Related Experiment Video
Updated: Apr 10, 2026
09:04
Lens-free Video Microscopy for the Dynamic and Quantitative Analysis of Adherent Cell Culture
Published on: February 23, 2018
10.1K
Resolving capacity of the circular Zernike polynomials
Optics Express
|June 16, 2015
Summary
This study analyzes circular Zernike polynomials for optical surface analysis. It provides formulas to calculate fitting errors and relates the spatial spectrum width to approximation order.
Area of Science:
- Optics and Optical Engineering
- Surface Metrology
Background:
- Circular Zernike polynomials are widely used for approximating and analyzing optical surfaces.
- Understanding the limitations of finite polynomial sets is crucial for accurate surface characterization.
Purpose of the Study:
- To analyze the lateral resolving capacity of circular Zernike polynomials.
- To determine the minimum number of polynomials needed for local deformation description.
- To investigate the relationship between polynomial count, spatial spectrum, and approximation error.
Main Methods:
- Analysis of approximation errors using finite sets of circular Zernike polynomials.
- Derivation of formulas for fitting error calculation.
- Investigation of the spatial spectrum characteristics of polynomial approximations.
Main Results:
- Formulas for calculating the error in fitting surface relief.
- Established connection between the spatial spectrum width and the order of approximation.
- Quantified the impact of approximation order on spatial frequency content.
Conclusions:
- The number of approximating polynomials directly influences the spatial spectrum and mean-square error.
- Accurate surface analysis requires careful selection of the polynomial set size based on deformation characteristics.
Related Concept Videos
01:29Complex Zeros
387
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...
387
01:27Real Zeros of Polynomials
294
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...
294
01:19Routh-Hurwitz Criterion II
1.3K
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...
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...
1.3K
01:22Polar and Cylindrical Coordinates
22.4K
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.
22.4K
01:26Introduction to Polynomial Functions
441
Polynomial functions are fundamental elements in algebra and calculus, defined by expressions that combine variables and constants through addition, subtraction, and multiplication, with the variable raised to nonnegative integer exponents. A general polynomial function of degree n is given byWhere an ≠ 0. The term anxn is the leading term, and an is the leading coefficient, while a0 is referred to as the constant term.Characteristics and ClassificationPolynomials are categorized by their...
441
01:20Inverse z-Transform by Partial Fraction Expansion
788
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...
788