A symmetry and bi-recursive algorithm of accurately computing Krawtchouk moments

Few scientific studies have discussed the accuracy of the Krawtchouk moments for the common case of p ≠ 0.5. In the paper, a novel symmetry and bi-recursive algorithm is proposed to accurately calculate the Krawtchouk moments for the case of p ∈  (0, 1). The numerical propagation error mechanism of direct recursively calculating the Krawtchouk moments is first analyzed. It reveals that the recursion coefficients and recurrence times of the three-term recurrence relations are the key factors of reducing the propagation error in the computation of the Krawtchouk moment of high order. Based on the analysis, the x − n plane is divided into four parts by x = n and x + n = N − 1. We use the n-ascending recurrence formula to calculate the polynomials in the domain of N − 1 − n ⩾ x ⩾ n ⩾ 0 and apply the n-descending recurrence relations in the domain of 0 ⩽ N − 1 − n ⩽ x ⩽ n. Thus the maximum recursion times are limited to N/2. Finally, with the help of the diagonal symmetry property on x = n, the Krawtchouk polynomial values of high precision in the whole x − n coordinates are obtained. The algorithm ensures that the maximum recursive numerical errors are within an acceptable range. An experiment on a large image of 400 × 400 pixels is designed to demonstrate the performance of the proposed algorithm against the classical method.