Note that the size of the linear system (number of equations and unknowns) corresponding to the image blurring process is proportional to the number of pixels in the image. Therefore, even for a meager 100 x 100 image we need to solve a system with 10000 equations and unknowns to deblur the image. Therefore, we will attempt to utilize a simple approximate, iterative approach - in the same spirit as the power method for eigenvalue/eigenvector computation. Given a strictly diagonally dominant n x n matrix A and a right hand side $ b \in \mathbf{R^n}$ (this will correspond to a blurred image in our case), we will construct a sequence of approximate solutions $x^k$ , k = 0, 1, 2, ... to the system Ax = b as follows.
Select an arbitrary $x^0 \in \mathbf{R^n} $
for k = 0,1,2...do
for i = 1,2..n do
Solve equation i independtly from all other equations: $$x_i^{k+1} = A_{ii}^{-1}(b_i-\sum_{j\ne i} A_{ij} x_j^k$$
end for
if $ \| x^{k+1} -x^k\| $ is small enough then
Stop, $x^{k+1}$ is an approximate solution
End if
End for
One can show that this algorithm, known as the Jacobi iteration, converges for strictly diagonally dominant matrices to the solution of the linear system Ax = b from an arbitrary initial guess $x^ (0)$.
Question
In the context of image deblurring, what would be a reasonable choice for the initial solution guess $x^{0}$ in this iterative algorithm?
Anyone have an idea for the answer to this question? Thank you in advance