Description of the Model
The linear model for microscope image is given by the imaging equation:
(1) where
![IJEHMC.2015010103.m02](https://igiprodst.blob.core.windows.net:443/source-content/9781466678613_118765/IJEHMC.2015010103.m02.png?sv=2015-12-11&sr=c&sig=%2Ff5U8zzwnY%2Bc69nMv1HJyBeZpx9B7ES%2B5xKXBhsfIgQ%3D&se=2019-11-19T16%3A48%3A07Z&sp=r)
represents the point spread function (PSF),
![IJEHMC.2015010103.m03](https://igiprodst.blob.core.windows.net:443/source-content/9781466678613_118765/IJEHMC.2015010103.m03.png?sv=2015-12-11&sr=c&sig=%2Ff5U8zzwnY%2Bc69nMv1HJyBeZpx9B7ES%2B5xKXBhsfIgQ%3D&se=2019-11-19T16%3A48%3A07Z&sp=r)
represents the recorded image,
![IJEHMC.2015010103.m04](https://igiprodst.blob.core.windows.net:443/source-content/9781466678613_118765/IJEHMC.2015010103.m04.png?sv=2015-12-11&sr=c&sig=%2Ff5U8zzwnY%2Bc69nMv1HJyBeZpx9B7ES%2B5xKXBhsfIgQ%3D&se=2019-11-19T16%3A48%3A07Z&sp=r)
represents the actual object, the constant
![IJEHMC.2015010103.m05](https://igiprodst.blob.core.windows.net:443/source-content/9781466678613_118765/IJEHMC.2015010103.m05.png?sv=2015-12-11&sr=c&sig=%2Ff5U8zzwnY%2Bc69nMv1HJyBeZpx9B7ES%2B5xKXBhsfIgQ%3D&se=2019-11-19T16%3A48%3A07Z&sp=r)
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
![IJEHMC.2015010103.m15](https://igiprodst.blob.core.windows.net:443/source-content/9781466678613_118765/IJEHMC.2015010103.m15.png?sv=2015-12-11&sr=c&sig=%2Ff5U8zzwnY%2Bc69nMv1HJyBeZpx9B7ES%2B5xKXBhsfIgQ%3D&se=2019-11-19T16%3A48%3A07Z&sp=r)
is a block-Toeplitz matrix constructed from the PSF kernel, which is discretized as a
![IJEHMC.2015010103.m16](https://igiprodst.blob.core.windows.net:443/source-content/9781466678613_118765/IJEHMC.2015010103.m16.png?sv=2015-12-11&sr=c&sig=%2Ff5U8zzwnY%2Bc69nMv1HJyBeZpx9B7ES%2B5xKXBhsfIgQ%3D&se=2019-11-19T16%3A48%3A07Z&sp=r)
matrix. Each row of
![IJEHMC.2015010103.m17](https://igiprodst.blob.core.windows.net:443/source-content/9781466678613_118765/IJEHMC.2015010103.m17.png?sv=2015-12-11&sr=c&sig=%2Ff5U8zzwnY%2Bc69nMv1HJyBeZpx9B7ES%2B5xKXBhsfIgQ%3D&se=2019-11-19T16%3A48%3A07Z&sp=r)
has only
![IJEHMC.2015010103.m18](https://igiprodst.blob.core.windows.net:443/source-content/9781466678613_118765/IJEHMC.2015010103.m18.png?sv=2015-12-11&sr=c&sig=%2Ff5U8zzwnY%2Bc69nMv1HJyBeZpx9B7ES%2B5xKXBhsfIgQ%3D&se=2019-11-19T16%3A48%3A07Z&sp=r)
nonzero elements corresponding to the PSF kernel, thus
![IJEHMC.2015010103.m19](https://igiprodst.blob.core.windows.net:443/source-content/9781466678613_118765/IJEHMC.2015010103.m19.png?sv=2015-12-11&sr=c&sig=%2Ff5U8zzwnY%2Bc69nMv1HJyBeZpx9B7ES%2B5xKXBhsfIgQ%3D&se=2019-11-19T16%3A48%3A07Z&sp=r)
is a
![IJEHMC.2015010103.m20](https://igiprodst.blob.core.windows.net:443/source-content/9781466678613_118765/IJEHMC.2015010103.m20.png?sv=2015-12-11&sr=c&sig=%2Ff5U8zzwnY%2Bc69nMv1HJyBeZpx9B7ES%2B5xKXBhsfIgQ%3D&se=2019-11-19T16%3A48%3A07Z&sp=r)
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
![IJEHMC.2015010103.m26](https://igiprodst.blob.core.windows.net:443/source-content/9781466678613_118765/IJEHMC.2015010103.m26.png?sv=2015-12-11&sr=c&sig=%2Ff5U8zzwnY%2Bc69nMv1HJyBeZpx9B7ES%2B5xKXBhsfIgQ%3D&se=2019-11-19T16%3A48%3A07Z&sp=r)
where
![IJEHMC.2015010103.m27](https://igiprodst.blob.core.windows.net:443/source-content/9781466678613_118765/IJEHMC.2015010103.m27.png?sv=2015-12-11&sr=c&sig=%2Ff5U8zzwnY%2Bc69nMv1HJyBeZpx9B7ES%2B5xKXBhsfIgQ%3D&se=2019-11-19T16%3A48%3A07Z&sp=r)
is a Laplacian matrix defining the smoothness regularization, and
![IJEHMC.2015010103.m28](https://igiprodst.blob.core.windows.net:443/source-content/9781466678613_118765/IJEHMC.2015010103.m28.png?sv=2015-12-11&sr=c&sig=%2Ff5U8zzwnY%2Bc69nMv1HJyBeZpx9B7ES%2B5xKXBhsfIgQ%3D&se=2019-11-19T16%3A48%3A07Z&sp=r)
is a positive diagonal matrix defining the sparseness regularization.
Similarity between spatial neighbors defined as:
where
![IJEHMC.2015010103.m30](https://igiprodst.blob.core.windows.net:443/source-content/9781466678613_118765/IJEHMC.2015010103.m30.png?sv=2015-12-11&sr=c&sig=%2Ff5U8zzwnY%2Bc69nMv1HJyBeZpx9B7ES%2B5xKXBhsfIgQ%3D&se=2019-11-19T16%3A48%3A07Z&sp=r)
and
![IJEHMC.2015010103.m31](https://igiprodst.blob.core.windows.net:443/source-content/9781466678613_118765/IJEHMC.2015010103.m31.png?sv=2015-12-11&sr=c&sig=%2Ff5U8zzwnY%2Bc69nMv1HJyBeZpx9B7ES%2B5xKXBhsfIgQ%3D&se=2019-11-19T16%3A48%3A07Z&sp=r)
denote intensities of neighboring pixels
![IJEHMC.2015010103.m32](https://igiprodst.blob.core.windows.net:443/source-content/9781466678613_118765/IJEHMC.2015010103.m32.png?sv=2015-12-11&sr=c&sig=%2Ff5U8zzwnY%2Bc69nMv1HJyBeZpx9B7ES%2B5xKXBhsfIgQ%3D&se=2019-11-19T16%3A48%3A07Z&sp=r)
and
![IJEHMC.2015010103.m33](https://igiprodst.blob.core.windows.net:443/source-content/9781466678613_118765/IJEHMC.2015010103.m33.png?sv=2015-12-11&sr=c&sig=%2Ff5U8zzwnY%2Bc69nMv1HJyBeZpx9B7ES%2B5xKXBhsfIgQ%3D&se=2019-11-19T16%3A48%3A07Z&sp=r)
, and
![IJEHMC.2015010103.m34](https://igiprodst.blob.core.windows.net:443/source-content/9781466678613_118765/IJEHMC.2015010103.m34.png?sv=2015-12-11&sr=c&sig=%2Ff5U8zzwnY%2Bc69nMv1HJyBeZpx9B7ES%2B5xKXBhsfIgQ%3D&se=2019-11-19T16%3A48%3A07Z&sp=r)
is the mean of all possible
![IJEHMC.2015010103.m35](https://igiprodst.blob.core.windows.net:443/source-content/9781466678613_118765/IJEHMC.2015010103.m35.png?sv=2015-12-11&sr=c&sig=%2Ff5U8zzwnY%2Bc69nMv1HJyBeZpx9B7ES%2B5xKXBhsfIgQ%3D&se=2019-11-19T16%3A48%3A07Z&sp=r)
’s in the image. The smoothness regularization
![IJEHMC.2015010103.m36](https://igiprodst.blob.core.windows.net:443/source-content/9781466678613_118765/IJEHMC.2015010103.m36.png?sv=2015-12-11&sr=c&sig=%2Ff5U8zzwnY%2Bc69nMv1HJyBeZpx9B7ES%2B5xKXBhsfIgQ%3D&se=2019-11-19T16%3A48%3A07Z&sp=r)
is defined as:
where
![IJEHMC.2015010103.m38](https://igiprodst.blob.core.windows.net:443/source-content/9781466678613_118765/IJEHMC.2015010103.m38.png?sv=2015-12-11&sr=c&sig=%2Ff5U8zzwnY%2Bc69nMv1HJyBeZpx9B7ES%2B5xKXBhsfIgQ%3D&se=2019-11-19T16%3A48%3A07Z&sp=r)
denote the spatial 8-connected neighborhood of pixel
![IJEHMC.2015010103.m39](https://igiprodst.blob.core.windows.net:443/source-content/9781466678613_118765/IJEHMC.2015010103.m39.png?sv=2015-12-11&sr=c&sig=%2Ff5U8zzwnY%2Bc69nMv1HJyBeZpx9B7ES%2B5xKXBhsfIgQ%3D&se=2019-11-19T16%3A48%3A07Z&sp=r)
. Thus,
![IJEHMC.2015010103.m40](https://igiprodst.blob.core.windows.net:443/source-content/9781466678613_118765/IJEHMC.2015010103.m40.png?sv=2015-12-11&sr=c&sig=%2Ff5U8zzwnY%2Bc69nMv1HJyBeZpx9B7ES%2B5xKXBhsfIgQ%3D&se=2019-11-19T16%3A48%3A07Z&sp=r)
where:
![IJEHMC.2015010103.m41](https://igiprodst.blob.core.windows.net:443/source-content/9781466678613_118765/IJEHMC.2015010103.m41.png?sv=2015-12-11&sr=c&sig=%2Ff5U8zzwnY%2Bc69nMv1HJyBeZpx9B7ES%2B5xKXBhsfIgQ%3D&se=2019-11-19T16%3A48%3A07Z&sp=r)
.
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
![IJEHMC.2015010103.m50](https://igiprodst.blob.core.windows.net:443/source-content/9781466678613_118765/IJEHMC.2015010103.m50.png?sv=2015-12-11&sr=c&sig=%2Ff5U8zzwnY%2Bc69nMv1HJyBeZpx9B7ES%2B5xKXBhsfIgQ%3D&se=2019-11-19T16%3A48%3A07Z&sp=r)
ℱ
![IJEHMC.2015010103.m51](https://igiprodst.blob.core.windows.net:443/source-content/9781466678613_118765/IJEHMC.2015010103.m51.png?sv=2015-12-11&sr=c&sig=%2Ff5U8zzwnY%2Bc69nMv1HJyBeZpx9B7ES%2B5xKXBhsfIgQ%3D&se=2019-11-19T16%3A48%3A07Z&sp=r)
,
![IJEHMC.2015010103.m52](https://igiprodst.blob.core.windows.net:443/source-content/9781466678613_118765/IJEHMC.2015010103.m52.png?sv=2015-12-11&sr=c&sig=%2Ff5U8zzwnY%2Bc69nMv1HJyBeZpx9B7ES%2B5xKXBhsfIgQ%3D&se=2019-11-19T16%3A48%3A07Z&sp=r)
denotes the diagonal vector of matrix
![IJEHMC.2015010103.m53](https://igiprodst.blob.core.windows.net:443/source-content/9781466678613_118765/IJEHMC.2015010103.m53.png?sv=2015-12-11&sr=c&sig=%2Ff5U8zzwnY%2Bc69nMv1HJyBeZpx9B7ES%2B5xKXBhsfIgQ%3D&se=2019-11-19T16%3A48%3A07Z&sp=r)
.
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
![IJEHMC.2015010103.m56](https://igiprodst.blob.core.windows.net:443/source-content/9781466678613_118765/IJEHMC.2015010103.m56.png?sv=2015-12-11&sr=c&sig=%2Ff5U8zzwnY%2Bc69nMv1HJyBeZpx9B7ES%2B5xKXBhsfIgQ%3D&se=2019-11-19T16%3A48%3A07Z&sp=r)
(5)where: