2010 Vol. XX No. XX, 000–000

11institutetext: Department of Astronomy, Peking University, Beijing 100871,
China<EMAIL_ADDRESS>
Received [year] [month] [day]; accepted [year] [month] [day]

# Comparison of Halo Detection from Noisy Weak Lensing Convergence Maps with
Gaussian Smoothing and MRLens Treatment

Y.-X. Jiao  H.-Y. Shan and Z.-H. Fan

###### Abstract

Taking into account the noise from intrinsic ellipticities of source galaxies,
we study the efficiency and completeness of halo detections from weak lensing
convergence maps. Particularly, with numerical simulations, we compare the
Gaussian filter with the so called MRLens treatment based on the modification
of the Maximum Entropy Method. For a pure noise field without lensing signals,
a Gaussian smoothing results a residual noise field that is approximately
Gaussian in statistics if a large enough number of galaxies are included in
the smoothing window. On the other hand, the noise field after the MRLens
treatment is significantly non-Gaussian, resulting complications in
characterizing the noise effects. Considering weak-lensing cluster detections,
although the MRLens treatment effectively deletes false peaks arising from
noise, it removes the real peaks heavily due to its inability to distinguish
real signals with relatively low amplitudes from noise in its restoration
process. The higher the noise level is, the larger the removal effects are for
the real peaks. For a survey with a source density $n_{g}\sim 30\hbox{
arcmin}^{-2}$, the number of peaks found in an area of $3\times 3\hbox{
deg}^{2}$ after MRLens filtering is only $\sim 50$ for the detection threshold
$\kappa=0.02$, while the number of halos with $M>5\times 10^{13}\hbox{
M}_{\odot}$ and with redshift $z\leq 2$ in the same area is expected to be
$\sim 530$. For the Gaussian smoothing treatment, the number of detections is
$\sim 260$, much larger than that of the MRLens. The Gaussianity of the noise
statistics in the Gaussian smoothing case adds further advantages for this
method to circumvent the problem of the relatively low efficiency in weak-
lensing cluster detections. Therefore, in studies aiming to construct large
cluster samples from weak-lensing surveys, the Gaussian smoothing method
performs significantly better than the MRLens.

###### keywords:

cosmology: theory - gravitation - dark matter - gravitational lensing

##  1 Introduction

The weak gravitational lensing effect provides a unique tool in measuring the
matter distribution in the universe (e.g., Bartelmann & Schneider 2001;
Hoekstra et al. 2006; Massey et al. 2007). Its additional dependence on the
distances to the source, to the lens and between the source and lens makes it
an excellent probe in cosmological studies of dark energy (e.g., Albrecht et
al. 2006; Benjamin et al. 2007; Kilbinger et al. 2009; Li et al. 2009). On the
other hand, however, different observational and physical effects can affect
the weak lensing analyses significantly. Being extracted from shape distortion
of background galaxies, the weak lensing effect on individual source galaxies
is severely contaminated by their intrinsic ellipticities. Therefore
statistical analyses on a large number of galaxies are necessary in weak
lensing studies. Even so, intrinsic shape alignments of galaxies, including
intrinsic-intrinsic and shear-intrinsic correlations, can be an important
source of error in cosmic shear correlation analyses. For cluster detections
from weak lensing convergence maps reconstructed from shear measurements
(e.g., Kaiser & Squires 1993; Bartelmann 1995; Kaiser 1995; Schneider & Seitz
1995; Squires & Kaiser 1996; Bridle et al.1998; Marshall et al. 2002), even
randomly orientated intrinsic ellipticities can result false peaks by their
chance alignments, which can reduce the efficiency of cluster detections
significantly (e.g., Schneider 1996; van Waerbeke 2000; White et al. 2002;
Hamana et al. 2004; Fan 2007). Thus further treatments for a convergence map
are normally required to suppress the noise effects.

The noise from intrinsic ellipticities of source galaxies is essentially shot
noise, and thus by averaging over a relatively large number of source galaxies
in weak lensing analyses, the residual noise can be effectively reduced. This
leads to the normal smoothing treatment. It is clear that the residual noise
depends on the form of the window function and the smoothing scale. For a
Gaussian smoothing with a window function of the form
$W(\theta)\propto\exp(-\theta^{2}/\theta_{G}^{2})$, the residual noise can be
estimated by
$\sigma^{2}_{0}\approx{(\sigma^{2}_{\epsilon}/2)}{[1/(2\pi\theta_{G}^{2}n_{g})]}$,
where $\sigma_{\epsilon}$ is the rms of the intrinsic ellipticity of
individual source galaxies, $\theta_{G}$ is the smoothing scale, and $n_{g}$
is the surface number density of source galaxies. For $\sigma_{\epsilon}=0.3$,
$n_{g}=30\hbox{ arcmin}^{-2}$ and $\theta_{G}=1\hbox{ arcmin}$, we have
$\sigma_{0}\approx 0.015$.

Recently, Starck et al. (2006) proposed the MRLens filtering technique, which
is based on the Bayesian analyses with a multi-scale entropy prior applied.
The False Detection Rate (FDR) method is used to select significant/non-
significant wavelet coefficients (e.g., Starck et al. 2006; Pires et al.
2009). The MRLens method suppresses noise adaptively according to the strength
of the noise itself. A more detailed description of the method is given in §4.

In this paper, with numerical simulations, we compare Gaussian smoothing with
MRLens treatment, paying particular attention to the completeness and the
efficiency of weak lensing halo detections from convergence maps. The rest of
the paper is organized as follows. In §2, we describe briefly the weak-lensing
convergence reconstruction and the Gaussian smoothing. In §3, we present the
important aspects of the MRLens treatment. Results are shown in §4. Section 5
contains summaries and discussions.

##  2 Weak lensing convergence reconstruction

In the weak lensing regime, the convergence $\kappa(\vec{\theta})$ is
essentially related to the weighted projection of density fluctuations
$\delta$ along the unperturbed light path. Specifically, we have

$\kappa(\vec{\theta})={3H_{0}^{2}\Omega_{0}\over
2}\int_{0}^{w_{H}}dw\bar{W}(w)f_{K}(w){\delta[f_{K}(w)\vec{\theta},w]\over
a(w)}$ (1)

where $H_{0}$ is the present Hubble constant, $\Omega_{0}$ is the present
matter density of the universe in unit of the critical density, ${w}$ is the
radial coordinate, $a(w)$ is the scale factor of the universe, and, with $K$
being the spatial curvature of the universe,

$\displaystyle f_{K}(w)$ $\displaystyle=|K|^{-1/2}\sin(|K|^{1/2}w)\quad\quad\
\ \ (K>0)$ (2) $\displaystyle=w\qquad\qquad\qquad\qquad\qquad\quad(K=0)$
$\displaystyle=|K|^{-1/2}\sinh(|K|^{1/2}w)\qquad(K<0)\ .$

The factor $\bar{W}(w)$ is the weighting function that is related to the
source galaxy distribution $G(w)$ by

$\bar{W}(w)=\int_{w}^{w_{H}}dw^{\prime}G(w^{\prime}){f_{K}(w^{\prime}-w)\over
f_{K}(w^{\prime})}\ .$ (3)

The lensing potential $\phi$ is related to $\kappa$ by

$\kappa={\nabla^{2}\phi\over 2}\ ,$ (4)

and the shears $\gamma_{1}$ and $\gamma_{2}$ are

$\gamma_{1}={\partial_{11}\phi-\partial_{22}\phi\over 2}\
,\quad\gamma_{2}=\partial_{12}\phi.$ (5)

Since both $\kappa$ and $\gamma_{i}$ are determined by the lensing potential,
they are mutually dependent of each other. In the Fourier space, we have
(Kaiser & Squires 1993)

$\kappa(\vec{k})=c_{1}(k)\gamma_{1}(\vec{k})+c_{2}(k)\gamma_{2}(\vec{k}),$ (6)

where $[c_{1},c_{2}]=[\cos(2\phi),\sin(2\phi)]$ with
$\vec{k}=k(\cos\phi,\sin\phi)$.

Observationally, the shear $\gamma$ can be extracted from the shape
measurement of source galaxy images. Under the condition $\kappa<<1$, we have

$\vec{e}^{\rm obs}\approx\vec{\gamma}+\vec{e}^{S}\ ,$ (7)

where $\vec{e}^{\rm obs}$ and $\vec{e}^{S}$ are the observed ellipticity, and
the intrinsic ellipticity of a source galaxy, respectively. Reconstructed from
$\vec{e}^{\rm obs}$, the convergence $\kappa_{n}(\vec{k})$ then contains noise
from the intrinsic part, i.e.,

$\kappa_{n}({\vec{k}})=c_{\alpha}(k)e^{\rm
obs}_{\alpha}({\vec{k}})=\kappa({\vec{k}})+c_{\alpha}(k)e^{S}_{\alpha}({\vec{k}}).$
(8)

With the transformation back to the real 2-D space and applying a smoothing
with the window function $W(\vec{\theta})$, we can obtain the smoothed
quantities (e.g., van Waerbeke 2000)

$\Sigma^{\rm
obs}(\vec{\theta})=\Gamma(\vec{\theta})+\frac{1}{n_{g}}\Sigma^{N_{g}}_{i=1}W(\vec{\theta}-\vec{\theta}_{i})e^{\rm
S}(\vec{\theta}_{i})$ (9)

and

$K_{N}(\vec{\theta})=\int
d\vec{k}e^{-i\vec{k}\cdot\vec{\theta}}c_{\alpha}(k)\Sigma_{\alpha}^{\rm
obs}(\vec{k}),$ (10)

where $\Sigma^{\rm obs}$, $\Gamma$, and $K_{N}$ are the smoothed $e^{\rm
obs}$, $\gamma$ and $\kappa_{n}$, respectively, and $n_{g}$ and $N_{g}$ are
the surface number density and the total number of source galaxies in the
field. The noise part of $K_{N}$ due to the intrinsic ellipticities is then

$N(\vec{\theta})=\frac{1}{n_{g}}\Sigma^{N_{g}}_{i=1}\int
d\vec{k}W(\vec{k})e^{-i\vec{k}\cdot(\vec{\theta}-\vec{\theta}_{i})}c_{\alpha}(k)e^{S}_{\alpha}(\vec{\theta}_{i}),$
(11)

where $W(\vec{k})$ is the Fourier transformation of the window function with
the form

$W(\vec{k})=\frac{1}{(2\pi)^{2}}\int
d\vec{\theta}e^{i\vec{k}\cdot\vec{\theta}}W(\vec{\theta}).$ (12)

Without considering the intrinsic alignment of $e^{S}$, it is expected from
the central limit theorem that the smoothed noise field $N(\vec{\theta})$ is
approximately Gaussian in statistics if the effective number of galaxies
included in the smoothing window is larger than about $10$ (e.g., van Waerbeke
2000). In this case, smoothing leads to correlations in $N(\vec{\theta})$, and
its two-point correlation function is approximately

$<N(\vec{\theta})N(\vec{\theta}^{\prime})>=\frac{\sigma_{\epsilon^{2}}}{2n_{g}}(2\pi)^{2}\int
d\vec{k}e^{i\vec{k}\cdot(\vec{\theta}^{\prime}-\vec{\theta})}|W(\vec{k})|^{2},$
(13)

where $\sigma_{\epsilon}$ is the intrinsic dispersion of $e^{\rm obs}$.

The approximate Gaussianity of $N(\vec{\theta})$ allows us to quantify the
noise effects straightforwardly. The noise effects on cluster mass
reconstruction and the noise peak statistics are analyzed in van Waerbeke
(2000). Even with weak alignments of intrinsic ellipticities,
$N(\vec{\theta})$ can still be approximately described by a Gaussian random
field with a modified two-point correlation function including the effects of
intrinsic alignments. The enhancement of the noise peak abundance due to the
weakly intrinsic alignments are analyzed in Fan (2007). In Fan et al. (2010),
the effects of the presence of real dark matter halos on the noise peak
statistics around them as well as the effects of the noise on the peak height
of real halos are investigated in detail. They further present a model to
calculate the total peak abundance in a large-scale convergence map, including
the peaks corresponding to real halos and the noise peaks from the chance
alignment of the intrinsic ellipticities of source galaxies. Such a model
makes it possible for us to use directly the peaks from convergence maps as
cosmological probes without the need to differentiate real and false peaks.

Due to its simple operational procedure and the Gaussian statistics of the
residual noise field, the smoothing treatment has been widely applied in weak
lensing analyses. Different smoothing functions have been used in different
studies. In this paper, we consider the Gaussian smoothing function $W_{G}$,
which is one of the most commonly adopted window functions. Specifically, we
have

$W_{G}=\frac{1}{\pi\theta_{G}^{2}}\exp\left(-\frac{\theta^{2}}{\theta_{G}^{2}}\right)\
,$ (14)

where $\theta_{G}$ is the smoothing scale. Then from Eq. (13), the rms of the
noise $\sigma_{0}$ after smoothing is given by

$\sigma_{0}^{2}={\sigma_{\epsilon}^{2}\over 2}{1\over
2\pi\theta_{G}^{2}n_{g}}\ .$ (15)

In our analyses, we choose $\sigma_{\epsilon}=0.3$, the typical value for
lensing source galaxies, and $\theta_{G}=1\hbox{ arcmin}$, which is the
optimal smoothing scale considering cluster-sized halos. Then for a lensing
survey with $n_{g}=30\hbox{ arcmin}^{-2}$, $\sigma_{0}\approx 0.015$, which is
about $20$ times lower than $\sigma_{\epsilon}$.

##  3 MRLens method

Starck et al. (2006) introduce a new reconstruction and filtering method,
namely, Multi-scale Entropy Restoration (MRLens). It is developed from the
Maximum Entropy Method. The basic idea is to use only ‘signals’ selected by
the so called False Discovery Rate (FDR) (Benjamini & Hochberg 1995) to
reconstruct the convergence field through a Multi-scale Entropy prior. In the
following, we present specific steps of MRLens.

###  3.1 Wavelet decomposition

For an original convergence map $\kappa_{obs}$ with $N=n\times n$ pixels, the
first step of MRLens is to decompose the image map into different components
representing fine structures of different scales.

To do this, we first initialize $j=0$ and set $C_{0}(k,l)=\kappa_{obs}(k,l)$,
i.e., $j=0$ corresponds to the unprocessed map with detailed structures. Then
we progressively go to higher $j$ to obtain smoother maps through (Starck et
al. 2001)

$C_{j+1}(k,l)=\sum_{m}\sum_{n}h_{1D}(m)h_{1D}(n)C_{j}(k+2^{j}m,l+2^{j}n),$
(16)

where $h_{1D}(m)={[1/16,4/16,6/16,4/16,1/16]}$ for $m=-2,-1,0,1,2$,
respectively. Defining

$w_{j+1}(k,l)=C_{j}(k,l)-C_{j+1}(k,l),$ (17)

we finally obtain

$\kappa_{obs}(k,l)=C_{J}(k,l)+\sum_{j=1}^{J}w_{j}(k.l),$ (18)

where $J$ is a chosen number determined by specific considerations on how
smooth we want to go. Here we set $J=7$. In our following analyses, each map
is $3\times 3\hbox{ deg}^{2}$ discretized into $1024\times 1024$ pixels. Thus
$2^{J}=128$ pixels corresponding to $\sim 22\hbox{ arcmin}$. Because we do not
expect to see significant structures resulting purely from noise on such a
large scale, $J=7$ is an appropriate choice.

It can be seen from Eq. (18) that $C_{J}(k,l)$ is the most smoothed version of
the original map $\kappa_{obs}$, and the terms in the summation contain ever
smaller-scale information with smaller $j$.

###  3.2 Multiscale Entropy

With the multi-scale wavelet decomposition, one can then construct an entropy
with the obtained wavelet coefficients $w_{j}(k,l)$ at each grid $(k,l)$ with
$j=1,2,...,J$. It can generally be written as

$H(\kappa)=\sum_{k,l}h[C_{J}(k,l)]+\sum_{j=1}^{J}\sum_{k,l}h[w_{j}(k,l)].$
(19)

For $h$, there are different definitions (e.g., Starck et al. 2006).

Here we follow Starck et al. (2001) to choose the entropy of NOISE-MSE $h_{n}$
in our considerations. At each scale $j$, the noise entropy at each grid
$(k,l)$ is derived by weighting the entropy with a probability that
$w_{j}(k,l)$ is contributed by noise. Specifically, we have

$h_{n}[w_{j}(k,l)]=\int_{0}^{\mid w_{j}(k,l)\mid}P_{n}[\mid
w_{j}(k,l)\mid-u]\frac{\partial h(x)}{\partial x}|_{x=u}du,$ (20)

where $P_{n}[w_{j}(k,l)]$ is the probability that the coefficient $w_{j}(k,l)$
can be due to noise, and is given by

$P_{n}[w_{j}(k,l)]=\mathrm{Prob}[W>\mid w_{j}(k,l)\mid].$ (21)

Eq. (20) essentially regards the information contained in $w_{j}(k,l)$ to be
built up from the summation of $dh(u)$. For each newly added $dh(u)$,
depending on the difference $|w_{j}(k,l)|-u$, there is a probability that it
is due to noise.

For Gaussian noise with rms $\sigma_{j}$ at scale $j$, we have

$\displaystyle P_{n}[w_{j}(k,l)]$ $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{2\pi}\sigma_{j}}\int_{\mid
w_{j}(k,l)\mid}^{+\infty}\exp(-W^{2}/2\sigma^{2}_{j})dW$ (22) $\displaystyle=$
$\displaystyle\mbox{erfc}\bigg{(}\frac{\mid
w_{j,k,l}\mid}{\sqrt{2}\sigma_{j}}\bigg{)}$

and thus

$h_{n}[w_{j}(k,l)]=\frac{1}{\sigma_{j}^{2}}\int_{0}^{\mid
w_{j}(k,l)\mid}u\mbox{ erfc}\bigg{(}\frac{\mid
w_{j}(k,l)\mid-u}{\sqrt{2}\sigma_{j}}\bigg{)}du.$ (23)

###  3.3 Selecting significant wavelet coefficients using the False Discovery
Rate (FDR)

The Multiscale Entropy method applies regularizations on wavelet coefficients
to minimize noise contributions while keeping the signal information. Thus for
those coefficients which are clearly signals, they should be kept unchanged.
Then a new Multiscale Entropy is defined as (e.g., Starck et al. 2006)

$\tilde{h}_{n}[w_{j}(k,l)]=\bar{M}_{j}(k,l)h_{n}[w_{j}(k,l)],$ (24)

where

$\bar{M}_{j}(k,l)=1-M_{j}(k,l),$ (25)

and $M$ is the multi-resolution support defined as (Starck et al. 1995)

$M_{j}(k,l)=\\{\begin{array}[]{ll}1&\textrm{if $w_{j}(k,l)$ is significant}\\\
0&\textrm{if $w_{j}(k,l)$ is not significant}\end{array}.$ (26)

Therefore $\tilde{h}_{n}$ means that we only need to regularize those wavelet
coefficients which are ′not significant′, that is, they are likely due to
noise.

For judging the significance of a wavelet coefficient, a commonly used
criterion is a ‘$k\sigma$’ threshold. If a coefficient is above the threshold,
it is defined to be ′significant′. This is equivalent to set a threshold for
the ratio of ′significant′ detections over the total number of pixels being
analyzed. Considering a Gaussian noise, a $2\sigma$ criterion corresponds to a
probability of $0.05$ for a noise coefficient being mis-classified as
′significant′. If we have totally $N$ pixels to consider, the number of false
discoveries is then on average $0.05N$. If the number of pixels related to
real signals in the analyses is comparable to $0.05N$, the false discovery
rate with respect to the number of real signals can be much higher than
$0.05$. Increasing $k$ can lower the number of false detections at the
expense, however, of the power of real detections. To overcome such
difficulties, an alternative thresholding technique, the False Discovery Rate
(FDR), has been proposed (Benjamini & Hochberg 1995; Miller et al. 2001;
Hopkins et al. 2002; Starck et al. 2006).

This method can effectively control, in an adaptive manner, the fraction of
false discoveries over the total number of discoveries, rather than over the
total number of pixels analyzed.

Let $P_{1},\ldots,P_{N}$ denote the p-values ordered from low to high for the
N pixels, where p-value is defined as

$p_{value}=\frac{1}{\sqrt{2\pi}\sigma_{j}}\int_{w_{j}(k,l)}^{\infty}\exp[-(w-\bar{w}_{j})^{2}/2\sigma^{2}_{j}]dw,$
(27)

with $\bar{w}_{j}$ the average of $w_{j}(k,l)$ for the scale $j$ over all the
pixels. Define

$d_{j}=\max\left\\{k_{j}:\ P_{k_{j}}<\frac{k_{j}\alpha_{j}}{c_{N}N}\right\\},$
(28)

then all the $w_{j}(k,l)$ with their values larger than $d_{j}$ are classified
as ′significant′. Here $c_{N}=1$ if all the pixels are statistically
independent. The meaning of $\alpha_{j}$ is approximately the pre-defined
false discovery rate at scale $j$ with respect to the total number of
detections. The larger the $\alpha_{j}$ value, the larger the fraction of
$w_{j}(k,l)$ defined to be ′significant′. In our analyses, we adopt FDR to
find the values of $M$ in Eq. (26). In the MRLens program, $\alpha_{0}$ is an
adjustable parameter, and $\alpha_{j}=\alpha_{0}\times 2^{j}$ (Starck et al.
2006).

###  3.4 Multi-scale Entropy Filtering algorithm

Given the discussions in the previous subsections, the Multi-scale Entropy
restoration method reduces to find the reconstructed $\kappa_{f}$ that
minimizes $I(\kappa_{f})$ defined as

$I({\kappa_{f}})=\frac{\parallel{\kappa_{obs}-\kappa_{f}\parallel^{2}}}{2\sigma_{n}^{2}}+\beta\sum_{j=1}^{J}\sum_{k,l}\tilde{h}_{n}[({\cal
W}{\kappa_{f}})_{j}(k,l)],$ (29)

where $\sigma_{n}$ is the rms of noise in the original convergence map
$\kappa_{obs}$, $J$ is the number of wavelet scales, $\cal W$ is the wavelet
transform operator and $\tilde{h}_{n}[({\cal W}{\kappa_{f}})_{j}(k,l)]$ is the
multi-scale entropy defined only for non-significant coefficients selected by
the FDR method. The $\beta$ parameter is calculated under the restriction that
the residual should have a standard deviation equal to the rms of noise. The
best $\kappa_{f}$ is then obtained by iterative calculations. Full details of
the minimization algorithm can be found in Starck et al. (2001).

It can be seen that the two terms in the right of Eq. (29) are balancing each
other. While the first term tends to keep the information in $\kappa_{f}$ the
most, the second term has the effect to lower the noise as much as possible.

##  4 Results

In this section, we present the results of our analyses. For weak-lensing
effects from large-scale structures in the universe, we use the publicly
available ray-tracing weak-lensing maps provided kindly by White & Vale
(2004). The specific set of lensing maps we analyze are generated from large-
scale N-body simulations with cosmological parameters $\Omega_{M}=0.296$,
$\Omega_{\Lambda}=0.704$, $w=-1.0$, $h=0.7$, and $\sigma_{8}=0.93$. The box
size is $300\hbox{ Mpc}~{}h^{-1}$, the number of particles is $512^{3}$ with
$m\approx 1.7\times 10^{10}M_{\odot}~{}h^{-1}$ for each, and the softening
length is $\approx 20\hbox{ kpc}~{}h^{-1}$. There are totally $16$ convergence
maps and each has a size of $3\times 3\hbox{ deg}^{2}$ pixelized into
$1024\times 1024$ pixels. The redshift distribution of source galaxies follows
$p(z)\propto z^{2}\exp[-(z/z_{0})^{3/2}]$ with $z_{0}=2/3$.

For each map, we add in Gaussian noise due to the intrinsic ellipticities of
source galaxies with the variance given by (e.g., Hamana et al. 2004),

$\sigma_{\rm pix}^{2}=\frac{\sigma_{\epsilon}^{2}}{2}\frac{1}{n_{g}\theta_{\rm
pix}^{2}},$ (30)

where $\theta_{\rm pix}$ is the pixel size of the simulated
convergence-$\kappa$ map, and $\sigma_{\epsilon}$ is the rms of the intrinsic
ellipticites taken to be $\sigma_{\epsilon}=0.3$. The surface number density
$n_{g}$ depends on specific observations. Here we consider $n_{g}=30\hbox{
arcmin}^{-2}$ which is typical for ground-based observations, and
$n_{g}=100\hbox{ arcmin}^{-2}$ expected from space observations, respectively.
Figure 1 presents one set of convergence maps without (left) and with (right)
noise. It can be seen very clearly that the noise from intrinsic ellipticities
of source galaxies dominates the map, and certain post-processing procedures
are necessary in order to extract weak-lensing signals embedded under noise.
Here we compare two such methods, namely, the normal smoothing method with a
Gaussian smoothing function, and the MRLens treatment, paying particular
attention to their effects on weak-lensing peak statistics.

Figure 1:  The convergence maps of $3\times 3\hbox{ deg}^{2}$. The left panel
is the noise-free map, and the right panel is the noisy convergence map with
$n_{g}=30\hbox{ arcmin}^{-2}$.

###  4.1 Statistical properties of residual noise

Post-processing procedures can reduce noise effectively. However, certain
levels of residual noise inevitably remain. It is thus important to understand
the statistical properties of the residual noise so that we can quantify their
effects on weak-lensing cosmological studies properly. For that, we first in
this subsection consider pure noise maps without including weak-lensing
signals. After applying Gaussian smoothing and MRLens, respectively, we
compare the residual noise-peak statistics in the two cases. This is highly
relevant to cosmological applications of weak-lensing cluster statistics, in
which, high peaks in convergence maps are thought to be related to clusters of
galaxies and their abundances contain important cosmological information. The
existence of residual noise can generate false peaks in convergence maps,
which in turn can contaminate the weak-lensing peak statistics significantly.

With Eq. (30), we generate a $3\times 3\hbox{ deg}^{2}$ noise map containing
$1024\times 1024$ pixels with $\theta_{pix}=0.176\hbox{ arcmin}$ and the
corresponding $\sigma_{pix}=0.22$ for $n_{g}=30\hbox{ arcmin}^{-2}$ and
$\sigma_{pix}=0.12$ for $n_{g}=100\hbox{ arcmin}^{-2}$. For Gaussian
smoothing, we take $\theta_{G}=1\hbox{ arcmin}$. For MRLens, we take
$\alpha_{0}=0.01$. In a smoothed map, a positive (maximum)/negative (minimum)
peak position is located if its value is above/below those of its eight
neighboring pixels (e.g., Jain & Van Waerbeke 2000; Miyazaki et al. 2002).

Figure 2 shows the probability distribution function (PDF) of peaks in the
residual noise field for the two cases, respectively, with the left for the
Gaussian smoothing and the right for the MRLens. In each panel, the solid and
dashed lines correspond to the results with $n_{g}=30\hbox{ arcmin}^{-2}$ and
$n_{g}=100\hbox{ arcmin}^{-2}$, respectively. The bin size is
$\Delta\kappa=0.005$. Both the positive and the negative peaks are counted in.
Two distinctly different distributions are seen. For the Gaussian smoothing
case, the peak number distribution has a double-peak behavior at
$\kappa/\sigma_{0}\sim\pm 1$, in good agreement with that expected for a
Gaussian random field (Bond & Efstathiou 1987; Van Waerbeke 2000). The rms of
the residual noise in this case is $\sigma_{0}\approx 0.016$ for
$n_{g}=30\hbox{ arcmin}^{-2}$ and $\sigma_{0}\approx 0.009$ for
$n_{g}=100\hbox{ arcmin}^{-2}$, in excellent agreement with the theoretical
value $0.015$ for $n_{g}=30\hbox{ arcmin}^{-2}$ and $\sigma_{0}\approx 0.008$
for $n_{g}=100\hbox{ arcmin}^{-2}$ calculated from Eq. (15). Considering
positive peaks that are relevant for weak-lensing analyses, the noise peaks
with $\kappa/\sigma_{0}\sim 1$, rather than with $\kappa/\sigma_{0}=0$, have
the highest occurrence probability. Such a property of noise can cause
statistically a positive shift for the peak height of a cluster measured from
noisy convergence maps. The shift depends on the density profile of the
cluster. This noise-induced shift can bias the cluster mass estimation from
weak-lensing observations. On the other hand, it can increase the weak-lensing
detectability of clusters, and thus affect the corresponding cosmological
studies significantly (Fan et al. 2010).

For the MRLens case, the residual noise after restoration treatment is low
with $\sigma_{0}\approx 0.0029$ for $n_{g}=30\hbox{ arcmin}^{-2}$ and
$\sigma_{0}\approx 0.0016$ for $n_{g}=100\hbox{ arcmin}^{-2}$, much less than
those of the Gaussian smoothing. However, the noise statistics is highly non-
Gaussian, which results significant complications in quantifying the noise
effect on weak-lensing signals. The number distribution of noise peaks is
narrowly concentrated around $\kappa=0$. Thus unlike the Gaussian smoothing,
it seems that we do not expect a systematic shift due to noise in weak-lensing
cluster peak measurement. It should be noted, however, in MRLens, the noise
filtering involves restoration procedures based on NOISE-MSE of Eq. (29). The
results depend on the noise properties [the second term in Eq. (29)] as well
as on the properties of signals we would like to detect [the first term in Eq.
(29)]. The higher the original noise is, the larger the fraction of the
wavelet coefficients that are suppressed. In such a treatment, the signals are
changed depending on the original noise level and their own properties.
Therefore considering the convergence peak for a cluster, the results after
MRLens restoration in the cases with and without noise are different. In this
sense, the existence of noise also induces a systematic bias for the peak
value of a cluster, though for a reason different from and much more
complicated than that of the Gaussian smoothing case. The quantitative
modeling of such a bias for MRLens needs to be further explored.

Figure 2: The probability distribution functions of peaks in pure noise maps.
The solid and dashed lines are for $n_{g}=30\hbox{ arcmin}^{-2}$ and
$n_{g}=100\hbox{ arcmin}^{-2}$, respectively. The left panel is for the
Gaussian smoothing with $\theta_{G}=1\hbox{ arcmin}$, and the right panel is
for the MRLens result with $\alpha_{0}=0.01$.

For MRLens, the $\alpha_{0}$ parameter in FDR affects the classification of
significant and non-significant wavelet coefficients. A smaller $\alpha_{0}$
results a more stringent criteria for the definition of a significant wavelet
coefficient, and thus stronger suppressions of noise. To test the
$\alpha_{0}$-dependence, we vary its value to obtain different restoration
results for pure noise maps. In Table 1, the rms of the residual noise for
different $\alpha_{0}$ and different $n_{g}$ are shown. With the increase of
$n_{g}$, the original noise level decreases with $(n_{g})^{-1/2}$. It is noted
that after MRLens treatment, the rms of the residual noise also approximately
follows $\sigma_{0}\propto(n_{g})^{-1/2}$. For the $\alpha_{0}$-dependence, as
expected, the residual noise decreases with the decrease of $\alpha_{0}$.
However, this dependence is rather weak. Changing $\alpha_{0}$ from $0.1$ to
$0.01$ only decreases $\sigma_{0}$ by $\sim 20\%$.

Table 1: Standard deviation of the reconstruction error with MRLens $\alpha_{0}$ |  $\sigma_{0}$($n_{g}=15$) |  $\sigma_{0}$($n_{g}=30$) |  $\sigma_{0}$($n_{g}=50$) |  $\sigma_{0}$($n_{g}=100$)
---|---|---|---|---
0.001 | 0.0038 | 0.0026 | 0.0021 | 0.0015
0.01 | 0.0041 | 0.0029 | 0.0023 | 0.0016
0.02 | 0.0041 | 0.0030 | 0.0023 | 0.0016
0.04 | 0.0044 | 0.0032 | 0.0024 | 0.0017
0.06 | 0.0045 | 0.0033 | 0.0026 | 0.0019
0.08 | 0.0047 | 0.0033 | 0.0027 | 0.0020
0.1 | 0.0050 | 0.0035 | 0.0027 | 0.0021
0.2 | 0.0051 | 0.0036 | 0.0029 | 0.0023

###  4.2 Peak statistics in noisy convergence maps

Now we consider peak statistics of noisy convergence maps. Figure 3 shows the
post-processed maps of the right panel of Figure 1 with Gaussian smoothing for
$\theta_{G}=1\hbox{ arcmin}$ (upper) and with MRLens for $\alpha_{0}=0.01$
(lower), respectively. The left panels are for $n_{g}=30\hbox{ arcmin}^{-2}$
and the right panels are for $n_{g}=100\hbox{ arcmin}^{-2}$.

Figure 3:  Noisy convergence maps of $3\times 3\hbox{ deg}^{2}$. The upper and
lower panels are for the Gaussian smoothing with $\theta_{G}=1\hbox{ arcmin}$
and MRLens with $\alpha_{0}=0.01$, respectively. The left and right panels are
for $n_{g}=30\hbox{ arcmin}^{-2}$ and $n_{g}=100\hbox{ arcmin}^{-2}$,
respectively.

Comparing to the maps in Figure 1, we see that the post-processing procedures
can indeed filter out much of the noise so that the real structures in the
large-scale mass distribution can be detected. For $n_{g}=30\hbox{
arcmin}^{-2}$, the MRLens map looks very smooth with only very massive
structures left. On the other hand, in the Gaussian smoothing case, small
structures can also be seen. However, it contains many more noise peaks than
that of the MRLens case. For $n_{g}=100\hbox{ arcmin}^{-2}$, the map is
smoother for the Gaussian smoothing case than that with $n_{g}=30\hbox{
arcmin}^{-2}$. The MRLens map, however, appears lumpier for the lower noise
case. Such opposite trends seen in the Gaussian smoothing and in MRLens
reflect clearly the different underlying filtering mechanisms between the two
smoothing schemes. For the Gaussian smoothing, the filtering is mainly
performed through an averaging procedure. Given a smoothing scale, the peak
signals of real clusters are more or less similar regardless the noise level.
Meanwhile, the noise peaks with relatively high $\kappa$ values are
significantly reduced if the noise level is lowered. Thus the smoother
appearance of the upper right panel is mainly due to the less number of high
noise peaks than that of the upper left panel. For MRLens, it involves a
restoration procedure that depends on the original noise level. The smaller
the original noise is, the lower the fraction is for the wavelet coefficients
to be suppressed. It is important to note that the suppression leads to the
removal of both noise peaks and true peaks of relatively low amplitudes. Thus
the lumpier structures seen in the lower right panel is largely attributed to
the lower level of removal of real structures than that of the lower left
panel.

In Figure 4 and Figure 5, we show the probability distribution function of
peaks for Gaussian smoothing and for MRLens, respectively. The results for
each case are obtained by averaging over $16$ simulated maps with noise added.

Figure 4: The probability distribution function of convergence peaks for the
case of Gaussian smoothing with $\theta_{G}=1\hbox{ arcmin}$. The black solid
line is for the result of the noise-free convergence peaks, the red dashed and
red solid lines are for the pure noise peaks and noisy convergence peaks with
$n_{g}=30\hbox{ arcmin}^{-2}$, respectively, and the blue dashed and blue
solid lines are for the pure noise peaks and noisy convergence peaks with
$n_{g}=100\hbox{ arcmin}^{-2}$, respectively. The left upper panel includes
both maximum and minimum peaks, and the left lower panel shows the
distribution function for maximum peaks only. The right panel is the zoom-in
version of the left lower panel. Figure 5: Same as Figure 4 but for the case
of MRLens with $\alpha_{0}=0.01$.

For the Gaussian smoothing results in Figure 4, the black, red dashed, red
solid, blue dashed, and blue solid lines are for the results of noise free
peaks, pure noise peaks with $n_{g}=30\hbox{ arcmin}^{-2}$, noisy convergence
peaks with $n_{g}=30\hbox{ arcmin}^{-2}$, pure noise peaks with
$n_{g}=100\hbox{ arcmin}^{-2}$, noisy convergence peaks with $n_{g}=100\hbox{
arcmin}^{-2}$, respectively. We can see that in the Gaussian smoothing cases,
the noise peaks dominate over the real peaks at $\kappa<3\sigma_{0}$. At
larger $\kappa>3\sigma_{0}$, real peaks can be detected with high
efficiencies. Comparing the blue solid line with the red solid line, we see
that by reducing the noise level from $\sigma_{0}\sim 0.015$ to
$\sigma_{0}\sim 0.008$, we effectively reduce the number of noise peaks with
$\kappa>0.025$, and thus increase the real peak detection efficiencies
significantly.

In Figure 5 for the MRLens results, the line styles are the same as those in
Figure 4. Different from that in the Gaussian smoothing cases, here the noise
peaks (red and blue dashed lines) contribute little to the total number of
peaks with $\kappa>0.02$ in comparison with the real peaks (black solid line).
However, the suppression process in the MRLens treatment mistakenly removes a
large number of real peaks with $\kappa<0.1$. Thus we expect a high efficiency
but a low completeness in weak-lensing peak detections after MRLens filtering.
Reducing the original noise level by increasing $n_{g}$ form $30\hbox{
arcmin}^{-2}$ to $100\hbox{ arcmin}^{-2}$ leads to a less suppression effect.
Therefore more peaks with $\kappa<0.1$ are kept and the completeness of peak
detections increases considerably.

In the next subsection, we investigate and compare explicitly the efficiency
and completeness of weak-lensing cluster detections in the two smoothing
treatments.

###  4.3 Efficiency and completeness of weak-lensing cluster detection

The existence of noise from intrinsic ellipticities of source galaxies results
false peaks in convergence maps, and thus lowers considerably the efficiency
of weak-lensing cluster detections. Increasing the detection threshold can
increase the efficiency, however at the expense of completeness. In this
section, we compare the weak-lensing cluster detection with Gaussian smoothing
and with the MRLens, respectively. Following Hamana et al. (2004), we define
the efficiency $f_{e}$ and completeness $f_{c}$ of cluster detection with
respect to the number of clusters (dark matter halos) above a certain mass
threshold. Specifically, we have

$f_{e}=\frac{N_{iii}}{N_{i}},$ (31) $f_{c}=\frac{N_{iii}}{N_{ii}},$ (32)

where $N_{i}$ denotes the number of convergence peaks with their heights above
a detection threshold, $N_{ii}$ represents the number of dark matter halo with
mass above a certain mass threshold, and $N_{iii}$ is the number of peaks that
have correspondences with dark matter halos among $N_{ii}$. A peak is defined
to be associated with its nearest dark matter halo if the location of the peak
is within a radius of $12$ pixels (corresponds to $2.11\hbox{ arcmin}$) around
the halo. If there are two or more peaks associated with a same halo, the
highest peak is defined to have the correspondence with the halo.

Figure 6: The efficiency $f_{e}$ and completeness $f_{c}$ as functions of the
peak detection threshold $\kappa$. The left and right panels are for
$n_{g}=30\hbox{ arcmin}^{-2}$ and $n_{g}=100\hbox{ arcmin}^{-2}$,
respectively. The upper two panels are for the Gaussian smoothing and the
lower two panels are for MRLens. The lines with symbols are for the
completeness, and the lines without symbols are for the efficiency. The red,
green and blue lines are for the results of halos with $M>5\times
10^{13}M_{\odot}$, $M>1\times 10^{14}M_{\odot}$, and $M>2\times
10^{14}M_{\odot}$, respectively.

Figure 6 shows the results of $f_{e}$ and $f_{c}$ for Gaussian smoothing
(upper panels) and MRLens (lower panels). The left panels are for
$n_{g}=30\hbox{ arcmin}^{-2}$, and the right panels are for $n_{g}=100\hbox{
arcmin}^{-2}$. In each panel, the red, green and blue lines are for halos with
mass $M>5\times 10^{13}M_{\odot}$, $M>1\times 10^{14}M_{\odot}$, and
$M>2\times 10^{14}M_{\odot}$, respectively. The lines with and without symbols
are, respectively, for the results of completeness and efficiency. The
horizontal axis in each panel is the peak detection threshold $\kappa$.

We first analyze the Gaussian smoothing cases. As we discuss previously, such
a smoothing process reserves more or less all the real peaks with scales above
the smoothing scale. At mean time, the number of noise peaks is large at
$\kappa<3\sigma_{0}$. Thus a high completeness and a low efficiency are
expected when the peak detection threshold is low. For $n_{g}=30\hbox{
arcmin}^{-2}$ (upper left), we have $\sigma_{0}\sim 0.015$. At the detection
threshold $\kappa=0.02\sim 1.3\sigma_{0}$, we have the completeness $f_{c}\sim
50\%,65\%$ and $70\%$ for $M>5\times 10^{13}\hbox{ M}_{\odot}$, $1\times
10^{14}\hbox{ M}_{\odot}$ and $2\times 10^{14}\hbox{ M}_{\odot}$,
respectively. The corresponding efficiencies are $30\%,10\%$ and $2\%$. When
the detection threshold $\kappa>3\sigma_{0}$, the number of noise peaks drops
significantly, leading to a large increase in the detection efficiency. On the
other hand, a considerable fraction of halos are missed due to the high
detection threshold, resulting a decrease in the completeness. Specifically,
at $\kappa=0.045\sim 3\sigma_{0}$, the completeness $f_{c}\sim 20\%,35\%$ and
$60\%$, and the efficiency $f_{e}\sim 50\%,25\%$ and $10\%$, for $M>5\times
10^{13}\hbox{ M}_{\odot}$, $1\times 10^{14}\hbox{ M}_{\odot}$ and $2\times
10^{14}\hbox{ M}_{\odot}$, respectively. With the increase of $n_{g}$ to
$n_{g}=100\hbox{ arcmin}^{-2}$ (upper right), the noise level $\sigma_{0}$
decreases by a factor of $\sqrt{100/30}$ to $\sigma_{0}\sim 0.008$. Thus
$3\sigma_{0}$ corresponds to $\kappa\sim 0.025$. At this detection threshold,
the number of noise peaks is smaller and correspondingly the efficiency is
higher than those with $n_{g}=30\hbox{ arcmin}^{-2}$. On the other hand, the
number of real peaks does not change much as the noise level decreases, and
thus the completeness is similar to that of $n_{g}=30\hbox{ arcmin}^{-2}$.
Quantitatively, at the threshold $\kappa=0.025$, the efficiency $f_{e}\sim
50\%,20\%$ and $8\%$, in comparison with $f_{e}\sim 35\%,12\%$ and $3\%$ in
the case of $n_{g}=30\hbox{ arcmin}^{-2}$, for $M>5\times 10^{13}\hbox{
M}_{\odot}$, $1\times 10^{14}\hbox{ M}_{\odot}$ and $2\times 10^{14}\hbox{
M}_{\odot}$, respectively. For the completeness, we have $f_{c}\sim 30\%,50\%$
and $80\%$ for $n_{g}=100\hbox{ arcmin}^{-2}$. For $n_{g}=30\hbox{
arcmin}^{-2}$, $f_{c}\sim 45\%,60\%$ and $70\%$. While being similar, $f_{c}$
decreases somewhat for $M>5\times 10^{13}\hbox{ M}_{\odot}$ and $1\times
10^{14}\hbox{ M}_{\odot}$ with the decrease of noise level. This is in
accordance with the analyses of Fan et al. (2010) where they find that the
existence of noise generates a systematic shift for the real peaks toward
higher amplitudes. The shift depends on the density profile of dark matter
halos associated with the real peaks, and can be as high as $\sim 1\sigma_{0}$
for NFW halos with low concentrations. In terms of $\kappa$ values, the shift
is larger for larger $\sigma_{0}$. Thus, in the case of $n_{g}=30\hbox{
arcmin}^{-2}$, the relatively large $\sigma_{0}$ leads to a large shift of the
real peak heights and consequently a larger number of real peaks above the
detection threshold than that in the case of $n_{g}=100\hbox{ arcmin}^{-2}$.

For MRLens, with $n_{g}=30\hbox{ arcmin}^{-2}$ (lower left), the completeness
of the weak-lensing cluster detection is very low, and $f_{c}\sim 10\%,20\%$
and $40\%$ at the threshold $\kappa=0.02$, in comparison with $f_{c}\sim
50\%,65\%$ and $70\%$ in the corresponding Gaussian smoothing case. This is
because the suppression of the wavelet coefficients aiming to reduce noise
removes a large fraction of real peaks in the range of $\kappa<0.1$ as seen
from Figure 5. The total number of peaks in $3\times 3\hbox{ deg}^{2}$ with
$\kappa\geq 0.02$ is only $\sim 53$, while the total number of halos in the
area with $M>5\times 10^{13}\hbox{ M}_{\odot}$ is $\sim 530$. Thus although
the efficiency in MRLens here is rather high ($\sim 80\%$ for $M>5\times
10^{13}\hbox{ M}_{\odot}$), the very few number of detected halos makes the
MRLens method be disadvantageous in comparing with that of simple Gaussian
smoothing method. For $n_{g}=100\hbox{ arcmin}^{-2}$, the noise level is lower
and thus the removal effect is less significant than the case of
$n_{g}=30\hbox{ arcmin}^{-2}$. Consequently, the completeness increases
considerably with $f_{c}\sim 20\%,40\%$ and $65\%$. Meanwhile, the efficiency
decreases somewhat. In this low noise case, the differences between the MRLens
and Gaussian smoothing in terms of the completeness and efficiency are less
than those of high noise case. But still, the completeness is lower for
MRLens, especially considering relatively low mass halos with $M>5\times
10^{13}\hbox{ M}_{\odot}$.

To further demonstrate the differences between the Gaussian smoothing and the
MRLens, in Figure 7, we show the peak-halo correspondences explicitly in $z-M$
plane for one of our $3\times 3\hbox{ deg}^{2}$ simulation maps, where $M$ is
the halo mass in unit of $10^{13}M_{\odot}$ and $z$ is the halo redshift from
simulations. The halos are the ones located in the solid angle of $3\times
3\hbox{ deg}^{2}$ in the considered direction and their redshift and mass are
taken directly from the halo catalogs constructed by White and Vale (2004).
The upper and lower panels are for the Gaussian smoothing and the MRLens,
respectively. The left and right panels correspond to $n_{g}=30\hbox{
arcmin}^{-2}$ and $n_{g}=100\hbox{ arcmin}^{-2}$, respectively. In each panel,
the ‘+’ symbols denote the dark matter halos identified in simulations with
$M\geq 5\times 10^{13}M_{\odot}$ and in the redshift range of $0\leq z\leq 2$.
There are very few halos extending to redshift beyond $z=2$. The green squares
represent those halos that have corresponding convergence peaks with
$\kappa\geq 0.02$. The differences between the two filtering methods are
strikingly seen. For MRLens with $n_{g}=30\hbox{ arcmin}^{-2}$ (lower left), a
majority of halos with $M<10^{14}\hbox{ M}_{\odot}$ or with $z>0.8$ are missed
in weak-lensing detections, consistent with its extremely low completeness
shown in Figure 6. Lowering the noise level by increasing $n_{g}$ to
$n_{g}=100\hbox{ arcmin}^{-2}$ increases the number of halos with associated
peaks by nearly a factor of $2$ (lower right). But the number is still much
less than that in the Gaussian smoothing case. Therefore in studies aiming to
detect a large number of clusters from blind surveys and subsequent
cosmological applications, the Gaussian smoothing method is clearly much
better than the MRLens. In addition, the noise field after a Gaussian
smoothing with $\theta_{G}\sim 1\hbox{ arcmin}$ is approximately Gaussian in
statistics, and thus its effects on weak-lensing cluster detection can be
modeled much easier than the case of MRLens where the left-over noise is
statistically highly non-Gaussian (Fan et al. 2010).

Figure 7: Peak-halo correspondences in $z-M$ plane. The upper and lower panels
are for the Gaussian smoothing and MRLens, respectively. The left and right
panels are for $n_{g}=30\hbox{ arcmin}^{-2}$ and $n_{g}=100\hbox{
arcmin}^{-2}$, respectively. In each panel, the ‘+’ symbols show all the halos
with $M\geq 5\times 10^{13}M_{\odot}$ and the squares show the halos with
corresponding convergence peaks with $\kappa\geq 0.02$.

In MRLens, the $\alpha_{0}$ parameter plays a crucial role in classifying
significant and non-significant wavelet coefficients. A larger $\alpha_{0}$
leads to a larger fraction of significant coefficients, and thus a less
suppression effect in MRLens restoration. To see if the problem of low
completeness in MRLens cluster detection can be largely improved by increasing
$\alpha_{0}$, we analyze the $\alpha_{0}$ dependence for the completeness
$f_{c}$ as well as for the efficiency $f_{e}$. The results are shown in Figure
8. The upper and lower panels are for $n_{g}=30\hbox{ arcmin}^{-2}$ and
$n_{g}=100\hbox{ arcmin}^{-2}$, respectively. The red, green and blue lines
are for $M\geq 5\times 10^{13}M_{\odot}$, $1\times 10^{14}M_{\odot}$, and
$2\times 10^{14}M_{\odot}$, respectively. The peak detection threshold is set
to be $\kappa=0.02$. We can see that both $f_{c}$ and $f_{e}$ are not very
sensitive to $\alpha_{0}$. Increasing $\alpha_{0}$ from $0.01$ to $0.1$
improves the completeness only by $\Delta f_{c}\sim 10\%$ for both
$n_{g}=30\hbox{ arcmin}^{-2}$, and $n_{g}=100\hbox{ arcmin}^{-2}$. Meanwhile,
the efficiency decreases by $\Delta f_{e}\sim 10\%-20\%$. Therefore increasing
$\alpha_{0}$ cannot overcome the shortcoming of MRLens considerably. Comparing
to the Gaussian smoothing cases, the completeness is still low for MRLens
weak-lensing cluster detection even with $\alpha_{0}=0.1$.

We then conclude that in probing cosmologies with weak-lensing cluster
abundance analyses, in which a large sample of clusters is needed, the
Gaussian smoothing method performs much better than the MRLens method. To
overcome the relatively low efficiency for low peaks in the Gaussian smoothing
treatment, a detection threshold $\kappa>3\sigma_{0}$ is normally set. In Fan
et al. (2010), the noise effects on convergence peak statistics can be
accurately modeled for the Gaussian smoothing method. Therefore it is
potentially possible to even include peaks with $\kappa<3\sigma_{0}$ in the
abundance analyses, which can increase the number of detected clusters greatly
so that to strengthen the derived cosmological constraints on different
parameters. This will be explored further in our future studies.

Figure 8: The $\alpha_{0}$ dependence of the completeness $f_{c}$ and the
efficiency $f_{e}$. The upper and lower panels are for $n_{g}=30\hbox{
arcmin}^{-2}$ and $n_{g}=100\hbox{ arcmin}^{-2}$ respectively. The peak
detection threshold is set to be $\kappa=0.02$. The red, green and blue lines
are for the results of halos with $M>5\times 10^{13}M_{\odot}$, $M>1\times
10^{14}M_{\odot}$, and $M>2\times 10^{14}M_{\odot}$, respectively. The lines
with and without symbols are for the completeness $f_{c}$ and the efficiency
$f_{e}$, respectively.

##  5 Summary

Constructing cluster samples through their weak-lensing effects has been an
important aspect of weak-lensing studies. Their statistical abundance contains
valuable cosmological information. Observations have shown the feasibility in
detecting clusters with weak-lensing effects (e.g., Wittman et al. 2006;
Dietrich et al. 2007; Gavazzi & Soucail 2007; Schirmer et al. 2007; Hamana et
al. 2009). In conjunction with optical observations, the detailed analyses on
the completeness and efficiency of weak-lensing selected cluster samples also
become possible (e.g., Geller et al. 2010). It is noted, however, the
efficiency and completeness depend on the method applied to reconstruct the
convergence field from shear measurements. Different methods can result
residual noise with different statistical properties, and can also change the
weak-lensing signals differently. In order to extract cosmological information
from observations, it is therefore crucial to understand how a particular
reconstruction method affects the results in detail.

In this paper, we systematically compare the Gaussian smoothing method and the
MRLens treatment to suppress noise from intrinsic ellipticities in convergence
maps. We concentrate on convergence peak statistics. It is found that while
the MRLens method can remove noise very effectively, it mistakenly removes a
large fraction of real peaks associated with clusters of galaxies. For
$n_{g}=30\hbox{ arcmin}^{-2}$, the number of peaks with $\kappa\geq 0.02$
after MRLens filtering is only $\sim 50$ in an area of $3\times 3\hbox{
deg}^{2}$ in comparison with $\sim 530$ for the number of halos of $M>5\times
10^{13}\hbox{ M}_{\odot}$. On the other hand, for the Gaussian smoothing
treatment, the number of detected clusters is $\sim 260$. Even with the
detection threshold $\kappa=3\sigma_{0}\sim 0.045$, which is normally set in
the Gaussian smoothing treatment to reduce the number of noise peaks in the
peak catalog and thus to increase the cluster detection efficiency, the number
of detected clusters is $\sim 100$, twice as many as that in the MRLens
filtering with the threshold $\kappa=0.02$. As the accuracy of statistical
abundance analyses depends crucially on the number of detected clusters, the
Gaussian smoothing method is therefore strongly favored to detect clusters as
many as possible. Furthermore, the Gaussian smoothing leads to a noise field
which is approximately Gaussian in statistics, while the residual noise from
MRLens filtering is highly non-Gaussian. Therefore the noise effects can be
modeled more straightforwardly for the Gaussian smoothing case than that of
MRLens (e.g., van Waerbeke 2000; Fan 2007; Fan et al. 2010). The recent
studies of Fan et al. (2010) on the weak-lensing peak statistics with noise
included provide an analytical model for the efficiency of peak detections in
the Gaussian smoothing case. Thus it is possible for us to include peaks with
$\kappa<3\sigma_{0}$ in the analyses. Then the number of detected clusters can
increase considerably, which in turn can lead to a significant improvement in
the cosmological constraints derived from weak-lensing cluster statistics.

###### Acknowledgements.

This research is supported in part by the NSFC of China under grants 10373001,
10533010 and 10773001, and the 973 program No.2007CB815401. HuanYuan Shan is
very grateful for the hospitality of CPPM.

## References

  * [Albrecht(2006)] Albrecht, A., Bernstein, G., Cahn, R., et al., 2006, astro-ph/0609591
  * [Benjaminet al(2007)] Benjamin, J., Heymans, C., Semboloni, E., et al., 2007, MNRAS, 381, 702
  * [Benjamini & Hochberg1995] Benjamini, Y. & Hochberg, Y., 1995, J.R.Stat.Soc.B, 57, 289
  * [Bartelmann(1995)] Bartelmann, M., 1995, A&A, 303, 643
  * [Bartelmannet al.(1996)] Bartelmann, M., Narayan, R., Seitz, S.,& Schneider, P., 1996,ApJ, 464, L115
  * [Bartelmann & Schneider(2001)] Bartelmann, M.,& Schneider, P., 2001, Physics Reports, 340, 291
  * [Bondet al(1987)] Bond, J. R., & Efstathiou, G., 1987, MNRAS, 226, 655
  * [Bridleet al.(1998)] Bridle, S. L., Hobson, M. P., Lasenby, A. N.,& Saunders, R., 1998, MNRAS, 299, 895
  * [Dietrichet al(2007)] Dietrich, J. P., Erben, T., Lamer, G., Schneider, P., Schwope, A., Hartlap, J., & Maturi, M., 2007, A&A, 470, 821
  * [Fan(2007)] Fan, Z. H. 2007, ApJ, 669, 10
  * [Fanet al(2010)] Fan, Z. H., Shan, H. Y., & Liu, J. Y., 2010, ApJ, 719, 1408
  * [Gavazziet al(2007)] Gavazzi, R., & Soucail, G., 2007, A&A, 462, 459
  * [Gelleret al(2010)] Geller, M. J.,Kurtz, M., J., Dell’Antonio, I. P., Ramella, M., & Fabricant, D. G., 2010, ApJ, 709, 832
  * [Hamanaet al.(2004)] Hamana, T., Takada, M.,& Yoshida, N., 2004, MNRAS, 350, 893
  * [Hamanaet al(2009)] Hamana, T.,Miyazaki, S., Kashikawa, N., Ellis, R. S., Massey, R. J., Refregier, A.,& Taylor, J. E., 2009, PASJ, 61, 833
  * [Hoekstraet al.(2006)] Hoekstra, H., Mellier, Y., van Waerbeke, L., et al., 2006, ApJ, 647, 116
  * [Hopkinset al.(2002)] Hopkins, A. M., Miller, C. J., Connolly, A. J., Genovese, C., Nichol, R. C., & Wasserman, L., 2002, AJ, 123, 1086
  * [Jain & Van Waerbeke(2000)] Jain B.,& Van Waerbeke L., 2000, ApJ, 530, L1
  * [Kaiser & Squires1993] Kaiser, N.,& Squires, G., 1993, ApJ, 404, 441
  * [Kaiser(1995)] Kaiser, N., 1995,ApJ, 439, L1
  * [Kilbingeret al(2009)] Kilbinger, M., Benabed, K., Guy, J., et al, 2009, A&A, 497, 677
  * [Liet al.(2009)] Li, R., Mo, H. J., Fan, Z. H., Cacciato, M., van de Bosch, F., Yang, X. H., & More, S., 2009, MNRAS, 394, 1016
  * [Marshallet al.(2002)] Marshall, P. J., Hobson, M. P., Gull, S. F.,& Bridle, S. L., 2002, MNRAS, 335, 1037
  * [Masseyet al(2007)] Massey, R., Heymans, C.,& Berge, J., et al. , 2007, MNRAS, 376, 13
  * [Milleret al.(2001)] Miller, C. J., Genovese, C., Nichol, R. C., et al., 2001,AJ, 122, 3492
  * [Miyazakiet al.(2002)] Miyazaki, S., Hamana, T., Shimasaku, K., et al., 2002, ApJ, 580, 97
  * [Pireset al.(2009)] Pires, S., Starck, J-L., Amara, A., Refregier, A.& Teyssier, R., 2009, A&A, 505, 969
  * [Schirmeret al(2007)] Schirmer, M., Erben, T., Hetterscheidt, M., & Schneider, P., 2007, A&A, 462, 875
  * [Schneider & Seitz(1995)] Schneider, P.& Seitz, C., 1995,A&A, 294, 411
  * [Schneider(1996)] Schneider, P., 1996, MNRAS, 283, 837
  * [Seitzet al.(1998)] Seitz, S., Schneider, P.,& Bartelmann, M., 1998,A&A, 337, 325
  * [Squires & Kaiser(1996)] Squires, G.,& Kaiser, N., 1996, 473, 65
  * [Starcket al.(1995)] Starck, J.-L., Murtagh, F., & Bijaoui, A., 1995,ASPC, 77, 279
  * [Starcket al.(2001)] Starck, J.-L., Murtagh, F., Querre, P.,& Bonnarel, F., 2001,A&A, 368, 730
  * [Starcket al.(2006)] Starck J.-L., Pires S. & Réfrégier A., 2006, A&A, 451, 1139
  * [van Waerbeke(2000)] van Waerbeke L., 2000, MNRAS, 313, 524
  * [Whiteet al(2002)] White, M., Van Waerbeke, L., & Jonathan, M., 2002, ApJ, 575, 640
  * [White & Vale(2003)] White, M., & Vale, C., 2004, Astroparticle Physics, 22, 19
  * [Wittmanet al(2006)] Wittman, D., Dell’Antonio, I. P., Hughes, J. P., Margoniner, V. E., Tyson, J. A., Cohen, J. G.,& Norman, D., 2006, ApJ, 643, 128