Orthogonal moments are used to represent digital images with minimum redundancy. Orthogonal moments with fractional-orders show better capabilities in digital image analysis than integer-order moments. In this work, the authors present new fractional-order shifted Gegenbauer polynomials. These new polynomials are used to defined a novel set of orthogonal fractional-order shifted Gegenbauer moments (FrSGMs). The proposed method is applied in gray-scale image analysis and recognition. The invariances to rotation, scaling and translation (RST), are achieved using invariant fractional-order geometric moments. Experiments are conducted to evaluate the proposed FrSGMs and compare with the classical orthogonal integerorder Gegenbauer moments (GMs) and the existing orthogonal fractional-order moments. The new FrSGMs outperformed GMs and the existing orthogonal fractional-order moments in terms of image recognition and reconstruction, RST invariance, and robustness to noise.

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In this paper we prove some common fixed point theorems for family of occasionally weakly compatible mappings in Menger space. Also improvement of the results of B. D. Pant and Sunny Chauhan [1] under relaxed conditions is given.

In this paper, we establish a common xed point theorem for two pairs of mappings satisfying an almost generalized contractive condition for comparable elements in a partially ordered metric space. Our results generalize and extend the main results in [2] to multivalued mappings. As corollaries we obtain the results in [8, 9], and many others.

The classical radial harmonic Fourier moments (RHFMs) and the quaternion radial harmonic Fourier moments (QRHFMs) are gray-scale and color image descriptors. The radial harmonic functions with integer orders are not able to extract fine features from the input images. In this paper, the authors derived novel fractional-order radial harmonic functions in polar coordinates. The obtained functions are used to defined novel multi-channel fractional-order radial harmonic moments (FrMRHFMs) for color image description and analysis. The invariants to geometric transformations for these new moments are derived. A theoretical comparison between FrMRHFMs and QRHFMs is performed from the aspects of kernel function and the spectrum analysis. Numerical simulation is carried out to test these new moments in terms of image reconstruction capabilities, invariance to the similarity transformations, color image recognition and the CPU computational times. The obtained theoretical and numerical results clearly show that the proposed FrMRHFMs is superior to the QRHFMs and the existing fractional-order orthogonal moments.

Orthogonal moments enable computer-based systems to discriminate between similar objects. Mathe- maticians proved that the orthogonal polynomials of fractional-orders outperformed their corresponding counterparts in representing the fine details of a given function. In this work, novel orthogonal fractional- order Legendre-Fourier moments are proposed for pattern recognition applications. The basis functions of these moments are defined and the essential mathematical equations for the recurrence relations, orthog- onality and the similarity transformations (rotation and scaling) are derived. The proposed new fractional- order moments are tested where their performance is compared with the existing orthogonal quaternion, multi-channel and fractional moments. New descriptors were found to be superior to the existing ones in terms of accuracy, stability, noise resistance, invariance to similarity transformations, recognition rates and computational times.