DETECTION OF COMPACT SOURCES IN CMB MAPS:
The Biparametric Scale Adaptive Filter
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2
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1
M. López-Caniego , D. Herranz , R. B. Barreiro , J. L. Sanz
(1) Instituto de Física de Cantabria, Santander (Spain); (2) CNR - Istituto di Elaborazione della Informacione, Pisa (Italy)
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INTRODUCTION
Interest: detection of compact sources (signal) embedded in a background (1D case).
Goal: linear filtering of the data to eliminate partially the background and estimation of the amplitude of the source.
Examples of filters currently used: Matched Filter (MF), Mexican Hat Wavelet (MH), Scale Adaptive Filter (SAF).
Assumptions: Profile of the source known and background represented by an homogeneous and isotropic Gaussian
random field with given power spectrum.
Our approach: Find a combination of optimal filter and detector such that the number of detections is maximum
for a fixed number of spurious sources
Local Peak Detection: introduces not only amplification but also the curvature of the peaks, i.e., Simple n”SIGMA”
thresholding is not the whole story!
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We introduce a new detector that uses amplitude and curvature information. The curvature of the
maxima of the background and that of the maxima of the background + source are very different.
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The detector is obtained with the Neyman-Pearson rule, fixing the number of spurious sources
and maximizing the number of true detections.
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The region of acceptance R* giving the highest number density of detections n*, for a given number
density of spurious n*b, is the region L(n,k), or equivalently, j(n,k)
L(n , k ) º
n(n , k )
³ L*
nb (n , k )
If L >= L* => signal is present
If L < L* => signal is absent
1 - ry s
y -r
ì
C2 = s 2
ïC1 =
1- r 2
1- r
ï
j (n , k ) º C1n + C 2 k í
2
ïy º k s r º s1
ï s n
s 0s 2
s
î
The Background and the Source
1D background represented by an homogeneous and isotropic Gaussian random field x(x) with zero average
value and power spectrum P(q), q=|Q|, Q being the Fourier mode.
! Distribution of maxima (Rice, 1954): expected number density of maxima per intervals (x,x+dx), (n,n+dn)
and (k,k+dk) is
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I. To filter or not to filter the data:
3s source before filtering
17s source after filtering
For Bright Sources -> No filtering may be ok...
the source can be easily
detected
1
nb (n,k) =
(n +k -2rnk )
2
nb
k
e 2(1-r )
2p 1- r2
2
2
nb º
1
n º
2pb r
x
s0
- x"
s0
bº
s2
s2
kº
rº
s 12
s 0s 2
2
and s n is the moment of order 2n associated with the field.
-
0.5s source before filtering
For Weak Sources
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3s source after filtering
-> Filtering improves the
conditions for detection
x2
For a Gaussian source, with profile given by t ( x) = e 2 R , embedded in the previous background, the
expected number density of maxima is (Barreiro et al. 2003)
n(n , k | n s ) =
nbk
-
2p (1 - r )
2
2
(n -n s ) 2 + (k -k s ) 2 - 2 r (n -n s )(k -k s )
2 (1- r 2 )
e
IV. Numerical Results
We have tested these ideas numerically for two distributions of weak sources:
In our approach, the filtering and detection processes are not independent. We look for
the optimal filter and the optimal detector such that the number of detections is maximum
fixing the number of spurious sources.
Uniform Distribution in the interval [0,2]s
->
p(n s ) =
1
, n s Î [0, n c ]
nc
II. The Filters: Biparametric Scale Adaptive Filter
-g
.- Power spectrum P(q)=Dq , g spectral index of the
background
.- Combination of MF + MHW for g = 0.
.- Conditions to obtain it:
- <w(R0,0)>=A unbiased estimator of the amplitude
- The variance of w(R0,0) has a minimum at R0
- w(R0,b) has a maximum in the filtered image at b=0
.- The BSAF has two free parameters c and a,
- c: parameter to be obtained numerically
- a: a > 0, modifies the filtering scale R.
1
(
)
- x2
~
Y µ x g e 2 1 + cx 2 , x º aqR
RD: Relative Difference between BSAF & MF
Improvement [%] in n* w.r.t. the
MF(a=1) vs. thespectral index g
Number of detections n* vs a
Filtering at scales other than R, the natural scale of the
source, can improve the number of detections, as shown by
Vielva et al. (2001) and López-Caniego et al. (2004a).
Scale-free Distribution in the interval [0.5,3]s and b =0.5 ->
p (n s ) µ
1
ns
b
[
, n Î n in , n
fin
]
We introduce the parameter a in all the filters, allowing to modify the filtering scale R, and use the same detection
criterion to compare them.
1
Mexican Hat Wavelet
- x2
~
Y µ x g e 12
- x2 æ
t
~
ö
Y µ x g e 2 ç1 + 2 x 2 ÷
m
ø
1 2 è
- x
~
Y µ x 2e 2
Biparametric SAF
~
Y µ xg e
Matched Filter
Scale Adaptive Filter
1
2
x
2
(1 + c x )
2
x º aqR
1+ g
2
1- g
t=
2
m=
RD: Relative Difference between BSAF & MF
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CONCLUSIONS
Detection of compact sources:
“Optimal Filter + Optimal Detector” -> maximum number of detections for a fixed number of spurious.
We introduce an optimal filter, the BSAF, with an extra degree of freedom that allow us to filter at
scales different from that of the source. This improves the number of detections significantly.
We introduce a linear detector based on the Neyman-Pearson rule that takes into account a priori
information of the pdf’s of the sources, the amplitude and the curvature of the maxima.
We have tested numerically and with simulations these ideas, comparing different filters (MF, MHW,
SAF and BSAF). We obtained that the number of detections is under certain circumstances superior to
the other filters. In the most favorable case, white noise, g=0, there is a 40% improvement with
respect to the standard Matched filter.
In the 2D case, we obtain similar results. For g = 0, the BSAF improves the MF by 40% (LC 2004c).
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R.B. Barreiro et al., 2003, MNRAS, 342,119
López-Caniego et al. 2004a, SPIE.5299..145L
López-Caniego et al. 2004b, submitted
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III. The Detector
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Uses the information in terms of the probability distribution functions (pdf’s)
H0: Null Hypothesis
-> background alone
H1: Alternate Hypothesis - > background + signal
The detector divides the space R in two subspaces:
R* : H0 is rejected -> A signal is present
R-: H1 is accepted -> A signal is absent
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(R* is the Region of acceptance)
Example of a simple detector: “Thresholding”. The space R is defined by the objects above or below a
certain arbitrary threshold ns.
High threshold - > small number of detections
Low threshold - > many detections with a high probability of spurious sources
Improvement [%] in n* w.r.t. the
MF(a=1) vs. thespectral index g
Number of detections n* vs a
REFERENCES
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López-Caniego et al. 2004c, submitted
Vielva et al., 2001, MNRAS 326,181