International Journal of Wavelets, Multiresolution and Information Processing

In this paper, we consider the bivariate real isotropic dilation matrices that are similar (up to constant factors) to rotation matrices; and we show that, in this case, the two-scale relations can be considered also as relations between not only dilated but also rotated scaling functions. We present sufficient conditions on a matrix to be similar to a rotation matrix, where the change-of-basis matrix is symmetric positive definite. Also we present a simple test that the real matrix with integer entries performs rotation by an incommensurable to $\pi$ angle. We show that if a dilation matrix is similar to a rotation matrix, then an ellipse defined by the change-of-basis matrix is invariant (up to a uniform dilation) under transformation by the dilation matrix.

Sufficient frame conditions for periodic dual wavelet are established. These conditions are given in terms of the Fourier coefficients. A way of constructing dual wavelet Riesz bases starting with trigonometric polynomials is provided.

The quality of olive fruit and its virgin olive oil is a main concern for consumers and fruit industrial companies. The effectiveness and fast detection of olive’s skin defects is the most decisive factor in determining its quality. It is necessary to design and implement image processing tools for segmentation and correct classification of the different fresh incoming olive batches. In this paper, we propose a new automatic image segmentation algorithm, based on discrete wavelets transform. The aim of the segmentation algorithm is to discriminate between olives and the background with the challenge of irregular and dispersive lesion borders, low contrast, artifacts in the olive fruit and variety of colors within the interest region. The second part of our work proposes a scheme for olive fruit classification. The classifier first identifies the olive fruit color and then, based upon discrete wavelets transform and Tamura statistical texture features, the healthy olive fruit is distinguished from the damaged one. The new texture feature vector is, then, compared with the robust Local Binary Pattern feature vector. The simplicity of our segmentation and classification algorithms makes them appropriate for designing a productive and profitable computer vision machine.

The area of nonlinear time series models has experienced a great development since the 1980s. Although there is a wide range of parametric nonlinear time series models, in general, we do not know if the postulated model is the most appropriated one for a specific data set. This situation highlights the importance of nonparametric models. An interesting nonparametric model to fit nonlinear time series is the well-known functional coefficient regression model. Nonparametric estimations by, e.g., local linear regression and splines, are developed in the literature. In this work, we study the estimation of such a model using wavelets. It is a proposal that takes into account both, classical and warped wavelets. We present the rates of convergence of the proposed estimators and carry out simulation studies to evaluate automatic procedures (among AIC, AICc and BIC) for selecting the coarsest and finest levels to be used during the estimation process. Moreover, we illustrate the methodology with an application to a real data set, where we also calculate multi-step-ahead forecasts and compare the results with other methods known in the literature.

A constructive algorithm based on the theory of spectral pairs for constructing nonuniform wavelet basis in $L^2(ℝ)$ was considered by Gabardo and Nashed [Nonuniform multiresolution analysis and spectral pairs, J. Funct. Anal. 158 (1998) 209–241]. In this setting, the associated translation set is a spectrum $\mathrm{\Lambda }$ which is not necessarily a group nor a uniform discrete set, given $\mathrm{\Lambda }=\{0,r/N\}+2\mathbb{Z},$ where $N\ge 1$ (an integer) and $r$ is an odd integer with $1\le r\le 2N-1$ such that $r$ and $N$ are relatively prime and $\mathbb{Z}$ is the set of all integers. The objective of this paper is to construct nonuniform wavelet frame on local fields. A necessary condition and four sufficient conditions for nonuniform wavelet frame on local fields are given.

Periodic sequences over finite fields, constructed by classical cyclotomic classes and generalized cyclotomic classes, have good pseudo-random properties. The linear complexity of a period sequence plays a fundamental role in the randomness of sequences. In this paper, we construct a new family of quaternary generalized cyclotomic sequences with order $2d$ and length $2p^m$, which generalize the sequences constructed by Ke et al. in 2012. In addition, we determine its linear complexity using cyclotomic theory. The conclusions reveal that these sequences have high linear complexity, which means they can resist linear attacks.

According to the theory of wavelet analysis on $R^2$, the wavelet analysis on a smooth surface $M$ with nonzero constant Gaussian curvature will be discussed systematically in this paper. First, a general area-preserving projection from a smooth surface to the plane will be presented by the Gaussian projection and the area-preserving projection on the sphere. Then the continuous wavelet transform and its inverse transform on a smooth surface $M$ with nonzero constant Gaussian curvature will be discussed by a general area-preserving projection, relative dilation operator and translation operator. Further, according to the multi-resolution analysis on a smooth surface, the discrete wavelet transform and relative properties will be investigated systematically, including the two-scale equations of the wavelet function, orthogonality and so on. Finally, two numerical examples will be given.

LSS-Target (the Low altitude, Slow speed and Small Target) is likely to be a threat to the observation platform, thus infrared LSS-Target detection is an urgent task. LSS-Target is a challenging issue due to the low Signal-to-Noise Ratio (SNR) and sophisticated background. Motivated by the analysis of infrared imaging characteristics, this paper proposes a novel fusion method for IR LSS-Target detection with complex urban background, which is suitable for precise guidance and self defense. First, an adaptive threshold segmentation based on accumulative histogram and maximum likelihood estimation are utilized to eliminate the clutter and improve SNR of the initial image. Second, a template is set up to identify the seed points in the image. Third, a constrained four criteria region growth algorithm is performed to separate the entire regions. Finally, the confidence measure is constructed, which can eliminate false targets and the background edges. Experimental results show that the method in this paper can screen out the real LSS-Target in real time with high accuracy under sophisticated background.

The wavelet packet transform as a mathematical tool has found recent application in spectrum sensing. The result of this application has produced very promising results. Primarily, wavelets were designed for edge detection in images. Recently, cognitive radio literature have reported on wavelet application to detect sub-band frequency edges in wide band spectrum. In this paper, we present the combination of the Hilbert transform and the wavelet packet transform with the aim of enhancing the detection of the sub-band frequency edges of a wavelet-packet-decomposed signal. The simulation results show the effectiveness of this approach. The new scheme detected sub-band frequency edges of the wavelet-packet-decomposed signal much better than the wavelet packet transform without combination with the Hilbert transform.

It is well known that the domain of Fourier transform (FT) can be extended to the Schwartz space $\mathcal{𝒮}(R)$ for convenience. As a generation of FT, it is necessary to detect the linear canonical transform (LCT) on a new space for obtaining the similar properties like FT on $\mathcal{𝒮}(R)$. Therefore, a space ${\mathcal{𝒮}}_{A_1}(R)$ generalized from $\mathcal{𝒮}(R)$ is introduced firstly, and further we prove that LCT is a homeomorphism from ${\mathcal{𝒮}}_{A_1}(R)$ onto itself. The linear canonical wavelet transform (LCWT) is a newly proposed transform based on the convolution theorem in LCT domain. Moreover, we propose an equivalent definition of LCWT associated with LCT and further study some properties of LCWT on ${\mathcal{𝒮}}_{A_1}(R)$. Based on these properties, we finally prove that LCWT is a linear continuous operator on the spaces of $L^{p,A_1}$ and $H_{A_1}^{s,p}$.