Zernike#

class deepinv.physics.generator.Zernike[source]#

Bases: object

Static utility class for Zernike polynomials (ANSI/Noll Nomenclature). All methods are static; no instantiation required.

Zernike polynomials are a sequence of polynomials that are orthogonal on the unit disk. They are commonly used in optical systems to describe wavefront aberrations, see Lakshminarayanan and Fleck[1] (or this link).

They are defined by two indices: the radial order \(n\) and the azimuthal frequency \(m\), with the constraints that \(n >= 0\), \(|m| <= n\), and \(n - |m|\) is even.

The Zernike polynomial \(Z_n^m\) can be expressed in polar coordinates \((\\rho, \\theta)\) as:

\[ \begin{align}\begin{aligned}Z_n^m(\rho, \theta) =\\\begin{split}\begin{cases} N_n^m R_n^m(\rho) \cos(m \theta) & \text{ if } m \geq 0 \\ N_n^m R_n^m(\rho) \sin(|m| \theta) & \text{ if } m < 0 \end{cases}\end{split}\end{aligned}\end{align} \]

where \(R_n^m(\rho)\) is the radial polynomial defined as:

\[R_n^m(\rho) = \sum_{k=0}^{(n - |m|)/2} (-1)^k \frac{(n - k)!}{k! \left(\frac{n + |m|}{2} - k\right)! \left(\frac{n - |m|}{2} - k\right)!} \rho^{n - 2k}\]

and \(N_n^m\) is the normalization constant (root-mean-square, ensuring the polynomials have unit energy) given by:

\[\begin{split}N_n^m = \begin{cases} \sqrt{n + 1} & \text{ if } m = 0 \\ \sqrt{2(n + 1)} & \text{ if } m \neq 0 \end{cases}\end{split}\]

References:

static cartesian_evaluate(n, m, x, y, use_mask=True)[source]#

Evaluates the Zernike polynomial \(Z_n^m\) on given Cartesian coordinates.

Parameters:

  • n (int) – the radial order

  • m (int) – the azimuthal frequency

  • x (torch.Tensor) – x-coordinates in Cartesian system

  • y (torch.Tensor) – y-coordinates in Cartesian system

  • use_mask (bool) – if True, values outside the unit disk are set to zero. Default to True

Returns:

Evaluated Zernike polynomial values as a tensor

Return type:

Tensor

static get_name(n, m)[source]#

Returns the ANSI standard name given indices \(n\) and \(m\).

Parameters:

  • n (int) – the radial order

  • m (int) – the azimuthal frequency

Returns:

The ANSI standard name or a default string if not found.

Return type:

str

static index_conversion(index, *, convention='ansi')[source]#

Converts a single index for Zernike polynomials between different conventions. For more details on the conventions, see wikipedia.

Parameters:

  • index (int) – the single index of the Zernike polynomial.

  • convention (str) – the convention to convert to. Currently only ‘ansi’ (for OSA/ANSI standard indices) and ‘noll’ (for Noll’s sequential indices) are implemented.

Single index for Zernike polynomials:

static normalization_constant(n, m)[source]#

Returns the Noll RMS normalization constant.

Parameters:

  • n (int) – the radial order

  • m (int) – the azimuthal frequency

Returns:

The normalization constant.

Return type:

float

static polar_evaluate(n, m, rho, theta, use_mask=True)[source]#

Evaluates the Zernike polynomial \(Z_n^m\) on given polar coordinates.

Parameters:

  • n (int) – the radial order

  • m (int) – the azimuthal frequency

  • rho (torch.Tensor) – radial coordinates in polar system

  • theta (torch.Tensor) – angular coordinates in polar system

  • use_mask (bool) – if True, values outside the unit disk are set to zero. Default to True

Returns:

Evaluated Zernike polynomial values as a tensor

Return type:

Tensor