skqfit package

Submodules

skqfit.asmjacp module

References:

G W Forbes, “Characterizing the shape of freeform optics”, Opt. Express 20(3), 2483-2499 (2012)

class skqfit.asmjacp.AsymJacobiP(nmax)

Bases: object

Generate asymmetric Jacobi like polynominals needed for the freeform fit as defined in the reference document [A.1]

The Jacobi P polynominals can generate large values that lead to a double overflow for large m and n values. It is tested to (1500, 1500) for (m,n). Exceeding these values may lead to overflow events.

The scipi.special.jacobi function is has a 1.5:1 performance advantage over the current implementation but it doesn’t support the normalization and can lead to an overflow condition for n or m over 500.

build_recursion(m)

Build the recurssion coefficients and saves them as a sequence of tuples. These are the coefficients to build the m type polynominals up to order n.

jmat_u_x(jmat, uv, xv)

Builds the asymmetric Jacobi P polynomial as defined in [2] A.1 for all of the x values with the scaling factor of u**m which extends the range of usable (m,n) values before an overflow condition occurs.

jmat_x(jmat, xv)

Builds the asymmetric Jacobi P polynomial as defined in [2] A.1 for all a vector of x values.

jvec_u_x(jvec, u, x)

Builds the asymmetric Jacobi P polynomial for a single x as defined in [2] A.1 for all of the x values with the scaling factor of u**m which extends the range of usable (m,n) values.

jvec_x(jvec, x)

Builds the sequence of jacobi polynomials for the value x based on [A.2-5]

skqfit.qspectre module

Reference documents

[1] G W Forbes, “Fitting freeform shapes with orthogonal bases”, Opt. Express 21, 19061-19081 (2013) [2] G W Forbes, “Characterizing the shape of freeform optics”, Opt. Express 20(3), 2483-2499 (2012) [3] G W Forbes, “Robust, efficient computational methods for axially symmetric optical aspheres”, Opt. Express 18(19), 19700-19712 (2010)

class skqfit.qspectre.QSpectrum(m_max=None, n_max=None)

Bases: object

Performs precomputation if Q spectrum limits are passed, otherwise it is delayed until the data is loaded. The class supports processing a data map or a pointer to a sag function that can be used for analytic testing.

Parameters: m_max, n_max: int

The azimuthal and radial spectrum order. Setting values above 1500 may lead to overflow events.

bfs_param()

Returns the fitted radius, curvature and centre.

Returns:

radius, curvature, centre: float, float, (x,y) tuple

build_map(x, y, a_nm, b_nm, curv=None, radius=None, centre=None, extend=1.0, interpolated=True, inc_deriv=False)

Creates a 2D topography map and optional x and y derivate maps using the x and y axis vectors and the Q-freeform parameters.

Parameters:

x, y: array

X, and Y axis values for the map to be created. The arrays must be sorted to increasing order.

a_nm, b_nm: 2D array

The cosine and sine terms for the Q freeform polynominal

curv: float

Nominal curvature for the part. If None uses the estimated value from the previous fit.

radius: float

Defines the circular domain from the centre. If None uses the estimated value from the previous fit.

centre: (cx, cy)

The centre of the part in axis coordinates. If None uses the estimated value from previous fit.

extend: float

Generate a map over extend * radius from the centre

interpolated: boolean

If True uses a high resolution regular polar grid to build the underlying data and a spline interpolation to extract the (x, y) grid, otherwise it evaluates each (x, y) point exactly. The non-interpolated solution is significanatly slower and only practical for smaller array sizes.

inc_deriv: boolean

Return the X and Y derivatives as additional maps

Returns:

zmap: 2-D array

Data map with shape (x.size, y.size)

xder: 2-D array

X derivative map with shape (x.size, y.size) if inc_deriv is True, else None

yder: 2-D array

Y derivative map with shape (x.size, y.size) if inc_deriv is True, else None

build_profile(xv, yv, a_nm, b_nm, curv=None, radius=None, centre=None, extend=1.0, inc_deriv=False)

Returns the nominal sag and optional x and y derivatives along a 1D trajectory of (x, y) coordinates.

Parameters:

x, y: arrays

Arrays of values representing the (x, y) coordinates.

a_nm, b_nm: 2D array

The cosine and sine terms for the Q freeform polynominal

curv: float

Nominal curvature for the part. If None uses the estimated value from the previous fit.

radius: float

Defines the circular domain from the centre. If None uses the estimated value from the previous fit.

centre: (cx, cy)

The centre of the part in axis coordinates. If None uses the estimated value from previous fit.

extend: float

Generate a map over extend * radius from the centre

inc_deriv: boolean

Return the X and Y derivatives as additional maps

Returns:

zval: array

Sag values for the (x, y) sequence

xder: array

X derivative map for the (x, y) sequence if inc_deriv is True, else None

yder: array

Y derivative map for the (x, y) sequence if inc_deriv is True, else None

build_q_spectrum(m_max=None, n_max=None)

Fits the departure from a best fit sphere to the Q-polynominals as defined in [1](1.1) and returns the root sum square of the azimuthal terms.

Parameters:

m_max, n_max: int

The azimuthal and radial spectrum order. If None it uses the previous values and if not defined it matches the values to the pixel resolution.

Returns:

2D array

The (m,n) matrix representation of the spectrum (sqrt(cos^2 + sin^2)) terms

data_map(x, y, zmap, centre=None, radius=None, shrink_pixels=7, bfs_curv=None)

Creates the spline interpolator for the map, determines the best fit sphere and minimum valid radius.

Parameters:

x, y: arrays

The arrays are the X and Y axis values for the data map. The arrays must be sorted to increasing order.

zmap: array_like

2-D array of data with shape (x.size,y.size).

centre: (cx, cy)

The centre of the part in axis coordinates. If None the centre is estimated by a centre of mass calculation

radius: float

Defines the circular domain from the centre. If None it determines the maximum radius from the centre that contains no invalids (NAN).

shrink_pixels: int

The estimated radius is reduced by 7 pixels to avoid edge effects with the spline interpolation. Ignored if the radius is specified.

q_fit(m_max=None, n_max=None)

Fits the departure from a best fit sphere to the Q-freeform polynominals as defined in [1](1.1) and returns the individual sine and cosine terms.

Parameters:

m_max, n_max: int

The azimuthal and radial spectrum order. If None it uses the previous values and if not defined it matches the values to the pixel resolution. The maximum resolution supported is (1500, 1500)

Returns:

a_nm, b_nm: 2D array

The (n,m) matrix representation of the cosine and sine terms

set_sag_fn(sag_fn, radius, bfs_curv=None)

A vectorized polar sag function that takes rho and theta as arguments.

This function is used to pass an analytic sag function to test the performance of the algorithm to a higher precision but can also be used to define the input map and bypass data_map().

skqfit.qspectre.qspec(x, y, zmap, m_max=None, n_max=None, centre=None, radius=None, shrink_pixels=7)

A wrapper function that creates the Q-spectrum object, loads and performs the fit of the data to the Q polynomials.

Parameters:

x, y: array_like

The interpolator uses grid points defined by the coordinate arrays x, y. The arrays must be sorted to increasing order.

zmap: array_like

2-D array of data with shape (x.size,y.size)

m_max, n_max: int

The azimuthal and radial spectrum order. If None, it uses the previous values and if not defined it matches the values to the pixel resolution.

centre: (cx, cy)

The centre of the part in axis coordinates. If None the centre is estimated by a centre of mass calculation

radius: float

Defines the circular domain from the centre. If None it determines the maximum radius from the centre that contains no invalids (NAN).

shrink_pixels: int

The estimated radius is reduced by 7 pixels to avoid edge effects with the spline interpolation. Ignored if the radius is specified.

Returns:

2D array

The (m,n) matrix representation of the spectrum (sqrt(cos^2 + sin^2)) terms

Module contents