Perhitungan Stabil Fungsi Krawtchouk dari Hubungan Triplet

Polynomial transformations are well-defined mathematical concepts, with the classical (continuous and discrete) orthogonal polynomials standing out as having a common basis and sharing multiple features.

 In the digital age, the discrete polynomials are especially popular and those with a finite support (discrete Chebyshev and Krawtchouk) have been used in various signal processing tasks, but dominantly for image processing. 

Examples of applications of Krawtchouk polynomials can be found in, e.g., [1–7]. The recurrence relation and difference equation associated with these polynomials seem to constitute a simple recipe for their numerical evaluation. However, the recursive character of these generating schemes makes them vulnerable to error accumulation, which, obviously, becomes increasingly pronounced at higher polynomial degrees and/or larger supports. 

For many applications, i.e., cases of small support and or low polynomial degree, this issue is of no consequence, and the attention is more focused on computational simplification rather than accuracy. In this paper, we focus on the fundamental issue of accuracy, not so much on the issue of computational parsimony; in fact, extra computational burden is introduced in order to control the accuracy in the function generation process. 

For the normalized discrete Chebyshev polynomials several mitigation schemes have been proposed; the latest one a computation scheme characterized by truncation of the iterations under the control of a user-defined deviation from unit norm [8]. 

The solutions defined for the discrete Chebyshev polynomials do not carry over since they make use of the (anti-)symmetry of the polynomials with respect to the mid of the domain, i.e., a property that does not generally hold for Krawtchouk polynomials. 

Nevertheless, a first mitigation was proposed in [9] building upon the notion that the direction of execution of the relations (i.e., decreasing/increasing index and polynomial degree) should be carefully chosen in order to reduce the error propagation. However, not all symmetries were exploited, nor does the scheme allow direct control over the accuracy like in [8].

selengkapnya : https://www.mdpi.com/2227-7390/9/16/1972