MLE) algorithm68 69 are classified as constrained iterative algorithms. The object estimation is
achieved in a iterative process by imposing various constraints. The positive constraint, a
primary and primitive one, ensures the estimate of the object, 6(x, y, z), is always positive. In
each iteration, a new estimate of object Ok+1(u, v, w) is calculated using I(u,v,w), H(u,v,w),
and Ok(u, v, w) from the last iteration. When the object converges or other criteria reach a preset
threshold, such as the maximum number of iterations is reached, the iteration process stops, and
the current O(u, v, w) is inverse Fourier transformed to the spatial domain. In practice, O(u, v, w)
from the inverse filter deconvolution algorithm is often used as an approximation in the first
iteration. The Jansson Van-Cittert, Gold, and MLE algorithms are different in the new object
that are generated in each iteration; and they are expressed as
(k+1(u, v, w) =K (k)(u,V, w) + W [I(u, ) (k) (, V,). 6(k) (u v, w) (B-7)
6(k+l)(u, v,w)=K6(k) (u,v, ) (u,vw) (B-8)
Hj(k) (u, v, w) (k) (u, v, w)
and,
(k+1) "(k) F________Y__Z__X, Z)((B -V9z)
o (x,y,z)= Ko (x, y,z) h^)((kz) i ,) (B-9)
(k) X, Z) o (x,y,z)
Here K is a normalized constant, and Wis a weight function. In all of these algorithms, the PSF
(H(k) or h(k)) remains the same in each iteration.
B.3 Blind Deconvolution
Blind deconvolution algorithms70 use the same principle as constrained iteration
algorithms, except that they also update the PSF in each iteration. This family of algorithms
excels when the PSF is unknown or is perhaps not accurately measured. In each iteration
Equations B-10, B-11, and B-12 below are paired with Equations B-7, B-8, and B-9,
respectively, to compute the PSF and the object. In the first iteration, the initial guess of the
object is usually set as the acquired image, and the initial guess of the PSF is simply set as the
one from a theoretical model or unity.
i(k+1) (u, v, w) = K {(k) (u, v, W) + W' I(, V, w) i(k), V, W). (k) (, v, w)]} (B-10)
H(k+)(u', v, w)=K'H(k)(u, v, W) ,vW) (B-1)
h(k+l) (X, Z) =K'h(k) (x, y, z) (k)(, y, z)"x i z) (B-12)
I//' (x, y,z)6o(k) X, ,Z)