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Projection Onto Convex Sets

Apart from the dirty image, the simplest class of Image Solvers are those that use the principle of Projection Onto Convex Sets. POCS is a simple but very general iterative algorithm. If one wants to solve a linear equation AX = Y then one uses an iterative algorithm given by successive and repeated applications of various projection operators:

$$\Delta$$X = T₁$$\left(\vphantom{ T_2 \left(T_3 ...T_N\left(B-AY \right) ...\right) }\right.$$T₂$$\left(\vphantom{T_3 ...T_N\left(B-AY \right) ...}\right.$$T₃...T_N$$\left(\vphantom{B-AY }\right.$$B - AY$$\left.\vphantom{B-AY }\right)$$...$$\left.\vphantom{T_3 ...T_N\left(B-AY \right) ...}\right)$$ $$\left.\vphantom{ T_2 \left(T_3 ...T_N\left(B-AY \right) ...\right) }\right)$$

where the projection operators T_i impose some form of a priori constraint e.g. positivity, finite support. CLEAN is a POCS algorithm. A POCS algorithm is guaranteed to converge for convex set.

In the context of the GI measurement equation, Y - AX is the residual image $\vec{\cal I}^{R}_{}$. The projection operators are open to choice. There are a number of possible operators:

Finite support

The finite support of the object can be represented by a projection operator, T^supp, that simply sets to zero all parts of estimate that are outside the region of support.

Positivity

The total intensity must be positive:

$$\geq$$ (5)

The corresponding projection operator, T^pos therefore projects invalid vectors onto the plane I = 0.

Fractional polarization

An analogy to positivity for the 4-vector $\vec{\cal I}\,$ is that the fractional polarization not exceed unity. Or:

$$\geq$$ (6)

This describes a hypercone in I, Q, U, V space. For points outside the cone, projection onto the surface of the cone can be done in a number of ways. The simplest operator, T^cone, sets to zero all elements outside the cone. Alternatively one could boost the I component until the point lies inside the cone, T^boost. The most appropriate operator, T^shrink, is probably that which projects perpendicular to the I axis so as to shrink Q, U, V to be less than or equal to I.

A terser, and perhaps useful equation expressing this constraint is that:

$$\vec{\cal I}^{T}_{} \vec{\cal I}\, \geq$$ (7)

where the matrix M is given by:

$$\left(\vphantom{{ \begin{array}{cccc} 1&0&0&0\\ 0&-1&0&0\\ 0&0&-1&0\\ 0&0&0&-1 \end{array} }}\right. \begin{array}{cccc} 1&0&0&0\\ 0&-1&0&0\\ 0&0&-1&0\\ 0&0&0&-1 \end{array} \left.\vphantom{{ \begin{array}{cccc} 1&0&0&0\\ 0&-1&0&0\\ 0&0&-1&0\\ 0&0&0&-1 \end{array} }}\right)$$ (8)

One can also think of polarized radiation in terms of the coherence matrix:

$$\cal {B} {1\over 2} \left(\vphantom{{ \begin{array}{cc} I+V&Q+iU\\ Q-iU&I-V \end{array} }}\right. \begin{array}{cc} I+V&Q+iU\\ Q-iU&I-V \end{array} \left.\vphantom{{ \begin{array}{cc} I+V&Q+iU\\ Q-iU&I-V \end{array} }}\right)$$ (9)

in which case the equivalent statement is that this matrix be positive semi-definite, and so the product of eigenvalues is non-negative:

$$\lambda_{1}^{} \lambda_{2}^{} \geq$$ (10)

Pointedness

The operator T^point returns the maximum as a delta function, suitably scaled. Used in conjuction with the finite support operator, this yields the Hogböm CLEAN algorithm. I will say more on CLEAN below.

Clumpiness

The operator T^clump returns a clump of components greater than some fraction of the peak. This is analogous to the Steer-Dewdney-Ito variant of CLEAN.