleteness results in an error larger than an acceptable chemical accuracy. The advantage of the MRMW complete basis set in vibrational analysis is also discussed.

The auditory brainstem response (ABR) is widely used as an index to assist hearing and brain function diagnoses. In particular, in clinical applications, the rapid detection of ABR peak characteristics is required. One approach to improving the speed of detection is to decrease the number of signal averaging procedures while denoising during the detection of ABR waveforms; another approach is to extract the characteristics of ABR waveform components. In our previous study, to represent ABR waveform components, we obtained not only the frequency characteristics of an ABR but also the frequency characteristics of each component of the ABR based on the time (latency). Using a one-dimensional discrete wavelet transform (DWT) in this latency-frequency analysis, we described an approximate method of reproducing ABR signals with a low SNR from observed values obtained with a smaller number of averaging procedures. At the same time, using this multiple-level frequency decomposition of ABR signals according to the known frequency content of the ABR, we extracted the peak latency of the fast component of the ABR using fewer averagings of the ABR data. From these decomposition and reconstruction results for ABR signals, we proposed the optimal decomposition level of the ABR and explained how we used the waveform of the ABR reconstructed by the inverse DWT (IDWT). In this paper, we propose a method of automated averaging of the ABR using the waveform reconstructed by discrete wavelet multiresolution analysis (MRA). Our proposed method will be useful for the fast detection of ABR latency characteristics in hearing screening test.

In this paper, we propose a design method for directional selection in the two-dimensional complex wavelet packet transform (2D-CWPT). Current two-dimensional complex discrete wavelet transforms (2D-CDWT) can extract directional components from images, but the number of directions is small, and the directions and resolutions are fixed. Thus the current 2D-CDWTs are not flexible enough. In this study, we propose a new design method of the directional filters that can detect desirable direction components. Additionally flexible directional selection is achieved because the directional filters are added to the 2D-CWPT. Finally, the proposed method is applied to defect detection in semiconductor wafer circuits and an encouraging result is obtained.