Page 1

VLA Scientific Memorandum No. 141

Can CLEAN be improved ?

by

T.J.Cornwell NRAO/VLA

March 1982

1. Introduction:

The one major drawback to the use of the CLEAN deconvolution

algorithm is its poor behaviour on regions of extended emission;

particularly as manifested in the appearence of stripe-like artifacts in

the CLEAN image. Advantages over more sophisticated deconvolution

algorithms such as MEM are its speed, particularly in the Clark algorithm

( although a similar two stage approach to MEM is possible ) and

simplicity ( especially of programming ). However, algorithms such as MEM

are designed to treat correctly regions of extended emission. It is clear

that an ideal deconvolution algorithm would merge the best attributes of

both CLEAN and MEM. In this memo I will present details of an initial

attempt at designing such an algorithm.

2. The CLEAN algorithm:

CLEAN utilises an iterative point source subtraction technique to

minimise a chi-squared term which, in the u,v plane, can be written :

x2 = z wk*[V Tk ]2

-equation (2.1)

where

= observed complex visibility at the kth sample point

T^ = predicted complex visibility at the kth sample point

Wk = Weight attached to the kth sample point

and [ ] represents the absolute value. Here, and below, repeated indices

are to be summed.

In the map plane X2 can be written as :

X2 = I p. .*f.*f.

i J

- 2 d.*f. + £ w. *[V, ]2

1 1

k k J

-equation (2.2)

where p. . = beam matrix

d^ = dirty map vector

f^ = predicted map vector

and the summations cover the entire map.

Choosing f to maximise -X2 we find the usual convolution equation :

2

The CLEAN algorithm chooses one of the possible solutions of this

equation. Preference is given to those images containing a number of

point sources in a mainly empty field. The uniqueness of the predicted

map and the asymptotic value of X2 both depend on the number of

independent sample points and number of beam areas of non-zero brightness

allowed in the CLEAN map ( see Schwarz 1978).

3. A modification of CLEAN :

One would like to alter CLEAN such that regions of extended emission

are treated properly and in particular so that stripes, not constrained

by the data, are removed. One approach, which we will adopt here, is to

change the dirty map or dirty beam in some way and then just use CLEAN as

usual.

In general we may do this by maximising a combination of -x2 and

some other function which measures "good" maps. Let H(f)* be this

function; we then maximise

0 = a*H(f) - X2

-equation (3.1)

where the variable a controls the balance between fitting the data and

obtaining a "good" map.

The predicted map is found by solving :

Z p. .*f.

= d. + ct*dH/df.

i

J

J

-equation (3.2)

In all interesting cases H will depend non-linearly on f. Perhaps

the easiest method of solution is to use CLEAN to solve equation (3.5)

and then calculate the correction, a*dH/df^, to the dirty map, iterating

until convergence is achieved.

The optimum value of a can be estimated by multiplying equation

(3.5) by f. and summing. We then find that :

0.5*(X2+(I f*p±'*f±- Z wk*[Vk]2))

= a* Z f.*dH/df.

l

l

-equation (3.3)

The difference on the left hand side is related to the discrepancy

in signal to noise of the observed and reconstructed visibilities. For a

reasonably unbiased image we will assume that this vanishes. If the

expected value of X2 is o2 per pixel then :

-equation (2.3)

3

o = a2/(2*<f *dH/df >)

where < > denotes the average value.

The only missing ingredient is the goodness measure H; this we now

consider.

What is desired of the goodness measure of a map ? Two attributes

seem important :

1. Only positive brightness should be allowed although, of course,

negative residuals will be permissible. This constraint should be

dropped for Q and U maps.

2. Images having low dispersion in pixel values should be preferred;

spurious stripes in the image should then be removed.

Infinitely many functions satisfy these criteria; the most

interesting the various entropy measures. I use the term entropy merely

to denote the lack of spread in pixel values, not any physical concept.

Some of the entropy measures are :

HI = - Z f.*ln(f.)

i

i

-equation (3.5)

H2 = Z ln(f±)

-equation (3.6)

H3 = - Z l./f.

l

-equation (3.7)

H4 = - Z l./f.2

i

-equation (3.8)

H5 = Z Af.)

l

-equation (3.9)

and their cousins Hi', formed by normalising f^ with respect to the total

flux in the image. ( Wernecke and D’Addario used H2 whereas Gull and

Daniell used HI1.)

All of these measures are maximised for images with low dispersions

in pixel values and all require positivity. If we drop the positivity

constraint then the smoothness measure S is available :

S = - Z f.2

i

-equation (3.10)

-equation (3.4)

We will now go on to consider the use of these goodness measures

in a practical CLEAN-based algorithm.

4

4. The Maximum smoothness Method

Unfortunately, if smoothness is used in place of H then all the

permissible solutions to equation (3.5) are very close to the principal

solution of the ordinary convolution equation. However, if we use the

CLEAN algorithm to find the solution then we introduce the extra

constraint that the map be composed of a number of point sources. We

should obtain a smoother map than the usual CLEAN solution and one which

does not have the sidelobes found in the principal solution. One very

convenient aspect is that to find the solution to this "maximum

smoothness method” (MSM) we only need to modify the beam to be :

p. . + o2*6. ./(2*<f2.>)

i

-equation (4.1)

If the sidelobes are reasonably small then the mean square signal

can be estimated from the dirty map.

Since negative pixel values are allowed the zero spacing flux is not

biased as it is if an entropy measure is used. The resolution is

invariant over the field of view.

5. The Maximum Entropy Method:

Of the entropy measures H2 is the most convenient since o is

independent of f ( see equation (5.2) ). We must then solve ( using CLEAN

) :

I p. .*f. = d. + o2/(2*f.)

i

J

J

-equation (5.1)

In practice we go through the following sequence :

1. CLEAN dirty map to obtain initial CLEAN map. We then use this map

to approximate the MEM map.

2. Correct dirty map using the MEM map, truncating below some

arbitrary level e.g. a to avoid the forbidden negative values.

3. CLEAN the corrected dirty map

4. Goto 2. unless convergence is attained

The CLEAN beam may be chosen at will but in practice superresolution

seems not to work well, and should be avoided just as it is in

conventional CLEAN.

5.1. Pros and Cons:

Several desirable aspects of this general approach to MEM are

apparent :

Al: Regions of good signal to noise ratio are less affected than the

weaker regions. Ignoring the effect of sidelobes we find that the map is

given by :

5

f. = 0.5*(d.+/(d.2+2*a2)

1

1 1

'

which tends to d_^ for good signal to noise and to a fixed level o/V(2) as

the signal vanishs.

A2: A stability analysis indicates that small sinusoids not

required by the data are removed.

A3: Positivity is strongly encouraged.

A4: In the case of H2 only one pass through the image is required to

find the correction to the dirty map.

A number of disadvantages are also involved :

Dl: The predicted image is slightly biased. This bias is about 0.7o

for weak points and vanishs for strong points. For example, the zero

spacing flux for an MEM map of a blank field is non-zero.

D2: The resolution varies with signal to noise, consequently simple

interpretation of the image may be difficult.

D3: Several passes through the entire cycle are required, however

each pass is only marginally more expensive than CLEAN.

D4: The r.m.s noise j*s a free parameter and can be chosen at will.

Large values produce a ridiculously smooth map whereas small values have

virtually no discernable effect. Such free parameters will appear in any

non-linear deconvolution method such as MEM or regularisation. In fact

the CLEAN windows play a similar role in CLEAN.

D5: The clipping below some arbitrary level is unsatisfactory in

that it strongly affects the positivity of the final map. I can see no

easy way in which this can be avoided in the present scheme.

Several of these disadvantages might affect the application of such

pseudo-MEM maps to the estimation of spectral indices, percentage

polarisation, optical depth etc. We will now examine these in further

detail.

First we consider the bias. From equation (5.2) we see that for a

signal of xo the bias is, ignoring sidelobes, :

0.5*(/(x2+2)-x)*o

-equation (5.3)

For a 5o detection the bias is then about O.lo and for a 3 a

detection the bias is about 0.4a; in most practical cases this effect

will be negligible. Also by virtue of the positivity constraint MEM

should provide a better estimate of the zero spacing flux than CLEAN and

hence one may gain .

Secondly, we consider the variable resolution. The use of the CLEAN

beam avoids the problems introduced by superresolution. The converse of

superresolution, subresolution, which occurs on weak features may be

more serious. Using equation (5.2) we find that, ignoring sidelobes, the

increase in width of a 5o (3) Gaussian is about 4.3 per cent (11.7 per

cent). Hence one must be careful in quoting apparent sizes of weak

-equation (5.2)

6

sources. In most cases I would expect that this effect to be negligible

compared to the uncertainties due to non-Gaussian profiles.

Thirdly, for maps of strong sources a will contain a contribution

due to the limited dynamic range; this is probably best estimated either

from a blank region of the I-map. If a stripe is present then convergence

can be hastened by initially setting o to the amplitude of the stripe and

subsequently decreasing it to the correct value.

The relative importance of the advantages and disadvantages will

vary from case to case as happens with CLEAN and self calibration.

6. How do these methods work ?:

Suppose that after CLEANing a map we find that a small sinusoid

corresponding to an unmeasured part of the uv plane is present in the

map. The concave nature of the entropy measures and the smoothness

measure ensures that the dirty map is altered by the addition of a small

sinusoid phase shifted by 180 degrees.

We can now see that equation (3.2) simply uses ordinary feedback

methods to stabilise the CLEAN algorithm and, as such, could be derived

with no mention of entropy or smoothness.

7. An example:

Fig. 1 shows a dirty map of SAG A at 20cm courtesy of R.D.Ekers and

J.van Gorkom. Figure 2 shows the CLEAN map (loop gain = 0.1,10000

iterations ). Stripes are present in the map running along pa 30 degrees

with an amplitude of about 10 to 20 mJy per beam. A slice taken on a

vertical line is shown in Fig. 3. I applied the pseudo-MEM algorithm to

this data using values for o of 10,20,50 mJy per beam. Slices from the

resulting maps are shown in Fig. 4. For o=50 the sinewave has, as

expected, been reversed in phase and amplified whereas for o=10 and 20 it

has decreased somewhat. After two more iterations with o=10 the slice is

as shown in Fig. 5. The zero level has changed by about 5-10 mJy per beam

and the stripes have diminished considerably. It can be seen that , with

the exception of the stripes, the final structure, shown in Fig 6, has

changed very little. In Fig. 7 I show the usual slice through the MSM map

made with 0=10. The smoothness seems comparable to that of the MEM map.

The full MSM map is shown in Fig. 8.

8. Does this really help ?:

The presence of stripes in a CLEAN map indicates that something has

gone awry with the algorithm we all love and trust. Does this mean that

we should rely on a completely unknown process to cobble together a

reasonable looking map ? Well maybe, and maybe not. We could only CLEAN

data which has no big holes in the uv coverage but this sort of

conservatism is that which would prevent any use of CLEAN or

selfcalibration.

It is possible that some other solution to the stripe problem exists

relying on, say, a variable loop gain or adaptive boxes; however for

7

those who would prefer this type of approach I would point out that these

pseudo-MEM and pseudo-MSM algorithms should be regarded simply as means

of stabilising the CLEAN algorithm. I have made no mention of the

canonical ensemble of monkeys usually invoked in discussions of MEM; in

fact I regard the various entropy measures and the smoothness measure as

more or less arbitrary functions which are chosen primarily to stabilise

CLEAN.

On purely practical grounds MSM appears to preferable over MEM,

mainly because at most two passes through CLEAN are necessary and because

no bias is introduced. It also treats Q and U maps correctly.

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