S. Bhatnagar and T. J. Cornwell

National Radio Astronomy Observatory

Socorro, New Mexico

Scale Sensitive Deconvolution

Deconvolution: Problem definition

Interferometers measure the data in Fourier space

Final product needed is the image

Measurement Equation:

F: Fourier Transform operator

The goal is to estimate the Model Image (IM), given the measurement (V) and an estimate of the PSF (B).

Inverse of the Beam Matrix does not exist. Direct methods for deconvolution are not practical

Represent : P is the Pixel Model and pk are the Degrees of Freedom (DOFs)

Deconvolution: As an optimization problem

is an optimal estimator for a gaussian random process (the noise).

Minimize: w.r.t.

Step size Residual image provides the update direction

is an optimal estimator provided the model for the data fundamentally separates signal from noise

Scale-less deconvolution:

Scale (correlation length) fundamentally separates the signal from the noise.

The image is decomposed into delta functions at discrete pixel locations (quantized). Ak is the only parameter.

Clean Iterations:

Each pixel is an independent DOF

Dimensionality of the search space: No. of pixels in the Box

Minimize along the axis of maximum derivative

Scale-less deconvolution:

Regularization:

- Box (limit the search space)

- Maximum number of components (limit over-fitting)

MEM: Constrained minimization:

Example: Clean (50K components)

Diagonal approximation: Hessian: H ~ Hii

Adaptive Scale Pixel (ASP) Model:

Interferometric PSF has widespread side lobes

Diagonal or band-diagonal approximation

of the Hessian is not sufficient

Decompose the image into a scale

sensitive basis (Aspen).

Minor cycle:

Find a set of 'active' Aspen {A}

Solve for the best-fit set {A + New Asp at Max:IR}={Ai}

Compute: IR=ID - B*{Ai}

Major cycle:

Compute: ViR=VO-FIiM and IiR=FViR

Asp deconvolution: Example

Minimizes the number of DOFs used

Iterations are not independent

Slower: Step size computation needs convolution

Sensitive to the local scale and SNR

Detects overlapping and well separated scales equally well

Uses a continuous range of scales and positions

Asp deconvolution: Features

Asp deconvolution: ...Features

Uses continuous range of scales and positions

Uses least DOFs: An order of magnitude less compared to Clean/MSClean (50000/8000 vs. 600)

Not very sensitive to boxing

Optimization: Ageing of Aspen

Not all Aspen remain significant/active

...Optimization: Dimensionality reduction

Adaptively determine the set of 'active' Aspen

Merger of Aspen

Use approx. PSF to determine the set of active Asp Approximate PSF =

Product and convolution of Aspen is another Asp. Approximate Hij can be analytically computed

...Optimization...

Work in progress:

Limits on inner and outer scales

Non-symmetric pixel model with tighter support

Full Hessian to determine the set of active Aspen

Include other constraints (e.g. flux at each pixel>0)

Integrate the code with AIPS++

Sanjay Bhatnagar

2003-10-04T15:50:00