Related Experiment Video
Updated: Jun 15, 2026
09:04
Lens-free Video Microscopy for the Dynamic and Quantitative Analysis of Adherent Cell Culture
Published on: February 23, 2018
Fast evaluation of equal-spaced Zernike polynomial expansion samples
1Sarver and Associates Inc, Carbondale, Ill, USA. ejsarver@aol.com
Journal of Refractive Surgery (Thorofare, N.J. : 1995)
|March 5, 2010
Summary
A new method uses difference equations to rapidly compute Zernike polynomial expansion samples on grids. This technique significantly speeds up calculations for optical system analysis and data display, offering high accuracy.
Area of Science:
- Optics
- Computational Science
Background:
- Zernike polynomial expansions are crucial for analyzing optical systems.
- Efficient computation of these expansions on various grids is essential for practical applications.
Purpose of the Study:
- To develop a fast and accurate method for calculating equal-spaced Zernike polynomial expansion samples.
- The method targets both rectangular and polar grids for analysis and display.
Main Methods:
- The study leverages the conversion of Zernike polynomials to Taylor polynomial expansions.
- Difference equations are employed for rapid calculation of polynomial samples.
- A software class was developed to implement this approach.
Main Results:
- The difference equation technique was over 400 times faster than direct calculation for a 10th order expansion on a 1000x1000 grid.
- Accuracy was maintained with negligible differences, achieving 11 digits of precision using double precision variables.
Conclusions:
- The difference equation approach provides a computationally efficient and accurate solution for Zernike polynomial sampling.
- This algorithm is valuable for optical system analysis and visualization.
Related Concept Videos
01:20Inverse 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...
To begin the process, the poles of the function are identified and the function is...
01:27Real 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,...
01:17Wald-Wolfowitz Runs Test II
The Wald-Wolfowitz runs test, commonly referred to as the runs test, is a nonparametric test used to assess the randomness of ordered data. The test evaluates the number of runs, which are consecutive sequences of similar elements within the data. If the number of runs is significantly higher or lower than expected, the data is considered non-random, indicating a detectable pattern or structure.
For binary data, runs are identified using symbols such as + and −, or equivalently, 1s and 0s. In...
For binary data, runs are identified using symbols such as + and −, or equivalently, 1s and 0s. In...
01:29Complex 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...
01:24Difference Equation Solution using z-Transform
The z-transform is a powerful tool for analyzing practical discrete-time systems, often represented by linear difference equations. Solving a higher-order difference equation requires knowledge of the input signal and the initial conditions up to one term less than the order of the equation.
The z-transform facilitates handling delayed signals by shifting the signal in the z-domain, which corresponds to delaying the signal in the time domain, and advancing signals by similarly shifting in the...
The z-transform facilitates handling delayed signals by shifting the signal in the z-domain, which corresponds to delaying the signal in the time domain, and advancing signals by similarly shifting in the...
01:06Introduction to z Scores
A z score (or standardized value) is measured in units of the standard deviation. It tells you how many standard deviations the value x is above (to the right of) or below (to the left of) the mean, μ. Values of x that are larger than the mean have positive z scores, and values of x that are smaller than the mean have negative z scores. If x equals the mean, then x has a zero z score. It is important to note that the mean of the z scores is zero, and the standard deviation is one.
z scores help...
z scores help...