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Some Comments to the W-projection algorithm. L. Kogan National Radio AstronomyObservatory Socorro, NM USA. - PowerPoint PPT Presentation
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June 2009 ARPG
Some Comments to the W-projection algorithm.
L. KoganNational Radio AstronomyObservatory
Socorro, NM USA
June 2009 ARPG
1.W PROJECTION: A NEW ALGORITHM FOR NON-COMPLANAR BASELINEST.J. Cornwell, K. Golap, and S. Bhatnagar, EVLA memo 67, December 2003 “MEMO”
2. The non-complanar baselines effect in radio interferometry:W-projection algorithmT.J. Cornwell, K. Golap, and S. BhatnagarJuly 2008 “PAPER”
June 2009 ARPG
.
The visibilities with W 0 can be recalculated to visibilities with W=0 with the help of convolution.
Frater R.H. was the first one discussing this idea (“Image Formation from Coherence Functions in Astronomy”, edited by C. Van Schooneveld, 1978)He considered the unpractical case of full UV coverage at each W layer.
W-projection is implementation of this idea for the general case including the poor UV coverage in some W layers.
June 2009 ARPG
The projection of the measured visibilities at W 0 to the desired visibilities at W = 0
ondistributi brightness source theis )(
vectorbaseline theofcomponent U theis
plane picture source at the coordinate theis :where
)2exp()exp()exp()()( 220
lB
U
l
dlUljWljWljlBuVisw
)exp(function theof ansformFourier tr theis G
nconvolutiofor stands :where
)(G )( G )(
2
111000
Wlj
dUUUUVisVisuVis
w
wwwww
June 2009 ARPG
How wide should be the gridding function ?)(UGw
This question is very important because the wider it This question is very important because the wider it is the more computation time is required.is the more computation time is required.
The W projection memo says: The W projection memo says: ““The support the W-dependent gridding function The support the W-dependent gridding function grows with both W and the field of view typically up grows with both W and the field of view typically up to a largest value of about 70 by 70 pixels.”to a largest value of about 70 by 70 pixels.”
I evaluated the convolution (gridding) function I evaluated the convolution (gridding) function analytically to verify this statement in particular.analytically to verify this statement in particular.
June 2009 ARPG
Analytical evaluation of the convolution function
function complex a is )(
radians;in view,of field full theis 2
gth;in wavelen sy' visibilit theof valueis
gth;in wavelenlayer W at the sy' visibilit theof valueis where
)2exp()exp( )(L
L-
2
UG
L
UU
WW
dlUljWljUG
w
w
June 2009 ARPG
gthsin wavelen is W radians;in view,of field of half is L
82U
41 pixel 1 pixelsin
0.5 level at thefunction n convolutio theof width full The
integrals. Frenels are 2
sin (x)
;2
cos C(x)
2 ;2 where
arctan)(
)]()([)]()([)(
2pix0.5
2
0
2
0
21
)()()()(
2
212
2122
1
12
12
WL
L
dttS
dtt
WW
ULxW
W
ULx
W
UUGPHAS
xSxSxCxCUG
x
x
xCxCxSxS
w
ww
June 2009 ARPG
The full width of the convolution function for the 4 VLA configurations at wavelength 4m
VLA,config D C B A
250 750 2500 7500
21 62 210 625
/maxW
5.02 pixU
June 2009 ARPG
00
2
111000
ies visibilitdesired the to ies visibilitmeasured theof
n)(projectioion recalculatfor base almathematic theisequation This
)exp(function theof ansformFourier tr theis G
nconvolutiofor stands :where
(*) )(G )( G )(
ww
w
wwwww
VisVis
Wlj
dUUUUVisVisuVis
• But the relation (*) is correct only if the measured visibilities are available everywhere.
• In reality only few points of can be available at one W layer especially for large W!!!
)( 10 UVisw
June 2009 ARPG
narrow).ly indefinite is DM beamdirty (The
coverage. UVfull theof case at theonly
completely canceled is W term theofeffect that showsequation The
) )(exp()()(
)exp( )exp()()(
)exp())]exp()(( )([
)( )]( )([ )()(
function) ( ies visibilitmeasuredactually oflocation
at the functions delta ofset theand tsmeasuremen ofset indefinite
theofproduct a as ies visibilitmeasured ofset thengrepresenti
map),(Dirty (*)equation of )( ansformFourier tr down the Write
2211
21
21
21
2
001
dlllWjllDBlB
WljdlWljlBllDB
WljWljlBlDB
GVisVislDM
w
w
w
wwwww
June 2009 ARPG
)exp( of ansformFourier tr 3.
nconvolutio theof FFT 2.
)exp( of nsform tra
Fourier with plane UVat the pixelby pixeln Convolutio 1.
:includes n timecomputatio algorithm projection- WThe
)exp(by pixelby pixeltion Multiplica 2.
)(DM calculate toFFT 1.
: includes n timecomputatio The
)exp())]exp()(( )([)(DM
)exp(by )(DMMultiply 2.
))exp()(( )()(DM
:locations baseline
at the sampled theof ansformFourier tr Take 1.
:layereach W for sopperation following thefollowing
obtained becan )( for theequation identical The
21
21
21
1
21
22
211
21
0
1
Wlj
Wlj
Wlj
l
WljWljlBlDBl
Wljl
WljlBlDBl
Vis
lDM
w
w
w
w
June 2009 ARPG
)2(l
lat diminished is W term theofeffect thereforeand
)(2)( pixels) (several narrow is )(
))(exp()()( )(
DM mapdirty summed for the expression following get the llwe'
),( severemost islayer each W at W term theofeffect the
gconsiderin and togetherlayers W all summarized Having
1
122
11
221max11
sum
max
l
llllllDB
dlllWjllDBlBlDM
W
sum
sumsum
June 2009 ARPG
The computation time comparison
• The authors of the W-projection declare its advantage in computation times 10-50 times.• But I heard this number as several times only!• The W-projection paper compares the computation time
with uvw-space facets algorithm existed inside of CASA.• The big number can reflect the bad quality of uvw-space
facets algorithm but not the good quality of the W-projection algorithm !
• It’d be more fare to compare with image-plane facet algorithm existed at AIPS.
