2.1.1. GrabCut Interactive Initialization and Probabilistic Modeling

The GrabCut algorithm is an interactive grayscale image segmentation technique proposed by Boykov and Jolly [14] (2001). To address the irregular edges and reflective surface characteristics of turbine disc fragments in aircraft engines, the GrabCut algorithm is employed to achieve robust separation between fragments and background. Pixels outside the bounding box Ω B are directly labeled as definite background ( α n = 0 ), while the interior region Ω F is designated as a potential foreground ( α n = 1 ). This weakly supervised mechanism significantly reduces the reliance on precise contour annotations inherent in traditional segmentation methods.

Based on the initial segmentation, the algorithm constructs separate Gaussian mixture models for the foreground (fragments) and background, each with K = 5 components (based on minimization of the BIC score) to accommodate the color distribution characteristics of metal fragments:

Among them:

α ∈ { 0 , 1 } denotes background/foreground labels,

π k is the mixture weight coefficient,

μ k and Σ k denote the mean and covariance matrix of the kth Gaussian component, respectively.

The algorithm achieves global optimization by constructing a Markov random field energy function comprising data terms E D and smoothing terms E S :

1) Data item

Quantifying the degree of color matching between pixels and GMMs:

Among them:

k n denotes the index of the optimal Gaussian component to which pixel n belongs.

2) Smoothing term

Using contrast-sensitive weighted Potts models to enhance spatial continuity while preserving edges:

Among them:

β = ( 2 ‖ z m − z n ‖ 2 ) − 1 is the adaptive parameter,

γ controls the smoothing intensity (typical values are 50 - 100).

Minimize energy by alternately performing the following steps:

1) GMM parameter estimation: Based on the current segmentation results, the Expectation-Maximization (EM) algorithm is employed to update the foreground/ background GMM parameters θ ( 1 ) , θ ( α ) , respectively, solving:

2) Image optimization: Based on the updated GMM parameters, construct the s-t graph network and apply the maximum flow/minimum cut algorithm to solve for the optimal label assignment α ^ , where:

Node weights are determined by the data item E D .

The edge weight corresponds to the smoothing term E S .

Eventually, the soft segmentation results are converted into hard segmentation masks through thresholding. Compared to traditional GraphCut methods, GrabCut in this paper improves robustness in textured scenes and reduces user interaction through probabilistic modeling and iterative optimization.

2.1.2. Fragment Separation Using the Eight-Neighborhood Algorithm

To address the problem of fragment adhesion in disks, this paper employs the eight-neighborhood algorithm for efficient segmentation. Adaptive binarization is first performed using the Otsu algorithm. Skeletons are extracted through connected component labeling and refinement with the Guo-Hall algorithm, with fracture constraints applied at branch points. By combining distance transformation and watershed algorithms, boundaries evolve along minimum energy paths driven by fragment centroid markers. Morphological opening and closing operations further eliminate noise and holes, while Canny edge detection enhances boundary integrity. Compared to the four-neighborhood approach, this algorithm demonstrates heightened sensitivity to diagonal connectivity, significantly improving segmentation accuracy for complex, adhered fragments.

2.1.3. Ellipse Fitting Algorithm Based on the Approximate Mean Squared Error

To address projection distortion and misalignment in turbine disk fragment images, an approximate mean-square (AMS) robust elliptical fitting algorithm is introduced. It reframes elliptical fitting as a maximum a posteriori estimation problem with regularization. Compared to traditional least-squares methods, the AMS algorithm integrates curvature consistency constraints and geometric symmetry priors, effectively suppressing noise in fragment edge defect regions and maintaining stable parameter estimation under low signal-to-noise ratios.

1) Mathematical model

In elliptical fitting, given a set of two-dimensional points { ( x i , y i ) } i = 1 N obtained

from edge detection, the general quadratic equation of an ellipse can be expressed as:

Among them:

A controls the coefficient of the x-squared term, influencing the opening width of the ellipse in the x-direction.

B controls the coefficient of the xy intersection term and determines the rotation angle of the ellipse (if b ≠ 0 , the ellipse is inclined).

C controls the coefficient of the y-squared term, influencing the opening width of the ellipse in the y direction.

D controls the coefficient of term x to determine the horizontal translation of the center of the ellipse.

E controls the coefficient of the y term to determine the vertical translation of the center of the ellipse.

F is a constant term that affects the position and size of the ellipse.

Satisfied with the elliptic constraint B 2 − 4 A C < 0 . To eliminate scale ambiguity, the normalization constraint A + C = 1 is typically imposed.

2) Maximum a posteriori estimation modeling

Assuming the observation noise follows a zero-mean Gaussian distribution ϵ i ~ Ν ( 0 , σ 2 ) , the posterior probability of the parameter vector θ = [ A , B , C , D , E , F ] T is:

The prior term p ( θ ) adopts the Tikhonov regularization form:

Among them:

R is the regularization matrix, and λ is the hyperparameter.

3) Regularization matrix design

The key innovation of the AMS method lies in designing an adaptive regularization matrix R, the construction rules of which are as follows.

First, apply curvature consistency constraints: force the fitted ellipse’s curvature at edge points to match the local geometric features. For the i-th point, compute its

curvature k i and construct a diagonal weighting matrix W = diag ( κ 1 2 , ⋯ , κ N 2 ) .

Second, geometric symmetry constraints: Punish asymmetric parameter combinations via the block matrix R s :

Final, regularization matrix final regularization matrix: Combining the above constraints, define R = α W + β R s , where α and β are balancing factors.

4) Iterative optimization solution

By maximizing the log-likelihood, the optimization problem is formulated as follows:

Among them:

The i-th row of the design matrix M ∈ R N × 6 is [ x i 2 , x i y i , y i 2 , x i , y i , 1 ] .

fitEllipseAMS is an ellipse fitting algorithm implemented in Open, subject to the constraint 4 A C − B 2 = 1 (ensuring an ellipse).

The solution steps are as follows:

1) Construct the design matrix M ∈ ℝ N × 6 , where the i-th row is [ x i 2 , x i y i , y i 2 , x i , y i , 1 ] ;

2) Solve the generalized eigenvalue problem M T M v = λ D v (where D is the constraint matrix) and select the eigenvector corresponding to the smallest eigenvalue;

3) Normalize the parameters;

4) Convert to geometric parameters (center, major and minor axes, and rotation angle).

This method provides a closed-form solution without iterations, and convergence is guaranteed by the eigenvalue decomposition.

This paper does not employ any annotated datasets; all algorithm validations are conducted through demonstrative verification based on a single turbine disk fragment image and its processing results. Import the original image of the aircraft engine rotor disk fragments into a C++ program. The original image is shown in Figure 1. Use the GrabCut algorithm to extract the foreground image, which is shown in Figure 2.

Figure 1. Initial image.

Figure 2. Foreground image processed by GrabCut.

1) Simulated restoration

The traditional three-point circle determination method locates the center by drawing perpendicular bisectors through any three points on an arc. Its core principle lies in leveraging the symmetry of the circle’s circumference to derive the geometric center. In digital scenarios, this logic translates into the following steps:

a) Edge detection: Acquire a binary image of the disk contour using the Canny edge detector or adaptive threshold segmentation.

b) Center positioning: Based on the extracted edges, identify relatively intact sector profiles to locate the center of the fitted circle.

2) Generating rotating arrays

An adaptive step size algorithm based on arc length feedback is employed. In implementation, spectral clustering is integrated to dynamically group sector vertices, preventing overlaps or gaps caused by fixed step sizes. Rotation step size formula:

Among them:

L is the arc length of the current sector.

Apply a Gaussian blur to smooth the image and reduce the jagged edges caused by the rotational array, as shown in Figure 3.

First, use the octant neighborhood algorithm to store the three fragments of the original image in the new image based on the pixel coordinates of the original image. As shown in Figure 4.

Iterate through all pixels in the original image, extract all contours using the octant neighborhood algorithm, calibrate the center points of contours based on circularity, area, and distance from the center, and fit the corresponding ellipses using the fitEllipseAMS algorithm.

Figure 3. Simulated repair diagram.

Figure 4. Single roulette fragment.

For precise contour calibration, the image is binarized. An octant-based boundary tracking algorithm extracts closed contours from the bottom-left corner, recording vertices. Based on the empirical distribution of turbine disk fragment contours and the principle of noise blob removal, abnormal contours with an area smaller than 10 pixels or circularity below 0.85 are discarded to ensure that only near-circular structures are retained. The fitEllipseAMS algorithm performs elliptical fitting using generalized eigenvalue decomposition under algebraic constraints, ensuring strictly elliptical results. Experiments confirm fitEllipseAMS maintains sub-pixel accuracy (MSE < 0.5 pixels) with outliers, outperforming standard methods. Due to insufficient fitting points in the smallest fragment, only the two largest fragments were fitted, as shown in Figure 5.

Figure 5. Elliptical fitting of fragment reference points.

Using the elliptical fitting parameters of the larger fragment on the right (major axis α r , minor axis b r , center ( x r , y r ), and rotation angle θ r ) as the reference, perform rotational adjustment and registration on the smaller fragment on the left. First, the elliptical parameters of the left fragment are extracted using the fitEllipseAMS algorithm. Through an affine transformation, the ratio of its major axis to minor axis is adjusted to the reference dimensions. It is then rotated by the angular difference Δ θ = θ r − θ l around its own center to align with the principal axis direction. Subsequently, calculate the translation vector ( Δ x , Δ y ) = ( x r − x C l , y r − y C l ) , move the left fragment to the position of the right circle’s center, and merge it with the right fragment, as shown in Figure 6.

Figure 6. Fragments are stitched according to the fitted ellipse.

The outer contour is extracted and subjected to an affine transformation, as shown in Figure 7.

Figure 7. Internal and external contours after assembling the fragmented parts.

By comparing the assembled contour of the turbine disc fragment with the simulated repair image, the precise location requiring repair is identified—whether due to bonding issues or other problems occurring during operation. The specific repair method is then determined based on an analysis of the size of the affected area. Using Boolean operations, the damaged rotor disc of the aero engine is processed to remove the bonded sections and achieve a binary representation of the core structure, as shown in Figure 8.

Figure 8. Damaged turbine disk body section of an aircraft engine.

To clearly demonstrate the full workflow of our proposed method and define the inputs/outputs for each stage, we provide the following end-to-end pseudocode.

Input: Original image I , hyperparameters (GMM K = 5 , circularity threshold 0.85, area threshold 10 px, 36 sectors, etc.).

Output: Aligned contours and defect measurements

1) Preprocessing: Input I , convert to HSV, and apply Gaussian blur ( σ = 3 ); Output: Preprocessed image.

2) Contour Segmentation: Input preprocessed image, use GMM ( K = 5 ) for foreground separation + Canny edge detection (thresholds 50–150) followed by contour finding; Output: Inner contour C i n (defect boundary) and outer contour C o u t (ellipse boundary).

3) Contour Filtering: Input C i n and C o u t , discard if circularity < 0.85 or area < 10 pixels; Output: Filtered contours.

4) Separate Ellipse Fitting: Input filtered contours, apply fitEllipseAMS to C o u t for outer ellipse E o u t ; fit to C i n for inner ellipse E i n ; Output: Fitted ellipse parameters.

5) Contour Alignment: Input fitted ellipses, sectorize into 36 sectors (10˚ each), match sectors by angular overlap + affine transform to optimize overlap distance < threshold; Output: Aligned contours.

6) Contour Fusion: Input aligned contours, merge C i n and C o u t via Gaussian blur, compute the final defect mask as the difference (fusing inner/outer contours into complete defect regions); Output: Fused contours and defect mask.

7) Defect Measurement: Input fused contours; for each defect region, compute the depth-to-half-width ratio r = d / ( w / 2 ) ; classify as V-type ( r < 30 % ), U-type ( 30 % < r ≤ 70 % ), flat-bottom type ( 70 % < r ≤ 100 % ), or inverted trumpet type ( r > 100 % ). Output: Defect measurements.

8) Return: Fused contours and measurements.

Create 36 sectors to obtain the coordinate points of the assembled fragments’ contour for the aircraft engine. Calculate the distance from each pixel point to the center, along with the radius and angle, then convert the angle to the 0˚ to 360˚ range. If the radius of the current point exceeds the maximum radius point of the sector, update the sector’s maximum radius point. Then traverse each sector, extract the maximum radius points from 10 sectors, and fit an ellipse using cv::fitEllipse. The fitting results are shown in Figure 9.

Figure 9. Fitting results for the outer circle of the broken wheel disc of the aero engine.

This paper presents a surface defect detection method using dual-color space analysis and distance metrics. In HSV space, a red reference region (hue: 0˚ - 10˚ and 160˚ - 180˚, saturation > 70, brightness > 50) and a white target region (saturation < 30, brightness > 200) are defined and segmented into binary masks. The masks are optimized via morphological operations. The white region’s maximum outer contour serves as the detection baseline, while red contour points are merged into a reference set. A FLANN-based KD-Tree accelerates nearest-neighbor searches. White contour points are traversed, and segments where the Euclidean distance to the nearest red reference point exceeds a threshold are flagged as defects, as shown in Figure 10. For defects not found counterclockwise, the start and end points are determined as the first points where the distance to the outer fitted circle exceeds and subsequently falls below the threshold (marked red and blue, respectively).

Figure 10. Defect marking diagram.

Define the width by connecting the start and end points. Locate the point with the maximum distance through a secondary search as the depth point (green marker), where its distance value represents the defect depth. The final output includes a quantified report detailing the defect location, width, depth, and key point coordinates. The report identified a total of 14 defects. Selected characteristic parameters for defects numbered 1 through 5 are shown in Table 1. The defect path is simultaneously visualized (yellow lines), with the overall measurement diagram shown in Figure 11. This method effectively achieves the automatic localization and quantitative assessment of surface deformation defects.

Table 1. Partial characterization parameters of defects numbered 1 - 5.

Figure 11. Marking of defects in the overall profile.