We use K-means clustering for image segmentation to find the optimal threshold, such that the image feature values of pixels on one side of the threshold are closer to their feature values’ mean than the distance between those feature values and the means on the other side of the threshold. This method is performed using the histogram of image intensity. We assume that the image intensities compose a vector space and try to find natural clustering in it. The pixels are clustered around centroid c_i, which are obtained by minimizing the objective function

(1)

The centroid for each cluster is iteratively obtained as follows,

(2)

where r is the image size in terms of pixel number, i iterates over all intensities, j iterates over all centroids, and μ_i are the centroid intensities. Using intensity value directly in microscopic cell image segmentation will not lead to the desired segmentation result due to the dynamic ranges, which vary in images. We propose a gray-level transformation function in the form T_r(I) = I ^r for the above algorithm to implement k-means segmentation in cell image I, where γ is a positive constancy.

The Expectation Maximization (EM) algorithm assumes that an image consists of a number of gray-level regions, which can be described by parametric data models. When the histogram of the gray levels is regarded as an estimate of the probability density function, the parameters of the function can be estimated for each gray-level region using the histogram. The objective of the EM algorithm is to find the maximum likelihood estimates of the parameters in the function. Correspondingly, EM consists of two steps: expectation and maximization. Using the same notations in Section 2.1, the mixture of probability density functions is as follows,

(3)

In the above, α _j is the proportion of the j-th density function in the mixture model, and is the j-th density function with parameter set θ_j. The Gaussian mixture model (GMM) is the most employed in practice, and has two parameters, mean µ _j and covariance ∑_j, such that θ _j = (µ_j, ∑ _j) . We consider GMM in our research. If we assume that θ ^t_j is the estimated value of parameters θ _j = (µ_j, ∑ _j), obtained at the t-th step, then can be obtained iteratively. The EM algorithm framework follows,

(4)

(5)

(6)

(7)

These equations state that the estimated parameters of the density function are updated according to the weighted average of the pixel values where the weights are obtained from the E step for this partition. The EM cycle starts at an initial setting of and updates the parameters using Equations ((4)-(7)) iteratively. The EM algorithm converges until its estimated parameters cannot change. Then, the final parameters, are applied in image segmentation by labeling pixels using Maximum Likelihood (ML). Pixel x_i is labeled using the following function,

(8)

Threshold segmentation is a method that separates an image into a number of meaningful regions through the selected threshold values. If the image is a grey image, thresholds are integers in the range of [0, L-1], where L-1 is the maximum intensity value. Normally, this method is used to segment an image into two regions: background and object, with one threshold. The following is the equation for threshold segmentation:

(9)

In the above equation, I_B is the segmentation resultant. The most famous threshold method was proposed by Otsu in [12]. The Otsu’s method finds the optimal threshold T among all the intensity values from 0 to L-1 and chooses the value that produces the minimum within-class variance as the optimal threshold value. Consequently, the optimal value of T_Opt is obtained through the following optimal computation,

(10)

In the whole image, variances σ² are made up of two parts: . Otsu shows that is the same as . Therefore, the optimal value of T can also be obtained through the following alternative optimization process:

(11)

Equation (11) is often used to find the optimal threshold value for simple calculation. Theoretically, is expressed in the following,

(12)

where are the probabilities of the two clusters separated by threshold T, and μ_i(T)_i=1,2 are the cluster means. ω_i(T)_i=1,2 and μ_i(T)_i=1,2 can be estimated using histogram h(x) as follows,

(13)

(14)

(15)

(16)

Using the above Equations (12)-(16), the optimal threshold T is exhaustively searched among [0, L-1] to meet the objective according to Equation (11).

We choose the global minimization of the active contour model (GMAC) [16] to analyze the implementation of active contour in cell-image segmentation. This method has a simple initialization and fast computation, and it can avoid being stuck at an undesired local minima. GMAC is based on Mumford and Shah’s (MS) function and the Chan and Vese’s model of active contours without edges (ACWE) [17]. GMAC improves ACWE by using weighted total variation and dual formulation of the TV form, which preserves the advantage of ACWE. We define GMAC and related concepts below.

(17)

where r₁ (x, c₁, c₂) = ((c₁-f(x))² - (c₂-f (x)²))dx, f(x) is the given image, and c₁ and c₂ are constants calculated for partitioning in iteration; e.g., if , µ* = arg minE² [µ,v,c₁,c₂], c₁ and c₂ are the means of pixels in two partitions and can be obtained using equations, θ > 0 is chosen small enough, λ > 0 is a parameter controlling scale related to the scale of observation of solution, and α is constant.

(18)

where g(x) is an edge indication function which gives a link between snake model and region terms. The minimization Equation (17) is solved using the following equations iteratively until convergence:

(19)

(20)

(21)

(22)

(23)

In Equation (21), δ t is the time step.

We use the traditional precision, recall, and F-score as the quantitative measures in pixel level. These measures are standard techniques used to evaluate the quality of the segmentation results against the ground truth. These measures quantify discrepancy between segmentation results and binary ground truth mask as follows:

(24)

(25)

(26)

where SR is the segmentation result and GT is the ground truth of images. The symbol ‘#’ refers to the pixel numbers in the sets.

3.1.1. Segmentation of Synthetic Data

Benchmark sets of synthetic cell population images with ground truth are simulated by P. Ruusuvuori in [18]. We select the second benchmark set which consists of multi-channel cell images because we do not have suitable real cell images with ground truth for evaluation. In this set, nuclei, cytoplasm, and subcellular components have been simulated by tuning parameters such as size, location, randomness of shape, and other background or fluorescence parameters (see details in [18]). The image sets are divided into two subsets: high quality and low quality (examples shown in Fig. 1), each consisting of 20 cell images. The second set has overlapping cells and a noisy background. Each image contains 50 cells. As each simulated image has a corresponding binary mask as ground truth, binary operations can easily calculate the quantitative measure defined above.

Fig. (1). Synthetic cell images a) (low quality) with noisy background and overlapping cells, b) (high quality) without noise in background and overlapping cells, c) ground truth of image a, d) ground truth of image b.

Fig. (2) shows the segmentation result of four methods for the low quality synthetic image data in Fig. (1a). Segmented images in Fig. (2) are compared and evaluated using the ground truth image in Fig. (1c). Fig. (3) shows the segmentation result of four methods for the high quality synthetic image data in Fig. (1b). Segmented images in Fig. (3) are compared and evaluated using the ground truth image in Fig. (1d).

Fig. (2). Segmentation result for synthetic cell images of low quality in Fig. (1a). a) K-means result, b) EM result, c) Otsu’s result, d) GMAC result.

Fig. (3). Segmentation result for synthetic cell image of high quality in Fig. (1b). a) K-means result, b) EM result, c) Otsu’s result, d) GMAC result.

Figs. (4-6) and Table 1 are the quality measure values for the segmentation results using subcellular images with low quality. Figs. (7-9) and Table 2 are the quality measure values for the segmentation results using subcellular images with high quality.

Table 2.

Average Measures of the Segmentation Methods Applied on High Quality Synthetic Cell Images

	F-Score	Precision	Recall
K-Means	0.9745	0.9726	0.9765
EM	0.9040	0.8267	0.9986
Otsu’s	0.9738	0.9798	0.9679
GMAC	0.9703	0.9874	0.9538

Table 1.

Average Measures of the Segmentation Methods Applied on Low Quality Synthetic Cell Images

	F-Score	Precision	Recall
K-Means	0.9350	0.9530	0.9180
EM	0.5331	0.3821	0.9915
Otsu’s	0.9269	0.9295	0.9259
GMAC	0.9445	0.9781	0.9133

Fig. (4). F-score of the four methods applied on low quality simulated cell images.

Fig. (5). Precision of the four methods applied on low quality simulated cell images.

Fig. (6). Recall of the four methods applied on low quality simulated cell images.

Fig. (7). F-score of the four methods applied on high quality simulated cell images.

Fig. (8). Precision of the four methods applied on high quality simulated cell images.

Fig. (9). Recall of the four methods applied on high quality simulated cell images.

We observe that the segmentation results of lower quality images, with noisier backgrounds and overlapping cells, have worse results than those in high quality images. K-means, Otsu’s threshold and GMAC obtain similar segmentation quality in both sets of images, measured by F-score, precision, and recall. Their performance is more robust against noises than EM. Moreover, the EM algorithm has lower precision, while keeping much higher recall values, especially for cell images with noisy backgrounds. To further understand these phenomena, real nucleus images are segmented in the next section.

3.1.2. Segmentation of Nucleus Images

Sixteen nucleus images were hand-outlined by an expert in the CellProfiler project [8, 15]. We use these images to evaluate the segmentation algorithms quantitatively. We obtain similar results, as shown in Figs. (10-13), as those we obtained in Section 3.1.1. We observe in Table 3 that EM maintains higher recall and lower precision values, even if its F-score values are as high as the other segmentation methods in several images. From Fig. (10d), we can see that the EM under-segments nucleus images strongly, which induces the high recall values. This under-segmentation is due to the presumed dual Gaussian mixture models in the calculation of EM. One model represents background, and the other refers to objects. When objects have much smaller grayness regions than background (as shown in Fig. 10a), the dual Gaussian mixture model leads to under-segmentation.

Table 3.

Average Quality Measures of the Segmentation Methods on Nucleus Images

	F-Score	Precision	Recall
K-Means	0.8714	0.8766	0.8668
EM	0.7473	0.6131	0.9664
Otsu’s	0.7976	0.9475	0.6910
GMAC	0.7880	0.8880	0.7148

Fig. (10). Segmentation of nucleus images: a) Nucleus images, b) Ground truth, c) K-means result, d) EM result, e) Otsu’s result, and f) GMAC result.

Fig. (11). F-score of the four methods applied on nucleus images.

Fig. (12). Precision of the four methods applied on nucleus images.

Fig. (13). Recall of the four methods applied on nucleus images.

Otsu’s method also has drawbacks. Although it performs well for nucleus segmentations, due to its fastness and simplicity in application, it cannot be proven the best segmentation method for nucleus images. As shown in Section 3.1.1, Otsu’s method shows stable precision and recall values even when it encounters arbitrarily defined noises. However, in the experiment using real nucleus images, the Otsu’s method recall value is significantly lower than its precision values, which means it has over-segmented the image.

GMAC is more robust and stable than the Otsu’s method in our experiments. GMAC depends on both image intensity distribution information (region) and gradient (edge) information. When the contrast between background and cells becomes light, and cells are hidden by noises, the combination of gradient and intensity information records better information than intensity alone does, e.g. in Otsu’s. In the k-means method, we choose k=2 to cluster some objects into one group and other segments into a background group. K-means performs the best in almost all experiments. Its good performance is due to the application of power function for the compensation of intensity transformation brought in by the microscopic device. In this research, we assume this power function is known, and we obtain it by choosing the optimal k-means result (smallest error between k-means segmentation result and ground truth). It demonstrates that the k-means method can obtain robust and precise segmentation results with the aid of power function.

In this evaluative study, brain cell images were captured using a computer controlled Microscope (Leica DMI 6000 Digital). The cell images are of a normal healthy astrocytes cell, which has been stained with Calcein AM, a vital dye that stains only living cells. The test images are 1040 ↔ 1392 pixels with 8-bit gray-levels. As no manual outlining has been performed on the images, the performance of segmentation methods is qualitatively evaluated.

As shown in Fig. (14a), brain cell images are dark, and cell contours are blurred. The segmentation results of k-means, Otsu’s, and GMAC (Fig. 14b, 14d, 14e) seem to be washed out. Using the nucleus, light areas, we can identify the existing cells in the image. The segmentation result of GMM EM (Fig. 14c) is still under-segmented. In Fig. (15), we can see that the background, denoted by the annotated line marked by ‘*’, has a narrow estimated intensity distribution while cells distribute in a wider intensity levels, denoted by the other annotated line marked by ‘+’. This presentation using standard Gaussian distribution leads to errors in the estimate of probability distribution. As shown in Fig. (16), the individual intensity distributions are summed to obtain the mixed Gaussian distribution, which is presented by the annotated line marked by square. The errors in estimation are accumulated in the sum procedure, which can be presented by the discrepancy between the areas covered by the estimated distribution and the true intensity distribution denoted by the annotated line marked by triangle. The other three segmentation results have cells split with the nucleus, although several cells are over-segmented. This over-segmentation can be explained as that these techniques consider image intensity and texture information in the segmentation process, while the spatial relation or some connection between pixels is missed. Moreover, compared to the synthetic images in Section 3.1.1, real cell images are more complex and difficult to segment.

Fig. (14). a) Brain cell microscopy image, b) K-means clustering result, c) EM clustering result, d) Otsu’s segmentation result, e) GMAC segmentation result.

Fig. (15). Estimated intensity distribution of image in Fig. (14a) using GMM EM model.

Fig. (16). The ground truth and estimation of the intensity distribution of image in Fig. (14a).