Numerical subgrid Bi-cubic methods of partial differential equations in image segmentation

Image segmentation is a core research in the image processing and computer vision. In this paper, we suggest a Bi-cubic spline phase transition potential and elaborate a Bi-Cubic spline phase transition potential development. In the image segmentation, we develop the new approach to apply the novel computational fluid dynamics in the boundary with subgrid. The numerical subgrid Bi-cubic method with Bi-Cubic spline for minimizing the piecewise constant energy functional is very efficient, robust and fast in the image segmentation with a multispecies multiphase segmentation models. The subgrid Bi-cubic spline is applied on the boundary with subgrid and the regular grid is applied on the non-boundary. The model generates a multispecies multiphase distribution with Bi-Cubic spline and we can extract the image segments with multispecies multiphase. Finally, we analyze the models and show the numerical results. Numerical results are presented with OCR (Optical Character Recognition) and the medical image.


Methods
The multiphase approximation model is proposed with K-phase fields from minimizing the Mumford-Shah functional and Allen-Chan equation is applied for the length of the curve C 1, 19 .Additionally, we proposed the new model with a Bi-cubic spline potential.This suggested model has a different Bi-cubic spline phase transition potential BiC( φ ) instead of the phase transition potential from the original model.The model of energy functional is the following: φ is phase-field which is meaning the mixing rate (Atwood number) in computational fluid dynamics 16,17,18 .φ is defined by φ − [φ] , [φ] : the largest integer not greater than φ .In the phase transition, we applied a Bi-cubic spline in the image domain and image intensity interface.We suggest and define a Bi-cubic spline phase transition potential BiC( φ ) as the following: From the energy funtional, the first term denomerator's ε is the gradient energy coefficient to the interfacial energy force.In the image application, we denote I for image intensity.I 0 is the normalized initial image inten- sity. is the image whole domain of all collections of pixels.From the modified model, the second term, |∇φ| 2 2 is variation of the transition with the spline (2) between smoothing levels and high oscillation levels.The third term is the source term to fit the model 1 . 1 sinc (φ) = sin(πφ)/(πφ), φ ∈ R. Refer to Fig. 1. is a nonnegative parameter.I 0 is the initial image after normalization.Also C k is the averages of I 0 in the regions and defined as the following 1 When φ flow attain to a steady state in the image intensity flow, the k + 1 level set will become the contour for each individual k th phase.For this purpose we get the following gradient descent flow equation: We show numerical subgrid Bi-cubic spline method for the Mumford-Shah functional.It is the modified model methods with the image intensity in 3D spline = 2D spline X 2D spline.
(1) www.nature.com/scientificreports/ The image intensity function is in three dimensional space.The image intensity function values is from 0 to 255 but it is rescaled from 0 to 1. First, we discretize two dimensional space of the domain, � = (a, b) × (c, d) as a evenly discretized regular domain setting.Let N x and N y be positive even integers, h = (b−a) N x be the uniform mesh size, and } be the set of the cell centers for the computational domain.Through the cell centers, we generate the subgrid on the domain.Let φ n ij be approximations of φ(x i , y j , n�t) , where �t = T/N t is the time step, T is the final time, and N t is the total number of time steps.By using these grid and subgrid on the boundary, we propose the following numerical subgrid Bi-cubic spline methods.
In Eq. ( 6), RHS' the first term with Bi-cubic spline can be transformed to the following: and Finally, the RHS of the Eq. ( 6) with Bi-cubic spline can be transformed to the following: Briefly, in our first numerical result, we will show that above computation results in the OCR (Optical Character Recognition) test, shown in Fig. 2. www.nature.com/scientificreports/ Figure 2 shows the application of the images by using our proposed model to OCR (Optical Character Recognition).After ten iterations, we have the segmentation image.The computational domain � = (0, 1) × (0, 1) with a 256 × 256 mesh.The interface parameter ε 2 and time step t = 5E−6 are used.
We show the suggested Bi-cubic spline algorithm procedure to solve the image segmentation problems (6) which is the following: Step 1. Normalize the image intensity values [0,1] from [0, 255].

Results
In this section, we present some numerical results by our proposed modified models with Bi-cubic spline.We applied this models, the numerical scheme and solutions to the medical image.This methods can be extended and applied to the panoramic image, the blurred image and the shaken image for our future works.It can be realized that our proposed scheme is fast, accurate, and robust.In our numerical experiments, we normalize the given image intensity I with the spline (2) as I 0 = I−I min I max −I min , where I max and I min are the maximum and the minimum values of the given image, respectively.Through this normalization with the spline (2), the initial image intensity function, I 0 ∈ [0, 1] .Then the mixing interfacial regions with RGB or Gray level ∈ [0, 1] .The interfacial concentration rate with the spline (2) varies from 0.1 to 0.9.The approximate distance measurement can be calculated by 4 √ 2ε tanh −1 (0.9) .We take the gradient energy parameter for the interfacial energy force, ε m = hm α , with α = 4 √ 2 tanh −1 (0.9) and m gird points for subgrid, h uniform grid steps.After we initialized φ = KI 0 with the spline (2) and K-phase, we obtain the reasonable numerical results.See the following figures.
Figure 3 shows the several images results with our model to the oil painting image.The original image is from the OpenCV Website 20 .The original is the open image.After ten iterations, we have the segmentation image.The computational domain � = (0, 1) × (0, 1) with a 256 × 256 mesh.The interface parameter ε 2 and time step t = 5x10 −6 .Figures 4 and 5 shows the application of the images by using our proposed model to a magnetic resonance imaging (MRI) brain image.

Conclusions and discussion
In this paper, we propose new image segmentation methods with the Bi-cubic spline (2) which is from the computational fluid dynamics scheme, the finite element scheme (FES).In the computational fluid dynamics (CFD), there are many novel approach to solve the partial differential equations.This is valuable time to apply image segmentation from the computational fluid dynamics.Our suggested methods are fast, robust and accurate.Therefore, we obtain the satisfactory results in the image segmentation.Especially, OCR (Optical Character Recognition) and the medical image are applied in the application image.From OCR images, we obtian the satisfactory results through adjusting φ values.From the brain images, we obtain the tumor part segmentation from the malignant image by adjusting φ values.We have some difficulty to obtain the real physical and medical images to investigate the data and methods comparison.This methods can be also apply to the satellite image.The several parameter of the suggested models can be adjusted to have the demanding results in the segmentation of image.This new image segmentation methods can be applied to develope the artificial intelligence about detecting several objects or obstacles.In the future works, we can develop the conservation law scheme with the volume fraction in image inpainting with the computational fluid dynamics.