On Computational Aspects of Krawtchouk Polynomials for High Orders.

Area of Science:

• Mathematics
• Computer Vision
• Image Processing

Background:

• Discrete Krawtchouk polynomials possess unique localization properties.
• These polynomials are vital for feature description in computer vision, particularly for images and video frames.
• Existing methods for computing Krawtchouk polynomial coefficients face challenges with computational efficiency and scalability.

Purpose of the Study:

• To develop a swift and efficient method for computing discrete Krawtchouk polynomial coefficients.
• To reduce the computational cost associated with generating these polynomials.
• To enhance their applicability in computer vision tasks, especially for large signal sizes.

Main Methods:

• A novel initial value is proposed to prevent zero-tending issues with increasing polynomial size.
• A combination of existing recurrence relations in the n- and x-directions is utilized to decrease computational complexity.
• Approximately 12.5% of coefficients are computed directly, with the rest derived using symmetry relations.

Main Results:

• The proposed method demonstrates superior computational efficiency compared to existing techniques.
• It enables the generation of larger polynomial sizes with reduced computational overhead.
• Image reconstruction error analysis confirms the method's effectiveness for large signal sizes.

Conclusions:

• The new method offers a significant improvement in computing discrete Krawtchouk polynomial coefficients.
• It provides a more efficient and scalable solution for computer vision applications.
• The findings pave the way for enhanced feature description and analysis in image and video processing.