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Let represent the image intensities, with , and the dimension of the image. Let S be the set of all functions of the form , We will represent the segmentation result as a function A pixel that corresponds to a nucleus, will be represented by one, whereas background pixels are represented by zero, i. The details on how to calculate this segmentation result, , will be thoroughly described in the segmentation subsection. Furthermore we will use the following gradient operator, inner product and norm notations: where represents the Kronecker delta, which is equal to one if and are equal and is equal to zero in all other cases.
Automatic cell nuclei detection is hampered by a number of factors such as non-uniform lighting, blur, clutter, etc. In order to improve the results of our method, we propose two preprocessing steps to minimize the influence of these degradations. First, the image is normalized in order to remove any differences in intensity: 2 where represents all pixels within a distance of the pixel.
The chosen distance depends on the image resolution and on the nuclei density. Furthermore, a gamma correction is applied, i.
This gamma correction suppresses low intensity dyeing due to cell apoptosis. Figure 1 shows an example of the preprocessing step. As can be seen in Fig. The combination of normalization and gamma correction results in good contrast, bright nuclei, while suppressing dye coming from dead cells Fig. A the raw image B the image with gamma correction applied, C The full prerprocessing applied, i. In  a segmentation method was proposed where the segmentation result is calculated by minimizing the following energy function: 4 With 5 where , are respectively the expected foreground and background intensity and is a weighting parameter used to tune the influence of the data-fit term in relation to the total variation regularization.
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Note that if and are constant, e. This allows to calculate a global optimizer using efficient optimization techniques such as Split Bregman or primal dual optimization . Chan et al. This results in the following optimization problem: 6 Furthermore, this formulation relates to the popular and widely used active contour without edges ACWE  , .
The steady state of the gradient flow corresponding to the energy function in eq. Note that is not necessarily unique, i. The function can have any value between 0 and 1, thus the found active contour does not have to represent a crisp segmentation. A binary segmentation result can be obtained by thresholding , i. In  ,  it is shown that itself is also a global minimizer for the energy in eq. The energy function in eq. This regularization is useful if pixels are incorrectly classified, i.
In microscopic images however, incorrectly detected nuclei are often caused by clutter in the image, e.
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This is not solved using total variation since these incorrectly detected nuclei are natural objects, i. Therefore, a number of shape based regularization terms have been proposed  — . However these shape priors are limited to images with only a single object of interest with a specific shape.
We propose a regularization term that exploits the regular shape of cell nuclei, penalizing segments which strongly deviate from the expected shape, while not constraining the number of nuclei. In this work, a nucleus is modeled as a disk.
For a given radius, r, and location, , we can calculate the ideal , i. We will refer to each of these possible segmentation results as atoms. In most applications however the image does contain multiple nuclei. Even the number of nuclei is typically not known.
Therefore we model the unknown segmentation as a superposition of disks, i. Note that this representation expects a predefined diameter of a cell nucleus. However the size of a nucleus is generally not fixed, but can be considered to an interval. Therefore we can extend eq.
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Note that this linear combination penalizes overlapping nuclei, which is desirable since a pixel can only correspond to a single nucleus. A good segmentation is one which consists of a small number of atoms.
This sparsity constraint can be used as a new regularization term: 10 with e the number of disc sizes considered, i. The energy term minimized by is based on a norm which comes down to calculating the number of nuclei, i. The norm is non-convex, hampering optimization. Fortunately the norm can be approximated by the norm which is the closest convex norm to.
In  it is shown that this approximation of an norm gives good results for the application of compressed sensing. This new prior results in the following active contour: In order to optimize the constrained problem in eq. Note that this only approximates the constraint in eq. Although there exists efficient techniques to enforce this constraint exactly, e. This allows the active contour to detect nuclei whose shape slightly deviates from the circular model or to detect partially overlapping nuclei.
Given the convexity of eq. The constraint that can be approximated by adding a barrier function to eq. For this purpose we propose the use of a piecewise linear barrier function  : This barrier function has the benefit of penalizing values out of the interval while not giving preference to any specific value inside the interval.
The piecewise linear barrier function is prone to overshooting using the Newton-Raphson method, i. However given the specific nature of this barrier function, with the minima corresponding to the roots of the function, it is possible to minimize this by searching for the roots using the Newton-Raphson optimization scheme. This results in the following updating step: 17 with The optimal in eq. For an optimal , the following optimality condition should be satisfied: 19 The solution of eq.
However minimizing eq. So if then the constrained optimum is either or , since a quadratic function is monotonic in an interval which does not contain its extremum. So the constrained optimum is given by: By adding a sparsity constraint on , we penalize representations which use more atoms than necessary.
The function can be used to detect the centroids of the cell nuclei.
However by using the norm as an approximation of the norm it is not possible to get the nuclei centroids explicitly from , a special detection algorithm is required. A schematic overview of the detection method is shown in Fig. Since disks of multiple sizes can add to the representation of a nucleus in the segmentation result, we first detect the dominant radius for each pixel. Each pixel has a number of corresponding atoms, each representing a nucleus of different size. The radius corresponding with the atom with the highest value for a specific pixel is considered the dominant radius of that pixel, i.
Since the most important information is the location of a cell nucleus, a merging step is applied such that all weights corresponding to the same pixel location are combined by summing them together. This is done in parallel with the calculation of the dominant radius.
1. Introduction to image processing | Digital Image Processing
The merging step results in a new image, , where high intensities occur at the centers of nuclei shown in Fig. In order to get crisp detections, the new image is converted to a binary image by thresholding it Fig. Due to small deviations in the shape of cell nuclei, it is possible that a single nucleus corresponds to multiple connected components in this binary image. Since these components are located in each other's vicinity, it is possible to overcome this problem by applying a morphological closing Fig.