- 13.1 Source Detection Using Wavelet Functions
- 13.2 Basic Algorithm
- 13.3 Source Pixel Identifications

- 13.4 Constructing the Source List

- 13.5 The Mexican Hat (MH) Function
- 13.6 Solutions to Integrals Involving the MH Function

- 13.7 Computation of Threshold Correlation Values
- 13.8 Normalized Background Map
- 13.9 References

13. wavdetect Theory

This chapter is a draft copy of the paper "A Wavelet-Based Algorithm
for the Spatial Analysis of Poisson Data'' by P.E. Freeman,
V. Kashyap, R. Rosner, D.Q. Lamb, *Ap. J. Suppl. Series*,
Vol. 138, p. 185, 2002. The published paper is available online via
`astro-ph`: `http://arXiv.org/abs/astro-ph/0108429` .

13.1 Source Detection Using Wavelet Functions

A wavelet is a member of a family of oscillatory scaleable functions which deviates from zero within a limited spatial regime, and has zero normalization. Perhaps the simplest example of such a function is a positively valued "top hat," with amplitude and width , flanked by two negatively valued troughs with total integrated area . (This in fact describes the Haar wavelet family.) Any function with zero normalization satisfying the property

(13.1) |

may be used as a wavelet function. is the

The importance of wavelet functions only began to be recognized in
early 1980s, when they were used in the analysis of seismic data
(Goupillaud *et al.* 1984). Wavelets are now used as tools in
atomic physics, the study of fractal structures, time series, sound,
and image analysis, *etc.* Slezak, Bijaoui, & Mars (1990) were
the first to apply wavelets in astronomy, applying them to galaxy
catalogue data to search for statistically significant clusters and
voids. The interested reader may find more information on wavelets in
Daubechies (1992) or Holschneider (1995).

Wavelet transforms, convolutions of an arbitrary analyzable function
with a set of suitably scaled wavelet functions, provides a superior
means to detect and characterize sources in an image, compared with
classical "sliding-box''^{13.1} and Fourier
methods. Because a wavelet is a function with limited spatial extent,
it can be used to determine source locations, like the
sliding-box. Unlike the sliding-box, however, it can also be used to
identify the dominant frequencies in the analyzed function, *i.e.*
to characterize source shape and extent. Such characterization could
also be done with Fourier analysis, but unlike Fourier methods,
wavelets determine *localized* frequency information, as a
consequence of their localization in both the spatial and Fourier
domains.

The program *wtransform* applies wavelets to the problem of detecting
sources in X-ray data, and the associated program *wrecon* analyzes
the detected sources and constructs the source list. While others
have developed codes with similar analysis goals (*e.g.*
Vikhlinin, Forman, & Jones 1994; Rosati *et al.* 1995; Grebenev,
Forman, & Jones 1995), *wtransform* and *wrecon* are considerably
more complex, *e.g.* correcting for exposure variation across the
field of view (FOV), and making error estimates, while not requiring
that the point spread function (PSF) have a particular functional form
(which is particularly important for analysis of *Chandra* images). Another program of similar complexity but using different
methods is the `palermo wdetect` program of Damiani *et
al.* (1997).

In §13.2 we describe the basic recipe one would follow to detect sources in data, without reference to any particular wavelet function. We do this is to underscore the fact that the basic details of the method are not dependent on the choice of wavelet function, so long as that function is simple, with one centralized positive amplitude mode. The fact that the PSF of X-ray detectors often has Gaussian-like shape motivates our use of the Marr wavelet, or "Mexican Hat" (MH) function, which we describe in Appendix 13.5.

The program *wtransform* uses the correlation of the MH function with
a given image to identify putative source pixels. In
§13.3 we describe in detail the steps that
*wtransform* follows, in particular discussing how the program
estimates the local background at each pixel, how it corrects for the
systematic shift that a pixel's correlation value will take if the
exposure varies within the limited spatial extent of the wavelet
function, and how it performs error analysis. In appendices which
relate to §13.3, we present derivations of
analytic quantities related to the MH function (Appendix
13.6), as well as describe the simulations used to
estimate the threshold correlation value for source detection in each
pixel (Appendix 13.7). In
§13.4 we discuss in detail how the source
list is constructed by *wrecon* , and in an appendix we describe how
*wrecon* computes a default background image for use in determining
source properties (Appendix 13.8).

13.2 Basic Algorithm

The algorithm by which one uses a simple unimodal wavelet function
to detect sources in a two-dimensional
image is conceptually simple.^{13.2} This function is
convolved with to produce a "correlation image" :

(Note that the data are assumed to have units counts, and not counts per second.) The expectation value of is zero, if there are no sources within the limited spatial extent of the wavelet function, and the background count rate is locally constant, because the normalization of is zero.

A clump of counts will manifest itself as a local maximum in a
correlation image if the scale sizes
are
approximately the same as, or larger than, the size of the clump. One
can see why this is so more easily if we write
, where and denote the positive and negative
amplitude portions of the wavelet function, respectively. If the
clump is contained completely within , then the contribution of
the positive term to outweighs that of the negative term,
producing a maximum.^{13.3} If the scale sizes are smaller, then
the wavelet function will extend over a smaller region within the
clump, within which the inferred counts amplitude will tend more and
more to a constant. Hence
. For larger scale sizes,
the correlation value tends asymptotically to a maximum,
.

Source detection boils down to the question: how large must the value become before we may safely accept that a source has contributed counts to the observed clump? To answer this, we compute the Type I error, or the probability that we would erroneously accept a source when the null hypothesis, that there is no source, is correct. We compute this quantity, also called the significance, in each image pixel :

is the inferred number of background counts within the limited spatial extent of the wavelet function, and is the probability sampling distribution for given . By definition, ; if a source is present, 0. If , a user-defined threshold (

If is estimated from the raw data themselves, this estimate will be biased if source counts are present, so that . Thus an iterative procedure is used to remove source counts from the image:

- estimate , the number of background counts in pixel , and compute ;
- compute ;
- determine which pixels are associated with (the strongest) sources;
- replace with in those pixels, and call the new data image ;
- estimate using , and compute ;
- compute ;
- determine which pixels are associated with (weaker) sources;
*etc.*

After this algorithm is used to determine lists of source pixels for many wavelet scale sizes, cross-identification of pixels across scales is performed to create the final source list. This cross-identification allows the weeding out of spurious sources which may be detected even if a high significance threshold is used.

13.3 Source Pixel Identifications

13.3.1 Background Estimation

In §13.2, we noted how the condition
0 can be interpreted to mean that the performs
a "background subtraction." It is thus natural to use a function of
to estimate in pixel , if it is *a
priori* unknown:

(13.4) |

is the normalization. Variations in exposure within the limited spatial extent of the wavelet function will affect the estimate of , which we estimate as . (If exposure variations may be safely ignored, then can be computed analytically; see Appendix 13.6.) The normalized (or flat-field) background is directly related to :

is an estimate of the number of counts that would be detected if there were no exposure variations; it differs from the number of photons seen from a similarly sized region of sky by a factor related to the efficiencies of the mirror assembly and detector. Note that is set to zero in pixels with exposure less than a user-specified threshold.

We stress that one should not interpret blindly as the
actual background amplitude
; it should only be
viewed as a computation made in order to select source pixels. For
instance, if a small wavelet scale is applied in a region with a large
PSF, then
around any source, because
the estimate of is unavoidably biased by the source counts
which lie in the region spanned by . In *wrecon* , we provide a
default method of combining information across scales to estimate
(which we discuss in Appendix
13.8); it is this quantity which is used to
help determine source properties.

13.3.2 Correlation Image

The wavelet is centered in pixel (

(13.8) |

The right side of this equation may be read as follows: "the inverse FFT of the product of a normalization factor , the FFT of , and the complex conjugate of the FFT of ." To avoid problems associated with FFT aliasing, we pad the image with zeros; an axis-length in the padded image is, in general, the next power of two larger than the length of that axis in the unpadded image (

13.3.2.1 Correcting for Exposure Variation

Variations in exposure within the limited spatial extent of the MH
function, *e.g.* caused by the shadows of mirror supports and
ribs, or by the edge of the FOV itself, will reduce the accuracy of
the estimate of . This quantity may be overestimated or
underestimated, depending upon the location of pixel relative
to regions of reduced exposure, but an overestimate is more damaging,
as it can lead to the detection of a spurious source. An overestimate
will occur if the exposure reduction primarily occurs within . To
see this, let us assume the counts in regions without exposure
variation to be a constant, and that the exposure reduction causes a
reduction in counts . Then:

In this example, 0. Thus exposure variations can lead to the detection of spurious sources near shadows (but not within them) and near the FOV edge. Below, we present two methods for correcting the estimate .

(13.9) |

We rewrite the second term in terms of the exposure and normalized background :

(13.10) |

We estimate what the correlation value would be if throughout the region spanned by the wavelet function:

As noted above, if exposure reductions primarily occur in . As a default, is set to one in each pixel, so that the edge-of-field correction may be made even if the exposure map is not read; this default array is then overwritten if the exposure map is entered. We note that the calculation of is prone to systematic error unless is extrapolated over regions of low exposure (where is set to zero) and beyond the edge of the FOV.

The source counts, , may also be altered by variations in exposure
if the source is extended; however, correcting requires knowledge
of
, which is *a priori* unknown. Also, our prime
motivation for correcting is to reduce the number of
spurious sources, and correcting does not help us in this regard.

This correction is deemed "fast" because

13.3.2.2 Calculation of Significance

In §13.2, we show how the significance is
computed in each pixel. However, for reasons discussed fully in
Appendix 13.7, *wtransform* does not compute
significances directly, but rather compares each correlation value
with a threshold correlation value . This
threshold is found by estimating the lower bound on the integral in
Eq. (13.3), given , a user-specified significance
threshold (*e.g.* 10), and
. If
, pixel
is identified as a possible source pixel.

In *wtransform* , the user actually specifies two threshold significance
values: , for source cleansing, and . During
iterations, is compared to , which may be set to
be much larger than (*e.g.* 10) to help reduce the
bias caused by source counts in estimates of .

13.3.3 Error Estimation

The analytic calculations of and involve the pixel-by-pixel summation of the independent Poisson-distributed random variable . To compute the variances of these quantities, we use the general formula

(13.13) |

where , and are the random variables (see,

If depends directly on , the covariance terms are zero, and the approximation formula simplies to the familiar formula .

The random variables have variance .

This quantity is calculated only after the data are cleansed of sources. It may be calculated analytically (Eq. [13.16]) or using FFTs (Eq. [13.17]).

This variance is computed once, after the first iteration, either analytically (Eq. [13.18]) or through the use of FFTs (Eq. [13.19]). To speed computation when using Fourier Transforms, we use the analytically-derived quantity . Details on its derivation may be found in Appendix 13.6.

The variance is then

(13.19) |

The presence of one summation over image pixels within another makes the computation time for any one given pixel 1 CPU-minute. Thus in

(13.20) |

Initial tests indicate that this formula is accurate away from sources, but greatly

13.4 Constructing the Source List

The program *wrecon* is used to construct the final source list after
*wtransform* has been run at all desired scales. To construct this
list, it uses:

- lists of correlation space maxima whose amplitudes are greater
than the threshold specified in
*wtransform*, created at each wavelet scale; - correlation images created by
*wtransform*at each wavelet scale; - and "flux images" computed within
*wrecon*itself.

13.4.1 Flux Image

The flux image is created by smoothing the raw data, then subtracting
the normalized background from the smoothed image^{13.5} on a pixel-by-pixel basis:

We use as the smoothing function because it has the desirable properties of being localized, and, for the particular case of the MH function, of mimicking the shape of a canonical Gaussian PSF.

The scale pairs at which to compute the flux images are user-specified; the number of pairs can be less than the number of wavelet scale pairs examined overall. Using fewer pairs reduces computation time, but may lead to a less-precise determination of source properties. Hereafter, we assume that , forcing symmetric wavelets.

13.4.1.1 The Source Cell

To determine source properties, it is necessary to define a "cell" on
the detector, in which some portion of a source's counts are
detected. If we choose the size and shape of the cell, then we need
detailed knowledge of the PSF to estimate the fraction of a source's
counts that are recorded in the cell. However, *wrecon* and the other
*Chandra* detection algorithms assume no details about a detector's PSF
except for its characteristic size
, which is computed
for *Chandra* data using the routine `psf_calc_size`; its
functional form is unknown. But even if we did have detailed knowledge
of the PSF, its use would provide a good estimate of the total source
counts only if the source were point-like.

*wrecon* uses the flux images to define the source cell. In regions
where there are no sources, the mean value of is zero; only in the
vicinity of sources does deviate markedly from zero. Hence
*wrecon* defines the source cell by

- using the flux image computed at the scale closest to the PSF size, by minimizing the function ;
- determining the local maximum in -space corresponding to , the estimated location of the putative source;
- and determining those pixels for which is the local maximum in -space. Pixels for which 0 are not considered.

A cell defined in this manner has advantageous properties:

- there are no
*a priori*assumptions about source shape and extent; - saddle points in -space allow the definition of boundaries between nearby sources;
- and the creation of the flux image involves smearing, or de-focusing, data, so that cell boundaries will lie outside the region where the source is evident to the eye. Hence, at least for isolated sources, nearly all source counts should have been detected within cell boundaries.

This approach is not always optimal when analyzing extended sources:
such sources may have more than one local maximum in counts space,
leading to multiple "sources" being listed. One way to counteract
this is to set
, as this will
increase the probability of the extended object being associated with
one maximum in -space (which will ensure a sufficiently large
cell). However, it also increases the probability that point sources
in the vicinity of the extended source will be "swallowed up" by the
larger source cell, so care must be exercised when interpreting
derived source properties such as count rate. This approach is also
not optimal if PSF is bimodal, as it is far off-axis for *Chandra* ; in
this case, *wtransform* will output two correlation maxima for a
well-sampled source, and *wrecon* will characterize each maximum
independently of the other. Only with the *a posteriori*
application of a detailed PSF can the two "sources" be properly
combined.

It is also possible for a source cell to contain two or more
underlying sources. This is indicated when there are multiple
correlation maxima in the source cell *at the PSF scale*. The
code indicates such a condition with a flag in the source list; it
does not currently attempt to "break up" the cell to refine source
property estimates.

13.4.1.2 Source Selection Algorithm

To create the source list, *wrecon* examines the lists of correlation
maxima in ascending order of wavelet size. If the location of a
maximum is not already associated with a source cell (see
below), it will be provisionally associated with a real source if

- 0;
- and the ellipse defined by

does not contain one or more sources first detected at scales smaller than the currently considered scale . This is an important check when , as sources in a crowded field merge, possibly creating "new" sources at new locations in the FOV.

After determining for the putative source its cell and properties,
*wrecon* first scans through the lists of all correlation maxima at
all scales, and marks as "examined" those which lie within the
current source cell, and then verifies that the putative source
appears at one or more scales
. It does this by
determining the scale size closest to the PSF size,
(by minimizing
),
then counting the number of correlation maxima that lie within the
source cell for
. If the number is
zero, *wrecon* rejects the putative source.

13.4.2 Source Properties

(13.22) | |||

(13.23) |

where , and represents the source cell. We note that the estimate of location may be biased if the source cell is truncated because of the presence of a nearby source.

A preferred weighting function would be the source flux , but its use greatly complicates the computation of variances and . (For similar reasons, is not used as a weighting function.) and will give similar estimates if the source is strong or if background is small.

The variance of the location estimate,
, is
estimated using Eq. (13.14):

(13.24) |

is computed similarly.

(13.25) |

It is possible to rewrite this equation in the form , which in theory makes the variance directly computable. However, this calculation is computationally intensive and is not performed by the current version of the program (see the discussion around Eq. [13.65] for details). The variance is computed using the formula

The missing covariance terms are positive, so Eq. (13.26) underestimates the true variance.

(13.27) |

is computed in an analogous manner to
, by replacing with
:

(13.29) |

The variance on the count rate estimate is

where and are given in eqs. (13.26) and (13.28) respectively. Equation (13.30) underestimates the true variance.

(13.31) |

The (underestimated) variance is given by

(13.32) |

(13.33) |

which has (underestimated) variance

(13.34) |

(13.35) |

where is the number of input scale pairs, and = 1 if the following conditions hold:

- 0;
- the associated local correlation maximum is identified as a
source pixel by
*wtransform*; - and the associated local correlation maximum is contained within a source cell.

13.5 The Mexican Hat (MH) Function

The source detection algorithm we present in this paper should not
depend on the details of function itself, at least for simple wavelets
which have one central positive mode. In this Appendix, we describe
the Marr wavelet function, or "Mexican Hat function,'' (MH) which is
used by *wtransform* and *wrecon* .

A wavelet function may be derived from a real-valued, non-negative,
infinitely differentiable function that satisfies the
condition
1 (see, *e.g.*, Holschneider
1995). One function that satisfies these requirements is the Gaussian:

(13.36) |

The relationship between and the wavelet function in two dimensions is

The integral of over all space is, by definition, zero. This allows us to disregard the factor in Eq. (13.37) when performing correlations.

The MH has a positive kernel surrounded by a negative annulus; the boundary between these two regions is an ellipse with axis-lengths and :

(13.38) |

The area of this ellipse, used to compute correlation thresholds, is . The amplitude of the MH function is 2 at its center, regardless of scale; its minimum amplitude is -0.27 on the ellipse defined by axis-lengths 2 and 2.

The MH function has several advantageous properties which motivate its use.

- When rotationally symmetric, its Gaussian-like positive kernel has similar shape to a canonical PSF.
- The kernel's limited extent allows only those sources with intrinsically or instrumentally broadened size to be strongly detected.
- It has an analytically-derivable Fourier Transform, speeding its correlation with data via the Correlation Theorem (Appendix 13.6).
- It has a limited extent in Fourier space, such that
limited, discrete sampling in and (
*e.g.*at values separated by factors of two) is sufficient to sample the entire frequency domain. - Its limited extent in both spatial and Fourier domains helps to minimize effects of aliasing.

13.6 Solutions to Integrals Involving the MH Function

13.6.1 Analytic Integration of the MH Function

13.6.1.1 Pixel-by-Pixel Integration

If the width of the wavelet function is of order the size of the image pixel, then using FFTs to compute and other related quantities becomes impractible. To determine , for instance, we integrate on a pixel-by-pixel basis, as shown in Eq. (13.7).

We substitute the form of in Eq. (13.37) into Eq. (13.7):

Here, represent coordinates relative to the center of pixel , and the integration is carried out over the pixel :

0 if are outside an ellipse with axis-lengths 5 and 5, limiting the range of integration.

If we expand Eq. (13.39), we find that there are two basic
integrals which must be performed. The first is:

where the substitution is made, and is the error function (see,

This integral can be solved by parts, by taking the derivative of and the integral of , leaving an integral of (solved above). Leaving out intermediate steps, we present the solution:

Taking the products of these solutions, and exchanging for when needed, we find:

(13.40) |

13.6.1.2 Integration of the Negative Annulus

If exposure variations and the FOV edge may be ignored in the computation of the background, then we may replace in Eq. (13.5) with a background normalization factor, which we derive here.

The overall normalization of the MH function is zero. Hence the background normalization factor may be derived by integrating over its positive core:

(13.41) |

The limits of integration reflect that the core extends over an ellipse with axes and . We reparametrize the integral using polar coordinates , after first mapping the boundary ellipse to a boundary circle using the transformation :

The determinant of the Jacobian of the transformation from is .

We evaluate the second integral using integration by parts:

The second integral in Eq. (13.45) cancels with the first integral in Eq. (13.44), leaving:

(13.46) |

13.6.2 Analytic Fourier Transform of the MH Function

The Fourier Transform of is

The wave-number equals , where is the pixel number in Fourier space, is the Nyquist wave-number, and is half the length of the relevant axis in the padded image. The fourth integral in Eq. (13.47) and the second integral in Eq. (13.48) are zero, as the integrands are products of even and odd functions.

and represent two integral solutions which one may find, e.g., in Gradshteyn & Ryzhik (1980; formulae 3.896-2 and 3.952-4 respectively):

and

Substituting these quantities into Eq. (13.49), one finds that

Substituting Eq. (13.50) into Eq. (13.48) and solving, we find:

To compute the error of the correlation values in each pixel, we calculate using an analytic Fourier Transform . The derivation of this function is similar to the derivation of . A new integral which appears is:

(13.52) |

One may solve this integral by parts, differentiating the term and integrating the term . The integral

(13.53) |

has solution (see, e.g., Gradshteyn & Ryzhik 1980, formula 3.952-5):

(13.54) |

The final solution is

(13.55) |

13.7 Computation of Threshold Correlation Values

An image pixel is associated with an astronomical source if the significance of its correlation value is greater than a user-defined threshold significance :

is the inferred number of background counts within the limited spatial extent of the wavelet function, and is the probability sampling distribution for given . This sampling distribution does not depend upon the scale size of the wavelet function. We assume that the background count rate is locally constant, so that instead of , we use the expected number of background counts in the positive kernel of the wavelet, a quantity which we denote . For the MH function, .

We find that
does not have analytic form for
those values of most likely to be seen in analyses of *Chandra* images, so we must estimate this distribution using simulations. The
computation time needed for such simulations means that we cannot
accurately assign significances to pixels if
. So
instead of computing the significance, *wtransform* compares the value
of in each pixel with a *threshold* correlation value
, defined by the equation

(13.56) |

We determine by:

- randomly selecting values from the range -10 3.25;
- creating 10241024 images, sampling data in each pixel from the Poisson distribution, with the expected number of counts in each pixel being , and with pixels;
- computing and in each image pixel;
- and recording in bins of size
= 0.2, for -6.9
3.1, with one bin being used
for all values of
-6.9.
^{13.6}From these distributions , we can determine .

By analyzing over 50,000 simulated images, we are able to determine 25
values of in each
bin, for values of 10, using the central 68% (17) of the values to estimate
variances. We fit simple functions to the estimated threshold values,
using the method and the estimated variances. We present our
results below. These functions describe the observed threshold values
well, except in the regime
and
, where a binned look-up table is used instead. We note that these
fits allow us in principle to compute threshold correlation values for
significance values below our computational lower limit
10 (such as for 10, the significance
corresponding to one false source pixel in an *Chandra* HRC image).

In the regime and , is well-described by the parabola:

(13.57) |

with

(Note that errors have not been estimated for these parameters.)

In the regime , is described by the function

a function used by Damiani

For values , we find that we must sometimes use the linear equation

(13.59) |

to represent the threshold, with

(13.60) |

and

If the probability sampling distribution is Gaussian with width
, then the significance is given by:

We find that if we use Eq. (13.58) in the regime 3, then the derived values of are larger than those predicted by Eq. (13.61) above. Because it is more conservative, we use Eq. (13.58) for all values 0.

13.8 Normalized Background Map

If the user does not provide a normalized background image, *wrecon* will create such an image by combining the normalized background
images that *wtransform* creates at each wavelet scale:

where

and is the number of scales used. "
"
denotes all
, along with the largest
. For instance, if
= 9 pixels,
1 for the standard progression values =
[8,8,16,...], and
0 for =
[...4,4]. We exclude the smallest scales because for
, the background is greatly overestimated in the
vicinity of sources. The weighting reflects that the area of the FOV
used to estimate the background goes as . The default
background for *wrecon* is then:

(13.64) |

We may algebraically manipulate the combination of
eqs. (13.5), (13.62), and (13.63), to
express the default normalized background as

The presence of the term makes it impossible to compute the variance of the default background estimate using FFTs. Hence the current version of the program computes the variance of the background estimate by summing variance estimates at each scale:

The missing covariance terms are positive, so Eq. (13.66) provides an

13.9 References

Damiani, F., Maggio, A., Micela, G., & Sciortino, S. 1997, ApJ 483, 350

Daubechies, I. 1992, Ten Lectures on Wavelets (Philadelphia: SIAM)

Eadie, W. T., Drijard, D., James, F. E., Roos, M., & Sadoulet, B. 1971, Statistical Methods in Experimental Physics (Amsterdam: North-Holland)

Goupillaud, P., Grossmann, A., & Morlet, J. 1984, Geoexploration 23, 85

Gradshteyn, I., & Ryzhik, I. 1980, Table of Integrals, Series, and Products (San Diego: Academic Press)

Grebenev, S. A., Forman, W., & Jones, C. 1995, ApJ 445, 607

Holschneider, M. 1995, Wavelets: An Analysis Tool (Oxford: Oxford University Press)

Micela, G., Sciortino, S., Kashyap, V., Harnden, F. R., Jr., & Rosner, R. 1996, ApJS 102, 75

Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P.1992, Numerical Recipes (Cambridge: Cambridge Univ. Press)

Rosati, P., Della Cecai, R., Burg, R., Norman, C., & Giacconi, R. 1995, ApJL 445, 11

Slezak, E., Bijaoui, A., & Mars, G. 1990, A&A 227, 301

Vikhlinin, A., Forman, W., & Jones, C. 1994, ApJ 435, 162

cxchelp@head.cfa.harvard.edu