A method for selective segmentation of gray uneven images based on offset field prior

Abstract

The present application belongs to the technical field of image processing, and particularly relates to a gray uneven image selective segmentation method based on offset field prior. Mainly comprising: inputting a gray uneven image to be segmented; manually adding an initial marker point set to the input image, and forming a corresponding initial contour line; constructing a gray uneven image selective segmentation model based on offset field prior according to the local constant and spatial smoothness of the offset field; converting the constructed model into an approximate approximation problem by using a hot kernel convolution, and designing an alternating minimization algorithm for numerical solution based on an alternating direction multiplier method; using the designed alternating minimization algorithm to perform selective segmentation on the gray uneven image, and obtaining a corresponding selective segmentation result; the gray uneven image can be selectively segmented well; the superiority of the method in segmentation accuracy and running speed is ensured, and the method has robustness to parameters and initial contours.

Description

Technical Field

[0001] This invention belongs to the field of image processing technology, specifically relating to a selective segmentation method for gray-scale non-uniform images based on offset field prior. Background Technology

[0002] Image segmentation is the process of dividing an image into several non-overlapping regions with similar properties. Selective segmentation, on the other hand, aims to segment specific regions within an image and has wide applications in many fields where complete segmentation of the entire image is not necessary. However, real-world images, especially most medical images, are typically non-uniform in grayscale. Factors contributing to this non-uniformity can be categorized into internal and external factors. The former primarily refers to the complex features of the object itself, such as complex textures, while the latter generally includes illumination shifts and defects in the imaging equipment. Non-uniformity in grayscale leads to inconsistencies in intensity values ​​across image regions and low contrast between objects and the background. Therefore, image segmentation for non-uniform grayscale images remains an important task.

[0003] In recent years, scholars have proposed many methods to segment images with gray-level non-uniformity caused by external factors. Gray-level non-uniformity caused by defects in imaging equipment can be handled using methods based on local image information, such as local Gaussian distribution fitting and local prefitting models. The LIC model proposes to approximate the offset field using local intensity clustering properties, which can effectively segment images with non-uniform gray-level and has good robustness to the initial contour position. In addition, Retinex theory can handle gray-level non-uniformity caused by illumination offset. Retinex theory represents a gray-level non-uniform image as the sum of the illumination offset part and the structural part. Currently, some segmentation methods based on Retinex theory have emerged, such as the CV-B model, CV-BS model, and RSS model.

[0004] Currently, most selective segmentation methods are based on the assumption of uniform image gray levels, which fails to segment images with non-uniform gray levels effectively. For example, Spencer and Chen introduced Euclidean distance as an independent constraint term into the energy functional, proposing a novel selective segmentation model (SC model). Subsequently, Roberts, Chen, and Irion proposed a new variational segmentation model (RCI model), replacing Euclidean distance with geodesic distance, achieving better segmentation results and robustness to the initial contour. Several models, such as the SMS model and the RSS model, are currently used for selective segmentation of images with non-uniform gray levels. However, existing selective segmentation models rarely consider local image information, and methods utilizing local image information for selective segmentation of images with non-uniform gray levels require further research. Summary of the Invention

[0005] To address the problems existing in the prior art, this invention proposes a selective segmentation method for gray-scale non-uniform images based on offset field prior.

[0006] First, unlike existing methods based on Retinex theory, this invention represents gray-level non-uniform images as the product of an offset field and a piecewise constant image. By introducing the local constant property of the offset field, this invention proposes a novel selective segmentation model for gray-level non-uniform images. The proposed model further incorporates a second-order regularization term of the offset field gradient to characterize its spatial smoothness. Then, hot kernel convolution is used to transform the model into an approximate problem, and an alternating minimization algorithm based on the alternating direction multiplier method is proposed to numerically solve the approximate problem. The proposed method can effectively selectively segment gray-level non-uniform images, has fewer iteration steps and shorter runtime, and is robust to parameters and initial contours.

[0007] This invention is specifically achieved through the following technical solution: a selective segmentation method for gray-level non-uniform images based on offset field prior.

[0008] Step 1: Input the image with uneven grayscale to be segmented;

[0009] Step 2: Manually add an initial set of marker points to the input image and form the corresponding initial contour lines;

[0010] Step 3: Based on the local constants of the offset field and the property of spatial smoothness, construct a selective segmentation model for gray-level non-uniform images based on the prior of the offset field.

[0011] Step 4: The model constructed in Step 3 is transformed into an approximate problem using hot kernel convolution, and an alternating minimization algorithm is designed based on the alternating direction multiplier method for numerical solution.

[0012] Step 5: The alternating minimization algorithm designed in Step 4 is used to selectively segment the image with uneven gray levels to obtain the corresponding selective segmentation results.

[0013] Preferably, step 3 involves constructing a selective segmentation model for gray-level non-uniform images based on the offset field prior, utilizing the local constants of the offset field and the property of spatial smoothness, as detailed below:

[0014] Based on the physical laws of imaging under multiple modes, let x = (x1, x2). A gray-scale non-uniform image I(x) is represented as the product of the offset field component b(x) and the piecewise constant component t(x): I(x) = b(x)t(x); where, the offset field... b It is usually assumed that the expression is smooth and slowly changing, and therefore is approximated as constant within a local window. That is, given a window radius r, for all x ∈ Ω, the expression in the neighborhood of point x is constant. The offset field at position y of all pixels within the area is approximately represented by the offset field at position x: By incorporating the local constant property of the offset field into existing selective segmentation models and further considering the spatial smoothness of the offset field, a selective segmentation model for gray-level non-uniform images based on the offset field prior is constructed:

[0015]

[0016] Where I represents the non-uniform grayscale image to be segmented; u represents the segmentation result image; b represents the offset field; c1 and c2 are the average pixel values ​​inside and outside the image segmentation region, respectively; and θ, λ1, λ2, and α are positive parameters. This represents the weighted total variation regularization term for the segmentation result image u; This is a geodetic distance constraint term; and For data fidelity items; It is a spatial smoothness prior term of the offset field;

[0017] Let ω be the edge detection function, where ω is a positive parameter; D is the difference operator; D G It is a geodesic distance function; It is a Gaussian kernel function with a positive parameter σ.

[0018] Preferred, For the standardized geodesic distance function, where satisfy P is the region enclosed by the initial set of marked points, ε D ,β G , It is a positive parameter. It is a standardized Euclidean distance function, and S is the equation The numerical solution obtained using Gauss-Seidel iteration, where ε S =10 -6 γ=5×10 -4 .

[0019] Preferably, the process of converting the model constructed in step 3 into an approximate problem using hot kernel convolution in step 4 is as follows:

[0020] First, for c1 and c2, the corresponding results are obtained by integration as follows:

[0021]

[0022] Then, the hot kernel convolution approximation is used.

[0023]

[0024] The proposed segmentation model is transformed into an approximate approximation problem:

[0025]

[0026] in It is a Gaussian kernel with standard deviation τ, and g is the edge detection function.

[0027] Preferably, the alternating direction multiplier method described in step 4 is used to design an alternating minimization algorithm for numerical solution. The specific steps are as follows: use the alternating minimization algorithm to solve the subproblem of model (3) to determine the final segmentation result u;

[0028] First, initialize the parameters, that is, set the iteration step k = 0 and the initial offset field b. (0) =1, u (0) The initial contour is formed by a set of manually entered marker points;

[0029] Repeat the iteration multiple times until the preset iteration termination condition is met:

[0030]

[0031] Among them, u (k) u represents the segmentation result obtained in the k-th iteration. (k+1) This represents the segmentation result obtained in the (k+1)th iteration, ε = 10 -4 This is the termination condition for the iteration;

[0032] In the process of repeatedly executing multiple iterations, for each iteration, that is, in the k-th iteration:

[0033] First, for a given b( k The following method is used to calculate u( k+1 ):

[0034]

[0035] in

[0036]

[0037] Let L(u; u) (k) ,b (k) (c1, c2) is the energy functional E u (u,b (k) c1, c2) in u (k) If the linear approximation is obtained at u, then u (k+1) The solution can be approximated using the following method:

[0038]

[0039] in

[0040]

[0041] Since formula (7) is a linear problem with respect to u,

[0042]

[0043] When solving b (k+1) When, for a given u (k+1) b is calculated by iteratively solving the following problem. (k+1) :

[0044]

[0045] For the given u (k+1) ,b (k+1) Calculated by the following formula

[0046]

[0047] Preferably, calculate b (k+1) The specific steps are as follows:

[0048] By introducing an auxiliary variable d, problem (10) is transformed into a constrained minimization problem:

[0049]

[0050] std(x) = b(x).

[0051] Using the Lagrange multiplier η, the constrained optimization problem is transformed into an unconstrained extremum problem, and the corresponding Lagrange function is:

[0052]

[0053] Where ρ is the penalty parameter;

[0054] The iterative solution using the alternating direction multiplier method is as follows:

[0055] First, initialize the execution parameters, that is, set the inner iteration step number l = 0, and the initial offset field for the inner iteration. Initial Lagrange multipliers

[0056] Repeat the iteration multiple times until the preset iteration termination condition is met:

[0057]

[0058] in, This represents the offset field obtained in the l-th iteration. This represents the number of offset fields obtained in the (l+1)th iteration, ε = 10 -4 This is the convergence condition for the iteration;

[0059] In the process of repeatedly executing multiple iterations, for each iteration, i.e., in the l-th iteration:

[0060] When given To calculate, solve the following problems.

[0061]

[0062] Calculated

[0063]

[0064] in

[0065] Using the calculated Calculated by the following formula

[0066]

[0067] The corresponding Euler-Lagrange equation is

[0068]

[0069] Using the Fast Fourier Transform, we can obtain:

[0070]

[0071] in and They represent the Fast Fourier Transform and its inverse transform, respectively; Δ is the Laplace operator.

[0072] Solving for the results Then, for a given Calculated by the following formula

[0073]

[0074] Preferably, the Dice similarity coefficient and Hausdorff distance are used as evaluation indicators to quantitatively measure the segmentation effect;

[0075] 1) Dice similarity coefficient

[0076]

[0077] Where S1 is the segmented object region and S2 is the corresponding gold standard object region;

[0078] 2) Hausdorf Distance

[0079]

[0080] Where R1 is the segmentation result image, R2 is the corresponding gold standard image; n is the number of elements in the object boundary set, and r1 and r2 are elements in R1 and R2, respectively.

[0081] In summary, the selective segmentation method for gray-level non-uniform images proposed in this invention effectively handles selective segmentation of gray-level non-uniform images by introducing the local constant property of the offset field and further using the second-order regularization term of the gradient to characterize the spatial smoothness of the offset field. The alternating minimization numerical solution algorithm designed based on hot kernel convolution and alternating direction multiplier method enables the method to achieve higher accuracy, faster running speed, and robustness to parameters and initial contours for selective segmentation of gray-level non-uniform images.

[0082] The present invention has the following beneficial effects: 1. Unlike the additive model I = B + S assumed by the existing Retinex theoretical method for gray-level non-uniform images, the present invention considers the multiplicative model I = b·t, introduces the local constant property of the offset field b into the selective segmentation model, and further considers the spatial smoothness of b, so that the proposed model can perform selective segmentation of gray-level non-uniform images very well.

[0083] 2. Based on hot kernel convolution and alternating direction multiplier method, this invention specifically designs an efficient and stable alternating minimization algorithm to numerically solve the proposed model, ensuring that the segmentation accuracy and running speed of the method in this invention are superior to existing models. It also exhibits robustness to parameters and initial contours. Attached Figure Description

[0084] Figure 1 This is a flowchart illustrating a selective segmentation method for gray-level non-uniform images based on offset field priors according to the present invention.

[0085] Figure 2 The image segmentation results of each model are shown in Figures 1-3 ((a) Gold standard segmentation image with initial contour, (b) SC model, (c) RCI model, (d) SMS model, (e) RSS model, (f) model of this invention).

[0086] Figure 3 The parameter mapping diagram for segmenting image 3 using the SC model;

[0087] Figure 4 The parameter mapping diagram for segmenting image 3 using the RCI model;

[0088] Figure 5 Parametric mapping for segmenting image 3 using the SMS model;

[0089] Figure 6 The parameter mapping diagram for segmenting image 3 using the RSS model;

[0090] Figure 7 This is a parameter mapping diagram of image 3 segmented by the model of the present invention;

[0091] Figure 8 The medical images 4-9 are segmentation results for each model ((a) gold standard segmentation image with initial contour, (b) SC model, (c) RCI model, (d) SMS model, (e) RSS model, (f) model of this invention). Detailed Implementation

[0092] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings:

[0093] This invention proposes a selective segmentation method for gray-level non-uniform images based on offset field priors. A flowchart is shown below. Figure 1 The specific steps are as follows:

[0094] The first step is to input a grayscale non-uniform image to be segmented, manually add an initial set of marker points to the input image, form the corresponding initial contour lines, and construct a selective segmentation model for grayscale non-uniform images.

[0095] Based on the physical laws of imaging under various modalities (e.g., cameras, MR imaging), let x = (x1, x2). A grayscale non-uniform image I(x) can be expressed as the product of the offset field component b(x) and the piecewise constant component t(x): I(x) = b(x)t(x). Here, the offset field b is usually assumed to be smooth and slowly changing, and therefore can be approximated as a constant within a local window. That is, given a window radius r, for all x ∈ Ω, in the neighborhood of point x... The offset field at all pixel positions y within the range can be approximately represented by the offset field at x: By incorporating the local constant property of the offset field into existing selective segmentation models and further considering the spatial smoothness of the offset field, a selective segmentation model for gray-level non-uniform images based on the offset field prior is constructed:

[0096]

[0097] Where I represents the non-uniform grayscale image to be segmented, u represents the segmented image, b represents the offset field, c1 and c2 are the average pixel values ​​inside and outside the segmented region of the image, respectively, and θ, λ1, λ2, and α are positive parameters. Let be the edge detection function, where ω is a positive parameter; D is the difference operator; It is a Gaussian kernel function with a positive parameter σ; This represents the weighted total variation regularization term of the segmentation result image u. For geodetic distance constraints, and For data fidelity items, It is a spatial smoothness prior term of the offset field. For the standardized geodesic distance function, where satisfy P is the region enclosed by the initial set of marked points, ε D ,β G , It is a positive parameter. It is a standardized Euclidean distance function, and S is the equation The numerical solution obtained using the Gauss-Seidel iteration, where ε S =10 -6 γ=5×10 -4 .

[0098] The second step is to design a numerical solution algorithm. The model of this invention is transformed into an approximate problem using hot kernel convolution, and an alternating minimization algorithm based on the alternating direction multiplier method is proposed for numerical solution. The specific steps are as follows:

[0099] First, we can find the values ​​of c1 and c2 by integration, and the results are as follows:

[0100]

[0101] Then, the hot kernel convolution approximation is used.

[0102]

[0103] The proposed segmentation model is transformed into an approximate approximation problem:

[0104]

[0105] Where g is the edge detection function. It is a Gaussian kernel with a standard deviation of τ.

[0106] Next, the alternating minimization algorithm is used to solve the subproblem of model (3) to determine the final segmentation result u. The specific process is as follows:

[0107] First, initialize the parameters, that is, set the iteration step k = 0 and the initial offset field b. (0) =1, u (0) The initial contour is formed by a set of manually entered marker points.

[0108] Repeat the iteration multiple times until the preset iteration termination condition is met:

[0109]

[0110] Among them, u (k) u represents the segmentation result obtained in the k-th iteration. (k+1) This represents the segmentation result obtained in the (k+1)th iteration, ε = 10 -4 This is the termination condition for the iteration.

[0111] In the process of repeatedly executing multiple iterations, for each iteration, that is, in the k-th iteration:

[0112] First, for a given b (k) u is calculated using the following method. (k+1) :

[0113]

[0114] in

[0115]

[0116] Let L(u; u) (k) ,b (k) (c1, c2) is the energy functional E u (u,b (k) c1, c2) in u (k) If the linear approximation is obtained at u, then u (k+1) The solution can be approximated using the following methods:

[0117]

[0118] in

[0119]

[0120] Note that (7) is a linear problem with respect to u, thus we obtain:

[0121]

[0122] When solving b (k+1) When, for a given u (k+1) b is calculated by iteratively solving the following problem. (k+1) :

[0123]

[0124] By introducing an auxiliary variable d, problem (10) is transformed into a constrained minimization problem:

[0125]

[0126] std(x) = b(x).

[0127] Using the Lagrange multiplier η, the constrained optimization problem is transformed into an unconstrained extremum problem, and the corresponding Lagrange function is:

[0128]

[0129] Where ρ is the penalty parameter.

[0130] The iterative solution using the alternating direction multiplier method is as follows:

[0131] First, initialize the execution parameters, that is, set the inner iteration step number l = 0, and the initial offset field for the inner iteration. Initial Lagrange multipliers

[0132] Repeat the iteration multiple times until the preset iteration termination condition is met:

[0133]

[0134] in, This represents the offset field obtained in the l-th iteration. This represents the offset field obtained in the (l+1)th iteration, ε = 10 -4 This is the convergence condition for the iteration.

[0135] In the process of repeatedly executing multiple iterations, for each iteration, i.e., in the l-th iteration:

[0136] When given To calculate, solve the following problems.

[0137]

[0138] Calculated

[0139]

[0140] in

[0141] Using the calculated Calculated by the following formula

[0142]

[0143] The corresponding Euler-Lagrange equation is

[0144]

[0145] Using the Fast Fourier Transform, we can obtain:

[0146]

[0147] Where Δ is the Laplace operator. and These represent the Fast Fourier Transform and its inverse transform, respectively.

[0148] Solving for the results Then, for a given Calculated by the following formula

[0149]

[0150] In fact, during each iteration of solving model (3), the inner iteration of subproblem b only needs to be performed once to obtain a relatively ideal final segmentation result.

[0151] For the given u (k+1) ,b (k+1) Calculated by the following formula

[0152]

[0153] The third step involves using the models and algorithms described in the first and second steps to selectively segment the input image with uneven grayscale, obtaining the selective segmentation results. The Dice similarity coefficient and Hausdorff distance are then used as evaluation metrics to quantitatively measure the segmentation performance.

[0154] 1) Dice similarity coefficient (DSC)

[0155]

[0156] Where S1 is the segmented object region and S2 is the corresponding gold standard object region;

[0157] 2) Hausdorf Distance (HD)

[0158]

[0159] Where R1 is the segmentation result image, R2 is the corresponding gold standard image; n is the number of elements in the object boundary set, and r1 and r2 are elements in R1 and R2, respectively.

[0160] Of the two evaluation metrics, the larger the Dice similarity coefficient, the better the segmentation effect; and the smaller the Hausdorff distance, the better the segmentation effect.

[0161] To better understand the content of the method of this invention, the invention will be further described below with reference to examples. This example uses MATLAB simulation experiments on two synthesized images with Gaussian noise of variance 0.0005 (sample1, sample2), one synthesized image with Gaussian noise of variance 0.001 (sample3), and six medical images (three CT images (CT1, CT2, CT3) and three MR images (MR1, MR2, MR3)). The computer used in the experiment had the following configuration: Processor: Intel(R) Core(TM) i7-7700HQ CPU @ 2.80GHz 2.81GHz; Memory: 16.00GB; System type: 64-bit operating system, x64-based processor. The model of this embodiment is compared with the SC model, RCI model, and RSS model. The segmentation effect is evaluated using the Dice similarity coefficient and Hausdorff distance. The results are shown in Tables 1 and 2.

[0162] Table 1 Comparison of selective segmentation effects of synthesized images

[0163]

[0164]

[0165] Table 2 Comparison of Selective Segmentation Results in Medical Images

[0166]

[0167] The data in the table shows that when the proposed model, along with the SC, RCI, SMS, and RSS models, were used to selectively segment different images, the Dice similarity coefficients of the segmentation results were significantly improved, and the Hausdorff distance was significantly reduced, indicating that the proposed model can indeed obtain more accurate selective segmentation results. Furthermore, except for images 3 and 7, the model proposed in this invention requires the least convergence time among all models, and the running time for images 3 and 7 is only slightly different from the shortest time. Except for image 3, the required number of iterations is also the fewest among all models, indicating that the proposed method has the highest running efficiency. In addition, from... Figure 3-7 Looking at the parameter mapping diagrams of various models: compared to the SC model and the RCI model, the parameter mapping diagram of the model proposed in this invention is red over a larger range, indicating that the model of this invention obtains accurate selective segmentation results over a wider range of parameters, and that the model of this invention has stronger parameter robustness.

[0168] The above description of the disclosed embodiments enables those skilled in the art to use the present invention. Furthermore, the above embodiments are merely illustrative of the technical concept of the present invention and should not be construed as limiting the scope of protection of the present invention. Any modifications made to the technical solutions based on the technical concept proposed in this invention fall within the scope of protection of this invention.

Claims

1. A selective segmentation method for gray-level non-uniform images based on offset field prior, characterized in that, Step 1: Input the image with uneven grayscale to be segmented; Step 2: Manually add an initial set of marker points to the input image and form the corresponding initial contour lines; Step 3: Based on the local constants of the offset field and the property of spatial smoothness, construct a selective segmentation model for gray-level non-uniform images based on the prior of the offset field, as follows: Based on the physical laws of imaging under multiple modalities, let... An image with uneven grayscale Represented as the offset field part and piecewise constant part The product of: Among them, the offset field It is usually assumed to be smooth and slowly changing, and therefore approximated as constant within a local window, i.e., given the window radius. For all At point Neighborhood All pixel positions within The offset field at that location is approximately determined by Offset field representation at: By incorporating the local constant property of the offset field into existing selective segmentation models and further considering the spatial smoothness of the offset field, a selective segmentation model for gray-level non-uniform images based on the offset field prior is constructed: ；（1） in, This represents an image with uneven grayscale to be segmented; This represents the segmentation result image; Indicates the offset field; and These are the average pixel values ​​inside and outside the image segmentation region, respectively. It is a positive parameter; Image representing the segmentation result The weighted total variation regularization term; This is a geodetic distance constraint term; and For data fidelity items; It is a spatial smoothness prior term of the offset field; Here is the edge detection function, where A is a positive parameter; D is the difference operator; It is a geodesic distance function; For those with positive parameters Gaussian kernel function; For the standardized geodesic distance function, where satisfy , It is the region enclosed by the initial set of marked points. It is a positive parameter. It is a standardized Euclidean distance function, and , For the equation The numerical solution obtained using Gauss-Seidel iteration, where ; Step 4: The model constructed in Step 3 is transformed into an approximate problem using hot kernel convolution, and an alternating minimization algorithm is designed based on the alternating direction multiplier method for numerical solution. Step 5: The alternating minimization algorithm designed in Step 4 is used to selectively segment the image with uneven gray levels to obtain the corresponding selective segmentation results.

2. The selective segmentation method for gray-level non-uniform images based on offset field prior according to claim 1, characterized in that, Step 4 involves using hot kernel convolution to transform the model constructed in step 3 into an approximate problem. The specific steps are as follows: First of all, for and The result obtained by integration is as follows: ； （2） Then, the hot kernel convolution approximation is used. ； The proposed segmentation model is transformed into an approximate approximation problem: ； （3） in The standard deviation is Gaussian kernel, This is the edge detection function.

3. The selective segmentation method for gray-level non-uniform images based on offset field prior according to claim 2, characterized in that, The alternating direction multiplier method described in step 4 is used to design an alternating minimization algorithm for numerical solution. The specific steps are as follows: Use the alternating minimization algorithm to solve the subproblem of model (3) to determine the final segmentation result. ; First, initialize the parameters, that is, set the number of iteration steps. Initial offset field , The initial contour is formed by a set of manually entered marker points; Repeat the iteration multiple times until the preset iteration termination condition is met: ；（4） in, Indicates the first The segmentation result obtained in the second iteration. Indicates the first The segmentation result obtained in the second iteration. This is the termination condition for the iteration; In the process of repeatedly executing multiple iterations, for each iteration, i.e., the th iteration... In the next iteration: First, for the given The following method is used to calculate : ； （5） in, ; (6) make For energy functionals exist The linear approximation at that point is then The solution can be approximated using the following method: ； （7） in, ; (8) Since formula (7) is about The linear problem, therefore ； （9） When solving When, for a given The following problem is calculated iteratively. : ； （10） For the given Calculated by the following formula : 。 4. The selective segmentation method for gray-level non-uniform images based on offset field prior according to claim 3, characterized in that, calculate The specific steps are as follows: Introducing auxiliary variables The problem (10) is transformed into a constrained minimization problem: ； （11） Using Lagrange multipliers The constrained optimization problem is transformed into an unconstrained extremum problem, and the corresponding Lagrangian function is: ； （12） in, For penalty parameters; The iterative solution using the alternating direction multiplier method is as follows: First, initialize the execution parameters, that is, set the number of inner iteration steps. Inner iteration initial offset field Initial Lagrange multipliers Repeat the iteration multiple times until the preset iteration termination condition is met: ； （13） in, Indicates the first The offset field obtained in the next iteration. Indicates the first The number of offset fields obtained in the next iteration. This is the convergence condition for the iteration; In the process of repeatedly executing multiple iterations, for each iteration, i.e., the th iteration... In the next iteration: When given To calculate, solve the following problems. : ； （14） The calculation yielded: ； （15） in ; Using the calculated Calculated by the following formula : ；（16） The corresponding Euler-Lagrange equations are: ； （17） Using the Fast Fourier Transform, we can obtain: ； （18） in, and These represent the Fast Fourier Transform and its inverse transform, respectively; For the Laplace operator; Solving for the results Then, for a given Calculated by the following formula : （19）。 5. The selective segmentation method for gray-level non-uniform images based on offset field prior according to claim 1, characterized in that, The Dice similarity coefficient and Hausdorff distance are used as evaluation metrics to quantitatively measure the segmentation effect. 1) Dice similarity coefficient: ; in, To segment the resulting object region. This corresponds to the gold standard object region; 2) Hausdorf distance: ； in, For the segmented image, The corresponding gold standard image; The number of elements in the object boundary set. They are respectively The elements in.

Images