Here’s a Python function to accompany the previous post. It’s not maximally efficient, but should make sense…

from scipy import stats
def gaussian_pixel(minxy, maxxy, sigma, meanxy=(0.,0.), norm=None):
    """Return the value of a pixel sampling a 2D Gaussian,
    normalized such that the area under the Gaussian is 1
    (default) or such that the peak is given by norm."""
    x1, y1 = minxy
    x2, y2 = maxxy
    x0, y0 = meanxy
    if norm is None:
        norm = 1. / 2 / math.pi / sigma ** 2
    return norm * 2 * math.pi * sigma ** 2 / (x2 - x1) / (y2 - y1) * (
        (1 - stats.erfc((x2 - x0) / math.sqrt(2) / sigma)) / 2.
        - (1 - stats.erfc((x1 - x0) / math.sqrt(2) / sigma)) / 2.) * (
        (1 - stats.erfc((y2 - y0) / math.sqrt(2) / sigma)) / 2.
        - (1 - stats.erfc((y1 - y0) / math.sqrt(2) / sigma)) / 2.)

If the pixels are larger than infinitesimal, as is generally the case, then the maximum value of the pixelated PRF will be the average value over the pixel, which will be less than 1.

For example, if the PRF is a two-dimensional Gaussian, centred on , with standard deviation , then the value in a pixel with and will be

Ugh. Let’s make that simpler. For a PRF centred on , and a pixel , this is

As an example, the fairly-Gaussian beam for the Herschel Space Observatory SPIRE instrument has an FWHM of around 18″, which corresponds to a standard deviation of around . If we make a Jy/beam map with pixel size 6″, then the peak value for a 1 Jy point source in the centre of a pixel will be

Easy.

First, find , which is the point response function (PRF), telling you what a point source of flux 1 will look like in the map. This may be normalized so that the peak is 1 (if your map is in Jy/beam or similar), or so that (if your map is in Jy/pixel or similar). If your map is in MJy/sr … well, figure it out and add a comment below. Basically, if you normalize your PRF correctly, you won’t need to worry about the map units in what follows. Phew.

Now the measured value of each pixel around the point source, , will be

Now the badness of the fit is measured by the , which is given by

Now just do this for each pixel in the map (corresponding to a point source centred on each pixel) and you’re done.

Worked example. is 0.5, 1.0 and 0.5, for three adjacent pixels (you’ll have realised that the map is in Jy/beam or similar), and is 1, 2 and 1 Jy/beam, for three adjacent pixels, with the same (tiny!) value of for each pixel (in this case, we can ignore the value of in what follows). So the flux at the central pixel is estimated to be

This is an example of a matched filter (I haven’t read the page, but hopefully including the link will make me look clever). And, given that point sources are under no particular obligation to align themselves with the centres of the pixels of your map, can easily be re-estimated for a source with a certain offset from the pixel centre.

Here’s the Python code

import numpy as np
from matplotlib import pyplot as plt
 
class FlickerImage(object):
    def __init__(self, im, err):
        self.im = im.copy()
        self.err = err.copy()
        finite = np.isfinite(self.im + self.err)
        self.vmin = (self.im - 2 * self.err)[finite].min()
        self.vmax = (self.im + 2 * self.err)[finite].max()
        self.im[np.invert(finite)] = self.vmax
        self.err[np.invert(finite)] = 0
    def flicker(self):
        fg = plt.imshow(np.zeros(self.im.shape),
                        interpolation='nearest',
                        vmin=self.vmin,
                        vmax=self.vmax)
        while True:
            ran = np.random.normal(size=im.shape)
            fg.set_data(im + err * ran)
            plt.draw()

And here’s an example script:

import pyfits
f = pyfits.open('file.fits')
im = f["IMAGE"].data
err = f["ERROR"].data
flicker_image = FlickerImage(im, err)
flicker_image.flicker()