Convolvables¶

Prysm features a rich implemention of Linear Shift Invariant (LSI) system theory. Under this mathematical ideal, the transfer function is the product of the Fourier transform of a cascade of components, and the spatial distribution of intensity is the convolution of a cascade of components. These features are usually used to blur objects or images with Point Spread Functions (PSFs), or model the transfer function of an opto-electronic system. Within prysm there is a class Convolvable which objects and PSFs inherit from. You should rarely need to use the base class, except when subclassing it with your own models or objects.

>>> from prysm.convolution import Convolvable

The built-in convolvable objects are Slits, Pinholes, Tilted Squares, and Siemens Stars. There are also two components, PixelAperture and OLPF, used for system modeling.

>>> from prysm import Slit, Pinhole, TiltedSquare, SiemensStar, PixelAperture, OLPF

Each is initialized with object-specific parameters,

>>> s = Slit(width=1, orientation='crossed')  # diameter, um
>>> p = Pinhole(width=1)
>>> t = TiltedSquare(angle=8, background='white', sample_spacing=0.05, samples=256))  # degrees
>>> star = SiemensStar(num_spokes=32, sinusoidal=False, background='white', sample_spacing=0.05, samples=256)
>>> pa = PixelAperture(width_x=5)  # diameter, um
>>> ol = OLPF(width_x=5*0.66)

Objects that take a background parameter will be black-on-white for background=white, or white-on-black for background=black. Two objects are convolved via the conv method, which returns self on a new Convolvable instance and is chainable,

>>> monstrosity = s.conv(p).conv(t).conv(star).conv(pa).conv(ol)

Some models require sample spacing and samples parameters while others do not. This is because prysm has many methods of executing an FFT-based Fourier domain convolution under the hood. If an object has a known analytical Fourier transform, the class has a method (Convolvable).analytic_ft which has abscissa units of reciprocal microns. If the analytic FT is present, it is used in lieu of numerical data. Models that have analytical Fourier transforms also accept sample_spacing and samples parameters, which are used to define a grid in the spatial domain. If two objects with analytical Fourier transforms are convolved, the output grid will have the finer sample spacing of the two inputs, and the larger span or window width of the two inputs.

The Convolvable constructor takes only four parameters,

>>> import numpy as np
>>> x = y = np.linspace(-20,20,256)
>>> z = np.random.uniform((256,256))
>>> c = Convolvable(data=z, unit_x=x, unit_y=y, has_analytic_ft=False)

has_analytic_ft has a default value of False.

Minimal labor is required to subclass Convolvable. For example, the Pinhole implemention is simply:

class Pinhole(Convolvable):
    def __init__(self, width, sample_spacing=0.025, samples=0):
        self.width = width

        # produce coordinate arrays
        if samples > 0:
            ext = samples / 2 * sample_spacing
            x, y = m.linspace(-ext, ext, samples), m.linspace(-ext, ext, samples)
            xv, yv = m.meshgrid(x, y)
            w = width / 2
            # paint a circle on a black background
            arr = m.zeros((samples, samples))
            arr[m.sqrt(xv**2 + yv**2) < w] = 1
        else:
            arr, x, y = None, m.zeros(2), m.zeros(2)

        super().__init__(data=arr, unit_x=x, unit_y=y, has_analytic_ft=True)

    def analytic_ft(self, unit_x, unit_y):
        xq, yq = m.meshgrid(unit_x, unit_y)
        # factor of pi corrects for jinc being modulo pi
        # factor of 2 converts radius to diameter
        rho = m.sqrt(xq**2 + yq**2) * self.width * 2 * m.pi
        return m.jinc(rho).astype(config.precision)

which is less than 20 lines long.

Convolvable objects have a few convenience properties and methods. (Convolvable).slice_x and its y variant exist and behave the same as slices on subclasses of OpticalPhase such as Pupil. (Convolvable).plot_slice_xy also works the same way. (Convolvable).shape is a convenience wrapper for (Convolvable).data.shape, an (Convolvable).support_x, .support_y, an .support mimic the equivalent diameter properties on OpticalPhase inheritants.

(Convolvable).show and (Convolvable).show_fourier behave the same way as plot2d methods found throughout prysm, except there are xlim and ylim parameters, which may be either single values, taken to be symmetric axis limits, or length-2 iterables of lower and upper limits.

Finally, Convolvable objects may be initialized from images,

>>> c = Convolvable.from_file(path_to_your_image, scale=1)  # plate scale in um

and written out as 8-bit images,

>>> p = 'foo.png'  # or jpg, any format imageio can handle
>>> c.save(save_path)

In practical use, one will generally only use the conv, from_file, and save methods with any degree of regularity. The complete API documentation is below. Attention should be paid to the docstring of conv, as it describes some of the details associated with convolutions in prysm, their accuracy, and when they are used.

class prysm.convolution.Convolvable(data, unit_x, unit_y, has_analytic_ft=False)[source]¶

Bases: object

A base class for convolvable objects to inherit from.

data : numpy.ndarray: numerical representation of object
has_analytic_ft : bool: whether this convolvable has an analytical Fourier transform
sample_spacing : float: center to center spacing of samples
unit_x : numpy.ndarray: x-axis unit
unit_y : numpy.ndarray: y-axis unit

conv(other)[source]¶

Convolves this convolvable with another.

other : Convolvable: A convolvable object

Convolvable: a convolvable that lacks an analytical fourier transform

The algoithm works according to the following cases:

Both self and other have analytical fourier transforms: - The analytic forms will be used to compute the output directly. - The output sample spacing will be the finer of the two inputs. - The output window will cover the same extent as the “wider”

input. If this window is not an integer number of samples wide, it will be enlarged symmetrically such that it is. This may mean the output array is not of the same size as either input.
- An input which contains a sample at (0,0) may not produce an output with a sample at (0,0) if the input samplings are not favorable. To ensure this does not happen confirm that the inputs are computed over identical grids containing 0 to begin with.
One of self and other have analytical fourier transforms: - The input which does NOT have an analytical fourier transform

will define the output grid.
- The available analytic FT will be used to do the convolution in Fourier space.
Neither input has an analytic fourier transform: 3.1, the two convolvables have the same sample spacing to within
a numerical precision of 0.1 nm:
- the fourier transform of both will be taken. If one has fewer samples, it will be upsampled in Fourier space
3.2, the two convolvables have different sample spacing:
- The fourier transform of both inputs will be taken. It is assumed that the more coarsely sampled signal is Nyquist sampled or better, and thus acts as a low-pass filter; the more finaly sampled input will be interpolated onto the same grid as the more coarsely sampled input. The higher frequency energy would be eliminated by multiplication with the Fourier spectrum of the more coarsely sampled input anyway.

The subroutines have the following properties with regard to accuracy:

Computes a perfect numerical representation of the continuous output, provided the output grid is capable of Nyquist sampling the result.
If the input that does not have an analytic FT is unaliased, computes a perfect numerical representation of the continuous output. If it does not, the input aliasing limits the output.
Accuracy of computation is dependent on how much energy is present at nyquist in the worse-sampled input; if this input is worse than Nyquist sampled, then the result will not be correct.

deconv(other, balance=1000, reg=None, is_real=True, clip=False, postnormalize=True)[source]¶

Perform the deconvolution of this convolvable object by another.

other : Convolvable: another convolvable object, used as the PSF in a Wiener deconvolution
balance : float, optional: regularization parameter; passed through to skimage
reg : numpy.ndarray, optional: regularization operator, passed through to skimage
is_real : bool, optional: True if self and other are both real
clip : bool, optional: clips self and other into (0,1)
postnormalize : bool, optional: normalize the result such that it falls in [0,1]

Convolvable: a new Convolable object

See skimage: http://scikit-image.org/docs/dev/api/skimage.restoration.html#skimage.restoration.wiener

static from_file(path, scale)[source]¶

Read a monochrome 8 bit per pixel file into a new Image instance.

path : string: path to a file
scale : float: pixel scale, in microns

Convolvable: a new image object

plot_slice_xy(axlim=20, lw=3, fig=None, ax=None)[source]¶

Create a plot of slices through the X and Y axes of the PSF.

axlim : float or int, optional: axis limits, in microns
lw : float, optional: linewidth provided directly to matplotlib
fig : matplotlib.figure.Figure, optional: Figure to draw plot in
ax : matplotlib.axes.Axis: Axis to draw plot in

fig : matplotlib.figure.Figure, optional: Figure containing the plot
ax : matplotlib.axes.Axis, optional: Axis containing the plot

save(path, nbits=8)[source]¶

Write the image to a png, jpg, tiff, etc.

path : string: path to write the image to
nbits : int: number of bits in the output image

shape¶

show(xlim=None, ylim=None, interp_method=None, power=1, show_colorbar=True, fig=None, ax=None)[source]¶

Display the image.

xlim : iterable, optional: x axis limits
ylim : iterable,: y axis limits
interp_method : string: interpolation technique used in display
power : float: inverse of power to stretch image by. E.g. power=2 will plot img ** (1/2)
show_colorbar : bool: whether to show the colorbar or not.
fig : matplotlib.figure.Figure, optional:: Figure containing the plot
ax : matplotlib.axes.Axis, optional:: Axis containing the plot

fig : matplotlib.figure.Figure, optional:: Figure containing the plot
ax : matplotlib.axes.Axis, optional:: Axis containing the plot

show_fourier(freq_x=None, freq_y=None, interp_method='lanczos', fig=None, ax=None)[source]¶

Display the fourier transform of the image.

interp_method : string: method used to interpolate the data for display.
freq_x : iterable: x frequencies to use for convolvable with analytical FT and no data
freq_y : iterable: y frequencies to use for convolvable with analytic FT and no data
fig : matplotlib.figure.Figure: Figure containing the plot
ax : matplotlib.axes.Axis: Axis containing the plot

fig : matplotlib.figure.Figure: Figure containing the plot
ax : matplotlib.axes.Axis: Axis containing the plot

freq_x and freq_y are unused when the convolvable has a .data field.

slice_x¶

Retrieve a slice through the x axis of the PSF.

self.unit_x : numpy.ndarray: ordinate data
self.data : numpy.ndarray: coordinate data

slice_y¶

Retrieve a slice through the y axis of the PSF.

self.unit_y : numpy.ndarray: ordinate data
self.data : numpy.ndarray: coordinate data

support¶

support_x¶

support_y¶

Convolvables¶

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