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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. [ABSTRACT FROM AUTHOR]

A constructive algorithm based on the theory of spectral pairs for constructing nonuniform wavelet basis in 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 which is not necessarily a group nor a uniform discrete set, given where (an integer) and is an odd integer with such that and are relatively prime and 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. [ABSTRACT FROM AUTHOR]

While in the process of decomposition of the Mallat discrete wavelet transform (DWT) fast algorithm, the algorithm has a drawback, that is, there is frequency distortion in the high frequency subband. In this paper, a new algorithm of decomposition and reconstruction in the discrete wavelet is presented. The algorithm can solve the frequency distortion in the high frequency subband in the process of decimation in each level. The simulation of numerical value example tests the validity of the algorithm. [ABSTRACT FROM AUTHOR]

By integrating the Fourier techniques and the edge information obtained using the radial symmetric functions, we propose in this paper an invariant feature extraction algorithm. Unlike the Gabor feature extraction method, the present method does not use direction dependent filters, nor does it use the images in polar form, for rotation invariance. Besides, the present Fourier-Radial invariant feature extraction algorithm, suitable for both the texture and non-texture images, has functional analogy with the Gabor feature extraction method, and hence, is easily implementable. It is mathematically proved, and justified through computations, that the method can generate the invariant and discriminative feature vectors. Our simulation results demonstrate that the method can be used for such applications as content-based image retrieval. [ABSTRACT FROM AUTHOR]

A multiresolution analysis (MRA) on local fields of positive characteristic was defined by Shah and Abdullah for which the translation set is a discrete set which is not a group. In this paper, we continue the study based on this nonstandard setting and introduce vector-valued nonuniform multiresolution analysis (VNUMRA) where the associated subspace V0 of L2(K, ℂM) has an orthonormal basis of the form {Φ (x - λ)}λ∈Λ where Λ = {0, r/N} + 풵, N ≥ 1 is an integer and r is an odd integer such that r and N are relatively prime and 풵 = {u(n) : n ∈ ℕ0}. We establish a necessary and sufficient condition for the existence of associated wavelets and derive an algorithm for the construction of VNUMRA on local fields starting from a vector refinement mask G(ξ) with appropriate conditions. Further, these results also hold for Cantor and Vilenkin groups. [ABSTRACT FROM AUTHOR]

In this paper, an efficient multi-scale image disparity estimation algorithm is proposed which estimates the local displacements needed to align different regions between a pair of images. This phase-based approach is based on fractional dual-tree complex wavelet transform (FrDTCWT). In the proposed FrDTCWT, initially we obtained the fractional Fourier transform of the image. We decomposed the fractional transformed image by dual-tree complex wavelet transform into real and imaginary parts. The complex analytic signal is obtained from the real-valued function which is obtained from this FrDTCWT. The FRDTCWT inherits the excellent mathematical properties of DTCWT and FrFT. Phase and amplitude of the image are computed from the complex analytic signal. The disparity is estimated as the optical flow using the phase difference method. The efficiency of the proposed algorithm is carried out by different experiments on synthetic as well as on realistic image sequences. [ABSTRACT FROM AUTHOR]

Weighted and controlled frames have been introduced recently to improve the numerical efficiency of iterative algorithms for inverting the frame operator. In this paper, we develop systematically these notions, including their mutual relationship. We will show that controlled frames are equivalent to standard frames and so this concept gives a generalized way to check the frame condition, while offering a numerical advantage in the sense of preconditioning. Next, we investigate weighted frames, in particular their relation to controlled frames. We consider the special case of semi-normalized weights, where the concepts of weighted frames and standard frames are interchangeable. We also make the connection with frame multipliers. Finally, we analyze weighted frames numerically. First, we investigate three possibilities for finding weights in order to tighten a given frame, i.e. decrease the frame bound ratio. Then, we examine Gabor frames and how well the canonical dual of a weighted frame is approximated by the inversely weighted dual frame. [ABSTRACT FROM AUTHOR]