Adaptive kernel smoothing
Adaptive kernel smoothing is a highly effective
technique for transforming the Poisson spatial point process of an ACIS
image into a scalar real field that represents more clearly the true distribution
of X-ray emission in the sky. Its advantage over standard Gaussian
or boxcar smoothing is that the width of the Gaussian kernel is self-adjusted
across the field to match the local density of events. The result
is a smoothed image that shows simultaneously point sources and diffuse
emission without smearing the former into the latter. See, e.g., Brian
Silverman's monograph "Nonparametric Density Estimation" (Chapman &
Hall, c. 1985) for a statistical discussion.
ACIS adaptive smoothing is achieved with a code developed by Harald Ebeling (Hawaii), and is implemented both within CIAO (asmooth) and
as an IDL widget (csmooth). The method is complicated and generates a
variety of ancillary files; users are strongly encouraged to read the CIAO manual
entry. The formal reference is H. Ebeling et
al. (2000, MNRAS).
An obvious complication with application to ACIS-I
images is nonuniform exposure across the field, particularly the chip gaps
which become smoothed along with the true photon distribution. The exposure
map must thus also be appropriately smoothed. ACIS scientists, in
communication with Harald, developed the following procedure:
-
Run adaptive smoothing on the ACIS-I image:
csmooth infile=acis.evt.fits outfile=acis.smooth
outsigfile=acis.sig outsclfile=acis.kernel sclmode=compute conmeth=slide
-
Smooth the exposure map with same spatial distribution
of kernel widths developed for the image map (i.e., the sclmap file produced
above is the input here):
csmooth infile=expmap.i0123 outfile=expmap.i0123.smooth
outsigfile=expmap.i0123.sig outsclfile=expmap.i0123.kernel sclmode=user
conmeth=slide sclmap=acis.kernel
-
Divide the two smoothed maps. CIAO dmimgcalc will preserve headers and ancillary
files:
dmimgcalc infile=acis.smooth infile2=expmap.i0123.smooth outfile=acis.smooth.norm
operation=div weight=1.0 weight2=1.0 verbose=1
clobber=yes
However, a variety of potential problems have been noted, for which more
investigation is needed:
Recall that the units of "exposure maps" may not be in seconds (see the
exposure Recipe page).
Linear features associated with chip gaps are sometimes
seen. It is not clear whether these are due to misalignment of the
original images or imprecisions in asmooth.
Faint diffuse structures often appear in the smoothed
image. It is not clear whether these are due to: true sky X-ray structures;
permanent structures in the ACIS detector (e.g. quantum efficiency variations);
temporary structures (e.g. remnants of cosmic ray blooms); or artifacts
of the asmooth algorithm. It seems likely that some features do not
have an astrophysical origin, as they run parallel to chip axis near chip
edges.
To assess this, P. Broos performed a test of asmooth using a synthesized
random image of 100,000 Poisson deviates using a realistic ACIS exposure map
(i.e. with gaps). The resulting smoothed image showed bright features in the
four corners of the array with amplitude of several percent, suggesting that
algorithmic artifacts
can be present. In light of this result, H. Ebeling suggests that
asmooth be run with the sliding-cell rather than FFT convolution option,
as the FFT will always produce artifacts around the field edges/corners
due to applied boundary conditions. The sliding-cell algorithm
is bias-free but makes much greater demands on CPU time. But a single
test of the sliding-cell option on a random image also showed diffuse features.
Note 1: Running csmooth on a full ACIS image seems to produce zeros
in the kernel map (sclmap output file) which causes a failure in the code.
On can check for and remove zeros as follows using CIAO and IDL:
dmstat fullfield_4.img.kernel
im=readfits('fullfield_4.img.kernel',head)
s=where(im EQ 0)
tvscl,im<0.1
im[s] = 0.1
writefits, 'fullfield_4.img.kernel.nozero', im, head