WIPE Deconvolution

The second part of WIPE essentially concerns the final reconstructed image, the Neat Map. As WIPE is slow when reconstructing large images, it can be initialized with a few Clean iterations. The program will start by prompting for the number of clean components and the clean gain to use:

clean_nbr_iter......: 10
clean_gain..........: .2

It will then select the clean_nbr_iter first components of clean and display the collected flux and the relative variation:

CLEAN (0) ....: component:[38,68], FLUX:1.500229E-01, RELATIVE_VARIATION:1.000000E+00 
CLEAN (1) ....: component:[39,74], FLUX:2.957820E-01, RELATIVE_VARIATION:4.927923E-01 
CLEAN (2) ....: component:[41,63], FLUX:4.298387E-01, RELATIVE_VARIATION:3.118767E-01 
CLEAN (3) ....: component:[48,61], FLUX:5.520994E-01, RELATIVE_VARIATION:2.214468E-01 
CLEAN (4) ....: component:[40,70], FLUX:6.701481E-01, RELATIVE_VARIATION:1.761532E-01 
CLEAN (5) ....: component:[41,77], FLUX:7.874965E-01, RELATIVE_VARIATION:1.490145E-01 
CLEAN (6) ....: component:[44,73], FLUX:8.867114E-01, RELATIVE_VARIATION:1.118909E-01 
CLEAN (7) ....: component:[36,66], FLUX:9.877205E-01, RELATIVE_VARIATION:1.022648E-01 
CLEAN (8) ....: component:[43,64], FLUX:1.088139E+00, RELATIVE_VARIATION:9.228493E-02 
CLEAN (9) ....: component:[37,77], FLUX:1.181291E+00, RELATIVE_VARIATION:7.885618E-02 
CLEAN (10) ....: component:[46,58], FLUX:1.271183E+00, RELATIVE_VARIATION:7.071504E-02

Selecting too many components for Clean can slow down WIPE later on, since WIPE will impose a positivity constraint and will have to process additional iterations of the conjugate gradients method to achieve it. By experience, we recommend a value between 5 to 20 components although it is problem dependant as one could imagine.

Another way to go, is to use Clean as usually, produce the expected Clean Map and then perform only one iteration of WIPE in order to check the value of the condition number $\kappa$.

At this point the program displays the current Clean Map and Clean Residu in the SOL window (cf. figure 3).

  

Figure 3: of a SOL window at the clean stage.

After this preliminary initialisation step, the user is prompted whether he (or she) wants to pursue the reconstruction. The user can come back to clean and add components or pass the hand over to WIPE:

Q-WIPE,  LEAVE CLEAN & WIPE ? [y,n] n
Q-WIPE,  CLEAN or WIPE ? [c,w] w

When switching to WIPE, the program starts by performing a first iteration of the WIPE process, optimizing the solution with the support corresponding to Clean, using the conjugate gradients methods. This last method, at the expense of a few additional iterations, allows one to exhibit an approximation of the condition number $\kappa$ of the mapping and gridding operator. $\kappa$ is a majorant of the amplification of the data error through the reconstruction process. Therefore it sets an upper bound for the reconstruction error and gives an idea of the stability and reliability of the reconstruction.

WIPE.......: wipe_iter=0 
WIPE(0)....: iter_gc=0 
WIPE(0)....: iter_gc=1 mu=1.346324E+00, mu'=2.088283E+00, kappa=1.245431E+00 
WIPE(0)....: iter_gc=2 mu=1.046280E+00, mu'=2.561411E+00, kappa=1.564645E+00 
WIPE(0)....: iter_gc=3 mu=7.935231E-01, mu'=2.902714E+00, kappa=1.912592E+00 
WIPE(0)....: iter_gc=4 mu=6.996743E-01, mu'=3.116523E+00, kappa=2.110509E+00 
WIPE(0)....: iter_gc=5 mu=6.120918E-01, mu'=3.296474E+00, kappa=2.320687E+00 
WIPE(0)....: iter_gc=6 mu=5.726386E-01, mu'=3.376165E+00, kappa=2.428127E+00 
WIPE(0)....: iter_gc=7 mu=5.535087E-01, mu'=3.404158E+00, kappa=2.479948E+00 
WIPE(0)....: iter_gc=8 mu=5.400138E-01, mu'=3.416653E+00, kappa=2.515348E+00 

I-WIPE,  FLUX:3.664394E+00; RELATIVE_VARIATION:6.530987E-01

After the first iteration of WIPE, the current solution is updated in the SOL window. The WIPE solution (Neat Map and Wipe residu) replaces the Clean one, as represented in the figure 4).

  

Figure 4: of a SOL window at the Wipe stage.

According to the value of $\kappa$ and the information the user is interested in (if it is still in the residu and we are above the noise level) the program prompts for the number of additional Wipe steps to perform. Each Wipe step will increase the effective support following an embedded heuristic (the increase is of the order of the diameter of the Neat Beam).

WIPE.......: wipe_nbr_iter = 5

At each Wipe iteration, the Neat Map and the residu are updated. The outputs are similar to the ones of the very first WIPE iteration. After performing the requested number of iteration, the program asks whether to increase the support or not:

Q-WIPE,  EXTENSION OF THE IMAGE SUPPORT ? [y, n] n

And one could go on like that with one step increase and decide precisely when to stop the process.

The next request offers a way to go back to WIPE:

Q-WIPE   DO YOU WANT TO STOP WIPE ? [y, n] y

If we really want to leave WIPE, a final smoothing of the object support, removing small regions (basically a morphological operation), is performed:

I-WIPE,  SMOOTHING OF THE IMAGE SUPPORT 
I-WIPE,  region 1 = 684 pixels 
I-WIPE,  region 2 = 41 pixels 
I-WIPE,  region 3 = 5 pixels 
I-WIPE,  2 Discarded connected region(s) 
I-WIPE,  MINIMIZATION ON THE SMOOTHED SUPPORT 
WIPE.......: wipe_iter=7 
WIPE(7)....: iter_gc=0 
WIPE(7)....: iter_gc=1 mu=1.074615E+00, mu'=2.813862E+00, kappa=1.618173E+00 
WIPE(7)....: iter_gc=2 mu=8.191533E-01, mu'=2.949595E+00, kappa=1.897573E+00 
WIPE(7)....: iter_gc=3 mu=6.177616E-01, mu'=3.341397E+00, kappa=2.325699E+00 
WIPE(7)....: iter_gc=4 mu=4.993783E-01, mu'=3.584762E+00, kappa=2.679263E+00 
WIPE(7)....: iter_gc=5 mu=4.009354E-01, mu'=3.667690E+00, kappa=3.024538E+00 
WIPE(7)....: iter_gc=6 mu=3.518280E-01, mu'=3.688257E+00, kappa=3.237765E+00  
WIPE(7)....: iter_gc=7 mu=3.114740E-01, mu'=3.705962E+00, kappa=3.449369E+00  
WIPE(7)....: iter_gc=8 mu=2.808061E-01, mu'=3.721767E+00, kappa=3.640586E+00  
WIPE(7)....: iter_gc=9 mu=2.511003E-01, mu'=3.735208E+00, kappa=3.856859E+00 
WIPE(7)....: iter_gc=10 mu=2.344449E-01, mu'=3.746117E+00, kappa=3.997333E+00 
WIPE(7)....: iter_gc=11 mu=2.241068E-01, mu'=3.756186E+00, kappa=4.093983E+00 
WIPE(7)....: iter_gc=12 mu=2.189550E-01, mu'=3.760925E+00, kappa=4.144478E+00  
WIPE(7)....: iter_gc=13 mu=2.166695E-01, mu'=3.763188E+00, kappa=4.167533E+00  
WIPE(7)....: iter_gc=14 mu=2.158919E-01, mu'=3.763985E+00, kappa=4.175474E+00  
WIPE(7)....: iter_gc=15 mu=2.154566E-01, mu'=3.764431E+00, kappa=4.179937E+00 

I-WIPE,  FLUX:5.370097E+00; RELATIVE_VARIATION:-5.083192E-02

At this step, we still have the chance to go back to Clean or WIPE:

Q-WIPE,  DO YOU WANT TO LEAVE CLEAN & WIPE ? [y,n] y

If we really want to exit Clean and WIPE, the solution reached at this point will be the final one. But it is not yet finished and one can move forward to error analysis in order to refine the computation of the condition number $\kappa$ and the Neat Map solution, as we shall see in the next section.