Thresholding for medical image segmentation for cancer using fuzzy entropy with level set algorithm

2.1 Selection cancer region based on a level set algorithm

In 1987, Osher and Sethian introduced a level set method, which utilizes and implements the dynamic variation boundary for medical image segmentation. Most medical image modalities are grayscale, and m = (x, y) is considered a point in the medical image. This point and the position of m(t) will evolve at the overtime, t is the time for the m(t) point on the surface; this point is expressed below [22].

The boundary segmentation for the medical image is defined as a part surface where the contour level set equals zero. The height surface is equal to the length from m = (x, y) to the nearest point on the contour [23].

Where x and y are the points on the image, t is the time, and d is the distance between x, y point and zero level set. The partial differential equations (PDE) function ∅(x, y, t = 0), and evaluation is possible by approximating the active contours by tracking zero level set for point and position of overtime m(t). The development of a curved surface is characterized by the various forces of the internal and external research archives. The height surface is equal to the distance from the nearest pixel on an active contour to (x, y), such that ∅(t, x, y,) < 0, the initial function ∅ with distance m(t) being negative inside the contour, if ∅(t, x, y,) > 0, the initial function ∅ with m(t) is positive outside the contour, and if ∅ (t, x, y,) ≌ 0 . the initial function ∅ of the level set matches the initial contour are getting closer to the target boundaries of cancer, the speed function is expected to gradually slow down to zero. [24]. The distance value of d outside the curve is positive, whereas the distance value of d inside the contour is negative; furthermore, the distance value of d on the boundary of the image is equal to zero [24]. The initial function ∅ can be the arbitrary function until the zero level set equals the initial contour. Thus, initial function ∅ = t = 0 or ∅t is overtime, and we can use motion equation ∂∅∂t with the chain rule.

If the motion equation is ∂∅∂t=∇∅ , and m(t) is the speed of normal force F to the surface, then m(t) = F(m(t))n, where n equals ∇∅|∇∅| . The motion equation is ∂∅∂t . The advancing force F must be regularized by an edge indication initial function ∅_t to stop the level set evolution near the optimal solution [24]. The equation can be rewritten as.

t = 0 is the time for the process, and defining the overtime for a motion to ∅(x, y, t) at any time t by evolving the initial ∅(x, y, t = 0) overtime is possible after defining the motion of ∅ in Eq. 4. The surface curvature by motion ∅ and a popular formulation for the level set method to find the part with cancer can be obtained by.

2.2 Medical cancer thresholding based on fuzzy entropy

Thresholding method is a technique that uses the classification pixels to segment colored and grayscale images. Many methods, such as scene processing, medical thresholding, document image analysis, and map processing have been proposed for image thresholding application. The simplest method for medical image segmentation is the image thresholding technique; this technique is the simplest method for implementing an image and thus accelerates the process. D = (i, j), where i = (0,1… M − 1), j = (0,1… N – 1). In these variables, M and N are two integer numbers that correspond to the width and height of a medical image. G = (0, 1… L – 1) is used to determine the number of grayscale in a medical image. I(x, y) is adopted to obtain the grayscale value of an image at a pixel (x, y), and DK = (x, y); moreover I(x, y) = k, where (x, y) = D and k = (0, 1… L – 1). We suggest that T1 and T2 be utilized as the thresholds for medical image segmentation; D is an original medical image domain and divided into three regions, namely, E_m, E_b, and E_d. The E_d region covers the pixels with less grayscale value than T1, E_m covers the pixels with a middle grayscale value between T1 and T2, and E_b covers the pixels with more grayscale value than T2 . W_a = E_m, E_b, and E_d is the unknown probabilistic distribution of D domain, which probability distribution is described in [25, 27].

The membership functions (μ) for E_d, E_m, and E_b correspond to μ_m, μ_b, and μ_d and requires six parameters for this process, that is, a1, b1, c1, a2, b2, and c2. accordance with the membership function (μ) the thresholds T1 and T2 are the variables for every pixel k = 0,1….,255 [26, 27].

The condition of a probability of E_m, E_b, and E_d to P_m|k, P_b|k and P_d|k of a pixel is divided into three classes, namely, dust, bright, and dark. The pixel pertains to D_k with p_m|k + P_b|k, + P_d|k = 1, k = 0,1,2 …255. then, the abovementioned equation can be rewritten as.

The grayscale value with a grade of pixels for k among the classes of dust(E_m), bright (E_b), and dark (E_d) can be determined as equivalent to the condition of the probabilities P_m|k, P_b|k, and P_d|k. The equation can be expressed as.

We selected the three parameters for fuzzy membership function, that is, U, S, and Z, where U = (k, a1, b1, c1, a2, b2, c2), S(k, a1, b1, c1, a2, b2, c2). and Z(k, a1, b1, c1, a2, b2, c2] as the membership function classes of dust μ_m(k), bright μ_b(k), and dark μ_d(k) [25, 26, 28], The three equations for membership functions can be rewritten as.

The six parameters must be within the range of (0 ≤ a1 < b1 < c1 < a2 < b2 < c2 ≤ 255) pixels. The function of fuzzy entropy for each class of dust, bright, and dark is defined as.

The total fuzzy entropy function for all classes can be computed and summarized.

The objective functions of the abovementioned equation apply optimization techniques to find the maximum value of fuzzy entropy for all classes. The threshold technique is calculated using the following equation.

On the basis of Eqs. (10–12), the thresholds T1 and T2 are considered the points of intersection for dust μ_m(k), bright μ_b(k), and dark μ_d (k). T1 and T2 can be calculated as follows.

2.3 Proposed method approach

Numerous researchers have worked and developed methods for solving cancer problems by using medical image segmentation. The proposed method is used to find multiple cancers in different medical images and conduct estimation analyses of these cancers. The goal of the proposed method is to output useful information for cancer boundary through medical image segmentation and be efficient in classifying cancer. This study also attempts to combine several methods to create an effective segmentation. A flowchart of this process is illustrated in Figure 1.

Figure 1

Proposed approach for extracting the part with cancer through medical image segmentation.

Our FELs thresholding proposed algorithm

Input: Medical image.

Output: Extracted the part with cancer and then segmented.

1. Read medical image.

2. Create a loop for reading the pixels of a medical image.

3. Initiate counter searching of a medical image in accordance with Eq. (4). If the force is positive (F =1), then the counter must be from inside the image. If the force is negative (F=–1), then the force must be outside the counter.

4. Evaluate the function ∅ by the numerical level set using Eqs. (2 and 3).

5. If the function ∅_t is approximately zero in a boundary by Eq. (4), then proceed to the next equation to select the part with cancer tissue range.

   k=∇∇∅|∇∅|=∅xx∅y2−2∅xy∅x∅y+∅yy∅x2(∅x2+∅y)1/2

6. Define the three parameters for fuzzy entropy membership functions, such as U, S, and Z, to initialize the membership function values, namely, dust μ_m(k), bright μ_b(k), and dark μ_d(k), using Eqs. (10–12), respectively.

7. Calculate double thresholding to find the lower and upper limits using Eq. (15).

8. Extract the part with cancer and segment the medical image in accordance with Eq. (16).

9. End.

This flowchart demonstrates the proposed method for a cancer region through medical image segmentation. This method consists of the following three stages: the first stage is using a level set algorithm to search in a medical image to find the part with cancer and select it. The second stage is using the fuzzy entropy to experiment and analyze the part with cancer and segment it after the selected previews of the algorithm. The final stage is using double thresholding to extract the part with cancer. The details of the proposed method are presented as follows.

1. The level set algorithm creates a loop to search for the medical image and find the parts with cancer tissue region and select it.

2. The background and foreground regions are calculated for every threshold grayscale level.

3. The pixels of the membership degree of the medical image are calculated for every threshold grayscale level.

4. The fuzzy entropy algorithm is calculated for every threshold grayscale level, and the smallest value of the threshold is selected.

5. The part with cancer is a threshold with the grayscale level based on the smallest value of the fuzzy entropy.

6. The cancer region is segmented and extracted.

3.1 Experiment the results

In this study, we present and explain the analysis of results with the initial counter that utilizes the FELs thresholding algorithm. The analysis is conducted by using three types of medical image, that is, ultrasound for breast cancer, brain MRI, and dermoscopy skin image. The results of the experiment show that the compact constraint of the proposed method exerts various effects on different degrees of compactness. Figures (2–4)a depict the original images. The searching and scanning have started to find the ground truth (GT) regions of cancer by decreasing and increasing the parameter value of the mask (M), as demonstrated in Figures (2–4)b; these processes continue pixel by pixel on the medical image until the nearest GT boundary of cancer. The level set method creates a red initial curve around the cancer region because this method is sensitive to this process, as displayed in Figures (2–4)c. The experimental images illustrated optimization steps of iterations until achevieing the optimum solution which showed the results in Figures 5, 6 and 7. Table 1 and Figure 8 summarize the performance of the FELs thresholding method, which is calculated for sensitivity, precision, specificity, and accuracy measures, for each medical image.

Figure 2

Fuzzy entropy with level set segmentation of ultrasound breast cancer: (a) Original ultrasound image with cancer, (b) is searching to find the cancer to region, (c) find the cancer and segmented after 130 iterations, (d) is initialization by thresholding (100-220,250-340) for extracted the cancer region.

Figure 3

Fuzzy entropy with level set segmentation of MRI Brain cancer: (a) Original MRI image with cancer, (b) is searching to find the cancer to region, (c) find the cancer and segmented after 130 iterations, (d) is initialization by thresholding (80-170,170-230) for extracted the cancer region.

Figure 4

Fuzzy entropy with level set segmentation of dermoscopy color image for skin cancer: (a) Original skin image with cancer, (b) is searching to find cancer to region, (c) find the cancer and segmented after 300 iterations, (d) is initialization by thresholding (120-230,170-310) for extracted the cancer region.

Figure 5

Illustrates iteration measures in ultrasound breast cancer segmentation.

Figure 6

Shows iteration measures in brain cancer segmentation.

Figure 7

Shows iteration measures of dermoscopy color image in skin cancer segmentation.

Figure 8

Results of performance measures segmentation for FELs proposed methods for medical image.

3.2 Segmentation result and comparison

The proposed (FELs) thresholding method is used in the experiment and test conducted on the three types of medical image to find the boundaries of the cancer regions; the results are segmented and extracted, and the (FELs) thresholding method is compared with the different segmentation methods for cancer. In 1945, Dice introduced a Dice coefficient method [29] and in 1912, Taccard introduced a Jaccard coefficient method [30], which utilizes for computing of the extent of spatial overlap between two binary images. It is commonly used in reporting to evaluate and perform the part of cancer segmentation or registration effects which can measure the overlap with the ground truth image. They can range between 0 (no overlap) and 1 (perfect agreement). Dice and Jaccard similarity coefficients are computed by measuring the similarity of the pixels in terms of the GT and cancer segmentation (CS) [31]. The performance of CS calculated by four factors is utilized. A true positive pixel indicates an existing and accurately detected cancer region, whereas a false positive pixel denotes a non-existing cancer case, but a result is detected. A true negative pixel indicates a non-existing cancer case, and the result is undetected, whereas a false negative pixel denotes an existing cancer case, but the result is undetected [33]. The parameter details are provided mathematically as follows:

Jaccard similarity: GT represents ground truth, and CS denotes cancer segmentation; Jaccard coefficient is defined as follows:

Dice similarity coefficient shows that.

Where CS is the cancer segmentation and GT is the ground truth image. The results showed that Jaccard and Dice coefficient was in between 0 and 1. The value of 0 corresponds to no similarity (no overlap), whereas the value of 1 indicates a similarity (perfect agreement). According to the results, Jaccard coefficient is implementing better than Dice coefficient for cancer segmentation. We test the proposed (FELs) thresholding method to measure the capability of the method in terms of sensitivity, precision, specificity, and accuracy measures by checking the negative CS results, positive CS results, positive predictive pixel values, and true CS results [33].

The accuracy results are defined as “correct” if the medical image is segmented correctly. If only a part of this region is segmented correctly, then the results are defined as “poor” results. To emphasize the accuracy results, we use the correct and poor rates, which are defined as

Where X is the total number of medical images. The FELs thresholding method has been applied to 84 images and divided into three types of medical images to determine the accuracy of the cancer image. The proposed method provides a correct rate of 96.4% and a poor rate of 3.6% after detecting 27 cancer cases from 28 ultrasound results for breast cancer; the correct rate of 96.9% and poor rate of 3.1% are obtained after detecting 31 cancer cases from 32 MRI brain cancer results; the correct rate of 95.8% and poor rate of 4.2% are achieved after detecting 23 cancer cases from 24 dermoscopy image results. The proposed algorithm has performed well and obtained a better result than the other approaches [32, 33]. Table 2 summarizes the comparison of the results.

Also, besides the cancer region, they have other types of the pathological lesion and a similar signal as the cancer region, such as traumatic brain injury (TBI). The study have applied (FELs) thresholding method to TBI image. According to the equations (19, 21 and 22) provided earlier, it might be possible to achieve the satisfied outcome as explained below:

Figures 10 and 11 show pathological lesions and similar to signal of the cancer region. The testing results for figure10 according to equations (1, 2 and 3) for sensitivity; specificity; and accuracy were (0, 0.2028, 0) respectively. Also for figure 11 was (0, 0.2869, 0) respectively.

Figure 9

Comparison the performance measures segmentation for our proposed method with other standard segmentation cancer methods according to Dice-coefficient, and Jaccard-coefficient.

Figure 10

Traumatic brain injury (TBI) with cancer, (a) Original image, (b) searching for injury with cancer, (c) find injury with cancer and segmented, (d) Thresholding and extracted the injury with cancer region.

Figure 11

Traumatic brain injury (TBI), (a) Original image, (b) searching for injury, (c) find injury and segmented, (d) Thresholding and extracted the injury region.