International Journal of Wavelets, Multiresolution and Information Processing

The paper presents a detailed analysis of algorithms used for the forward and the inverse discrete wavelet transform (DTWT) of finite-length signals. The paper provides answers to questions such as "how many wavelet coefficients are computed from the signal at a given depth of the decomposition" or conversely, "how many signal samples are needed to compute a single wavelet coefficient at a given depth of the decomposition" or "how many coefficients at a given depth are influenced by the selected type of boundary treatment" or "how many samples of the input signal simultaneously influence two neighboring wavelet coefficients at a given depth of the decomposition". As a byproduct, the rigorous analysis of the algorithms gives details needed for the implementation. The paper is accompanied by several Matlab functions.

Koenker⁹ studied a studentized version of Neyman's score statistic and obtained theoretical results which indicated that the studentized version will outperform the score test under a linear model if the data is from a heavy-tailed t-distribution. However, the author failed to examine the size and power performances of the studentized test through a simulation study. Subsequently, Cai, Hurvich and Tsai⁷ after a simulation study in a nonparametric setting, found that even when the data is from a normal population the score test was biased in estimating a pre-assigned level of significance. Thus, he recommended that the studentized score test should be used in all situations. Several authors have, however, shown earlier that when the data is from a normal population, Neyman's partial score test is asymptotically unbiased in estimating a pre-assigned level of significance. As a result in this paper, we obtain the partial score statistic and the studentized version under various models but conduct our simulation studies under the special case considered by Cai et al.⁷ in order to examine the studentized test. We found, in our simulation studies, that when the model of interest is nonparametric with uncorrelated errors, the power of the score test is generally higher than that of the studentized test. The difference in power performances becomes more pronounced under the heavy-tailed t-distribution. In the normal case, both the partial score test and its studentized version performed well in controlling the size of the test. We also found that if the score statistic is constructed based on the underlying distribution of the data, then the score statistic will always outperform the studentized test in both power and size.

Ramanujan Sums (RS) have been found to be very successful in signal processing recently. However, as far as we know, the RS have not been applied to image analysis. In this paper, we propose two novel algorithms for image analysis, including moment invariants and pattern recognition. Our algorithms are invariant to the translation, rotation and scaling of the 2D shapes. The RS are robust to Gaussian white noise and occlusion as well. Our algorithms compare favourably to the dual-tree complex wavelet (DTCWT) moments and the Zernike's moments in terms of correct classification rates for three well-known shape datasets.

In this note we prove a stability estimate for an inverse heat source problem. Based on the obtained stability estimate, we present a generalized Tikhonov regularization method and obtain the error estimate. Numerical experiment shows that the generalized Tikhonov regularization works well.

Based on the theory of the discrete multi-wavelets in the space L²(R), the theory of the discrete multi-wavelets in the space L²(C) is presented properly in this paper, where C denotes a smooth plane curve. Firstly, the length-preserving projection is constructed, and by the length-preserving projection, the multiplicity multi-resolution analysis in the space L²(C) is defined properly and we define the dilation operator and translation operator in the space L²(C). Then, the two-scale refinement equations of multi-scaling function and multi-wavelet in the space L²(C) is deduced by using length-preserving mapping, the orthogonality is discussed, and the decomposition and reconstruction algorithm is computed. Finally, the example is given.

Tests to determine the dependence or independence of random variables X and Y are well established. Recently, criteria based on reproducing kernel Hilbert spaces has received much attentions. They are developed in the setting of norm of Hilbert spaces. In this paper we propose tests in the setting of constraints of coefficients of functions. Some estimates of tests are constructed. In particular the error between the test of constrained covariance and the estimate is bounded.

We define distribution spaces in ℝ^d via ℓ^q(L^p) norms of a sequence of convolutions of with smooth functions, the shearlet system. Then, we define associated sequence spaces and prove characterization with the shearlet coefficients. We also prove continuous embeddings between some shear anisotropic inhomogeneous spaces and between classical (dyadic isotropic) inhomogeneous spaces and shear anisotropic inhomogeneous spaces.

In this paper, we investigate coefficient-based regularized least squares regression problem in a data dependent hypothesis space. The learning algorithm is implemented with samples drawn by unbounded sampling processes and the error analysis is performed by a stepping-stone technique. A new error decomposition technique is proposed for the error analysis. The regularization parameters in our setting provide much more flexibility and adaptivity. Sharp learning rates are addressed by means of l²-empirical covering numbers under a moment hypothesis condition.

In this paper, a wavelet multifractal analysis is developed which permits to characterize simultaneous singularities for a vector of functions. An associated multifractal formalism is introduced and checked for the case of functions involving self similar aspects.

In this paper, text detection in moving business cards for helping visually impaired persons using a wearable camera is presented. The current methods to help visually impaired persons to read the natural scene text, menus and book covers have three problems. First, they assumed that the blind persons are standing still and the captured scene is not moving. Second, the blind persons do not know that the menu or the book cover is captured by the camera. Third, these methods cannot "see" business cards to help the blind persons. The proposed method includes moving detection, thumb detection motion blur detection and text detection methods. Experimental results show that the proposed method can reduce the time complexity. The reduced rates for training and testing sets are 83.18% and 77.08%, respectively. The text detection rates for training and testing sets are 93.44% and 94.58%, respectively. The fps is 53 for 320 × 240 video frames. The program size is 102KB and can be run on mobile devices.