Description of the Model
The linear model for microscope image is given by the imaging equation:
(1) where
represents the point spread function (PSF),
represents the recorded image,
represents the actual object, the constant
corresponds to an image background and * indicates that 2D convolution.
The PSF for the phase contrast microscope is:
Let 
be the 
matrix, here 
take on only integer values. Then, the matrix 
can be converted to a 
column vector by lexicographic ordering. Let this column vector be
:
Then the linear model (1) can be expressed using matrix notation:
(2) where
is a block-Toeplitz matrix constructed from the PSF kernel, which is discretized as a
matrix. Each row of
has only
nonzero elements corresponding to the PSF kernel, thus
is a
symmetric sparse matrix.
The first step to restore image 
from equation (2) is to remove non-uniform background. We refer to the (Wu et al., 2008) to estimate background. The corrected image is computed by subtracting the estimated background image from the recorded image. Thus, new linear model is
(3)Due to the ill-conditioned nature of the image restoration problem, an attempt to solve 
from equation (3) by inverse transformation can result in undesirable effect in the solution.
Instead, we consider the following constrained quadratic programming problem to restore
, which was proposed in (Yin et al., 2012).
(4) Subject to
where
is a Laplacian matrix defining the smoothness regularization, and
is a positive diagonal matrix defining the sparseness regularization.
Similarity between spatial neighbors defined as:
where
and
denote intensities of neighboring pixels
and
, and
is the mean of all possible
’s in the image. The smoothness regularization
is defined as:
where
denote the spatial 8-connected neighborhood of pixel
. Thus,
where:
.
2D Fourier transform, ℱ, on image 
is 
ℱ
. 
is an image with complex values, where the 
is the magnitude, 
is the phase. The sparseness regularization 
is defined as
where
ℱ
,
denotes the diagonal vector of matrix
.
It can be easily checked that the function 
defined by (4) is strictly convex quadratic function.
Problem (4) can be rewritten as follows:
subject to
(5)where: