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
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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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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


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 . is an image with complex values, where the is the magnitude, is the phase. The sparseness regularization is defined as

where , 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:

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