The content of this repository is related to the project that helps eliminate numerical instability and high-order orthogonal moment calculation error while preserving it's orthogonality, which is one of the most important properties of these types of functions.

The high-order orthogonal polynomial base causes a numerical instability that makes it difficult to calculate the moments. In this project, we analyze the calculation of the following discrete orthogonal polynomials,

• Tchebycheff polynomials
• Krawtchouk polynomials
• Hahn polynomials
• Meixner polynomials
• Charlier polynomials

The source codes of the project are described in the chapter 3: "Computation of 2D and 3D high-order discrete orthogonal moments" of the book "Recent Progress in Moments and Moment Invariants" GCSR Volume 7 Edited by Dr. George A. Papakostas.

If you find these codes useful as a support tool.

You can access the information about the book through the following link: https://sciencegatepub.com/sgp-books/gcsr/gcsr_vol7/

or you can directly download the chapter using the following link: https://sciencegatepub.com/?smd_process_download=1&download_id=3179

• José S. Rivera-Lopez, César Camacho-Bello, and Lucia Gutiérrez-Lazcano, Chapter 3. “Computation of 2D and 3D High-order Discrete Orthogonal Moments”. Recent Progress in Image Moments and Moment Invariants, GCSR Volume 7 (2021), 53-74, DOI: 10.15579/gcsr.vol7.ch3.

  Bibtex:
  @article{Rivera2021Recent,
  title={Recent Progress in Image Moments and Moment Invariants},
  author={José S. Rivera-Lopez and César Camacho-Bello and Lucia Gutiérrez-Lazcano},
  journal={Science Gate},
  volume={7},
  pages={53--74},
  year={2021},
  URL={https://sciencegatepub.com/sgp-books/gcsr/gcsr_vol7/},
  ISBN={2241-9063}
  }
  

• C. Camacho-Bello and J. S. Rivera-Lopez, “Some computational aspects of tchebichef moments for higher orders,”Pattern Recognition Letters, vol. 112, pp. 332–339, 2018.

  Bibtex:
  @article{camacho2018some,
    title={Some computational aspects of Tchebichef moments for higher orders},
    author={Camacho-Bello, C{\'e}sar and Rivera-Lopez, Jos{\'e} S},
    journal={Pattern Recognition Letters},
    volume={112},
    pages={332--339},
    year={2018},
    publisher={Elsevier}
   }
  

1. Download this repository.
2. To obtain discrete ortogonal moments of image test run the file "Discrete_orthogonal_moments.m" using Matlab.
3. By default the value assigned for the order of calculation of the moment is the maximum possible. Change the value of "n" in line 66 from "Discrete_orthogonal_moments.m", to obtain different orders of discrete orthogonal moments form image test.
4. By default set discrete orthogonal polynomials is Tchebichef polynomial. For obtain different set polynomial modify line 69 from "Discrete_orthogonal_moments.m" corresponding to the calculation of the polynomial base.