Fingerprint image expansion method and system based on topological phase field and trusted check

Abstract

The application discloses a fingerprint image expansion method and system based on a topological phase field and a trusted check, divides a small-area fingerprint embedded canvas into a known real area and a to-be-generated area, estimates a known area fingerprint structure feature and extrapolates to the to-be-generated area, reconstructs a phase potential based on the known area structure feature, takes a trace as a boundary phase, constructs a topological phase field in the to-be-generated area, and extracts a ridge skeleton probability graph; structural conditions containing the topological phase field and the skeleton probability graph are input into a generation network to output a to-be-generated area gray-scale image, and a real gray scale is forced to be written back; a candidate image is subjected to minutia point conservation check and credibility evaluation, and an expansion result with a confidence graph and a matching weight graph is output. The application converts pixel guessing into an interpretable structure field extrapolation, guarantees that an expansion area structure is continuous and natural, avoids that a central real feature is tampered, explicitly suppresses a false minutia point to avoid identity drift, and provides a downstream security weight reduction matching, and improves recognition robustness.

Description

Technical Field

[0001] This embodiment belongs to the field of fingerprint recognition technology, specifically involving a fingerprint image expansion method and system based on topological phase field and trusted verification. Background Technology

[0002] In devices such as mobile terminals, embedded access control systems, and small data acquisition modules, the effective acquisition area of ​​fingerprint sensors is severely limited. This limited acquisition area directly results in an insufficient number of extractable ridges and minutiae, significantly reducing the separability of matches and thus increasing the false rejection rate and false acceptance rate. To improve the robustness of small-area fingerprint recognition, image enlargement processing is typically required to increase the number of feature points and the effective comparison area.

[0003] Existing image enlargement methods have inherent flaws. Direct interpolation only changes the sampling grid and does not add any new biometric information. From an information theory perspective, reversible resampling does not change the information entropy contained in the image, thus failing to substantially improve recognition performance. Conventional methods such as copy filling or reflection filling create unnatural textures that conflict with the true ridge orientation. On the other hand, while general image extrapolation models can generate visually continuous textures, fingerprints have extremely strong structural regularity. General models lack explicit constraints on the fingerprint orientation field, ridge frequency, and minutiae topology, making them prone to producing breaks, pseudo-forks, or pseudo-endpoints at boundaries. Once these false generated features are mistakenly identified as genuine features by downstream recognition systems, it will cause serious identity drift and security risks.

[0004] Furthermore, existing generative mapping methods typically perform complex generation calculations across the entire fingerprint area, resulting in slow inference speeds and difficulty in deployment on resource-constrained edge devices. More critically, current technologies generally output the mapping results directly to the downstream matcher without explicitly labeling the generation attributes and reliability of the expanded region. Since generated regions inevitably contain structural errors, treating them the same as the actual collected regions in the downstream matching system will introduce uncontrollable matching noise. Therefore, there is an urgent need in this field for a mapping method specifically designed for small-area fingerprints. This method should not only ensure a natural structural connection between the expanded and original regions and suppress the generation of false features, but also explicitly output the reliability of the expanded region for downstream matching with reduced weighting, thereby improving recognition robustness while avoiding security risks. Summary of the Invention

[0005] The technical problem to be solved by the present invention is to overcome the shortcomings of insufficient reliability in the existing fingerprint image expansion methods, thereby providing a fingerprint image expansion method and system based on topological phase field and reliable verification.

[0006] A fingerprint image augmentation method based on topological phase field and trusted verification includes: A small area fingerprint image is embedded in the center of the target canvas to form a center-locked image, and the known real area and the area to be generated are divided. Estimate fingerprint structural features in the known real region and extrapolate the fingerprint structural features to the region to be generated; Based on the fingerprint structure features of the known real area, a phase potential is reconstructed, and its trace at the boundary between the known real area and the area to be generated is taken as the boundary phase. A topological phase field is constructed in the area to be generated. The topological phase field is a scalar field, and the gradient direction of the topological phase field is consistent with the ridge normal and the gradient magnitude is proportional to the ridge frequency. A ridge skeleton probability map is extracted based on the topological phase field. The ridge skeleton probability map is a probability distribution map used to represent the probability that each pixel of the fingerprint image belongs to the ridge skeleton. The structural conditions, including the topological phase field and the ridge skeleton probability map, are input into the generator network, which outputs a grayscale image of the region to be generated. The true grayscale of the known real region is forcibly written back to the corresponding position of the output image to form a candidate image. The generated candidate images are subjected to minutiae conservation verification and confidence evaluation, and the expanded image results with confidence map and matching weight map are output.

[0007] Furthermore, the fingerprint structural features include orientation field, frequency field, and quality map; Extrapolating the fingerprint structural features to the region to be generated includes: Harmonic extrapolation of the direction field and frequency field is performed on the dual-angle channel. The estimated value of the boundary between the known real region and the region to be generated is used as the Dirichlet boundary condition, and the outer boundary of the region to be generated is used as the zero-flux von Neumann condition for solving. The quality map is extrapolated exponentially based on its distance from the known real area, so that the quality weight of each pixel in the area to be generated decreases as it moves further away from the known real area.

[0008] Furthermore, constructing a topological phase field in the region to be generated includes: An energy functional is constructed consisting of an area integral term for the region to be generated, a boundary curve integral term, and a second-order smoothing term. The area integral term includes a mass weight to make the phase gradient approximate the target vector field determined by the frequency field and the ridge normal. The boundary integral term constrains the phase of the region to be generated to inherit the boundary phase. Variational calculus of the energy functional yields a partial differential equation containing a biharmonic operator. The clamping condition of inherited phase and normal phase gradient is taken at the boundary, and the natural boundary condition is taken at the outer boundary of the region to be generated. Solving the partial differential equation yields the optimal topological phase field.

[0009] Furthermore, the ridge skeleton probability map is extracted based on the topological phase field, including: The cosine of the obtained optimal topological phase field is used to obtain the phase contour distribution, and the difference between the cosine value and the ridge determination threshold is soft-thresholded by the Sigmoid function to obtain a probability map representing the ridge peak position as the ridge skeleton probability map.

[0010] Furthermore, the step of inputting the structural conditions, including the topological phase field and the ridge skeleton probability map, into the generator network, outputting a grayscale image of the region to be generated, and forcibly writing back the true grayscale of the known real region to the corresponding position in the output image includes: The center-locked map, the mask of the region to be generated, the distance map, the extrapolated orientation field, frequency field, quality map, optimal topology phase field and ridge skeleton probability map, and random perturbation variables are input into the structural condition generation network, and the grayscale image of the candidate generation region is output. Using a hard compositing method, the true grayscale of the center-locked image is directly output in the known real area, and the grayscale image of the candidate generation area is output in the area to be generated, and then combined into a complete image.

[0011] Furthermore, the generated candidate images undergo minutiae conservation verification, including: Extract the endpoints of the boundary ridge line at the junction of the known real area and the area to be generated, and construct a dimensionless matching cost based on the positional distance between the two endpoints, the difference in the directional angle after dual-angle processing, and the local frequency difference. The optimal pairing of endpoints is obtained by solving the assignment problem. The topology gating score is calculated by combining the number of pseudo minutiae and the number of seam breaks in the candidate image to suppress pseudo minutiae that do not meet the endpoint pairing conditions in the candidate image.

[0012] Furthermore, the credibility of the generated candidate images is evaluated, including: Multiple candidate images are generated, and the phase consistency cost, boundary connectivity cost, identity consistency cost, and quality cost of each candidate image are calculated respectively. The candidate image with the smallest total cost is selected as the final expanded image based on the weighted sum of each cost. The identity consistency cost is the cosine distance between the candidate image and the center-locked image in the pre-trained matching network embedding space.

[0013] Furthermore, the output includes a confidence plot, including: Collect error terms for candidate images, including orientation field residual, phase residual, boundary connectivity error, minutiae anomaly error, and identity embedding error; The weighted sum of the error terms is input into the Sigmoid function, and the temperature parameter is solved by minimizing the negative log-likelihood on the validation set. The probability output by the Sigmoid function is then calibrated for temperature, resulting in a calibrated pixel-level confidence map.

[0014] Furthermore, the output includes the expanded map result with matching weights, including: Based on the mask identifier, the matching weight of the known real area is set to a fixed value, and the matching weight of the area to be generated is set to the product of the global weight reduction coefficient and the corresponding pixel value of the confidence map, forming a pixel-by-pixel matching weight map, which is used by the downstream recognition system to perform weight reduction matching on the features of the area to be generated.

[0015] A fingerprint image augmentation system based on topological phase field and trusted verification includes: The canvas embedding and region segmentation module is used to embed a small area fingerprint image into the center of the target canvas to form a center-locked map, and to divide the known real area and the area to be generated. The structure field estimation and extrapolation module is used to estimate the fingerprint structure features in the known real region and extrapolate the fingerprint structure features to the region to be generated; The topological phase field solving module is used to reconstruct the phase potential based on the fingerprint structure features of the known real area and take its trace at the boundary as the boundary phase. It constructs a topological phase field in the area to be generated so that the gradient direction is consistent with the ridge normal and the gradient magnitude is proportional to the ridge frequency. It also extracts the ridge skeleton probability map based on the topological phase field. The structural condition texture generation module is used to input the structural conditions containing the topological phase field and the ridge skeleton probability map into the generation network, output the grayscale image of the area to be generated, and force the real grayscale of the known real area to be written back to the corresponding position of the output image. The confidence verification and output module is used to perform minutiae conservation verification and confidence evaluation on the generated candidate images, and output the expanded image results with confidence map and matching weight map.

[0016] Beneficial Effects: This invention provides a complete and self-consistent prior structural constraint for fingerprint expansion by estimating fingerprint structural features from known real areas and extrapolating them to the area to be generated. Based on the known area structural features, the phase potential is reconstructed and a topological phase field is constructed, transforming uncontrollable pixel guesses into interpretable structural field extrapolation. This allows the ridges of the area to be generated to extend continuously in the form of phase contour lines, ensuring a natural connection between the expanded and original areas in direction and frequency, effectively avoiding boundary breaks and texture conflicts that are prone to occur in traditional generation models. During the image output stage, the known real areas are forcibly rewritten with true grayscale values, preventing network generation errors from damaging the central, genuine fingerprint and ensuring the absolute reliability of the original biometric features. Simultaneously, through minutiae conservation verification, unfounded pseudo-forks and pseudo-endpoints can be explicitly suppressed, preventing downstream recognition systems from misusing false features and causing identity drift. Finally, a confidence map and a matching weight map are output simultaneously, ensuring that the expanded area is no longer disguised as completely genuine collected features. Downstream recognition systems can use this information to reduce the weight of the generated area or perform verification, improving the robustness of small-area fingerprint recognition while fundamentally avoiding security risks. Attached Figure Description

[0017] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0018] Figure 1 This is a flowchart illustrating the overall method steps of the present invention; Figure 2 This is a schematic flowchart of the method of the present invention; Figure 3 This is a schematic diagram of the fingerprint region division according to the present invention; Figure 4 This is a schematic diagram of the branch structure of the method of the present invention; Figure 5 This is a schematic block diagram of the system structure of the present invention. Detailed Implementation

[0019] To make the above-mentioned objectives, features, and advantages of this application more apparent and understandable, the specific embodiments of this application are described in detail below with reference to the accompanying drawings. Many specific details are set forth in the following description to provide a thorough understanding of this application. However, this application can be implemented in many other ways different from those described herein, and those skilled in the art can make similar modifications without departing from the spirit of this application. Therefore, this application is not limited to the specific embodiments disclosed below.

[0020] In the description of this application, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of indicated technical features. Thus, a feature defined as "first" or "second" may explicitly or implicitly include at least one of that feature. In the description of this application, "multiple" means at least two, such as two, three, etc., unless otherwise explicitly specified.

[0021] Example 1: Reference Figures 1-4 As shown, this embodiment provides a fingerprint image expansion method based on topological phase field and trusted verification.

[0022] In mobile terminals, embedded access control systems, and small data acquisition modules, the effective acquisition area of ​​fingerprint sensors is severely limited, with common input resolutions of only 96×96 pixels or smaller. This limited acquisition area directly results in an insufficient number of extractable ridges and minutiae, significantly reducing the separability of matches and thus increasing the false rejection rate and false acceptance rate.

[0023] The small-area fingerprint expansion task is defined as expanding the generated band outside the central ground truth region. Comparative analysis of existing technologies shows that general image extrapolation models such as U-Net are prone to boundary breakage or generating false minutiae; directional deviations in two-stage orientation field methods are propagated to textures; diffusion models have slow inference and high deployment costs; VAR models lack explicit control over minutiae topology; and frequency-domain lightweight models have weak control over local ridge orientation. This embodiment first estimates the structure field based on the known fingerprint region, replacing pixel guessing with interpretable structure field extrapolation. This method models small-area fingerprint expansion as a constrained generation problem with boundary conditions, topological constraints, and reliable output, specifically including steps S1 to S5.

[0024] Step S1: Embed the small-area fingerprint image into the center of the target canvas to form a center-locked image, and divide it into a known ground truth region and a region to be generated. The outer frame represents the entire canvas, the inner frame represents the known ground truth region, the dashed line represents the boundary, and the annular band represents the region to be generated. Given a small-area fingerprint image, its size is the product of its height and width. Taking the expansion to a large-size canvas as an example, place the input image in the center of the target canvas, forming a region with a width of... The generation boundary band of a pixel, the output size, and the embedding offset satisfy the following relationship: ; in, , These represent the height and width of the output image, respectively. , These represent the height and width of the input image, respectively. This indicates the pixel width of the expansion on each side; This represents the top-left offset vector of the input image embedded in the canvas.

[0025] As a preferred embodiment, , ,but Generates output pixels with a certain size. .

[0026] From an information theory perspective, direct interpolation amplification only changes the sampling raster and does not add new biometric information, thus satisfying the Shannon entropy invariance principle. ; in, Shannon entropy is used to measure the average amount of information contained in an image. This indicates the final expanded map output; This represents a center-locked diagram; This represents the content to be completed in the generated region; This represents the conditional entropy of the region generated under the condition of a known center-locked map, which is the amount of information actually added by the map expansion; This indicates that the input minimap is reversibly resampled; This represents the input of a small-area fingerprint image. This formula indicates that this embodiment achieves evidence-based generative image expansion by injecting conditional information through prior knowledge and structural constraints.

[0027] Let the target canvas coordinates be two-dimensional vectors, and the known region mask be defined as: ; in, Representing coordinates The mask of the known region at the location; Represents the pixel coordinates on the target canvas; Indicates the known real area that was actually collected; This indicates the area to be generated; a value of 1 indicates that the pixel belongs to the actual acquisition area, and a value of 0 indicates that the pixel belongs to the area to be generated.

[0028] Based on this, the entire canvas area is divided into two complementary subsets: the known real area and the area to be generated. ; in, Represents a known real region; Indicates the region to be generated; The target canvas area is represented by the colon; the colon represents all pixels that satisfy the condition; the backslash \ represents the set difference, i.e. Indicates from the global domain deducting from The remaining set after that. Two subsets satisfy... and .

[0029] Embed the input thumbnail in the center of the canvas, fill only the known areas with true grayscale, and set the rest to zero, to obtain a center-locked image: ; in, This indicates that the center-locked graph is in coordinates grayscale at the location; This indicates the input of a small-area fingerprint image; Represents the canvas coordinates Subtract embedding offset The obtained local coordinates of the input image; Represents the embedded offset vector; Indicates an indicator function on a region; Represents the canvas pixel coordinates; This represents the entire canvas area. This formula indicates that the known area is filled with the true grayscale value. , generation area .

[0030] The indicator function is defined as follows: ; in, Indicator functions for sets; Represents pixel coordinates; Represents a known real region. (Symbol) Indicates "does not belong to". When belong Time to take Otherwise take .

[0031] The output of the expanded map model must not rewrite the known real region; therefore, the final output adopts a hard composite method. ; in, This indicates that the final expanded map output is in coordinates. grayscale at the location; Represents the real area mask; This represents a center-locked diagram; Indicates the generation area mask (the generation area is 1); This represents the candidate gray levels given by the generator. This formula guarantees that: within the known region ( The output is always equal to the true grayscale. In the generation region ( Output the generated result Therefore, any network error will not change the true fingerprint collected by the center.

[0032] Candidate grayscale values ​​are generated by structural conditional network Given under various structural field conditions: ; in, This represents the grayscale value of the candidate expanded image output by the generator; Represents a structural conditional generative network; This represents the trainable parameters of the network; This represents a center-locked diagram; Indicates a mask; Represents a distance graph; Indicates the direction field; Represents the frequency field; Represents the phase field; Represents the probability map of the ridge skeleton; Represents a local quality map; This indicates a candidate random perturbation or a candidate number.

[0033] Distance map The measure is the nearest distance from each pixel in the generated region to the known region, used to control the attenuation of generation intensity as the generation zone deepens: ; in, Indicates the distance from the coordinates to the known area; For distance operators; This indicates taking the minimum value; Indicates in The pixel coordinates of the inner traversal; express and The Euclidean (between) )distance; Represents the canvas pixel coordinates; This represents a known real region.

[0034] In summary, small-area fingerprint expansion is formalized as a constrained optimization problem with hard constraints. While keeping the central ground truth region unchanged, the network parameters are learned to minimize the expected loss. ; in, Indicates the optimal network parameters; This represents taking the parameter that minimizes the objective. ; Represents the mathematical expectation; This represents a training sample pair (small input image and large real target image). Indicates that the sample was taken from; This represents the actual distribution of fingerprint data; Indicates output With the real target Total losses between; Indicates expanded image output; Represents the actual target image; "subject to" means "subject to" or "constraint". This means that the output in the known area must be equal to the true gray level; Indicates "for any"; This represents a known real region.

[0035] Step S2: Estimate the fingerprint structural features in the known real region and extrapolate these features to the region to be generated. The fingerprint structural features include the orientation field, frequency field, and quality map. Perform block structure tensor analysis or Gabor response analysis on the known region to obtain the orientation field. Ridge frequency field ROI Prospect Probability Local quality diagram and the set of endpoints of the boundary ridge .

[0036] First, construct the structure tensor from the image grayscale gradient, and denote the first-order gradient of the image along the horizontal and vertical directions as: ; in, , These represent center-locked diagrams. In coordinates Along the horizontal direction and vertical direction The first-order partial derivative (grayscale gradient component); , Indicates to Find the partial derivative; This represents pixel coordinates. In implementation, it can be approximated using the Sobel equal difference operator.

[0037] Gaussian smoothing of the outer product of the gradients yields a scale of... The structure tensor (a symmetric positive semidefinite) matrix): ; in, Representing coordinates The scale is The structure tensor; The standard deviation is expressed as Two-dimensional Gaussian kernel; This represents the convolution operation; , Represents the square of the gradient component; This represents the product of two gradient components; Indicates the Gaussian smoothing scale; , , Represents the three independent elements of the structure tensor after smoothing. , , ).

[0038] The two eigenvalues ​​of the structure tensor characterize the strength of local anisotropy, with the larger eigenvalue corresponding to the direction of the fastest gray-level change (i.e., the ridge normal): ; in, , Represents the structure tensor Two eigenvalues ​​( For large eigenvalues, (for small eigenvalues) , , Represents the structural tensor elements; This means taking the plus sign to get Taking the minus sign gives ; This indicates that the structure tensor is positive semidefinite. The directions of large eigenvalues ​​are the principal gradient directions (ridge normals), and the directions of small eigenvalues ​​are the ridge tangents.

[0039] Directional coherence is defined by two eigenvalues, which measures the reliability of the local ridge direction: ; in, Representing coordinates directional coherence at that location; , Represents two eigenvalues; This represents the smallest positive number that prevents the denominator from being zero (e.g.) The square root term of the numerator equals ; The closer to 1, the more consistent the local direction (clear ridge lines); the closer to 0, the more chaotic the direction (noise or singularities).

[0040] The ridge tangent angle is given by the structure tensor elements via the arctangent of the four quadrants, and then rotated. Convert to ridge direction: ; in, Representing coordinates The gray-level gradient normal angle obtained from the structure tensor at that location; Represents the arctangent function in the four quadrants (range) ); , , Represents the structural tensor elements; coefficients Originating from a two-angle average; Indicates the tangential angle of the ridge line; Indicates by Take the mold, so that This eliminates the direction of the ridge line. The equivalence ambiguity.

[0041] To avoid and The angular regression discontinuity caused by equivalence is addressed by using a dual-angle, dual-channel representation of the direction field, while simultaneously defining the ridge tangential and normal unit vectors: ; in, A two-channel vector representation of the direction field; , This represents the cosine and sine components after taking two angles with respect to the tangential angle of the ridge line; Represents the tangential unit vector of the ridge line; This represents the unit vector normal to the ridge (orthogonal to the tangent). Indicates the tangential angle of the ridge line; superscript Represents the transpose of a vector (column vector); using make sure and Mapping to the same direction eliminates directional ambiguity.

[0042] The ridge frequency field can be obtained from the response of the Gabor filter bank or from one-dimensional spectral analysis along the normal. The Gabor filter kernel is defined as: ; in, Indicates the center direction as The center frequency is Gabor filter kernel; , Represents the local coordinates of the kernel; , Indicates will Rotate to a coordinate system aligned with the ridge line; , This represents the standard deviation of the Gaussian envelope along the tangential and normal directions of the ridge line; This indicates the ridge frequency corresponding to the nucleus; Represents the natural exponential function; This represents the cosine function. This kernel produces the strongest response to a ridge whose direction and frequency match its own.

[0043] The coordinate rotation relationship is as follows: ; in, , Represents the local coordinates after rotation; , Represents the original local coordinates; Indicates the tangential angle of the ridge line; , This represents trigonometric functions. The rotation makes... Tangential along the ridge line Along the ridge line normal.

[0044] To eliminate the influence of the DC component, the frequency estimation first performs mean subtraction within a local window. The mean gray value of the window is defined as: ; in, Indicates local window centered Inside The average gray level; Display window The number of pixels within; This indicates summing of pixels within the window; Indicates the pixel coordinates within the window; This indicates that the center-locked graph is in The grayscale value at that location.

[0045] The local frequency response is then calculated along the ridge normal, using relative coordinates. To avoid the result depending on the global origin of the image: ; in, Representing coordinates In candidate angular frequency Complex frequency response; Indicates the window Sum of inner pixels; Indicates the pixel coordinates within the window; Indicates the gray level at that location; Indicates the window mean; Indicates Gaussian weights of scale ; Represents an exponential function; Represents the imaginary unit ( ); Indicates the candidate angular frequency; Representing coordinates The ridge normal unit vector at the location; superscript Indicates transpose; Represents the relative displacement vector; This represents the projection of the relative displacement onto the normal direction.

[0046] Take the angular frequency that maximizes the response amplitude and convert it into a ridge frequency measured per period per pixel: ; in, This represents the angular frequency that maximizes the amplitude of the local response; This indicates the variable that makes the following expression the largest; Indicates the candidate angular frequency; , This indicates the lower and upper limits of the allowed angular frequencies (limiting the ridge frequency within a reasonable range for the fingerprint). Indicates complex response The magnitude (amplitude); Representing coordinates Ridge frequency at the location; This is the conversion constant from angular frequency to ordinary frequency. This constraint can prevent the formation of overly dense or sparse lines in the generated region.

[0047] The local quality map is jointly given by foreground probability, gray-level variance significance, spectral peak significance, and directional coherence: ; in, Representing coordinates Local quality / reliability weights at the location; express The probability of it belonging to the foreground of the fingerprint; This means truncating the value to a range. The function; Display window Inside The grayscale variance; , Indicates low and high variance thresholds (satisfying) ); , These represent the first and second peak amplitudes of the local spectrum, respectively. Represents the smallest positive number that is protected against division by zero; This represents directional coherence. The multiplication of each factor represents the combination of multiple reliability constraints; a lower value for any factor indicates lower reliability. Low.

[0048] Extrapolating the fingerprint structural features to the region to be generated includes: harmonic extrapolating the orientation field and frequency field in the dual-angle channel, using the estimated value of the boundary between the known real region and the region to be generated as the Dirichlet boundary condition, and using the outer boundary of the region to be generated as the zero-flux von Neumann condition for solving.

[0049] Structural tensor and Gabor analysis rely on the actual gray-level gradient and gray-level values, therefore the orientation field Frequency field With quality diagram Only in the known real area The phase field is directly estimated; however, subsequent phase field solutions need to be obtained in the generation region. The frequency of each point is known. With the ridge line normal Therefore, with The estimated values ​​above are boundary conditions, and the direction field and frequency field are smoothly extrapolated to... Based on this, the mass field of the generated region is defined, so that the orientation field, frequency field and mass field are defined on the entire canvas.

[0050] First, a harmonic extrapolation is performed on the direction field and frequency field. Considering that the ridge direction has... Equivalence, extrapolation in dual-angle channels Proceed through each channel; Two channels and frequencies Within the generating region, it is a harmonic function, and it is defined by the boundary. The estimated values ​​are Dirichlet boundary conditions used to generate the outer boundary of the region. The zero-flux Neumann condition: ; in, Represents the Laplace operator ( ); The dual-angle representation of the direction field, with its two channels. , Each to their own satisfaction ; Represents the ridge frequency field; Represents pixel coordinates; , , Let represent the known real region, the region to be generated, and their boundary, respectively. Harmonic extrapolation is equivalent to taking the Dirichlet energy from all extensions that satisfy the boundary value. The smallest (i.e., the smoothest extension) can naturally extend the known direction and density of ridges into the generation region without introducing any spurious mutations.

[0051] On a uniform grid, using the five-point difference to discretize the Laplacian operator, the iterative update formula is: ; in, This represents the scalar channel to be extrapolated, taking the two components of the two angles of the direction field in sequence. , With frequency field (Three channels are decoupled and can run in parallel); superscript , Indicates the number of iterations; subscript , For grid row and column indexes; , This represents the current value of the four adjacent grid points (up, down, left, and right); coefficient. From the five-point difference scheme (due to the right end) (Updated to the mean of the four neighboring regions); boundary The grid points on the outer boundary are fixed to the Dirichlet condition estimates during the iteration. Maintain zero throughput with mirror nodes (Neumann condition).

[0052] This system of equations can be solved iteratively using the Gauss-Seidel, Successive Over-Relaxation (SOR), or Conjugate Gradient (CG) method: Since the coefficient matrix is ​​diagonally dominant and positive definite, the iteration converges linearly with a contraction factor less than 1 until the relative residual is below a threshold (e.g., ...). Stop immediately; with multigrids, near-linear complexity can be achieved. ( (where is the number of pixels in the generated area), and the three channels are independent and can be solved in parallel.

[0053] The solution process is essentially an iterative smoothing process subject to boundary constraints, and the resulting discrete harmonic solution satisfies the extremum principle: ; in, , They represent channels respectively. At the border The infimum and supremum of the upper estimate; Represents any point in the generation region extrapolated value at; This represents the generating region. This formula is the extremum principle of discrete harmonic functions, indicating that the extrapolated value is always bounded by the known boundary value, and there will be no out-of-bounds maxima or minima.

[0054] The extrapolated directions and frequencies are defined by the values ​​of the known region, and will not generate denser or sparser lines out of thin air, nor will it introduce directions that do not appear in the known region. That is, it achieves only extension and not conjecture, which is consistent with the overall goal of this embodiment to replace pixel conjecture with interpretable structural field extrapolation to ensure reliability. In addition, the global expansion uses the same five-point difference and iterator to solve the phase field, with the only difference being the boundary conditions and the right-hand side (which is zero here). Therefore, it does not introduce additional algorithms or significantly increase the computing power.

[0055] extrapolated Because of smoothing, it is generally no longer a unit vector and must be normalized first. Then restore the ridge tangent angle to obtain the direction of each point in the generated region: ; in, Represents the generation region Tangential angle of the ridgeline at that location; For the four quadrants, the arctangent is used; , They represent the normalized directional fields, respectively. The first and second channels (corresponding) , );coefficient It originates from dual-angle reconstruction. That is, the ridge line normal is obtained. Thus, the generation area has been obtained. Directional field ( , ) and frequency field .

[0056] Since the generated region does not contain realistic textures, its reliability should decrease as one moves deeper into the generated zone. Therefore, the mass field is extrapolated from the boundary towards the generated region with exponential decay based on the distance to the known region: ; in, Represents the generation region Quality / reliability weights at the location; Indicates boundary The quality of the above (take the value of the nearest known point or the mean of its neighborhood); It is a natural exponential function; express Distance to the known region; Indicates the decay scale (preferably several ridge periods); This represents the generated region. This extrapolation automatically assigns low quality to generated pixels far from the known region, consistent with the confidence modeling in this embodiment where the further away from the known region, the lower the confidence level. It also automatically reduces the weight of unreliable regions in the subsequent phase field solution data items.

[0057] The directional field, frequency field, and mass field have all been defined on the entire canvas. This provides a complete and self-consistent structural field input for solving the phase field.

[0058] Step S3: Reconstruct the phase potential based on the fingerprint structure features of the known real area, and take its trace at the boundary between the known real area and the area to be generated as the boundary phase. Construct a topological phase field in the area to be generated. The topological phase field is a scalar field, and the gradient direction of the topological phase field is consistent with the ridge normal and the gradient magnitude is proportional to the ridge frequency. Extract the ridge skeleton probability map based on the topological phase field. The ridge skeleton probability map is a probability distribution map used to represent the probability that each pixel of the fingerprint image belongs to the ridge skeleton.

[0059] In this step, the required orientation field is solved. Frequency field With mass field The boundary phase has been given by global expansion. Constructed by this section. The idea is: in the region to be generated... Solving a phase field This ensures that the gradient direction aligns with the ridge normal, the gradient magnitude is proportional to the ridge frequency, and that the gradient is aligned with the ridge frequency at the boundary. The phase of a known ridge line is inherited at the location; the ridge line is represented as... contour lines ( (the peak).

[0060] Before solving for the phase field, it is necessary to first determine the boundary. Determine its Dirichlet boundary phase Due to the target vector field It is generally not an irrotational (conservative) field. Not always zero, along different paths Line integral The result depends on the integration path; direct path integration is ill-conditioned and produces multiple values. To avoid path dependence, this embodiment does not perform path integration, but instead integrates within the known region. The phase potential is solved using the least squares approach, and its trace on the boundary is taken as... : ; in, Indicates the area in the known region The phase potential reconstructed using the least squares (Poisson) method; For the Laplace operator; Let be the divergence of the target vector field; Represents a known real region; This represents the boundary phase, i.e., the Dirichlet data required for phase field solving at the boundary. Indicates a boundary; Choose any anchor point and fix its phase (e.g.) Once determined, the result is unique and independent of the integration path, thus completely eliminating the need for further analysis. Path dependence and multivalued problems caused by non-integrability.

[0061] Constructing a topological phase field in the region to be generated includes: constructing an energy functional consisting of an area integral term, a boundary curve integral term, and a second-order smoothing term for the region to be generated, wherein the area integral term includes a mass weight to make the phase gradient approximate the target vector field determined by the frequency field and the ridge normal, and the boundary integral term constrains the phase of the region to be generated to inherit the boundary phase. ; in, Represents the optimal topological phase field; Represents the phase field that minimizes the energy functional. The first item is a data item: This represents the area integral over the generating region. For quality weight, Represents the phase gradient. Indicated by frequency The determined target gradient magnitude, Indicates the ridge normal. This represents the square of the second norm of a vector. This term represents the area integral infinitesimal element—it requires the phase gradient to approximate the target vector field. The second term is the boundary term: For boundary constraint weights, This represents the integral along the boundary curve. This represents the squared deviation of the phase from the known boundary phase. Indicates the boundary phase inherited from the known region. The third term represents the arc length infinitesimal element; the fourth term is the smoothing term. To smooth out the weights, It represents the square of the phase Laplace (second derivative), used to suppress unfounded local distortions.

[0062] Find the variational expression for the energy functional (take the first-order variational expression). This yields the Euler–Lagrange equations, which are partial differential equations containing biharmonic operators: ; in, Represents the divergence operator; Indicates quality weight; Indicates the phase gradient; Represents the target gradient vector field; Indicates smoothing weights; The biharmonic operator for the phase field (Laplace's Laplace, derived from the variation of the smoothing term); Represents pixel coordinates; Indicates the generated region; The '' indicates that the equation holds true everywhere within the generating region. This equation is a stability condition for the sum of the variational values ​​of the data terms and the smoothing terms to be zero.

[0063] This equation requires boundary conditions to obtain a definite solution, at the boundary. We inherit the known phase using Dirichlet conditions, and apply Neumann natural boundary conditions at the outer boundary of the generation region: ; in, Represents the phase field; This indicates the phase inherited by the known region at the boundary; This represents the boundary between the known region and the generated region (here, the Dirichlet condition). Represents the gradient residual vector; Indicates the outer boundary of the generation region The unit's external normal direction; Represents the dot product of vectors; This indicates the removal of boundaries in the generated region. The latter part (here, the Neumann natural boundary condition is used, and the normal component of the residual vector is zero).

[0064] The second set of clamping and free boundary conditions is as follows: ; in, Indicates phase Along the outer normal directional derivative; Indicates the target phase gradient at The projection onto the ridge (clamping condition, such that the phase slope at the intersection also inherits the known ridge line); Indicates a boundary; Indicates the phase Laplace; This indicates the outer boundary of the generated region. Note: Taking small values ​​between 0.1 and 1.0, the smoothing term mainly functions as a numerical regularization term; when... When the fourth-order term disappears, it strictly degenerates into a second-order Poisson equation.

[0065] When the mass weights are approximately uniform and the smoothness weights are small, the Euler-Lagrange equation degenerates into a Poisson equation with a source term: ; in, The Laplace operator representing the phase field ( ); Represents the source terms on the right-hand side of the Poisson equation; Represents the divergence of the target vector field; It is a constant; Indicates divergence; Indicates the ridge frequency; This represents the ridge normal. This equation is the standard form for reconstructing the scalar potential from the gradient field, and its solution... The gradient approximates the least squares value. .

[0066] Generally non-conservative, therefore no scalar field exists. make Precision everywhere equals Poisson solution In fact The irrotational (gradient) components in the Helmholtz–Hodge orthogonal decomposition are optimally approximated in the least squares sense. : ; in, Represents the target gradient vector field; Indicates the Poisson solution The gradient (irrotational component); This represents the curl (residual) component with zero divergence, i.e. The non-integrable part that cannot be represented by any phase; condition show Make residual Orthogonal to the gradient subspace, therefore it is The optimal approximation in the sense of the word.

[0067] residual Only The curl is significant (high curvature regions, singularity neighborhoods) and cannot be ignored; its magnitude is proportional to the local circulation. This embodiment uses quality weights. Reduce the contribution of the low reliability region and smooth the terms slightly. Suppressing residual oscillations limits ridge spacing and orientation deviations to sub-pixel levels, with the deviations decaying rapidly with increasing distance from singularities. This residual deviation is further compensated for by phase consistency. With confidence level Explicitly measuring and disclosing downstream ensures that the data is not mistaken for the true structure, which aligns with the overall goal of this embodiment of trusted graph expansion.

[0068] The phase field is solved using the same five-point difference discretization and iterative method as harmonic extrapolation, the only difference being that the right-hand side is a non-zero source here. When solving for the same uniform grid (step size) A large sparse linear system of equations is obtained on ) It can be solved iteratively using the Gauss-Seidel method or the conjugate gradient method, with the point-by-point update formula as follows: ; in, Indicates the first Grid points in the next iteration Phase value at; superscript , Indicates the number of iterations; subscript , Indicates grid row and column indexes; , This indicates the current phase of the four adjacent grid points (up, down, left, and right). Indicates the grid step size (pixel spacing); Represents grid points Source term value at; coefficient and Derived from a five-point difference scheme. The known grid points are fixed in the inherited phase (Dirichlet) during iteration.

[0069] Iterative convergence is determined by using relative residuals as the stopping criterion: ; in, Indicates the first The relative residual of the next iteration; Represents the discrete Laplace coefficient matrix; Indicates the first Solution in vector form for the next iteration; Represents a discrete source term vector; express Norm; Represents the smallest positive number that is protected against division by zero; Indicates the given convergence threshold (e.g.) ); Right-hand side Error is expressed as a shrinkage factor Linear convergence; Indicates an exact solution; For contraction factor (depending on) The spectral radius is used to ensure iterative convergence.

[0070] For ease of subsequent explanation, the Sigmoid function is defined as follows: ; in, This represents the standard Sigmoid function; This represents the real number of the independent variable; Represents the natural constant; its range is , is used to compress any real number into a probability.

[0071] Extracting a ridge skeleton probability map based on the topological phase field includes: taking the cosine of the obtained optimal topological phase field to obtain the phase contour distribution, and performing soft thresholding on the difference between the cosine value and the ridge determination threshold using the Sigmoid function to obtain a probability map representing the ridge peak position as the ridge skeleton probability map. ; in, Representing coordinates The probability that the location belongs to the ridge skeleton; Represents the Sigmoid function; The slope coefficient represents the degree of softening of the control threshold (the larger the slope, the closer it is to the hard threshold). Represents the cosine function; This indicates the optimal phase obtained. Indicates the threshold for ridge line determination; Represents coordinates; Indicates the generated region. When approaching 1 (at the phase contour line) , corresponding to the ridge peak.

[0072] Step S4: Input the structural conditions containing the topological phase field and the ridge skeleton probability map into the generator network, output the grayscale image of the region to be generated, and force the real grayscale of the known real region to be written back to the corresponding position of the output image to form a candidate image.

[0073] This embodiment includes structural field extrapolation and texture rendering branches. The step of inputting structural conditions, including the topological phase field and ridge skeleton probability map, into the generation network, outputting a grayscale image of the region to be generated, and forcibly writing back the true grayscale of the known real region to the corresponding position in the output image includes: inputting the center-locked map, the mask of the region to be generated, the distance map, the extrapolated orientation field, frequency field, quality map, the optimal topological phase field and ridge skeleton probability map, and random perturbation variables into the structural condition generation network, outputting grayscale images of candidate generated regions; using a hard synthesis method, directly outputting the true grayscale of the center-locked map in the known real region, and outputting the grayscale images of the candidate generated regions in the region to be generated, combining them into a complete image.

[0074] The formula for generating multiple candidates is: ; in, Indicates the first One candidate generated graph; Represents a structural conditional generative network; Indicates network parameters; , , , , , , , This represents the center-locked map, mask, distance map, orientation field, frequency field, optimal phase field, skeleton map, and quality map as conditions. Indicates the first One candidate random perturbation (latent variable); Indicates the candidate sequence number; This indicates the total number of candidates.

[0075] To ensure that the above-mentioned generative network has reliable graph expansion capabilities, model training is required.

[0076] Supervisory pairs are constructed by cropping from real large-scale fingerprint images. The target block is cropped from a real large-scale fingerprint image as the ground truth, and then a smaller central block is cropped from it as input to construct the supervisory pair. ; in, This represents the true target image obtained by sampling from the fingerprint data distribution; Indicates that the sample was taken from; This represents the actual distribution of fingerprint data; Indicates by The small area input obtained by cropping; Indicates the size is The clipping operator; Indicates a mask; Indicates the region of knowledge Operators for generating masks; Indicates a known region; This represents a center-locked diagram; This indicates pixel-wise (Hadamard) multiplication.

[0077] The preferred joint loss function is a weighted sum of seven terms: ; in, Indicates the total training loss; to Indicates the weight of each sub-loss; Indicates pixel reconstruction loss; Indicates the directional field loss; Indicates frequency loss; This represents the phase constraint loss; Indicates the boundary connectivity loss; This indicates a loss of identity consistency. This indicates resistance to texture loss.

[0078] Pixel reconstruction terms are calculated only in the generated region, and a Charbonnier robust penalty is used to balance noise resistance and differentiability. ; in, Indicates pixel reconstruction loss; Indicates the number of pixels in the generated area; Indicates the generation region Sum of inner pixels; Represents pixel coordinates; This represents the Charbonnier robust penalty function; This indicates that the expanded image outputs grayscale. Represents the true target grayscale; That is, grayscale residual.

[0079] The robust penalty function and its derivative are defined as follows: ; in, This represents the Charbonnier penalty function; Represents the grayscale residual (must be related to the number of iterations). distinguish); Represents the smoothing constant (to prevent in) It is not differentiable at this point and must be related to the scale of the structure tensor. distinguish); This represents its first derivative; the right-hand side shows that the derivative is bounded to... Therefore, this loss is smooth and has a stable gradient throughout the entire domain, and also possesses... Anti-outlier and Differentiability.

[0080] The orientation field and frequency field losses respectively constrain the ridge direction and ridge spacing, where the orientation difference is processed by a double-angle inner product. Periodicity: ; in, Indicates the directional field loss; Indicates the number of pixels in the generated area; To express summation; Represents pixel coordinates; This represents the two-channel vector of the direction field of the generated map; Represents the orientation field of the real target image; superscript Represents the vector dot product; since For a unit biangular vector, the inner product is... , Subtract the inner product The more consistent the direction, the smaller the loss.

[0081] ; in, Indicates frequency loss; Indicates the number of pixels in the generated area; To express summation; Represents pixel coordinates; Represents the frequency field of the generated graph; Represents the frequency field of the real target image; This represents the absolute value. This constraint generates ridges with a density consistent with reality.

[0082] The phase term requires that the phase gradient derived from the generated image be consistent with the structure field: ; in, This represents the phase constraint loss; Indicates the number of pixels in the generated area; To express summation; Represents pixel coordinates; This represents the phase gradient derived from the generated graph; This represents the target gradient vector of the structural field; This represents the square of the L2 norm.

[0083] Boundary connectivity terms constrain the distance between ridge endpoints within the seam band; identity terms employ a pre-trained matching network. Embedded cosine distance: ; in, Indicates the boundary connectivity loss; Indicates the actual number of seam endpoints; This represents a very small positive number that prevents the denominator from being zero when the endpoint set is empty; Indicates the actual endpoint Summation; Indicates the actual seam endpoint; Represents the set of actual seam endpoints; This indicates that the nearest generated endpoint is selected; Indicates the generation of endpoints; This indicates the generation of a set of seam endpoints; This represents the Euclidean distance between the actual endpoint and the most recently generated endpoint.

[0084] ; in, This indicates a loss of identity consistency. This represents the embedding function of the pre-trained fingerprint matching network; , Represents the embedding vector between the generated expanded graph and the real target graph; superscript Represents the dot product of vectors; , The L2 norm of the embedding vector; This indicates a very small positive number that prevents the denominator from being zero; Subtracting the cosine similarity gives the cosine distance.

[0085] The adversarial loss uses a least-squares form, only identifying local textures that generate boundary bands: ; in, This represents the adversarial texture loss of the generator; Represents the coordinates of the generated region Take the expected value; Indicates the coordinates of the center of a local plot; Indicates the generation region; The parameter is The local discriminator (must be related to angular frequency) distinguish); Indicates the discriminator parameters; Indicates Centered on A partial image patch of the generated area extracted from the image; Indicates radius as Plotting operator; Indicates the radius of the block; For LSGAN generator terms, it prompts the discriminator to classify the generated patch as "true (1)".

[0086] The discriminator objective optimization formula is: ; The network parameters are updated using an optimizer, and their first and second moment estimates are as follows: ; in, Indicates the first The loss gradient of the step; Represents the total loss with respect to the parameter The gradient; It represents the first moment (the exponential moving average of the gradient). This represents the second moment (the exponential moving average of the squared gradient). , The attenuation coefficients representing the first and second moments (typically) , ); , This represents the moment estimate from the previous step; This represents the element-wise square of the gradient.

[0087] The parameter update formula is: ; in, , These represent the first and second moments after bias correction. , Indicates the attenuation coefficient Power; , Indicates the network parameters before and after the update; Indicates the learning rate; This represents taking the square root of the second moment element by element; Represents the smallest positive number that is prevented from being divided by zero (typically) This formula represents the standard Adam parameter update.

[0088] Step S5: Perform minutiae conservation verification and confidence evaluation on the generated candidate images, and output the expanded image results with confidence map and matching weight map.

[0089] The generated candidate image undergoes a minutiae conservation check, including: extracting the endpoints of the boundary ridges at the junction of the known ground truth region and the region to be generated; and constructing a dimensionless matching cost based on the positional distance between the endpoints, the difference in orientation angles after dual-angle processing, and the local frequency difference. ; in, Endpoints and The topological matching cost (dimensionless). , Indicates the endpoints of two boundary ridges; , , These represent the weights of the differences in position, direction, and frequency, respectively. , Indicates the positions of the two endpoints; Indicate its Euclidean distance; The normalized scale of the location is indicated (preferably the average ridge period). , Indicates the tangential angle of the ridge line at both endpoints; For processing Directional equivalence; , Indicates the local ridge frequency at both ends; This indicates taking the larger of the two values; This represents the smallest positive number that prevents the denominator from being zero when normalizing the frequency. This represents the absolute value. The three terms respectively penalize mismatches in position, direction, and frequency.

[0090] The optimal endpoint pairing is obtained by solving the assignment problem. A topology gating score is calculated by combining the number of pseudo-minutions and seam breaks in the candidate images to suppress pseudo-minutions in the candidate images that do not meet the endpoint pairing conditions. The optimal pairing and average cost are defined as follows: ; in, This indicates the optimal one-to-one pairing (assignment). Represents all feasible pairings. Choose the option that minimizes the total cost. This represents a pairing scheme; This represents the set of all possible one-to-one pairs (including the "no pair / virtual endpoint" option); This indicates summation over the endpoints; Indicates the total number of endpoints; Endpoints Its paired endpoint The matching cost; In the plan Below and Paired endpoints; Indicates the first Candidates The average endpoint matching cost; This represents the number of pairs in the optimal pairing; This indicates traversing the endpoint pairs in the optimal pairing.

[0091] The formula for calculating the topology-gated score is: ; in, Indicates the first Topological gating scores of candidate expanded graphs; Indicates the first One candidate map expansion; Represents an exponential function; , , This indicates the weights of the three penalty terms; Indicates candidate The average endpoint matching cost; Indicates candidate The number of pseudo-decrement points that do not meet the endpoint, direction, and frequency conditions; Indicates candidate Number of ridge line breaks at the seam; score The closer the value is to 1, the more natural the topology. This gating makes the expanded map region tend to naturally continue existing ridges, rather than adding details that could lead to mismatches.

[0092] The credibility evaluation of the generated candidate images includes: generating multiple candidate images and calculating the phase consistency cost, boundary connectivity cost, identity consistency cost, and quality cost of each candidate image; sorting them according to the weighted sum of each cost and selecting the candidate image with the smallest total cost as the final expanded image; wherein, the identity consistency cost is the cosine distance between the candidate image and the center-locked image in the pre-trained matching network embedding space.

[0093] The candidate ranking formula is: ; in, Indicates the final selected candidate number; Indicates in Choose the candidate that minimizes the total cost. Indicates the candidate sequence number; Indicates the total number of candidates; , , , Indicates the weights of the four costs; , , , These represent phase consistency cost, boundary connectivity cost, identity consistency cost, and quality cost, respectively. Indicates the first A candidate graph was generated.

[0094] Phase consistency cost measures the deviation between the candidate grayscale inverse phase gradient and the structure field: ; in, Indicates candidate Phase consistency cost; Indicates the number of pixels in the generated area; Indicates to Sum of inner pixels; Represents pixel coordinates; Indicates the candidate graph The phase gradient obtained by reverse calculation; This represents the target gradient vector obtained from the structure field; This represents the squared L2 norm. The smaller this cost, the better the candidate texture matches the predicted orientation / frequency structure.

[0095] Boundary connectivity cost measures the shortest distance from the actual seam endpoint to the candidate endpoint: ; in, Indicates candidate Boundary connectivity cost; Indicates the actual number of seam endpoints; Indicates the end point of the butt joint Summation; Indicates the actual seam endpoint; Represents the set of actual seam endpoints; This means selecting the nearest endpoint from the set of candidate seam endpoints; Indicates the candidate joint endpoint; Represents the set of candidate seam endpoints; Indicates the actual endpoint location With candidate endpoint location The Euclidean distance.

[0096] Identity consistency cost is calculated using the cosine distance between candidate and known regions in the pre-trained matching network embedding space: ; in, Indicates candidate The cost of identity consistency; This represents the embedding function of the pre-trained fingerprint matching network; , Represents the embedding vector between the candidate graph and the center-locked graph; superscript Represents the dot product of vectors; , The L2 norm of the embedding vector; This indicates the smallest positive number whose denominator is zero; the fraction is cosine similarity. Subtracting it gives the cosine distance; the smaller the distance, the more consistent the identities.

[0097] Quality penalty for low-quality candidate generation regions: ; in, Indicates candidate The quality cost; Indicates the number of pixels in the generated area; Indicates to Sum of inner pixels; Represents pixel coordinates; Indicates candidate graph in The local quality of the generated region. The higher the average quality of the generated region, the lower the cost.

[0098] The output includes a confidence map, comprising: collecting error terms from candidate images, including orientation field residuals, phase residuals, boundary connectivity errors, minutiae anomaly errors, and identity embedding errors; weighting and summing the error terms and inputting them into the Sigmoid function, and using the negative log-likelihood on the validation set to solve for the temperature parameter, calibrating the probability output by the Sigmoid function with temperature, and obtaining the calibrated pixel-level generation confidence map.

[0099] The original confidence level calculation formula is: ; in, Representing coordinates Generation confidence at the location; Represents the Sigmoid function; This indicates a confidence baseline bias (making zero-error candidates approach the high confidence interval); to Represents the weights of the five error terms (all non-negative); Indicates the direction field error; Indicates phase error; Indicates the boundary connectivity error; This indicates abnormal errors at minute points; This represents the identity embedding error. The larger the error, the worse the identity embedding error. The lower.

[0100] Temperature calibration performs monotonic calibration on its logarithmic probability: ; in, Indicates the confidence level after calibration; Represents the Sigmoid function; Indicates the original confidence level logit; Indicates temperature parameter ( ); Represents logarithmic probability; Represents the natural logarithm; This represents the original confidence level. To make the probability more "conservative". To make them more "confident".

[0101] The temperature parameter is obtained by minimizing the negative log-likelihood on the validation set: ; in, Indicates the optimal temperature; This indicates taking the value that minimizes the subsequent expression at a positive temperature. This indicates summation over the validation samples; Indicates the number of validation samples; Indicates the first The ground truth label of each sample (whether the generated area pixels are consistent with the real ones); Indicates the first The calibration confidence level of each sample; The expression represents the natural logarithm; the expression within square brackets represents the binary cross-entropy (i.e., the negative log-likelihood). This formula represents the standard probability calibration target.

[0102] The output includes an expanded image result with a matching weight map, which includes: setting the matching weight of the known real area to a fixed value according to the mask identifier, and setting the matching weight of the area to be generated to the product of the global weight reduction coefficient and the corresponding pixel value of the confidence map, forming a pixel-by-pixel matching weight map for the downstream recognition system to perform weighted matching on the features of the area to be generated.

[0103] The downstream matching weight map is determined by both the mask and the confidence level. The weight of the real region is always 1, while the weight of the generated region is determined by the confidence level and the global weight reduction coefficient. Joint decision: ; in, Representing coordinates Matching weight at the location; Represents the real area mask; Indicates the generation region mask; This represents the global weighting coefficient of the generation region; Indicates the confidence level generated; express The value range ensures that the weight of the generated region does not exceed that of the real region. When the confidence level is low, the generated region will not dominate the matching score.

[0104] From the perspective of Bayesian testing, the optimal pixel-wise weighting is precisely the reliability weighting, and the matching decision can be written as the weighted log-likelihood ratio: ; in, fingerprints and The weighted log-likelihood ratio matching statistic; This indicates summation over pixels (or minutiae); Indicates location; Indicates the matching weight; Represents the natural logarithm; Indicates in Local features / residuals extracted at the location; This indicates a true match (same finger) assumption. Down The likelihood; This indicates the assumption of impersonation (different fingers). The likelihood of the following; , These represent the hypotheses of a true match and a spoofing, respectively. Low-confidence pixels... After the weighting is reduced, the contribution to the decision is weakened accordingly, which is consistent with the optimality of reliability-weighted Bayesian methods.

[0105] If the downstream uses detail matching, the confidence-weighted matching score is used: ; in, Indicating two fingerprints , The weighted matching score (must be related to the skeletal skeleton probability) distinguish); This represents the summation of detail pairs over two images; express The Middle A few details; express The Middle A few details; , This indicates the matching weight for the corresponding minutiae position; The similarity kernel for a single pair of detail points (must be equal to the total number of candidates) distinguish); This represents the smallest positive number that prevents the denominator from being zero. The denominator is the weighted sum, which normalizes the fraction to zero. .

[0106] Among them, the similarity kernels at the details show consistency in position, direction, and type: ; in, Indicate details and Similarity; Represents an exponential function; , Indicates the positions of two detail points; This represents the square of its Euclidean distance; Indicates the location tolerance scale; The cosine of the direction difference between two detail points; Normalize directional consistency to ; The type consistency indicator function (1 if the endpoints / forks are of the same type, 0 otherwise); , This indicates the type of two detail points. Three-factor multiplication requires that the position, direction, and type all match simultaneously.

[0107] This approach limits the costly generation and solving processes to the generation boundary band, which accounts for only about 20% of the output. As shown in the complexity analysis table: Table 1: Complexity Analysis Table ; The complexity of structure field estimation (Gabor filter bank / structure tensor) is: ; in, Indicates the time complexity of structural field estimation; Indicates large asymptotic notation; Indicates the total number of pixels on the canvas; Indicates the number of channels in the direction (or frequency) (e.g., 16); Indicates the side length of the filtering window; Indicates the number of pixels in the window.

[0108] The iterative solution complexity of the topological phase field is: ; in, This indicates the complexity of solving the phase field; Indicates the number of iterations.

[0109] The complexity of the structural condition generation network in the forward pass of the boundary band is: ; in, This indicates the complexity of solving the phase field; Indicates asymptotic notation; Indicates the number of iterations; Indicates the number of pixels in the generated area; Indicates the grid step size; This represents the convergence threshold; the right-hand side shows that the number of iterations increases logarithmically with the accuracy requirement and inversely with the grid step size (multigrids can further reduce this to an approximate value). ).

[0110] Compared to diffusion-based methods that repeatedly denoise the entire image, this approach offers a speedup of: ; in, Indicates the relative acceleration ratio; Indicates the complexity of the full-graph diffusion method; This indicates the generation complexity of the proposed solution. This indicates the number of diffusion sampling steps (usually tens to hundreds). Indicates the total number of pixels on the canvas; Indicates the number of pixels in the generated area; This indicates a value much greater than 1. (By...) and With the combined amplification of the two factors, the computing power advantage of this scheme is significant.

[0111] The parameter configurations differ depending on the deployment scenario. Refer to Table 2 for details. Table 2: Example Parameter Configuration ; Example 1 is for The scene is expanded around the image, with a generation bandwidth of 8 pixels, 16 directional channels, and 1 candidate. The generation network uses a lightweight residual network with approximately 3 million parameters and a weighting factor of 0.6.

[0112] Example 2 targets low-quality, small-area fingerprint expansion, generating 4 to 8 candidate fingerprints from the network output. The verification module sorts the candidates by endpoint connectivity, number of pseudo-minimum points, and matcher embedding distance, selecting the candidate with the highest confidence. The weighting coefficient is between 0.4 and 0.6.

[0113] Example 3 is for embedded deployment, with the following inputs and outputs: The generation bandwidth is 16 pixels, the directional channels are 8, the number of candidates is 1, the generation network uses depthwise separable convolution, the number of parameters is controlled at around 1 million, and the weighting factor is 0.5.

[0114] Example 2: Reference Figure 5 As shown, this embodiment provides a fingerprint image augmentation system based on topological phase field and trusted verification. This system corresponds to the method in Embodiment 1. A fingerprint image augmentation system based on topological phase field and trusted verification includes: The canvas embedding and region segmentation module is used to embed a small area fingerprint image into the center of the target canvas to form a center-locked map, and to divide the known real area and the area to be generated. The structure field estimation and extrapolation module is used to estimate the fingerprint structure features in the known real region and extrapolate the fingerprint structure features to the region to be generated; The topological phase field solving module is used to reconstruct the phase potential based on the fingerprint structure features of the known real area and take its trace at the boundary as the boundary phase. It constructs a topological phase field in the area to be generated so that the gradient direction is consistent with the ridge normal and the gradient magnitude is proportional to the ridge frequency. It also extracts the ridge skeleton probability map based on the topological phase field. The structural condition texture generation module is used to input the structural conditions containing the topological phase field and the ridge skeleton probability map into the generation network, output the grayscale image of the area to be generated, and force the real grayscale of the known real area to be written back to the corresponding position of the output image. The confidence verification and output module is used to perform minutiae conservation verification and confidence evaluation on the generated candidate images, and output the expanded image results with confidence map and matching weight map.

[0115] Through the coordinated operation of its modules, the system transforms uncontrollable pixel guesses into interpretable structural field extrapolation. While ensuring the continuity and naturalness of the expanded image area structure, it explicitly suppresses false minutiae and outputs credibility for downstream safe weighted matching.

[0116] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0117] The embodiments described above are merely illustrative of several implementation methods of this application, and while the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the patent application. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of this application, and these all fall within the protection scope of this application. Therefore, the protection scope of this patent application should be determined by the appended claims.

Claims

1. A fingerprint image augmentation method based on topological phase field and reliable verification, characterized in that, include: The fingerprint image is embedded in the center of the target canvas to form a center-locked image, and the known real area and the area to be generated are divided. Estimate fingerprint structural features in the known real region and extrapolate the fingerprint structural features to the region to be generated; Based on the fingerprint structure features of the known real area, the phase potential is reconstructed, and its trace at the boundary between the known real area and the area to be generated is taken as the boundary phase. A topological phase field is constructed in the area to be generated. The topological phase field is a scalar field, and the gradient direction of the topological phase field is consistent with the ridge normal and the gradient magnitude is proportional to the ridge frequency. Based on the topological phase field, a ridge skeleton probability map is extracted. The ridge skeleton probability map is a probability distribution map, which represents the probability that each pixel of the fingerprint image belongs to the ridge skeleton. Constructing a topological phase field in the region to be generated includes: An energy functional is constructed consisting of an area integral term for the region to be generated, a boundary curve integral term, and a second-order smoothing term. The area integral term includes a mass weight to make the phase gradient approximate the target vector field determined by the frequency field and the ridge normal. The boundary integral term constrains the phase of the region to be generated to inherit the boundary phase. Variational calculus of the energy functional yields a partial differential equation containing a biharmonic operator. The clamping condition of inherited phase and normal phase gradient is taken at the boundary, and the natural boundary condition is taken at the outer boundary of the region to be generated. Solving the partial differential equation yields the optimal topological phase field. The structural conditions, including the topological phase field and the ridge skeleton probability map, are input into the generator network, which outputs a grayscale image of the region to be generated. The real grayscale of the known real region is forcibly written back to the corresponding position of the output image to form a candidate image. Perform minutiae conservation checks and confidence assessments on the generated candidate images, and output the expanded image results with confidence maps and matching weight maps; The generated candidate images are subjected to minutiae conservation checks, including: Extract the endpoints of the boundary ridge line at the junction of the known real area and the area to be generated, and construct a dimensionless matching cost based on the positional distance between the two endpoints, the difference in the directional angle after dual-angle processing, and the local frequency difference. The optimal pairing of endpoints is obtained by solving the assignment problem. The topology gating score is calculated by combining the number of pseudo minutiae and the number of seam breaks in the candidate image to suppress pseudo minutiae that do not meet the endpoint pairing conditions in the candidate image.

2. The fingerprint image augmentation method based on topological phase field and reliable verification according to claim 1, characterized in that, The fingerprint structural features include orientation field, frequency field, and quality map; Extrapolating the fingerprint structural features to the region to be generated includes: Harmonic extrapolation of the direction field and frequency field is performed on the dual-angle channel. The estimated value of the boundary between the known real region and the region to be generated is used as the Dirichlet boundary condition, and the outer boundary of the region to be generated is used as the zero-flux von Neumann condition for solving. The quality map is extrapolated exponentially based on its distance from the known real area, so that the quality weight of each pixel in the area to be generated decreases as it moves further away from the known real area.

3. The fingerprint image augmentation method based on topological phase field and reliable verification according to claim 1, characterized in that, Based on the topological phase field, a ridge skeleton probability map is extracted, including: The cosine of the obtained optimal topological phase field is used to obtain the phase contour distribution, and the difference between the cosine value and the ridge determination threshold is soft-thresholded by the Sigmoid function to obtain a probability map representing the ridge peak position as the ridge skeleton probability map.

4. The fingerprint image augmentation method based on topological phase field and reliable verification according to claim 1, characterized in that, The step of inputting structural conditions containing the topological phase field and ridge skeleton probability map into the generator network, outputting a grayscale image of the region to be generated, and forcibly writing back the true grayscale of the known real region to the corresponding position in the output image includes: The center-locked map, the mask of the region to be generated, the distance map, the extrapolated orientation field, frequency field, quality map, optimal topology phase field and ridge skeleton probability map, and random perturbation variables are input into the structural condition generation network, and the grayscale image of the candidate generation region is output. Using a hard compositing method, the true grayscale of the center-locked image is directly output in the known real area, and the grayscale image of the candidate generation area is output in the area to be generated, and then combined into a complete image.

5. The fingerprint image augmentation method based on topological phase field and reliable verification according to claim 1, characterized in that, The credibility of the generated candidate images is evaluated, including: Multiple candidate images are generated, and the phase consistency cost, boundary connectivity cost, identity consistency cost, and quality cost of each candidate image are calculated respectively. The candidate image with the smallest total cost is selected as the final expanded image based on the weighted sum of each cost. The identity consistency cost is the cosine distance between the candidate image and the center-locked image in the pre-trained matching network embedding space.

6. The fingerprint image augmentation method based on topological phase field and reliable verification according to claim 5, characterized in that, The output includes a confidence plot, including: Collect error terms for candidate images, including orientation field residual, phase residual, boundary connectivity error, minutiae anomaly error, and identity embedding error; The weighted sum of each error term is input into the Sigmoid function, and the temperature parameter is solved by minimizing the negative log-likelihood on the validation set. The probability output by the Sigmoid function is then calibrated for temperature, resulting in a calibrated pixel-level confidence map.

7. The fingerprint image augmentation method based on topological phase field and reliable verification according to claim 6, characterized in that, Output the expanded map results with matching weighted graphs, including: Based on the mask identifier, the matching weight of the known real area is set to a fixed value, and the matching weight of the area to be generated is set to the product of the global weight reduction coefficient and the corresponding pixel value of the confidence map, forming a pixel-by-pixel matching weight map, which is used by the downstream recognition system to perform weight reduction matching on the features of the area to be generated.

8. A fingerprint image augmentation system based on topological phase field and reliable verification, characterized in that, include: The canvas embedding and region segmentation module is used to embed a small area fingerprint image into the center of the target canvas to form a center-locked map, and to divide the known real area and the area to be generated. The structure field estimation and extrapolation module is used to estimate the fingerprint structure features in the known real region and extrapolate the fingerprint structure features to the region to be generated; The topological phase field solving module is used to reconstruct the phase potential based on the fingerprint structure features of the known real area and take its trace at the boundary as the boundary phase. It constructs a topological phase field in the area to be generated so that the gradient direction is consistent with the ridge normal and the gradient magnitude is proportional to the ridge frequency. It also extracts the ridge skeleton probability map based on the topological phase field. Constructing a topological phase field in the region to be generated includes: An energy functional is constructed consisting of an area integral term for the region to be generated, a boundary curve integral term, and a second-order smoothing term. The area integral term includes a mass weight to make the phase gradient approximate the target vector field determined by the frequency field and the ridge normal. The boundary integral term constrains the phase of the region to be generated to inherit the boundary phase. Variational calculus of the energy functional yields a partial differential equation containing a biharmonic operator. The clamping condition of inherited phase and normal phase gradient is taken at the boundary, and the natural boundary condition is taken at the outer boundary of the region to be generated. Solving the partial differential equation yields the optimal topological phase field. The structural condition texture generation module is used to input the structural conditions containing the topological phase field and the ridge skeleton probability map into the generation network, output the grayscale image of the area to be generated, and force the real grayscale of the known real area to be written back to the corresponding position of the output image. The confidence verification and output module is used to perform minutiae conservation verification and confidence evaluation on the generated candidate images, and output the expanded image results with confidence map and matching weight map; The generated candidate images are subjected to minutiae conservation checks, including: Extract the endpoints of the boundary ridge line at the junction of the known real area and the area to be generated, and construct a dimensionless matching cost based on the positional distance between the two endpoints, the difference in the directional angle after dual-angle processing, and the local frequency difference. The optimal pairing of endpoints is obtained by solving the assignment problem. The topology gating score is calculated by combining the number of pseudo minutiae and the number of seam breaks in the candidate image to suppress pseudo minutiae that do not meet the endpoint pairing conditions in the candidate image.

Images