Circular Functions

Zernike Modes

aotools.functions.zernike.makegammas(nzrad)

Make “Gamma” matrices which can be used to determine first derivative of Zernike matrices (Noll 1976).

Parameters:

nzrad – Number of Zernike radial orders to calculate Gamma matrices for

Returns:

Array with x, then y gamma matrices

Return type:

ndarray

aotools.functions.zernike.phaseFromZernikes(zCoeffs, size, norm='noll', rot=0)

Creates an array of the sum of zernike polynomials with specified coefficeints

Parameters:

• zCoeffs (list) – zernike Coefficients

• size (int) – Diameter of returned array

• norm (string, optional) – The normalisation of Zernike modes. Can be "noll", "p2v" (peak to valley), or "rms". default is "noll".

• rot (float) – Rotates the Zernike mode by rot radians around its centre. Defaults to zero.

Returns:

a size x size array of summed Zernike polynomials

Return type:

ndarray

aotools.functions.zernike.zernIndex(j)

Find the [n,m] list giving the radial order n and azimuthal order of the Zernike polynomial of Noll index j.

Parameters:

j (int) – The Noll index for Zernike polynomials

Returns:

n, m values

Return type:

list

aotools.functions.zernike.zernikeArray(J, N, norm='noll', rot=0)

Creates an array of Zernike Polynomials

Parameters:

• maxJ (int or list) – Max Zernike polynomial to create, or list of zernikes J indices to create

• N (int) – size of created arrays

• norm (string, optional) – The normalisation of Zernike modes. Can be "noll", "p2v" (peak to valley), or "rms". default is "noll".

• rot (float) – Rotates the zernike modes by rot radians around their centre. Defaults to 0.

Returns:

array of Zernike Polynomials

Return type:

ndarray

aotools.functions.zernike.zernikeRadialFunc(n, m, r)

Fucntion to calculate the Zernike radial function

Parameters:

• n (int) – Zernike radial order

• m (int) – Zernike azimuthal order

• r (ndarray) – 2-d array of radii from the centre the array

Returns:

The Zernike radial function

Return type:

ndarray

aotools.functions.zernike.zernike_nm(n, m, N, rot=0)

Creates the Zernike polynomial with radial index, n, and azimuthal index, m.

Parameters:

• n (int) – The radial order of the zernike mode

• m (int) – The azimuthal order of the zernike mode

• N (int) – The diameter of the zernike more in pixels

• rot (float) – Rotates the zernike by rot radians around its centre. Defaults to 0.

Returns:

The Zernike mode

Return type:

ndarray

aotools.functions.zernike.zernike_noll(j, N, rot=0)

Creates the Zernike polynomial with mode index j, where j = 1 corresponds to piston.

Parameters:

• j (int) – The noll j number of the zernike mode

• N (int) – The diameter of the zernike more in pixels

• rot (float) – Rotates the Zernike mode by rot radians around its centre. Defaults to zero.

Returns:

The Zernike mode

Return type:

ndarray

Karhunen Loeve Modes

Collection of routines related to Karhunen-Loeve modes.

The theory is based on the paper “Optimal bases for wave-front simulation and reconstruction on annular apertures”, Robert C. Cannon, 1996, JOSAA, 13, 4

The present implementation is based on the IDL package of R. Cannon (wavefront modelling and reconstruction). A closely similar implementation can also be find in Yorick in the YAO package.

Usage

Main routine is ‘make_kl’ to generate KL basis of dimension [dim, dim, nmax].

For Kolmogorov statistics, e.g.

kl, _, _, _ = make_kl(150, 128, ri = 0.2, stf='kolmogorov')

Warning

make_kl with von Karman stf fails. It has been implemented but KL generation failed in ‘while loop’ of gkl_fcom…

date:

November 2017

aotools.functions.karhunenLoeve.gkl_azimuthal(nord, npp)

Compute the azimuthal function of the KL basis.

Parameters:

• nord (int) – number of azimuthal orders

• npp (int) – grid of point sampling the azimuthal coordinate

Returns:

gklazi – azimuthal function of the KL basis.

Return type:

ndarray

aotools.functions.karhunenLoeve.gkl_basis(ri=0.25, nr=40, npp=None, nfunc=500, stf='kolstf', outerscale=None)

Wrapper to create the radial and azimuthal K-L functions.

Parameters:

• ri (float) – normalized internal radius

• nr (int) – number of radial resolution elements

• np (int) – number of azimuthal resolution elements

• nfunc (int) – number of generated K-L function

• stf (string) – structure function tag describing the atmospheric statistics

Returns:

gklbasis – dictionary containing the radial and azimuthal basis + other relevant information

Return type:

dic

aotools.functions.karhunenLoeve.gkl_fcom(ri, kernels, nfunc, verbose=False)

Computation of the radial eigenvectors of the KL basis.

Obtained by taking the eigenvectors from the matrix L^p. The final function corresponds to the ‘nfunc’ largest eigenvalues. See eq. 16-18 in Cannon 1996.

Parameters:

• ri (float) – radius of central obscuration radius (normalized by D/2; <1)

• kernels (ndarray) – kernel L^p of dimension (nr, nr, nth), where nth is the azimuthal discretization (5 * nr)

• nfunc (int) – number of final KL functions

Returns:

• evals (1darray) – eigenvalues

• nord (int) – resulting number of azimuthal orders

• npo (int)

• ord (1darray)

• rabas (ndarray) – radial eigenvectors of the KL basis

aotools.functions.karhunenLoeve.gkl_kernel(ri, nr, rad, stfunc='kolmogorov', outerscale=None)

Calculation of the kernel L^p

The kernel constructed here should be simply a discretization of the continuous kernel. It needs rescaling before it is treated as a matrix for finding the eigen-values

Parameters:

• ri (float) – radius of central obscuration radius (normalized by D/2; <1)

• nr (int) – number of resolution elements

• rad (1d-array) – grid of points where the kernel is evaluated

• stfunc (string) – string tag of the structure function on which the kernel are computed

• outerscale (float) – in unit of telescope diameter. Outer-scale for von Karman structure function.

Returns:

L^p – kernel L^p of dimension (nr, nr, nth), where nth is the azimuthal discretization (5 * nr)

Return type:

ndarray

aotools.functions.karhunenLoeve.gkl_radii(ri, nr)

Generate n points evenly spaced in r^2 between r_i^2 and 1

Parameters:

• nr (int) – number of resolution elements

• ri (float) – radius of central obscuration radius (normalized; <1)

Returns:

r – grid of point to calculate the appropriate kernels. correspond to ‘sqrt(s)’ wrt the paper.

Return type:

1d-array, float

aotools.functions.karhunenLoeve.gkl_sfi(kl_basis, i)

return the i’th function from the generalized KL basis ‘bas’. ‘bas’ must be generated by ‘gkl_basis’

aotools.functions.karhunenLoeve.make_kl(nmax, dim, ri=0.0, nr=40, stf='kolmogorov', outerscale=None, mask=True)

Main routine to generatre a KL basis of dimension [nmax, dim, dim].

For Kolmogorov statistics, e.g.

kl, _, _, _ = make_kl(150, 128, ri = 0.2, stf='kolmogorov')

As a rule of thumb

nr x npp = 50 x 250 is fine up to 500 functions

60 x 300 for a thousand

80 x 400 for three thousands.

Parameters:

• nmax (int) – number of KL function to generate

• dim (int) – size of the KL arrays

• ri (float) – radial central obscuration normalized by D/2

• nr (int) – number of point on radius. npp (number of azimuthal pts) is =2 pi * nr

• stf (string) – structure function tag. Default is ‘kolmogorov’

• outerscale (float) – outer scale in units of telescope diameter. Releveant if von vonKarman stf. (not implemented yet)

• mask (bool) – pupil masking. Default is True

Returns:

• kl (ndarray (nmax, dim, dim)) – KL basis in cartesian coordinates

• varKL (1darray) – associated variance

• pupil (2darray) – pupil

• polar_base (dictionary) – polar base dictionary used for the KL basis computation. as returned by ‘gkl_basis’

See also

gkl_basis, set_pctr, pol2car

aotools.functions.karhunenLoeve.pcgeom(nr, npp, ncp, ri, ncmar)

This routine builds a geom dic.

Parameters:

• px (int) – the x, y coordinates of points in the polar arrays.

• py (int) – the x, y coordinates of points in the polar arrays.

• cr – the r, phi coordinates of points in the cartesian grids.

• cp – the r, phi coordinates of points in the cartesian grids.

• ncmar – allows the possibility that there is a margin of ncmar points in the cartesian arrays outside the region of interest.

aotools.functions.karhunenLoeve.piston_orth(nr)

Unitary matrix used to filter out piston term. Eq. 19 in Cannon 1996.

Parameters:

nr (int) – number of resolution elements

Returns:

U – Unitary matrix

Return type:

2d array

aotools.functions.karhunenLoeve.pol2car(cpgeom, pol, mask=False)

Polar to cartesian conversion.

(points not in the aperture are treated as though they were at the first or last radial polar value…)

aotools.functions.karhunenLoeve.polang(r)

Generate an array with the same dimensions as r, but containing the azimuthal values for a polar coordinate system.

aotools.functions.karhunenLoeve.radii(nr, npp, ri)

Use to generate a polar coordinate system.

Generate an nr x npp array with npp copies of the radial coordinate array. Radial coordinate span the range from r=ri to r=1 with successive annuli having equal areas.

ie, the area between ri and 1 is divided into nr equal rings, and the points are positioned at the half-area mark on each ring; there are no points on the border

See also

polang

aotools.functions.karhunenLoeve.rebin(a, newshape)

Rebin an array to a new shape. See scipy cookbook. It is intended to be similar to ‘rebin’ of IDL

aotools.functions.karhunenLoeve.set_pctr(bas, ncp=None, ncmar=None)

call pcgeom to build the dic geom_struct with the right initializtions.

Parameters:

bas (dic) – gkl_basis dic built with the gkl_bas routine

aotools.functions.karhunenLoeve.setpincs(ax, ay, px, py, ri)

determine a set of squares for interpolating from cartesian to polar coordinates, using only those points with ri <= r <= 1

aotools.functions.karhunenLoeve.stf_kolmogorov(r)

Kolmogorov structure function with r = (D/r0)

aotools.functions.karhunenLoeve.stf_vonKarman(r, L0)

von Karman structure function with r = (D / r0) L0 is in unit of telescope diameter, typically a few (3; or 20m)

aotools.functions.karhunenLoeve.stf_vonKarman_yao(r, L)