This routine removes a fringe pattern with repetitive characteristics from a flat field.
A typical flat field array from a VLT observation using the ISAAC instrument is shown in Figure 1. In the upper frame a periodic fringe pattern is clearly visible. It can also be seen that the periodicity is not constant throughout the array but it changes visibly, for example, in the upper right quadrant.
This change in periodicity gets confirmed in the lower frame of figure 1 where only shortwards of pixel ~400 averaging brings up the signal-to-noise, and longwards of that same pixel the average gets flatten because the fringe period changes in that region.
Also, the smooth change in period inmediately explains while you
cannot use methods like Fast Fourier Transform (FFT) where only one
frequency per row can be found.
Figure 1: Upper frame is ISAAC's flat field image obtained after
subtraction of dark frame with same exposure time. Lower frame is
the average of every row.
Template id 'ISAACSW_spec_cal_NightCalib'. Integration time: '4
seg'. FITS file 'flat.fits' is supplied with defringeflat package.
A more detailed, quantitative introduction to the wavelet transform is given by C. Torrence and G. Compo, only a brief introduction is presented below.
In short, the wavelet transform calculates the frequency spectrum in a region around each point of a one-dimensional array, to give the spectrum's variation with location in the array. The size of the region scales with period, which is the main difference from the windowed Fourier transform.
Figure 2 shows an example of a periodic function (a sine function whose period changes linearly from 70 to 40 pixels) and its wavelet transform. The lower frame shows the amplitude of the wavelet transform. The region in the shaded area is inside the "cone of influence" of the edges of the data, which signals that these values cannot be trusted.
A trace in the transform that peaks at the expected frequency is
clearly visible. However, this trace has a shape with some
non-zero width. As can be seen in cross-section (Figure 3), the shape is not quite a Gaussian, but it
gives a fair approximation.
Figure 2. Upper frame: sinusoidal function whose period and
amplitude change from left to right. Lower frame: amplitude of
a Morlet wavelet transform of the spectrum.
Figure 3. Vertical cross section of the wavelet transform's
amplitude (solid line) and the best-fit Gaussian (dotted line).
The routine defringeflat uses the following procedure to compute the fringe pattern of a flat field.
To improve the signal-to-noise ratio and remove "wild" pixels, a 1-D median average of a specified number of rows is computed for each of the rows (Figure 4).
For each of the filtered rows, the slope is subtracted through a linear fit inside the specified trim limits, every point outside the trim limits is replaced with zeroes, a wavelet transform is performed according to specified wavelet parameters, and then possibly interpolated (Figure 5).
We then identify the trace of the wavelet amplitudes that corresponds to the fringe system. A specified function (see next section: Trace fitting methods) is fit to the trace profile using a user-identified fringe period (at some reference image coordinates) as the initial guess (Figure 6). Then we follow the local maximum along the columns. Repeating this procedure for each row, we obtain a fitted set of parameters (e.g., center, width, and amplitude of a Gaussian) for each pixel in the array.
If specified, each parameter array is compared to a median-filtered version of itself to identify and replace outliers, and then mean-smoothed (Figure 7).
Then, the wavelet transform is reconstructed from the
(smoothed) parameters obtaining the fringe's wavelet transform(Figure 8). Finally, an inverse wavelet transform is
performed to obtain the computed fringe pattern (Figure
9), which is then removed from the flat(Figure
10).
Figure 4. Flat field in Fig. 1 after being median averaged in the
vertical direction. In this particular example, each row is the median
average of the 40 rows surrounding it.
Figure 5. Upper frame shows middle row of Fig. 4. Middle frame shows the
amplitude of the wavelet transform in the relevant periods. Lower
frame: Same as middle frame but interpolated in period to 10 times the
original period resolution.
Figure 6. Cross section of wavelet transform's middle column, original
points are always represented by small crosses. Upper frame shows
original points and the interpolated profile (solid line). Middle and
lower frames show the profiles fitted (solid line) when using methods
'Gaussian' and 'true shape', respectively.
Figure 7. Parameter smoothing. From left to right the parameters
are amplitude, center, width, and background of a Gaussian fit of
the trace. From top to bottom, they are shown as raw
fitted, patched (bad pixels replaced), and smoothed parameters.
Figure 8. Fringe's wavelet. Amplitude of the reconstructed fringe
trace.
Figure 9. Fringe obtained from inverse transform of Figure 8.
Figure 10. Flat field from Figure 1 after being cleaned by
defringeflat.
To run this example you would need the wavelet routines from C. Torrence and G. Compo and the astrolib routines from NASA.
IDL> flat = readfits('example/flat.fits', h)
IDL> func = 'gauss'
IDL> clean = defringeflat(flat, 40, binw=40, trim=[10,145,950,950], $
osamp=10, /smooth, fitname=func, fringe=fringe)
IDL> tvscl, flat
IDL> tvscl, fringe
IDL> tvscl, clean
It is difficult to estimate the error given that the real shape of the fringes is unknown. However, the effectiveness of this method can be seen by how much does the fringe pattern disappear.
Let's take a look at the average of the rows from the VLT flat field. As it was discussed regarding Fig. 1, in the first third of the columns the period of the fringes remain constant, and hence the average just increases the signal-to-noise ratio.
Figure 11 shows columns 30 to 250 of the average
of all rows before and after having the fringe pattern removed by
defringeflat routine using 'true shape' and Gaussian fitting
functions. It is visible to the naked eye as well as in their power
spectrum that the fringe pattern was fully removed. True shape seems to
remove more throughfully, but there is more risk that it is also
removing real flat fields artifacts.
Figure 11: Fringe removal efficiency. Upper frame is original average of
rows. Upper left: Subsample of the first 400 columns of the plot in
Figure 1. Middle left: Same subsample after Gaussian fringe
cleaning. Lower left: Same subsample after 'true shape' fringe
cleaning. Right panels: FFT power spectra of data in corresponding left
panels.