Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

Real Zeros of Polynomials01:27

Real Zeros of Polynomials

Polynomials are algebraic expressions of terms with variables raised to non-negative integer powers. A central aspect of analyzing polynomial functions is determining their real zeros—values of the variable for which the polynomial evaluates to zero. These values represent the x-intercepts of the polynomial’s graph.The Rational Zeros Theorem lists possible rational solutions for a polynomial equation with integer coefficients. If f(x)=anxn+....+a0​, then every rational zero is of the form p/q​,...
Complex Zeros01:29

Complex Zeros

Complex zeros are the solutions to polynomial equations that include imaginary numbers, specifically, numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit defined by i2=-1. These zeros satisfy the equation P(x) = 0, where P(x) is a polynomial with real or complex coefficients. Since the complex number system includes all real numbers, it provides a complete framework for analyzing all possible roots of a polynomial.Every polynomial of degree n≥1 can be...
Polar and Cylindrical Coordinates01:22

Polar and Cylindrical Coordinates

The Cartesian coordinate system is a very convenient tool to use when describing the displacements and velocities of objects and the forces acting on them. However, it becomes cumbersome when we need to describe the rotation of objects. So, when describing rotation, the polar coordinate system is generally used.
Inverse z-Transform by Partial Fraction Expansion01:20

Inverse z-Transform by Partial Fraction Expansion

The inverse z-transform is a crucial technique for converting a function from its z-domain representation back to the time domain. One effective method for finding the inverse z-transform is the Partial Fraction Method, which involves decomposing a function into simpler fractions with distinct coefficients. These fractions correspond to known z-transform pairs, facilitating the inverse transformation process.
To begin the process, the poles of the function are identified and the function is...
Routh-Hurwitz Criterion II01:19

Routh-Hurwitz Criterion II

In the application of the Routh-Hurwitz criterion, two specific scenarios can arise that complicate stability analysis.
The first scenario occurs when a singular zero appears in the first column of the Routh table. This situation creates a division by zero issues. To resolve this, a small positive or negative number, denoted as epsilon (∈), is substituted for the zero. The stability analysis proceeds by assuming a sign for ∈. If ∈ is positive, any sign change in the first column of the Routh...
Graphs of Polar Equations01:17

Graphs of Polar Equations

The polar coordinate system represents points using a distance from a central point (the pole) and an angle from a reference direction (the polar axis). Unlike rectangular coordinates, polar coordinates are ideal for graphing curves with radial symmetry or periodic behavior.Some general forms of graphs in polar coordinates include the following:Equation of a Circle (Centered at the Pole):A graph where the radius remains constant for all angles traces a circle centered at the pole:Equation of a...

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same author

Cervical Cancer Classification From Pap Smear Images Using Deep Convolutional Neural Network Models.

Interdisciplinary sciences, computational life sciences·2023
Same author

Developments in the detection of diabetic retinopathy: a state-of-the-art review of computer-aided diagnosis and machine learning methods.

Artificial intelligence review·2022
Same author

Tchebichef moment based restoration of Gaussian blurred images.

Applied optics·2016
Same author

Correlation between subjective and objective assessment of magnetic resonance (MR) images.

Magnetic resonance imaging·2016
Same author

Real-Time Head Pose Tracking with Online Face Template Reconstruction.

IEEE transactions on pattern analysis and machine intelligence·2015
Same author

Visual Quality Evaluation of Image Object Segmentation: Subjective Assessment and Objective Measure.

IEEE transactions on image processing : a publication of the IEEE Signal Processing Society·2015
Same journal

Gaussian-modulated continuous-variable quantum key distribution over 60 km fiber using an integrated silicon photonic receiver.

Optics letters·2026
Same journal

E2E-OCT: end-to-end joint learning model using optical coherence tomography images for vocal cord leukoplakia diagnosis.

Optics letters·2026
Same journal

Holographic generation of panoramic 3D scenes by concave ellipsoidal mirror reflection.

Optics letters·2026
Same journal

Dual-pilot phase recovery with pair-wise maximum-ratio combining for coherent PONs.

Optics letters·2026
Same journal

Mapping the whispering gallery modes of a CaF<sub>2</sub> disk resonator with half-tapered fibers to estimate the fundamental mode volume.

Optics letters·2026
Same journal

Quantitative estimation of deep-subwavelength scale via dark-field scattering axial energy concentration decay profiles.

Optics letters·2026
See all related articles

Related Experiment Video

Updated: May 8, 2026

Assessment of Zebrafish Lens Nucleus Localization and Sutural Integrity
07:16

Assessment of Zebrafish Lens Nucleus Localization and Sutural Integrity

Published on: May 6, 2019

Recursive formula to compute Zernike radial polynomials.

Barmak Honarvar Shakibaei1, Raveendran Paramesran

  • 1Department of Electrical Engineering, University of Malaya, Kuala Lumpur, Malaysia. barmak.honarvar@gmail.com

Optics Letters
|August 14, 2013
PubMed
Summary
This summary is machine-generated.

This study introduces a new recursive formula for calculating Zernike radial polynomials, simplifying wavefront sensing and aberration analysis. The method reduces computational complexity by avoiding dependence on polynomial degree or azimuthal order.

More Related Videos

Lens-free Video Microscopy for the Dynamic and Quantitative Analysis of Adherent Cell Culture
09:04

Lens-free Video Microscopy for the Dynamic and Quantitative Analysis of Adherent Cell Culture

Published on: February 23, 2018

Liquid-cell Transmission Electron Microscopy for Tracking Self-assembly of Nanoparticles
08:39

Liquid-cell Transmission Electron Microscopy for Tracking Self-assembly of Nanoparticles

Published on: October 16, 2017

Related Experiment Videos

Last Updated: May 8, 2026

Assessment of Zebrafish Lens Nucleus Localization and Sutural Integrity
07:16

Assessment of Zebrafish Lens Nucleus Localization and Sutural Integrity

Published on: May 6, 2019

Lens-free Video Microscopy for the Dynamic and Quantitative Analysis of Adherent Cell Culture
09:04

Lens-free Video Microscopy for the Dynamic and Quantitative Analysis of Adherent Cell Culture

Published on: February 23, 2018

Liquid-cell Transmission Electron Microscopy for Tracking Self-assembly of Nanoparticles
08:39

Liquid-cell Transmission Electron Microscopy for Tracking Self-assembly of Nanoparticles

Published on: October 16, 2017

Area of Science:

  • Optics and Photonics
  • Computational Mathematics

Background:

  • Zernike polynomials are fundamental in optical testing, wavefront sensing, and aberration theory.
  • Their orthogonality over the unit circle and finite boundary conditions are key properties.
  • Existing computational methods can be complex and dependent on polynomial order.

Purpose of the Study:

  • To present a novel recursive formula for computing Zernike radial polynomials.
  • To simplify the calculation of these polynomials for optical applications.
  • To reduce the computational complexity associated with Zernike polynomial evaluation.

Main Methods:

  • Derivation of a recursive formula based on the relationship between Zernike radial polynomials and Chebyshev polynomials of the second kind.
  • The formula is independent of the radial polynomial's degree and azimuthal order.
  • Implementation and analysis of the computational efficiency of the new recurrence relation.

Main Results:

  • A new, efficient recursive formula for Zernike radial polynomials has been successfully derived.
  • The computational complexity is significantly reduced compared to existing algorithms.
  • The formula's independence from degree and azimuthal order offers greater flexibility.

Conclusions:

  • The developed recursive formula provides a computationally efficient method for calculating Zernike radial polynomials.
  • This advancement can benefit optical testing, wavefront sensing, and aberration theory applications.
  • The simplified computation enhances the practical utility of Zernike polynomials in optical system analysis.