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NAME

explain_detect -- description of detect tasks and algorithms

USAGE

This document describes tasks and algorithms used in the detect package.

DESCRIPTION


Summary
-------

This document provides a brief description of the point source
detection tasks currently available in the DETECT package, and offers
suggestions on their proper use and the interpretation of their
results.


I. Overview
-----------

The DETECT package provides a simple set of tools for detecting
discrete sources in x-ray images. The package uses techniques similar
to those used in the standard processing of Einstein Observatory IPC
and HRI data and in the Level 1 processing of ROSAT HRI data.  The
current DETECT package is intended to provide a basic source detection
capability in regions of x-ray images where the point response
function and the exposure change only slowly.  DETECT takes advantage
of existing IRAF or PROS tasks to provide a set of simple tools which
can be combined flexibly by the user to meet different source
detection needs. An IRAF task (ldetect) is also provided to execute
DETECT tasks in sequence.

New features in this version (PROS 2.2) of DETECT include:

   Revised algorithm for calculating signal-to-noise (see section II
   below) in SNRMAP.

   New task LMATCHSRC, that matches sources between and within different 
   detect cell sizes.

   Added 6 & 9 arc-second detect cell sizes for ROSAT HRI and Eintein HRI.

   Updated the soft/hard and broad band definitions for the ROSAT PSPC.

In the remainder of this document we discuss in more detail the basic
algorithms employed by the current DETECT tasks.  We offer some sug-
gestions for choosing parameters for the tasks and interpreting
results.  In the final section we discuss limitations of the current
implementation.


II. Basic Algorithms
--------------------

DETECT scans an x-ray image with a square "detect cell" of side d,
placed at locations separated by d/3 in each dimension. At each
location, the signal-to-noise ratio of a source assumed to be centered
in the detect cell is calculated, with background estimated from a
frame of width d/3 immediately surrounding the detect cell.  Locations
containing local peaks in the signal-to-noise ratio above a threshold
are identified, and maximum likelihood positions for sources
corresponding to each local peak are determined. A final
signal-to-noise ratio in a detect cell centered on each final position
is calculated, and all locations with signal-to-noise ratios greater
than the snr threshold (bepos.snr_thresh) are labeled as sources.


    There are 5 basic tasks in the current DETECT package:

        CELLMAP -       prepares x-ray qp file for DETECT. The qp file
                        is blocked into an image of resolution d/3. An
                        X-ray PI or PHA filter along with an image section 
                        to keep the central 512x512 image are also applied.

        SNRMAP  -       computes the signal-to-noise ratio (snr) map
                        for the image prepared by CELLMAP. 

        LPEAKS  -       scans the snr map line by line for local peaks
                        above a user-specified threshold.

        BKDEN   -       computes the mean background density in
                        counts/sq. arcmin. in a user-defined region of
                        the image prepared by CELLMAP. Image pixels with
                        values which deviate from the mean by more than
                        a user-defined threshold are rejected.

        BEPOS   -       determines the maximum likelihood position for
                        each peak identified by LPEAKS, using the original
                        qp file. Signal-to-noise ratios in detect cells
                        centered on the final positions are computed and
                        positions with snr's greater than a user-specified
                        threshold are identified as sources.

In addition, the following auxiliary tasks are provided:

        LDETECT  -      runs the above 5 tasks in sequence, for a single
                        detect cell size and a single snr threshold.

        DETMKREG -      makes a region file with LPEAKS and BEPOS positions.
                        The file, when overlayed onto SAOIMAGE is a visual
                        tool indicating where source detection in the image 
                        was determined, along with the cellsize as indicated 
                        by the size of the region.

        LBMAP    -      generates background and background variance maps
                        at the same resolution as in CELLMAP. These maps
                        are for illustrative purposes only and are not
                        used in other DETECT tasks.  The equations used to
                        build the background map use the same algorithms as 
                        implemented in bepos.


    More detailed help on running each task may be found in the
    individual task help files. We describe below the important
    details in SNRMAP, BEPOS, and BKDEN.


SNRMAP Algorithm
----------------

This task borrows heavily from the algebra derived for the Einstein
IPC Rev. 1 LDETECT programs, as described in Harnden et al. (1984, SAO
Special Report 393). The input x-ray image is that prepared in
CELLMAP. Defining

        S       = Expected total source counts,
        B       = Expected background counts in detect cell,
        C       = Total counts in square detect cell of size d
                  (a 3x3 box in image of resolution d/3),
        T       = total counts in detect cell plus surrounding
                  background frame (a 5x5 box at d/3 resolution),
        Q       = T - C,
        alpha   = Integral of point response function over detect cell,
        beta    = Integral of point response function over detect
                  cell plus surrounding background frame,

    we have

        C       = alpha * S + B
        T       = beta  * S + (25/9) * B.


    Solving for B,

        B  = [(alpha-beta)*C+alpha*Q]/[(25/9)*alpha-beta],	(1)


    and

        var(B) = [(alpha-beta)**2*var(C)+alpha**2*var(Q)] /	(2)
                 [(25/9)*alpha-beta]**2.

    Solving for S,

         S  = [(16/9)*C - Q]/[(25/9)*alpha-beta], 		(3)

    and

         snr = S/sqrt[var(S)]					(4a) 

For bright sources, C and Q may be sufficiently large (> ~40) that the
Gaussian limits var(C)=(C) and var(Q)=(Q) may be used in determining
var(S). However, for fainter sources, Poisson rather than Gaussian
statistics apply, and the above definitions for var(C) and var(Q) are 
underestimates, leading to an overestimate of the snr.  This is 
particularly important for Q, which is likely to contain only a few 
(mostly background) counts for faint sources in short exposures.  To 
properly deal with small number statistics requires a more sophisticated 
treatment than is possible with our simple snr algorithm. We attempt to 
compensate for the deficiency, as was done in the Einstein IPC reprocessing 
system, by defining

	var(Q) = QP

where

	QP = (SQRT(Q) + 1)**2		Q > 40

	QP = [q84 - Q]**2		Q<= 40

and q84 is the 84.13% confidence level quantile for a Poisson distribution
with a mean of Q. Values of q84 are tabulated in, for example, 
Regner, V.H. 1951, Phys. Rev., 84, 161L. Thus,
 
	snr = [(16/9)*C - Q]/sqrt[(16/9)**2*C + QP] 		(4b)


As can be seen above, calculation of S and B depend on the values of
alpha and beta, which may be poorly modeled. However, determination
of the signal-to-noise is independent of these quantities.
SNRMAP uses the IRAF IMAGES CONVOLVE task to generate the C and T
maps, by convolving the input image with 3x3 and 5x5 boxcar functions
of amplitude 1.0.  The B, var(B), and snr maps are calculated using
image arithmetic with the PROS IMCALC task.


BEPOS Algorithm
---------------

BEPOS uses the likelihood ratio parameter estimation technique (Cash,
1979, Ap. J., v. 228, 939) to determine the best estimate and 90%
confidence uncertainty in source position. BEPOS uses code similar to
that used in the ROSAT Level 1 HRI processing system.  A Gaussian
point response function is assumed. At present, the sigma of the point
may be input by the user if known. Otherwise, the value is estimated
from the position of the source in the image, using the relations
discussed in section III. Currently, the off-axis angle is calculated
from the image center.

Once the value of sigma has been determined, the maximum likelihood
position is determined from events in a circular region of radius
ROPT*sigma, where ROPT is a user-defined parameter (OPT_RADIUS_SIGMA).
This region is calculated once for each input position (output of lpeaks).

An initial estimate of the background is required for the position
calculation. BEPOS uses a constant (i.e. field average) value for
this quantity, as determined from BKDEN.

BEPOS also estimates the position uncertainty by incrementing the x,y
positions about the maximum likelihood position until the C-statistic
(-2*ln(likelihood ratio) has increased by the appropriate amount for
the desired confidence level.

BEPOS calculates the signal-to-noise ratio at the final source position 
(snr), using the counts in a detect cell centered on that position and 
the counts in the background frame surrounding that cell.  The 
signal-to-noise ratio is defined in the same manner as in SNRMAP, 
i.e., as in eq. 4b above.

Sources with snr below a user-defined threshold are rejected.  A figure 
of merit is also calculated (pconf and fsflag), to describe how well the 
photon distribution matches that expected for the point response function.  
Sources which do not match the point response function well are identified 
in the output.

Finally, BEPOS estimates the total source counts (srccnts) and background 
counts (bkcnts) in the detect cell, using eqns. 1 and 3 above. The 
quantities alpha and beta, previously required as input, are now calculated 
by BEPOS from the PRF sigma, under the assumption of a gaussian PRF.


BKDEN Algorithm
---------------

BKDEN calculates the average background in a user-specified region of
the image prepared by CELLMAP. A simple average of the values in all
image pixels in the region is calculated and pixels with values
greater than a threshold are rejected. The process is then repeated
for all unrejected pixels until either no further pixels are rejected
or a maximum number of iterations is reached. The threshold is
determined by calculating the probability, P, of a pixel having more
that T counts, assuming the values are distributed according to
Poisson statistics with a mean given by the average for the iteration.
The product of P and N, the number of pixels in the region, will be
required to be less than F, the number of acceptable fluctuations
greater than T in the region. For the full field, F defaults to 1.

If M is the mean pixel value in a given iteration, T is determined by
the relation

                m=T
       N * [1 - SUM{exp(-M)*M**m/m!}] < F,                            (5)
                m=0

    or
                m=T
                SUM{M**m/m!} > exp(M) * (N-F)/N.                      (6)
                m=0


III. Choosing Detect Cell Sizes
-------------------------------

Experience with the Einstein Observatory detection software indicates
that the optimum cell size is approximately 3.6 times the sigma of the
point response function (Harnden, 1984).

The ROSAT instrument+mirror point response function varies strongly
with THETA, the distance from the optical axis (off-axis angle). In
addition, the PSPC instrument point response function varies with
energy. Therefore, different detect cell sizes may be appropriate for
different regions or energy bands. Although DETECT will, by default,
operate on the entire image, the user is advised to run DETECT
separately with different detect cell sizes for different annular
regions or energy bands (if appropriate), or, at the least, to
consider the results as valid only for the region or energy band
appropriate to the detect cell used.

We have used the following parameters to describe the ROSAT point
response function vs. energy, E, and off-axis angle, THETA (obtained
in late '91 from Martin Zombeck [SAO] and Gunther Hasinger [MPE]).
For both instruments, we assume a single gaussian point response
function. More accurate models have since become available (see ROSAT
Status Reports #22 and 23) but at present these models only apply to
sources on-axis. The adequacy of the single gaussian model will be
discussed further in sections VI and VII below.

ROSAT HRI:
----------
	
    FWHM = 2.355*sqrt[sigma(HRI)**2 + sigma(aspect)**2 +          (7)
               (sigma(mirror)+0.0205*THETA**2.349)**2]

       	sigma(HRI)    = 0.74"
       	sigma(aspect) = 1.0"
       	sigma(mirror) = 1.3"


ROSAT PSPC:
-----------

       	FWHM(PSPC)    = sqrt(409.65/E + 69.28*E**2.88 + 66.29)
       	FWHM(mirror)  = 0.29*THETA**1.74
       	FWHM(aspect)  = 2.355*1.5

    Total FWHM = sqrt(FWHM(PSPC)**2 + FWHM(mirror)**2 + FWHM(aspect)**2)  (8)


Here, THETA is in arc-minutes, E in keV, and the result in arc-seconds.

We use these functions to calculate an average point response function
sigma for various annular regions. For the PSPC, we use a Crab-like
spectrum to calculate the average energy in the standard PSPC energy
bands, and use that to calculate FWHM(PSPC). The results, shown in the
following table, should guide the user on the appropriate regions and
energy bands for various detect cell sizes. The detect cell data for
the Einstein HRI and IPC are shown for comparison. We have also
included ALPHA and BETA, the fractions of the point response functions
contained in the detect cell and in the cell and surrounding
background frame. The PSPC energy bands are 0.1-0.4 keV (soft), 0.4 -
2.4 keV (hard), and 0.1 - 2.4 keV (broad).


    ROSAT HRI:
    ==========

    cell    prfsig  region                  ALPHA           BETA
    ----------------------------------------------------------------
    9"x9"   3.3"    0<THETA<10'             0.684           0.954
    ----------------------------------------------------------------
    12      3.3     0<THETA<10              0.867           0.995
    ----------------------------------------------------------------
    24      7.6     9<THETA<13.5            0.783           0.983
    ----------------------------------------------------------------
    36      12.0    12.5<THETA<16.0         0.751           0.975
    ----------------------------------------------------------------
    48      16.0    15.0<THETA<18.0         0.753           0.976
    ----------------------------------------------------------------

    ROSAT PSPC:
    ===========

    cell    band    prfsig  region          ALPHA           BETA
    ----------------------------------------------------------------
    45      s       14.9"   0<THETA<10'     0.755           0.976
    ----------------------------------------------------------------
    30      h/b     10.4    0<THETA<10      0.724           0.968
    ----------------------------------------------------------------
    60      s       20.6    10<THETA<20     0.731           0.970
    ----------------------------------------------------------------
    60      h/b     17.4    10<THETA<20     0.838           0.992
    ----------------------------------------------------------------
    120     s       36.8    20<THETA<30     0.805           0.987
    ----------------------------------------------------------------
    120     h/b     35.2    20<THETA<30     0.831           0.991


    ----------------------------------------------------------------

    Einstein HRI:
    =============

    cell    region          ALPHA           BETA
    ------------------------------------------------
    12      0<THETA<5'      0.623           0.745
    ------------------------------------------------
    24      0<THETA<5       0.774           0.853
    ------------------------------------------------
    48      0<THETA<5       0.882           0.956
    ------------------------------------------------

    Einstein IPC:
    =============

    cell    pi bands        ALPHA           BETA
    ------------------------------------------------
    144     h/b             0.883           1.0
    ------------------------------------------------
    240     s               0.883           1.0
    ------------------------------------------------


IV. Setting DETECT thresholds
-----------------------------

The user must set signal-to-noise thresholds in LPEAKS and BEPOS.  We
recommend that thresholds be set to the values between 2.75 and 3.0
sigma for LPEAKS (default = 2.75) and 3 sigma for BEPOS (default=3.0).
The threshold in LPEAKS is defaulted below that in BEPOS, since LPEAKS 
sources of lesser significance may yield a more significant source when 
the detect cell is centered on the final source position.  Our experience 
indicates that an LPEAKS threshold of 2.75 will include most sources with 
a final BEPOS signal-to-noise ratio of 3.5 or more.  (The user may, of 
course, set a lower LPEAKS threshold, but doing so will increase run-time.)

Without applicable simulations, the detection efficiency at threshold
is poorly determined.  Thresholds for Einstein and ROSAT Level-1
processing were set on the basis of simulations that allowed on
average one false source per field, but because of differences in
instruments and algorithms, these results are not universally
applicable.  In view of this, the user may wish to explore the
significance of some sources (especially faint ones) with
circular/annular regions, as opposed to the square cells used by
DETECT, using a tool such as the xray.ximage task imcnts.


V. Choosing the Background Region
---------------------------------

In general, the ROSAT background changes only slowly, so BKDEN can be
run in any region. However, in observations which include regions of
diffuse emission, BKDEN should be run in the same region in which the
user wishes to detect sources, to obtain background values consistent
with the "local" background values used in SNRMAP. The threshold
number of fluctuations, used to reject source pixels, should be set to
1, at most.


VI. Estimating Source Intensities
---------------------------------

BEPOS uses eq. 3 above to estimate total source counts for each
source.  To do this, the program must determine alpha and beta, the
integrals of the PRF over the detect cell and the cell plus background
frame.  Previously, these quantities were required as user input, but
in the current release, they are calculated, assuming a gaussian PRF
with a sigma which is either input by the user, or calculated from
eqns. 7 or 8 above.

At least for the HRI (both ROSAT and Einstein), this assumption may
lead to significant errors in computing total source counts. We have
examined ROSAT HRI calibration sequences for AR Lac at various off-
axis angles. We find that the PRF may be roughly modeled by a core
gaussian with sigma of 2.83", independent of off-axis angle, and
(presumably gaussian) wings, which increase in size and amplitude with
off-axis angle. The PRF model in eq. 7 includes contributions from
both components.  However, we find that neither this model nor a core
gaussian alone can accurately reproduce alpha and beta for all detect
cells and off-axis angles, as determined empirically from the AR Lac
data.  Therefore, we offer the following tables of multiplicative
correction factors to apply to total source counts as determined by
BEPOS for the ROSAT HRI.

If BEPOS is run with a constant PRF sigma of 2.83" (core gaussian):


			Detect Cell
                        -----------

        Theta       9"     12"     24"     36"     48"
        -----     ----    ----    ----    ----    ----
        2'        1.06    1.14    1.10    1.09    1.08
        3'        1.05    1.11    1.10    1.09    1.09
        6'        1.00    1.11    1.10    1.09    1.08
        8'        1.78    1.48    1.12    1.10    1.09
        10'       2.58    2.46    1.24    1.09    1.08
        12'       3.19    3.14    1.76    1.24    1.12
        14'       5.59    5.33    3.00    1.67    1.23
        16'       7.50    7.51    4.44    2.69    1.58
        18'      16.11   21.24   12.18    4.87    2.61


If BEPOS calculates the PRF sigma as a function of off-axis angle:


			Detect Cell
			-----------
        Theta       9"     12"     24"     36"     48"
        -----     ----    ----    ----    ----    ----
        2'        1.55    1.27    1.10    1.09    1.08
        3'        1.43    1.23    1.10    1.09    1.09
        6'        0.95    1.08    1.10    1.09    1.08
        8'        0.84    0.95    1.10    1.10    1.09
        10'       0.45    0.73    1.07    1.08    1.08
        12'       0.18    0.35    1.02    1.11    1.10
        14'       0.10    0.21    0.90    1.12    1.10
        16'       0.04    0.11    0.61    1.08    1.06
        18'       0.04    0.11    0.74    1.04    1.14


The user should multiply the value of source counts determined by
BEPOS by the value in the appropriate table for the detect cell and
off-axis angle of interest. Columns of the tables may be interpolated
for other off-axis angles.

A similar table is provided for the Einstein HRI. In this case, the
default PRF sigma calculated by BEPOS is 4.25" at all off-axis angles
(chosen to yield approximately correct source intensities for the 12"
detect cell).

			Detect Cell
                        -----------
            Theta      12"     24"     48"
            -----     ----    ----    ----
            All       1.02    1.35    1.19


VII. Limitations
----------------

i. Assumption of a Gaussian PRF 
------------------------------- 

As mentioned above, this approximation may only be valid for the ROSAT
HRI on-axis. Although the approximation is not expected to affect the
final source position significantly, it may affect the calculated
uncertainty or the determination of how well the source distribution
matches the point response function. In the current implementation, it
also requires ad hoc procedures, as discussed in the previous section,
to determine source intensities. In particular, the description of the
ROSAT HRI PRF vs. off-axis angle suffers from this problem. The model
described in eq. 7 accurately describes the 50 percent power radius
vs. off-axis angle. Under the assumption of a gaussian PRF, this model
may be recast as PRF sigma vs. off-axis angle. However, as discussed
above, the integrals of that model PRF (e.g. alpha or beta) do not
agree with the integrals of the true PRF for that off-axis angle.

Extensive analysis of the PSPC PRF on-axis indicates that it too
contains more than a single gaussian component, although at 1 keV,
approximately 85% of the photons may be described by a single gaussian
(see ROSAT Status Report #23, Newsletter 7, and Office for Guest
Investigator Programs calibration memo CAL/ROS/92-001). However, the
applicability of a single gaussian model for the PSPC PRF off-axis and
the implication for the calculation of source intensities remains to
be fully explored.

ii. Determination of Off-axis Angle
-----------------------------------

At present, the off-axis angle is defined to be the angular separation
of the source from the image center. Boresight offsets are not
accounted for and, if large, may lead to significant errors in PRF
sigma, especially for sources far off-axis, where the PRF varies
rapidly with off-axis angle.

iii. Resolution of Nearby Sources
---------------------------------

A task (lmatchsrc) is provided to match nearby sources from either the 
same or different LDETECT runs using the same QPOE file.  Detection
efficiency as a functions of angular separation for nearby sources
will be examined as part of a future simulation study.

iv. Contamination from Near-by Sources
--------------------------------------

No correction is made in BEPOS for such contamination. If accurate
source intensities are desired in crowded regions, the user is advised
to calculate them separately, using a source region large enough to
include all source counts despite uncertainties in the point response
function, and to exclude regions where source contamination exists.
The IMCNTS task (xray.ximages) is recommended.

v. Detection in Regions of Rapidly Varying Exposure
---------------------------------------------------

No attempt is made to handle such cases, and users are advised to
treat sources detected near instrument edges or shadows due to rib
support structures with caution.

vi. Determination of Background Counts in Detect Cell
-----------------------------------------------------

This quantity is determined using eq. 1. However, it also suffers from
the difficulties in calculating alpha and beta assuming a gaussian
PRF. It should therefore be considered unreliable in the current
implementation.  


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