Abstract
In recent years, there has been a renewed interest in finding methods to construct orthogonal transforms. This interest is driven by the large number of applications of the orthogonal transforms in image analysis and compression, especially for colour images. Inspired by this motivation, this paper first introduces a new orthogonal transform known as a discrete fractional COSHAD (FrCOSHAD) using the Kronecker product of eigenvectors and the eigenvalues of the COSHAD kernel functions. Next, this study discusses the properties of the FrCOSHAD kernel function, such as angle additivity. Using the algebra of quaternions, the study presents quaternion COSHAD/FrCOSHAD transforms to represent colour images in a holistic manner. This paper also develops an inverse polynomial reconstruction method (IPRM) in the discrete COSHAD/FrCOSHAD domains. This method can effectively recover a piecewise smooth signal from the finite set of its COSHAD/FrCOSHAD coefficients, with high accuracy. The convergence theorem has proved that the partial sum of COSHAD provides a spectrally accurate approximation to the underlying piecewise smooth signal. The experimental results verify the numerical stability and accuracy of the proposed methods.
Highlights
With the development of fast algorithms, the discrete cosine transform (DCT) and the Hadamard transform have been widely used in signal processing and image processing, most prominently in the compressed representations of images [1,2,3]
Kernel function, an FrCOSHAD transform with a fractional order p1 operated on by an FrCOSHAD transform with a fractional order p2 will be an FrCOSHAD transform with fractional order (p1 + p2). To evaluate this one basic feature of the FrCOSHAD kernel function, we investigate the performance of the 1D FrCOSHAD transform using a discrete rectangular window function defined as x (t) = {10, for |t| ≤ 50, for 51 ≤ |t| ≤ 120, (18)
We introduce the definition of the right-side form of the Q-FrCOSHAD transform of a function with two variables
Summary
With the development of fast algorithms, the discrete cosine transform (DCT) and the Hadamard transform have been widely used in signal processing and image processing, most prominently in the compressed representations of images [1,2,3]. The Hadamard transform is highly practical for representing signals, images, and mobile communications, mainly because the elements of the Hadamard matrix are either +1 or −1. The computation of the transform of a signal consists of additions and subtractions of the signal samples. The Hadamard transform and many of its variations, such as the sequency-ordered complex Hadamard transform (SCHT) [4, 5] and the Jacket transform [6], have been proposed, and their applications to image processing and communications have been reported [7]
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