Cell Segmentation in Phase Contrast Microscopy by Constrained Optimization

Cell Segmentation in Phase Contrast Microscopy by Constrained Optimization

Enkhbolor Adiya (Inje University, Gimhae, South Korea), Bouasone Vongphachah (Inje University, Gimhae, South Korea), Alaaddin Al-Shidaifat (Inje University, Gimhae, South Korea), Enkhbat Rentsen (National University of Mongolia, Ulan Bator, Mongolia) and Heung-Kook Choi (Inje University, Gimhae, South Korea)
Copyright: © 2015 |Pages: 12
DOI: 10.4018/IJEHMC.2015010103

Abstract

Utilization of automatic cell segmentation process is difficult to identify in a cell due to halo and shade-off distortions when observing the phase contrast microscopy images. Therefore, it is an important step to restore artifact-free images made ready for segmentation process. The main focus of this paper is to define a gradient projection algorithm to restore images based on the minimization problem of quadratic objective function with non-negative constraints. The proposed algorithm converges to a global minimum solution independent on initialization. The experimental result shows that the proposed algorithm can restore artifact-free images, which could produce high quality segmentation results using a simple thresholding method.
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Method

Description of the Model

The linear model for microscope image is given by the imaging equation:

IJEHMC.2015010103.m01
(1) where IJEHMC.2015010103.m02 represents the point spread function (PSF), IJEHMC.2015010103.m03 represents the recorded image, IJEHMC.2015010103.m04 represents the actual object, the constant IJEHMC.2015010103.m05 corresponds to an image background and * indicates that 2D convolution.

The PSF for the phase contrast microscope is:

IJEHMC.2015010103.m06

Let IJEHMC.2015010103.m07 be the IJEHMC.2015010103.m08 matrix, here IJEHMC.2015010103.m09 take on only integer values. Then, the matrix IJEHMC.2015010103.m10 can be converted to a IJEHMC.2015010103.m11 column vector by lexicographic ordering. Let this column vector beIJEHMC.2015010103.m12:

IJEHMC.2015010103.m13

Then the linear model (1) can be expressed using matrix notation:

IJEHMC.2015010103.m14
(2) where IJEHMC.2015010103.m15 is a block-Toeplitz matrix constructed from the PSF kernel, which is discretized as a IJEHMC.2015010103.m16 matrix. Each row of IJEHMC.2015010103.m17 has only IJEHMC.2015010103.m18 nonzero elements corresponding to the PSF kernel, thus IJEHMC.2015010103.m19 is a IJEHMC.2015010103.m20 symmetric sparse matrix.

The first step to restore image IJEHMC.2015010103.m21 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

IJEHMC.2015010103.m22
(3)

Due to the ill-conditioned nature of the image restoration problem, an attempt to solve IJEHMC.2015010103.m23 from equation (3) by inverse transformation can result in undesirable effect in the solution.

Instead, we consider the following constrained quadratic programming problem to restoreIJEHMC.2015010103.m24, which was proposed in (Yin et al., 2012).

IJEHMC.2015010103.m25
(4) Subject to IJEHMC.2015010103.m26 where IJEHMC.2015010103.m27 is a Laplacian matrix defining the smoothness regularization, andIJEHMC.2015010103.m28 is a positive diagonal matrix defining the sparseness regularization.

Similarity between spatial neighbors defined as:

IJEHMC.2015010103.m29
where IJEHMC.2015010103.m30 and IJEHMC.2015010103.m31denote intensities of neighboring pixels IJEHMC.2015010103.m32 and IJEHMC.2015010103.m33, and IJEHMC.2015010103.m34is the mean of all possibleIJEHMC.2015010103.m35’s in the image. The smoothness regularization IJEHMC.2015010103.m36 is defined as:
IJEHMC.2015010103.m37
where IJEHMC.2015010103.m38 denote the spatial 8-connected neighborhood of pixel IJEHMC.2015010103.m39. Thus, IJEHMC.2015010103.m40 where:IJEHMC.2015010103.m41.

2D Fourier transform, ℱ, on image IJEHMC.2015010103.m42 is IJEHMC.2015010103.m43IJEHMC.2015010103.m44. IJEHMC.2015010103.m45 is an image with complex values, where the IJEHMC.2015010103.m46 is the magnitude, IJEHMC.2015010103.m47 is the phase. The sparseness regularization IJEHMC.2015010103.m48is defined as

IJEHMC.2015010103.m49
where IJEHMC.2015010103.m50IJEHMC.2015010103.m51, IJEHMC.2015010103.m52 denotes the diagonal vector of matrix IJEHMC.2015010103.m53.

It can be easily checked that the function IJEHMC.2015010103.m54 defined by (4) is strictly convex quadratic function.

Problem (4) can be rewritten as follows:

IJEHMC.2015010103.m55
subject to IJEHMC.2015010103.m56 (5)where:

IJEHMC.2015010103.m57
IJEHMC.2015010103.m58

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