Edge enhanced spatially adaptive fractional order total variation super-resolution reconstruction method

Abstract

The edge-enhanced spatial adaptive fractional order total variation super-resolution reconstruction method relates to the technical field of laser radar imaging, and solves the problems of the existing technology, such as the smooth bias and energy dissipation caused by the traditional integer order variation model in processing depth jump, and the low resolution and edge blur of the GM-APD laser radar distance image. The present application introduces the inverse distance weighted interpolation to perform high-fidelity initialization on the low-resolution distance image; a spatial adaptive FOTV regularization model is constructed, and an alternating direction multiplier method is used to split the original objective function into an optimization subproblem about the high-resolution image u, an optimization subproblem about the auxiliary variable d, and an update of the dual variable b, and the like. The present application effectively improves the resolution of the laser radar distance image, solves the problem of edge blur of the distance image, reduces the root mean square error, improves the structural similarity and peak signal-to-noise ratio, and makes the target distance image clearer.

Description

Technical Field

[0001] This invention relates to the field of lidar imaging technology, and specifically to an edge-enhanced spatially adaptive fractional-order fully variable super-resolution reconstruction algorithm. Background Technology

[0002] Due to limitations in detector manufacturing processes, GM-APD lidar has a limited array size, and the raw distance images it acquires often exhibit low resolution and edge degradation, which restricts the accuracy of scene perception.

[0003] Currently, methods for reconstructing the range image resolution of GM-APD LiDAR can be mainly divided into three categories. The first is interpolation-based methods, such as bicubic interpolation, improved Kriging interpolation, and various edge-aware interpolation algorithms. These methods have low computational complexity and good real-time performance, but due to the lack of constraints on global priors for the image, they are prone to severe interpolation blurring and geometric distortion when dealing with the complex depth jumps in GM-APD range images. The second is deep learning-based super-resolution reconstruction methods, which have achieved significant improvements in image super-resolution by constructing deep nonlinear mapping networks. However, these data-driven methods heavily rely on large-scale and high-quality paired datasets for model training. For GM-APD LiDAR imaging systems, it is extremely difficult to obtain large-scale, high-resolution range images from real-world scenes as ground truth references, which greatly limits the direct deployment of deep neural networks in practical ranging tasks. The third approach is a super-resolution reconstruction method based on a reconstruction model. By transforming prior knowledge into regularization constraints, it achieves high generalization and strong noise resistance without training. It also extends variational thinking to 3D range images, effectively reconstructing fine structures. Combining anisotropic second-order total variational analysis with guided filtering improves edge fidelity, and leveraging local adaptive kernel functions enhances adaptability to non-stationary noise. Furthermore, by modeling the physical characteristics of the detector, the model can still accurately characterize the target topology in noisy environments even with sparse data and random detection. However, existing variational models face bottlenecks in large-scale inference. Fractional calculus, due to its nonlocal long-range memory, has been introduced into the SR domain, but the underlying model uses isotropic diffusion, which can easily lead to over-smoothing of edges. Additionally, the small scope of integer-order gradient operators can easily cause geometric distortion and staircase artifacts. Moreover, when existing algorithms handle complex scenes, the lack of geometric prior coupling can lead to the loss of weak signal details during the energy minimization process. Even worse, the inherent smoothing bias and energy dissipation of variational functionals will continuously attenuate high-frequency geometric components during iteration, resulting in a lack of physical sharpness in super-resolution images. Summary of the Invention

[0004] This invention provides an edge-enhanced spatially adaptive fractional total variational super-resolution reconstruction method to overcome the problems of smoothing bias and energy dissipation caused by traditional integer-order variational models when dealing with depth jumps, as well as the inherent low resolution and edge blurring of GM-APD lidar range images.

[0005] An edge-enhanced spatially adaptive fractional-order total variational super-resolution reconstruction method includes the following steps:

[0006] Step 1: Use inverse distance weighted interpolation to perform spatial mapping and initial upsampling on the low-resolution range image to obtain the initial estimate of the high-resolution range image to be estimated;

[0007] Step 2: Construct an edge-enhanced spatial adaptive fractional-order total variational super-resolution reconstruction model;

[0008] Construct an edge-enhanced spatially adaptive fractional total variational energy functional based on the initial estimates from step one; that is, construct the objective function using a data fidelity term, a spatially adaptive fractional regularization term, and an edge-enhancing penalty term.

[0009] Step 3: Using the alternating direction multiplier method, the objective function is split into an optimization subproblem concerning the high-resolution range image u to be estimated, an optimization subproblem concerning the auxiliary variable d, and an update of the dual variable b; thus achieving super-resolution reconstruction of the high-resolution range image.

[0010] The beneficial effects of this invention are:

[0011] 1. The method described in this invention constructs an edge-enhanced spatially adaptive fractional total variational (SFOTV) regularization model. Fractional calculus is introduced to characterize the nonlocal long-range correlation of signals, effectively overcoming the edge over-smoothing and energy dissipation problems that traditional integer-order variational models easily produce in deeply abrupt regions, significantly improving the representation ability of complex geometric structures and continuously changing regions. Furthermore, an edge-adaptive damping weight and a structure complexity-aware dual-weight matrix are introduced and deeply coupled with a fractional-order gradient field, enabling the model to adaptively adjust the regularization intensity according to local spatial structural characteristics. This achieves a balance between boundary preservation and detailed structural fidelity while improving overall reconstruction accuracy.

[0012] 2. The method described in this invention designs an edge enhancement mechanism based on residual feedback. By extracting high-frequency residuals and dynamically feeding them back to the energy functional, the inherent smooth bias in the variational approximation process is broken. In the super-resolution iteration, the sharpness of the target physical boundary is actively restored and maintained, eliminating step artifacts and edge cross-boundary blurring. Attached Figure Description

[0013] Figure 1This is a flowchart of the edge-enhanced spatially adaptive fractional total variational super-resolution reconstruction method described in this invention;

[0014] Figure 2 This is a rendering of the original resolution distance image;

[0015] Figure 3 Super-resolution reconstruction effect diagrams for different models are shown; (a) is the effect diagram of model A, (b) is a local magnified view of model A, (c) is the effect diagram of model B, (d) is a local magnified view of model B, (e) is the effect diagram of model C, (f) is a local magnified view of model C, (g) is the effect diagram of the model of the present invention, and (h) is a local magnified view of the model of the present invention.

[0016] Figure 4 The graphs show the changes of the three core indicators RMSE, PSNR, and SSIM with the first derivative of order v; (a) is the graph of RMSE, (b) is the graph of RMSE derivative (root mean square error derivative), (c) is the graph of PSNR, (d) is the graph of PSNR derivative (peak signal-to-noise ratio derivative), (e) is the graph of SSIM, and (f) is the graph of SSIM derivative (structural similarity derivative).

[0017] Figure 5 The images show a visual comparison of the reconstruction results of each algorithm under a 2x magnification factor; where (a) is a low-resolution image, (b) is a magnified view of (a), (c) is the true range image, (d) is a magnified view of (c), (e) is the cubic interpolation range image, (f) is a magnified view of (e), (g) is the guided filter range image, (h) is a magnified view of (g), (i) is the TGV range image, (j) is a magnified view of (i), (k) is the range image reconstructed by the method of this invention, and (l) is a magnified view of (k).

[0018] Figure 6 The images show the imaging experiment scene and local feature maps; where (a) is the experimental scene map and (b) is the local feature map.

[0019] Figure 7 The images show the results of different algorithms; (a) is a low-resolution image, (b) is a magnified view of (a), (c) is the true range image, (d) is a magnified view of (c), (e) is the cubic interpolation range image, (f) is a magnified view of (e), (g) is the guided filter range image, (h) is the guided filter range image, (g) is a magnified view of (h), (i) is the TGV range image, (j) is a magnified view of (i), (k) is the range image reconstructed by the method in this invention, and (l) is a magnified view of (k). Detailed Implementation

[0020] Specific Implementation Method 1: Combination Figure 1 This embodiment describes an edge-enhanced spatial adaptive fractional total variational super-resolution reconstruction method, which includes a data input and preprocessing module, an edge-enhanced spatial adaptive fractional total variational energy functional module, and an alternating direction multiplier method module.

[0021] The data input and preprocessing module is used to degrade and blur downsample the input low-resolution range image, and estimate the initial value through inverse distance weighted interpolation (IDW);

[0022] The edge-enhanced spatial adaptive fractional total variational energy functional module includes a data fidelity term, an edge-enhancing penalty term, and a spatial adaptive fractional regularization term.

[0023] The spatial adaptive fractional regularization term is implemented through steps such as constructing a fractional calculus operator, a fractional total variation (FOTV) regularization term, an edge adaptive damping and structural complexity-aware dual weight matrix, and a spatial adaptive fractional total variation regularization term.

[0024] The alternating direction multiplier method module uses the idea of ​​the alternating direction multiplier method to split the original objective function into an optimization subproblem concerning the high-resolution range image u, an optimization subproblem concerning the auxiliary variable d, and an update of the dual variable b. The split Bregman method is then used to efficiently iteratively solve the energy functional.

[0025] The method described in this embodiment first introduces inverse distance-weighted IDW interpolation to perform high-fidelity initialization of the low-resolution range image. Second, it constructs an edge-enhanced SFOTV super-resolution reconstruction model, using the nonlocal long-range correlation of fractional calculus to improve global reconstruction accuracy and the ability to represent complex geometric structures and continuously changing regions. Third, it dynamically compensates for high-frequency feature loss during iteration through a residual feedback edge enhancement mechanism. Finally, it employs the alternating direction multiplier method to split the original objective function into an optimization subproblem concerning the high-resolution range image u, an optimization subproblem concerning the auxiliary variable d, and an update of the dual variable b. Finally, it uses the split Bregman method to efficiently iteratively solve the energy functional. The specific implementation process is as follows:

[0026] Step 1. Degrade and blur downsample the low-resolution range image acquired by the GM-APD lidar sensor, and introduce inverse range-weighted interpolation (IDW) to estimate the initial value of the low-resolution range image; the specific steps are as follows:

[0027] Step 11. Use bilateral IDW to directly perform spatial mapping and initial upsampling on the observed low-resolution range image vector y;

[0028] In this embodiment, the core idea of ​​IDW is that the depth value of any empty position in the high-resolution grid is determined by the weighted average of the known low-resolution distance pixels in its spatial neighborhood, and the influence weight is inversely proportional to the Euclidean distance.

[0029] For the initial high-resolution range image to be estimated any of the first 1 pixel, its initial estimate The calculation formula is:

[0030]

[0031] In the formula, Low-resolution range image vector The k-th known depth pixel value in the data; Let K be the set of neighborhoods of the K known pixels that are closest to the i-th low-resolution pixel in the mapping space. For comprehensive weighting;

[0032] In this embodiment, the comprehensive weight Based on spatial distance weights With depth similarity weight The decision is made jointly and is expressed by the following formula:

[0033]

[0034] In the formula, The known depth value of the nearest neighbor of the mapping center point. To control for the standard deviation of depth similarity decay, p is the distance power parameter. A very small smoothing constant is used to ensure the numerical stability of the algorithm. After mapping the low-resolution range image to a grid, compare it with the known high-resolution range image pixels to be estimated. Reverse projection low-resolution distance image pixels The relative Euclidean distance between them:

[0035]

[0036] In the formula, Let be the two-dimensional spatial coordinates of the i-th pixel in the high-resolution range image. Let be the two-dimensional spatial coordinates of the k-th pixel in the low-resolution range image, and be the spatial scale scaling factor. , This represents the total number of pixels after scaling. The original total number of pixels. Through the above mapping, IDW can maintain the local rigid topological structure of the depth distribution while enlarging the image scale, providing excellent initial values ​​for subsequent fractional-order total variational reconstruction.

[0037] Step 2. Construct an edge-enhanced spatially adaptive fractional total variational (SFOTV) super-resolution reconstruction model;

[0038] In this embodiment, the model enhances global reconstruction accuracy and the ability to represent complex geometric structures and continuously changing regions by leveraging the nonlocal long-range correlation of fractional calculus. Furthermore, it dynamically compensates for high-frequency feature loss during iteration through an edge enhancement mechanism based on residual feedback. Addressing the ill-posed inverse problem of range image super-resolution, it constructs an edge-enhanced spatially adaptive fractional-order total variational energy functional composed of a data fidelity term, an edge enhancement regularization term, and a spatially adaptive fractional-order regularization term. The specific process is as follows:

[0039] In this embodiment, in order to infer a high-resolution range image from a low-resolution range image vector y containing noise and degraded resolution, a regularized energy functional with a strict minimum lower bound must be constructed. This embodiment constructs an edge-enhanced spatially adaptive fractional total variation (SFOTV) energy functional based on the fusion of inverse distance-weighted initial estimates: that is, the constructed objective function is as follows:

[0040]

[0041] in, As a data fidelity term, this term establishes a physical connection between the reconstructed high-resolution range image u and the low-resolution range image range vector y; This is an edge enhancement penalty term used to compensate for high-frequency feature loss during the iteration process; Here, H represents the spatial adaptive fractional total variational regularization term used to constrain the geometric structure during the super-resolution upsampling process; H is the optical fuzziness matrix of the system; and D is the spatial downsampling operator matrix. For regularization parameters, Enhance weights at the edges;

[0042] In this embodiment, the construction process of the spatial adaptive fractional total variation regularization term is as follows:

[0043] In this implementation, to address the block effect inherent in integer-order total variational models due to limited local scope, fractional-order calculus reconstruction regularization constraints are introduced. The unique nonlocal memory of fractional operators is utilized to establish long-range correlations between pixels, thereby extracting richer global geometric priors and improving structural fidelity while effectively suppressing block effects. Starting from the definition of the Grünwald-Letnikov fractional-order calculus operator and combining it with matrix approximation derivation of two-dimensional discrete meshes, basic fractional-order total variational regularization terms are constructed. Furthermore, to address the issue of edge blurring caused by applying completely equal intensity of smoothing and diffusion in all spatial locations and directions of the fractional-order model, local spatial statistical features are introduced to construct a spatially adaptive weight matrix, ultimately achieving deep fusion optimization of the two. The specific steps are as follows:

[0044] Step A1. Construction of fractional calculus operators; using the GL fractional calculus definition which is easy to numerically discretize, the one-dimensional signal is extended to the two-dimensional image signal, and the fractional calculus formulas in the x-axis and y-axis directions are derived.

[0045] For a given real function , ,and Then the definition of the v-th order GL fractional calculus is:

[0046]

[0047] In the formula, For adaptive weights of fractional gradients, The function after the fractional calculus operator is applied, where h represents the integration step size and g represents the discrete step size;

[0048] The equivalent expression for the v-th order fractional calculus of a one-dimensional signal is:

[0049]

[0050] In the formula, Г is the gamma function, which is a generalization of the factorial function in the real number field, satisfying Г(n+1)=n!

[0051] Two-dimensional image signal Dividing the pixels into equal parts with a pixel interval h=1, we can obtain the fractional calculus formulas for the x-axis and y-axis:

[0052]

[0053]

[0054] Summation indices, where i and j represent the number of steps moved from the current point in the past direction; P represents the number of polynomial terms, and let the coefficients of the v-order fractional calculus be... Then we have:

[0055]

[0056] Step A2. Based on the fractional finite forward difference formula, apply the matrix approximation method to construct the fractional total variation (FOTV) regularization term;

[0057] The fractional-order finite forward difference formula is defined as follows:

[0058]

[0059] in, and These are fractional difference weighting coefficients;

[0060] Applying matrix approximation, the above equation can be written in the following form:

[0061]

[0062] in, Let D represent the fractional-order differential linear operators in the horizontal and vertical directions, respectively. The spatial downsampling operator matrix D has the following form:

[0063]

[0064] Step A3. Introduce local spatial statistical properties to construct a dual-weight matrix of edge adaptive damping and structural complexity awareness; the specific implementation process is as follows:

[0065] In this embodiment, to overcome the diffusion mechanism of the edge-enhanced FOTV super-resolution reconstruction model, a globally consistent smoothing constraint is applied at all spatial locations. This spatial invariance causes the model to be unable to effectively distinguish between flat bases and abrupt edges in the image, resulting in edge blurring and loss of minor geometric features. Therefore, this embodiment constructs an edge adaptive damping weight matrix. And the structural complexity-aware weight matrix T.

[0066] Constructing the edge adaptive damping weight matrix Extract the local first-order spatial difference of the high-resolution image u to be estimated. Let the image size be... For any coordinates Pixel, differential gradient values ​​in the horizontal and vertical directions and They are respectively:

[0067]

[0068]

[0069] Therefore, the local gradient magnitude at that pixel can be calculated. :

[0070]

[0071] The edge adaptive damping weight matrix S can be expressed as:

[0072]

[0073] Its elements The definition is as follows:

[0074]

[0075] in, A scaling constant used to control the edge diffusion range, adjusted to regulate the decay rate. This is particularly relevant in the edge regions where depth jumps occur. Much larger hour, Approaching 0, while When it approaches 0, Approaching 1.

[0076] Constructing a structural complexity-aware weight matrix T: FOTV-based image processing methods often impose globally consistent regularization penalties, which inevitably leads to the loss of minute geometric features in the GM-APD lidar range image, making it difficult to simultaneously achieve background denoising and preservation of microscopic details. To quantify the complexity of the local geometric structure, the standard deviation distribution of the current high-resolution range image u to be estimated is extracted within the local neighborhood. For any coordinate in the grid... The pixels, define its Spatial Local Sliding Window First, calculate the average depth within the neighborhood. :

[0077]

[0078] Calculate the local standard deviation at this pixel. :

[0079]

[0080] In the formula, This represents the depth value at the distance from pixel (i,j) in the image.

[0081] Structural complexity-aware weights This can be represented by matrix T as follows:

[0082]

[0083] Using an exponential decay function, element The definition is as follows:

[0084]

[0085] By arranging the calculated local standard deviations of the entire image, we can obtain a local standard deviation matrix with the same dimensions as the original image. ,in This represents the maximum value of the global and local standard deviations. It is a very small positive constant, designed to ensure that the denominator is not zero and to guarantee the numerical stability of the algorithm;

[0086] exist Larger areas rich in detail It exhibits exponential decay. This mechanism allows the model to adaptively reduce the regularization smoothing constraint in this region, thereby effectively protecting the microscopic geometric features of the range image from over-penalization.

[0087] Step A4. To effectively assign the local gradient prior and spatial variance prior to fractional operators, the Hadamard product is used to adaptively dampen the edge weight matrix. With structural complexity-aware weight matrix Pixel-by-pixel coupling is performed to construct a comprehensive spatial adaptive control matrix. By combining the fractional difference matrices in the horizontal and vertical directions for approximate expression, the control matrix is... Applying this to the fractional gradient field, a spatially adaptive fractional total variational (SFOTV) regularization term is constructed. Its matrix operations and scalar expansion are as follows:

[0088]

[0089] This spatially adaptive mechanism enables the global smoothing model to be aware of content characteristics. In super-resolution inference, the weight matrix can adaptively adjust the diffusion intensity based on local topological properties: in smooth base regions, the long-range memory of fractional operators is fully preserved to smooth noise; in physical edges or regions with weak details, the regularization penalty is adaptively weakened or even truncated. This design fundamentally ensures that the distance image can still maintain sharp structural boundaries and high-fidelity features after large-scale sampling.

[0090] In this embodiment, an edge enhancement penalty term is constructed, and a two-dimensional Gaussian smoothing kernel is used to perform low-pass filtering on the current estimated image to extract a smooth low-frequency reference surface. The construction of the edge enhancement penalty term consists of three parts: low-frequency reference surface extraction, separation of high-frequency residual signals, and synthesis of the edge enhancement penalty term.

[0091] Step B1. Extraction of the low-frequency reference surface;

[0092] Using a two-dimensional Gaussian smoothing kernel As a low-pass filter, it extracts the smooth basis in the currently estimated image u. The mathematical expression for it is defined as:

[0093]

[0094] Among them, standard deviation This determines the scale range of spatial filtering. A two-dimensional convolution operation is performed between a Gaussian kernel and the current image to obtain a pure low-frequency reference surface after filtering out high-frequency fluctuations.

[0095] Step B2. Separation of high-frequency residual signals;

[0096] The current estimated image is differentially analyzed with its low-frequency reference surface to accurately extract the high-frequency residual signal containing edge, noise, and geometric transition features.

[0097]

[0098] Step B3. Synthesis of edge enhancement penalty terms;

[0099] A gain adjustment parameter is introduced into the high-frequency residual to construct a complete edge enhancement penalty term:

[0100]

[0101] in, The hyperparameters for controlling the edge compensation intensity.

[0102] In this embodiment, the edge enhancement regularization term effectively breaks the inherent smoothing bias in the variational approximation process through dynamic feedback of high-frequency residuals. This enables the model to actively restore and maintain the sharpness of the target's physical boundary while significantly improving the range image space resolution, thereby achieving high-fidelity super-resolution reconstruction.

[0103] Step 3. The reconstructed image, denoising variables, and residuals are updated alternately using the split Bregman method and the alternating direction multiplier method, respectively, thereby achieving super-resolution reconstruction of the distance image. In this embodiment, the alternating direction multiplier method is used to split the objective function into an optimization subproblem concerning the high-resolution image u, an optimization subproblem concerning the auxiliary variable d, and an update of the dual variable b. The specific process is as follows:

[0104] Step 31. Under the premise of fixing the auxiliary variable d and the dual variable b, the optimization problem of u is a smooth quadratic differentiable functional. When solving the optimization subproblem of u, a strategy of single-step gradient descent approximation combined with back projection is adopted.

[0105] make Since the Fréchet derivative of the estimated high-resolution range image u is zero, the exact gradient flow equation can be obtained using the chain rule:

[0106]

[0107] Where k is the k-th iteration, For the image to be reconstructed in the k-th iteration, Overall optimization objective function For the objective function with respect to gradient, For the system observation matrix, These are the weight parameters for high-frequency constraint terms. For the fixed auxiliary variable after the k-th iteration, These are the dual variables after the k-th iteration;

[0108] Using line search step size Gradient descent method for updating main variables:

[0109]

[0110] The optimization subproblem of step 32.d is:

[0111] Given fixed u and b, the optimization subproblem for d is:

[0112]

[0113] Based on the spatial adaptive weight matrix The principle behind its update formula is as follows:

[0114]

[0115] In the formula, shrink() is the shrinkage operator; and at the same time, because The adaptive adjustment effectively protects the physical depth mutation edge from excessive decay; This is a regularization parameter that controls the overall strength of the norm term; For penalty parameters;

[0116] Step 33. Breqman Iteration Parameter Update: After obtaining the latest estimated image with auxiliary variables Then, a simple additive residual update is performed on the dual variable b:

[0117]

[0118] This step continuously accumulates the current fractional gradient error, enabling the algorithm to strongly approximate the constraints. This ensures the rapid and stable convergence of the overall optimization model. Finally, when the change between two iterations is less than a set threshold or the maximum number of iterations is reached, the computation stops and the final high-resolution distance image is output.

[0119] Specific Implementation Method Two: Combination Figures 2 to 5 As shown, this embodiment is a simulation experiment of the edge-enhanced spatially adaptive fractional total variation super-resolution reconstruction method described in Specific Embodiment 1:

[0120] In this embodiment, the Monte Carlo photon counting simulation method is used to generate three-dimensional spatiotemporal echo photon distribution data based on a reference image. Furthermore, the peak thresholding method is employed to extract the time-of-flight of the echo signal, synthesizing the original high-resolution distance image value, such as... Figure 2 As shown.

[0121] In terms of quantitative performance evaluation, this invention selects Peak Signal-to-Noise Ratio (PSNR), Structural Similarity Indicator (SSIM), and Root Mean Square Error (RMSE) as full-reference evaluation metrics. RMSE and PSNR aim to quantify the gains in edge sharpness and numerical accuracy of the reconstructed image at the pixel level; while SSIM focuses on evaluating the similarity between the reconstructed image and the ground truth in local geometric topology and structural properties, thus comprehensively measuring the algorithm's ability to restore detailed features of the distance image. This invention selects Bicubic Interpolation, Guided Filter (GF), and Second-Order Generalized Variation (TGV) regularized super-resolution reconstruction algorithms as comparison algorithms.

[0122] To further explore and verify the independent contributions and synergistic effects of key components in the edge-enhanced spatially adaptive SFOTV super-resolution reconstruction model of this invention, a systematic ablation experiment was designed for the X2x super-resolution reconstruction task. This invention constructed three variant models—Model A (basic FOTV model), Model B (FOTV model with edge enhancement), and Model C (spatially adaptive FOTV model with dual weights)—for comparative analysis with the complete model of this invention (Proposed Mode). The comparison figures are shown below. Figure 3 As shown.

[0123] Figure 3 This paper showcases the super-resolution reconstruction results and local magnified details of different models in the Conces scene. Model A exhibits severe smoothing in depth discontinuities, resulting in blurred edges and loss of high-frequency details. Model B, while restoring global contrast, suffers from noticeable stepped jagged distortion at sloping edges. Model C suppresses jaggedness through edge enhancement but introduces unnatural artifacts at complex depth boundaries. In contrast, the proposed model D demonstrates the best performance in both global feature recovery and local detail preservation, exhibiting sharp and continuous edges, rich details such as facial contours and dot arrays in the background, and an overall visual effect highly consistent with the real-world distance image, validating its significant advantages in edge preservation and super-resolution quality.

[0124] The experimental results of each group were quantitatively evaluated based on the selected evaluation indicators RMSE, PSNR, and SSIM, as shown in Table 1.

[0125] Table 1

[0126]

[0127] The results show that the complementary mechanism formed by the dual weight matrix and the edge enhancement regularization term can effectively improve the super-resolution reconstruction quality of the range image.

[0128] To systematically analyze the specific impact of different fractional orders on the super-resolution reconstruction performance of GM-APD range images, this invention uses the Conces scene as a test case and conducts a sensitivity analysis experiment on the fractional order parameter v. Curves showing the variation of the three core performance indicators (RMSE, PSNR, and SSIM) of the reconstructed image with order v are plotted. Furthermore, curves showing the variation of the first derivative of these performance indicators with respect to v are introduced, as shown below. Figure 4 As shown.

[0129] from Figure 4 It can be seen that as the fractional order v increases, the RMSE of the reconstructed image decreases significantly, while PSNR and SSIM steadily increase. Around v=0.6, the first derivative of RMSE approaches zero, indicating that the error enters a minimum convergence plateau, while the first derivatives of PSNR and SSIM tend to reach their maximum values.

[0130] To comprehensively evaluate reconstruction performance, bicubic interpolation, guided filter (GF), and second-order generalized total variation (TGV) regularized super-resolution reconstruction algorithms were selected as comparison algorithms, and their processing results were compared with those of the algorithm proposed in this invention. Figure 5 The visual comparison results of the reconstruction results of each algorithm under the X2 magnification factor are shown.

[0131] from Figure 5 As can be seen, Bicubic is overly smoothed, resulting in the loss of high-frequency details; GF has good overall sharpness, but edge degradation occurs at depth transitions; TGV, while maintaining the piecewise constant, introduces texture artifacts. In comparison, the method of this invention reconstructs the sharpest edges, achieves the best geometric fidelity, and is closest to the true values.

[0132] To further objectively evaluate the reconstruction accuracy and robustness of each algorithm, this invention uses three image quality evaluation indicators—RMSE, PSNR, and SSIM—for quantitative analysis. Detailed evaluation results are shown in Table 2, which compares the reference image quality evaluation indicators for different methods.

[0133] Table 2

[0134]

[0135] As can be observed from the comparison in Table 2, the quantitative evaluation results are consistent with the visual comparison. At magnifications of X2, X4, and X8, the algorithm of this invention outperforms the comparative methods in all indicators. Compared with the best-performing TGV, RMSE is reduced by 2.76%, 3.72%, and 5.93%, respectively; PSNR is improved by 0.24 dB, 0.33 dB, and 0.53 dB, respectively; and SSIM is improved by 0.019 at high magnification (×8). This series of significant numerical improvements fully verifies that the method of this invention has higher reconstruction accuracy and stronger microstructure preservation ability in depth image super-resolution reconstruction tasks.

[0136] Specific Implementation Method Three: Combination Figure 6 and Figure 7 This embodiment describes the performance verification of the edge-enhanced spatially adaptive fractional total variational super-resolution reconstruction method described in Specific Embodiment 1. The specific process is as follows:

[0137] This experiment builds a Gm-APD lidar system to experimentally verify the performance of the method of the present invention. The main components of the experimental platform consist of the following core units:

[0138] Construction of GM-APD lidar experimental platform;

[0139] A Gm-APD lidar system was built to experimentally verify the performance of the method of the present invention. The main components of the experimental platform consist of the following core units:

[0140] 1. Laser: The light source uses a narrow linewidth fiber laser with a center wavelength of 1064nm, equipped with a light intensity modulation module with an adjustable repetition frequency (1-100 kHz), and the single pulse energy is set to 100µJ, the pulse duration is 5ns, and the repetition emission frequency is 20kHz.

[0141] 2. Emission system: The laser beam is collimated by the emission lens group and projected onto the target area. The echo signal reflected from the target surface is focused onto the Gm-APD pixel array by the receiving optical system.

[0142] 3. Receiving system: The time-to-digital converter (TDC) immediately terminates its timing process when the detector captures the first valid echo photon, and the readout circuit performs time-to-digital conversion on the flight time of the laser photon;

[0143] 4. GM-APD Array Detector: A 64×64 pixel GM-APD focal plane array detector, achieving optimized matching of 80% optical efficiency at the transmitting end and 90% transmission efficiency at the receiving end through a transmit / receive branch optical path design. Both the receiving and transmitting field of view are set to 0.8°×0.8°. It can detect and image targets at a distance of 425m-440m. Target imaging is as follows... Figure 6 As shown.

[0144] To verify the superiority of the method of this invention, a 64×64 high-resolution range image was obtained, and downsampled by 2 times to obtain a low-resolution range image. The result processed by the algorithm proposed in this invention was compared with the results processed by bicubic interpolation, guided filtering, and the TGV algorithm. The processing results are as follows: Figure 7 As shown.

[0145] from Figure 7 It can be seen that by combining the subjective visual effects of the magnified local images, the limitations of each algorithm can be further discovered: bicubic interpolation produces severe blurring at the edges of abrupt changes in target depth; although guided filtering improves the situation slightly, there is still obvious blurring across depth boundaries at the edges; the TGV algorithm, due to the excessive smoothing caused by total variation regularization, results in unnatural step-like artifacts at the target edges.

[0146] The method of this invention, through the synergistic effect of spatial adaptive weights and edge enhancement compensation terms, not only achieves excellent smoothing and noise reduction in flat areas, but also perfectly preserves the sharpness of target edges and true depth steps, completely eliminating ringing and blur artifacts at the edges. The results are shown in Table 3, which compares the quality evaluation metrics of different methods with reference images.

[0147] Table 3

[0148]

[0149] Using RMSE, PSNR, and SSIM for quantitative evaluation, the method of this invention significantly outperforms bicubic interpolation, guided filtering, and TGV algorithms in all indicators: RMSE is greatly reduced to 13.5007, PSNR is increased to 30.2342 dB, and SSIM approaches the ideal value. Subjective visual effects and objective evaluation indicators show that the method in this paper achieves better super-resolution reconstruction results for simulated images and real distance images than bicubic interpolation, guided filtering, and TGV.

[0150] 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.

[0151] The embodiments described above are merely illustrative of several implementations of the present invention, and while the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the invention patent. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these all fall within the protection scope of the present invention. Therefore, the protection scope of this invention patent should be determined by the appended claims.

Claims

1. An edge-enhanced, spatially adaptive fractional-order total variational super-resolution reconstruction method, characterized by: This method is implemented by the following steps: Step 1: Use inverse distance weighted interpolation to perform spatial mapping and initial upsampling on the low-resolution range image to obtain the initial estimate of the high-resolution range image to be estimated; Step 2: Construct an edge-enhanced spatial adaptive fractional-order total variational super-resolution reconstruction model; Construct an edge-enhanced spatially adaptive fractional total variational energy functional based on the initial estimates from step one; that is, construct the objective function using a data fidelity term, a spatially adaptive fractional regularization term, and an edge-enhancing penalty term. Step 3: Using the alternating direction multiplier method, the objective function is split into an optimization subproblem concerning the high-resolution range image u to be estimated, an optimization subproblem concerning the auxiliary variable d, and an update of the dual variable b; thus achieving super-resolution reconstruction of the high-resolution range image.

2. The edge-enhanced spatially adaptive fractional-order total variational super-resolution reconstruction method according to claim 1, characterized in that: In step one, the initial estimate of the high-resolution range image to be estimated. The formula is: ； In the formula, Low-resolution range image vector The k-th known depth pixel value in the data; Let K be the set of neighborhoods of the K known pixels that are closest to the i-th low-resolution pixel in the mapping space. This is the overall weight.

3. The edge-enhanced spatially adaptive fractional-order total variational super-resolution reconstruction method according to claim 2, characterized in that: In step two, the objective function is as follows: ； In the formula, For data fidelity items; Enhance the penalty term for the edge; denoted as the spatial adaptive fractional total variation regularization term; H is the optical fuzziness matrix of the system; D is the spatial downsampling operator matrix; u is the high-resolution range image to be estimated; For regularization parameters; Enhance the weights at the edges.

4. The edge-enhanced spatially adaptive fractional-order total variational super-resolution reconstruction method according to claim 3, characterized in that: The construction process of the spatial adaptive fractional total variation regularization term is as follows: Step A1: Using the GL definition of fractional calculus, extend the one-dimensional signal to the two-dimensional image signal, and derive the fractional calculus formulas in the x-axis and y-axis directions; Step A2: Construct fractional total variation regularization terms; Step A3: Introduce local spatial statistical properties to construct an edge adaptive damping weight matrix and a structural complexity-aware weight matrix; Step A4: Use the Hadamard product to couple the edge adaptive damping weight matrix and the structural complexity-aware weight matrix pixel by pixel to construct a comprehensive spatial adaptive control matrix; combine the fractional difference matrix approximation in the horizontal and vertical directions, apply the control matrix to the fractional gradient field to construct a spatial adaptive fractional total variation regularization term.

5. The edge-enhanced spatially adaptive fractional total variational super-resolution reconstruction method according to claim 4, characterized in that: In step A3, the process of constructing the edge adaptive damping weight matrix is ​​as follows: Set the size of the high-resolution image u to be estimated as For any coordinates Pixel, differential gradient values ​​in the horizontal and vertical directions and They are respectively: ； ； Calculate Local gradient magnitude at pixel : ； The edge adaptive damping weight matrix S is then expressed as: ； Its elements The definition is as follows: ； In the formula, A scaling constant used to control the edge diffusion range and adjust the decay rate.

6. The edge-enhanced spatially adaptive fractional total variational super-resolution reconstruction method according to claim 4, characterized in that: In step A3, the process of constructing the structure complexity-aware weight matrix is ​​as follows: Extract the standard deviation distribution of the current high-resolution distance image u to be estimated within the local neighborhood; for any coordinates pixels, defined Spatial Local Sliding Window ; Calculate the average depth within the local neighborhood. : ； Calculate pixels Local standard deviation : ； In the formula, For the pixels in the high-resolution distance image to be estimated The depth value at that location; Structural complexity-aware weights Represented by matrix T as follows: ； Using an exponential decay function, the elements in the matrix The definition is as follows: ； In the formula, It is a local standard deviation matrix; This represents the maximum value of the global and local standard deviations. It is a positive number; For pixels The local standard deviation at a given location.

7. The edge-enhanced spatially adaptive fractional-order total variational super-resolution reconstruction method according to claim 6, characterized in that: In step A4, the matrix operations and scalar expansion of the spatial adaptive fractional total variational regularization term are as follows: ； In the formula, and These are elements in the edge adaptive damping weight matrix S and the structure complexity-aware weight matrix T, respectively. It is a fractional-order differential linear operator.

8. The edge-enhanced spatially adaptive fractional-order total variational super-resolution reconstruction method according to claim 3, characterized in that: The edge enhancement penalty term consists of three parts: low-frequency reference surface extraction, high-frequency residual signal separation, and edge enhancement penalty term synthesis. The low-frequency reference surface extraction process is as follows: Using a two-dimensional Gaussian smoothing kernel As a low-pass filter, a smooth basis is extracted from the high-resolution range image u to be estimated. The mathematical expression for it is defined as: ； In the formula, standard deviation The scale range of spatial filtering is determined by performing a two-dimensional convolution operation between a Gaussian kernel and the current image to obtain the low-frequency reference surface after filtering out high-frequency fluctuations. ; The separation process of the high-frequency residual signal is as follows: The high-resolution distance image to be estimated is compared with the low-frequency reference surface by performing a difference operation to extract the high-frequency residual signal containing edge, noise, and geometric transition features: ； The complete edge enhancement penalty term is constructed as follows: ； In the formula, The hyperparameters for controlling the edge compensation intensity.

9. The edge-enhanced spatially adaptive fractional total variational super-resolution reconstruction method according to any one of claims 1-8, characterized in that: In step three, with the auxiliary variable d and the dual variable b fixed, a strategy of single-step gradient descent approximation combined with back projection is adopted when solving the optimization subproblem of u. make Since the Fréchet derivative of the estimated high-resolution range image u is zero, the gradient flow equation is obtained using the chain rule: ； In the formula, k is the number of iterations. For the image to be reconstructed in the k-th iteration, For the objective function with respect to gradient, The optical fuzzy matrix of the system, To enhance the weights at the edges, Let this be the auxiliary variable after the k-th iteration. These are the dual variables after the k-th iteration; Using line search step size Gradient descent is used for variable updates: ； In the formula, L( Let be the objective function for the k-th iteration. The image to be reconstructed is the image from the (k+1)th iteration.

10. The edge-enhanced spatially adaptive fractional-order total variational super-resolution reconstruction method according to claim 9, characterized in that: Given fixed u and b, the optimization subproblem for d is: ； According to the spatial adaptive control matrix The updated formula is: ； In the formula, shrink() is the shrinkage operator; For regularization parameters; This is the regularization parameter, i.e., the penalty parameter; The high-resolution range image to be estimated is updated using the Breqman iteration parameters after k+1 iterations. Auxiliary variables in k+1 iterations Then, an additive residual update is performed on the dual variable b: ； In the formula, Let be the dual variable for the (k+1)th iteration.

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