Image Representation and Analysis by Continuous Chebyshev Polynomials

This paper proposes a spline-like Chebyshev polynomial image representation obtained by block-based compressed sampling (CS) for image analysis and processing. Due to orthogonality and close relation with the discrete cosine transform, Chebyshev polynomials have near-zero redundancy measure and possess great compression properties. Consequently, they fit perfectly into CS paradigm. We propose a spline-like model, that equalizes a desired number of derivatives on each block boundary, to avoid blocking artifacts. This can be seen as an additional constraint in the standard CS optimization problem. Still, the proposed model provides a system of equations that is underdetermined for achieving sparse reconstruction using the l
₁
optimization. The proposed framework offers polynomial representation of the acquired image and allows further analysis and processing conducted on Chebyshev polynomial coefficients without the need of converting them into samples. In this paper, we demonstrate the efficiency of the proposed model by implementing an analytic Chebyshev polynomial gradient operator that can be used instead of discrete gradient approximations in edge detection methods. The proposed operator is compared to Sobel operator and we prove its potential by implementing it into an edge detection algorithm.