# On Computational Aspects of Krawtchouk Polynomials for High Orders

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## Abstract

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## 1. Introduction

## 2. Preliminaries

## 3. Proposed Recurrence Algorithm

- The Initial value ${\mathbf{\U0001d4da}}_{0}^{0.5}\left(\right)open="("\; close=")">N/2-1$ is computed using Equation (9).
- The value of ${\mathbf{\U0001d4da}}_{0}^{0.5}\left(\right)open="("\; close=")">N/2$ is computed from ${\mathbf{\U0001d4da}}_{0}^{0.5}\left(\right)open="("\; close=")">N/2-1$ using Equation (10).
- The first set of initial used for TTRA is computed in the range of $n=2,3,\dots ,N/2-1$ and $x=N/2-1$ using Equation (11).
- The first set of initial used for TTRA is computed in the range of $n=2,3,\dots ,N/2-1$ and $x=N/2$ using Equation (12).
- The modified recurrence algorithm, Equation (13), is used to compute the values of the coefficients in the range $n=0,1,\dots ,N/2-1$ and $x=N/2,\dots ,N-n-2$. It should be noted that, for each x, when ${\mathbf{\U0001d4da}}_{n}^{0.5}\left(\right)open="("\; close=")">x+1$ and ${\mathbf{\U0001d4da}}_{n}^{0.5}\left(x\right)<{10}^{-5}$, the recurrence relation is terminated and n is increased by 1.

## 4. Performance Evaluation of the Proposed Method

- A test image, $\mathbf{I}$, is used.
- The test image is resized to a small size ${N}_{s}\times {N}_{s}$,
- DKP, $\mathbf{R}$, is generated with signal size and order of (${N}_{s}\times {N}_{s}$),
- The moment, $\mathbf{M}$, is computed using $\mathbf{M}=\mathbf{R}\times \mathbf{I}\times {\mathbf{R}}^{T}$,
- The test image is then reconstructed back from the moment domain using ${\mathbf{I}}_{\mathbf{r}}={\mathbf{R}}^{T}\times \mathbf{M}\times \mathbf{R}$,
- The normalized mean square error (NMSE) is computed between $\mathbf{I}$ and ${\mathbf{I}}_{\mathbf{r}}$,
- If the $NMSE<2.5\times {10}^{-3}$, the size of the image and DKP is increased by 2, i.e., ${N}_{s}={N}_{s}+2$ and repeat step 1 to 7.
- If the $NMSE>2.5\times {10}^{-3}$, the maximum size can be generated by the polynomial is reached and reported.
- The maximum size considered in the experiment is set to 12,288.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DKP | discrete Krawtchouk polynomials |

DKPC | discrete Krawtchouk polynomial Coefficient |

TTRA | three-term recurrence algorithm |

TTRAn | three-term recurrence algorithm in the n-direction |

TTRAx | three-term recurrence algorithm in the x-direction |

SRBM | symmetry relation-based method |

NMSE | normalized mean square error |

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**Figure 2.**Plot of ${\mathbf{\U0001d4da}}_{0}^{0.5}\left(x\right)$ for different values of polynomial size.

**Figure 9.**Construction analysis for the proposed method using sinusoidal Siemens start with $\omega =50$. (

**a**) NMSE, image reconstructed using number of moment equal to (

**b**) 800, (

**c**) 1600, (

**d**) 2400, (

**e**) 3200, (

**f**) 4000, (

**g**) 4800, (

**h**) 5600, (

**i**) 6400, and (

**j**) 7200.

**Figure 10.**Construction analysis for the proposed method using sinusoidal Siemens start with $\omega =100$. (

**a**) NMSE, image reconstructed using number of moment equal to (

**b**) 800, (

**c**) 1600, (

**d**) 2400, (

**e**) 3200, (

**f**) 4000, (

**g**) 4800, (

**h**) 5600, (

**i**) 6400, and (

**j**) 7200.

**Figure 11.**Construction analysis for the proposed method using sinusoidal Siemens start with $\omega =200$. (

**a**) NMSE, image reconstructed using number of moment equal to (

**b**) 800, (

**c**) 1600, (

**d**) 2400, (

**e**) 3200, (

**f**) 4000, (

**g**) 4800, (

**h**) 5600, (

**i**) 6400, and (

**j**) 7200.

**Figure 12.**Construction analysis for the proposed method using Lena image. (

**a**) NMSE, image reconstructed using number of moment equal to (

**b**) 800, (

**c**) 1600, (

**d**) 2400, (

**e**) 3200, (

**f**) 4000, (

**g**) 4800, (

**h**) 5600, (

**i**) 6400, and (

**j**) 7200.

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**MDPI and ACS Style**

Mahmmod, B.M.; Abdul-Hadi, A.M.; Abdulhussain, S.H.; Hussien, A.
On Computational Aspects of Krawtchouk Polynomials for High Orders. *J. Imaging* **2020**, *6*, 81.
https://doi.org/10.3390/jimaging6080081

**AMA Style**

Mahmmod BM, Abdul-Hadi AM, Abdulhussain SH, Hussien A.
On Computational Aspects of Krawtchouk Polynomials for High Orders. *Journal of Imaging*. 2020; 6(8):81.
https://doi.org/10.3390/jimaging6080081

**Chicago/Turabian Style**

Mahmmod, Basheera M., Alaa M. Abdul-Hadi, Sadiq H. Abdulhussain, and Aseel Hussien.
2020. "On Computational Aspects of Krawtchouk Polynomials for High Orders" *Journal of Imaging* 6, no. 8: 81.
https://doi.org/10.3390/jimaging6080081