Striped tube lidar stripe echo signal enhancement restoration method based on sagan

Abstract

The application discloses a streak tube laser radar streak echo signal enhancement and restoration method based on SAGAN, and the method is as follows: noise signals are added to original ground signal streaks to simulate underwater signal streaks; the obtained simulated underwater signal streaks are input into a generator network as noise signals; output results of the generator and the ground streaks are input into a discriminator network; an adversarial generation network is trained by using a mixed precision floating point number method until the discriminator network cannot distinguish the output results of the generator from the original ground streaks; parameters of the trained generator network are obtained; the original underwater signal streaks are input into the trained generator network, and a restored and enhanced underwater signal streak can be obtained. The method restores and enhances echo signals of underwater targets by introducing an adversarial generation network (SAGAN) with a self-attention mechanism.

Description

Technical Field

[0001] This invention relates to a method for enhancing and restoring stripe echo signals of a stripe tube lidar, specifically a method for enhancing and restoring stripe echo signals of a stripe tube lidar based on a generative adversarial network (SAGAN). Background Technology

[0002] Compared to traditional acoustic and magnetic detection methods, airborne lidar is a typical cross-medium detection method with unparalleled advantages in mobility, safety, and adaptability. The advantages of airborne lidar for detecting small targets in shallow water are mainly reflected in the following aspects: First, superior mobility; thanks to its integrated design with the flight platform and high cruising speed, it can quickly reach and cover the operational sea area. Second, excellent safety; unaffected by shallow water reefs, underwater obstacles, etc., it can conduct detection operations day and night and has strong anti-interference capabilities. Third, high operational efficiency; benefiting from the technical characteristics of cross-medium detection, it has a fast detection speed and a high effective action time ratio (characterized by no sensor deployment and recovery, and flexible maneuverability), enabling rapid search and detection of large areas of water.

[0003] Compared to traditional microwave radar, lidar boasts advantages such as high precision, high resolution, high detection sensitivity, high security, small size, and light weight, making it suitable for airborne and shipborne deployment. Furthermore, due to its different operating mechanism, the laser pulses emitted by lidar exhibit stronger anti-interference capabilities and acquire richer data information compared to traditional microwave radar signals, resulting in superior detection and identification capabilities. Stripe tube-based array lidar features high imaging frame rate, full waveform sampling, high detection sensitivity, and high range resolution, showing great promise in airborne radar mapping and military applications.

[0004] Airborne lidar systems have significant advantages in target detection, but they also face a series of technical challenges when dealing with shallow water targets, such as insufficient resolution for detecting small targets, a significant decrease in measurement depth due to the influence of water on the echo signal, and the need to further improve the coverage range of a single flight.

[0005] Currently, lidar detection of underwater targets is typically based on helicopters or small unmanned aerial vehicle (UAV) platforms. Examples include the US AN / AES-1 Airborne Laser Mine Detection System (ALMDS), Canada's SHOALS (Scanning Hydrographic Operational Airborne Lidar Survey System), and the CZMIL (Coastal Zone Mapping and Imaging Lidar) system designed to address measurement challenges in turbid and shallow water areas. These lidar systems for underwater target detection are mounted on helicopter platforms, typically flying at altitudes of only 400-500 meters, with a detection width usually less than 300 meters.

[0006] Due to its unique operating mechanism, the airborne stripe tube lidar (ASTIL) can represent the acquired echo signals as two-dimensional single-channel digital images. Compared to the signals acquired by traditional single-point detection LiDAR, the echo signals acquired by ASTIL can contain the surface cross-sectional morphology within a detection area of ​​hundreds of meters, as well as echo intensity information of each target. Furthermore, the radar platform typically flies at an altitude of several kilometers above the ground. Its echo signals are rich in semantic information, possessing the potential for direct ground feature and area identification based on a single echo and single data source, and a single flight can cover a vast area of ​​targets.

[0007] Researchers hoped to leverage ASTIL's powerful detection capabilities to simultaneously perform underwater mapping. However, when ASTIL detects an underwater target, the echo signal stripes received by ASTIL from the water body are typically weak due to the unavoidable influence of the water during laser propagation.

[0008] Figure 1 The images show typical fringe patterns for underwater and ground targets. When ASTIL detects an underwater target, the laser signal suffers from quality degradation and intensity attenuation due to multiple reflections and refractions in the water, as well as absorption and scattering by the water and impurities. The resulting echo fringe pattern typically has weak signal strength and a low signal-to-noise ratio, making subsequent image processing and point cloud recognition difficult.

[0009] Currently, image enhancement methods can be mainly categorized into those based on traditional feature calculation, those based on computer graphics, and those based on neural networks.

[0010] Traditional feature-based calculation methods typically require estimating the distance between the underwater target and the ASTIL, as well as the water depth, to calculate the effects of laser reflection, refraction, and scattering in the water. However, these conditions are often difficult to meet in practical applications, therefore these methods are generally not used.

[0011] Image enhancement methods based on computer graphics primarily rely on traditional image processing techniques such as mathematical transformations, morphological operations, and filter design. Their core principle is to achieve contrast adjustment, noise suppression, and detail enhancement through spatial or frequency domain analysis of the image, using changes in the spatial and frequency domains to enhance the image. Computer graphics-based methods offer advantages such as strong algorithmic interpretability and high computational efficiency. However, their ability to handle complex scenes is limited, they are prone to losing image details, and the processing effect often depends on parameter selection, requiring manual intervention to achieve the desired result. The choice of parameters typically has a significant impact on the final processing outcome.

[0012] Neural network-based methods mainly include two types: diffusion models and generative adversarial networks (GANs). These methods are driven by large amounts of data, have strong global optimization capabilities, and can handle more complex degradation patterns and generate high-quality results.

[0013] The core idea of ​​the diffusion model is to generate data through two main processes: forward diffusion and backward diffusion. Forward diffusion is a parameterized Markov chain that gradually adds noise to the original data until it becomes pure noise. Backward diffusion is the reverse of the forward process; it starts with pure noise and, through the learning of a neural network, gradually removes the noise to recover the original data. As backward diffusion progresses, noise is gradually removed, data features are gradually recovered, and ultimately, a generated sample that closely approximates the original data is obtained.

[0014] A Generative Adversarial Network (GAN) consists of two parts: a generator and a discriminator. The generator is responsible for generating samples that are as realistic as possible, while the discriminator is responsible for distinguishing between generated and real samples. The generator's goal is to convert low-quality LiDAR signals into high-quality signals, while the discriminator's goal is to determine whether the signal comes from real data or was generated by the generator. Through continuous adversarial training, the generator can learn how to generate high-quality data that is closer to the real signal, thereby achieving signal enhancement.

[0015] Compared to diffusion models, GANs generate samples with higher detail and greater diversity. Secondly, during training, GANs have a simpler and more direct training objective, resulting in higher training efficiency compared to diffusion models. Furthermore, GAN models are more flexible and can be easily combined with other models, facilitating subsequent improvements to meet the needs of practical applications.

[0016] However, conventional GANs are weak at distinguishing noise and edge details in image enhancement methods. The resulting images may over-suppress noise or lose detail information, leading to unrealistic or inaccurate images that fail to meet the practical needs of signal enhancement. Furthermore, GANs are unstable during training, prone to gradient vanishing and exploding, requiring network improvements to prevent the waste of computational resources caused by gradient explosion and vanishing during training.

[0017] Due to the significant loss and attenuation of echo signals from underwater targets during the detection process based on airborne stripe tube lidar, the point cloud density in the water area is significantly reduced compared to the point cloud density on the ground during point cloud reconstruction. Summary of the Invention

[0018] To address the aforementioned problems, this invention provides a method for enhancing and restoring stripe echo signals from a stripe tube lidar based on SAGAN. This method introduces a Generative Adversarial Network (SAGAN) with a self-attention mechanism to restore and enhance the echo signals of underwater targets.

[0019] The objective of this invention is achieved through the following technical solution:

[0020] A method for enhancing and restoring stripe echo signals from a stripe tube lidar based on SAGAN includes the following steps:

[0021] Step 1: Save the ground signal stripe pattern to be processed and the original underwater signal stripe pattern to the relevant folders respectively;

[0022] Step 2: Add noise signals to the original ground signal stripe pattern from Step 1 to simulate an underwater signal stripe pattern. The specific steps are as follows:

[0023] Step 2-1: Add reverberant noise and random Gaussian noise to the original ground signal fringe pattern;

[0024] Step 2-2: Add random attenuation to the original ground signal stripe pattern;

[0025] Step 3: Improve the original GAN. The generator network uses the U-Net architecture. Introduce residual modules into the generator to build an adversarial generative network with a self-attention mechanism.

[0026] Step 4: The size of the simulated underwater signal stripe pattern obtained in Step 2 is reduced by a proportional ratio from 512×1024 to 256×512, and it is input into the generator network as a noise signal. The output of the generator and the ground stripe pattern in Step 1 are input into the discriminator network.

[0027] Step 5: Train the adversarial generative network using a mixed-precision floating-point method until the discriminator network can no longer distinguish between the generator's output and the original ground stripe pattern. Specifically, single-precision floating-point methods are used in the normalization layer and during model saving, while half-precision floating-point methods are used in other layers and during training.

[0028] Step 6: Obtain the parameters of the generator network trained in Step 5, and input the original underwater signal stripe pattern into the trained generator network to obtain the restored and enhanced underwater signal stripe pattern.

[0029] Compared with the prior art, the present invention has the following advantages:

[0030] 1. This invention enhances and restores waveform sampling lidar echo signals based on SAGAN. Through the organic combination of structural innovation and training strategy, it exhibits excellent detail recovery capability and robustness in stripe pattern image enhancement, solving the problem that the stripe pattern of echo signals from water bodies received by waveform sampling lidar is usually weak and has a low signal-to-noise ratio.

[0031] 2. During the training process, the image size is reduced to one-quarter of its original size, and mixed-precision floating-point numbers are used to significantly improve the training speed of the model while maintaining the computational accuracy of the model. Attached Figure Description

[0032] Figure 1 The images show typical stripe patterns; the left image shows a typical stripe pattern for an underwater target, and the right image shows a typical stripe pattern for a ground target.

[0033] Figure 2 This is a schematic diagram of the overall process for enhancing and restoring the stripe echo signal of a stripe tube lidar based on SAGAN.

[0034] Figure 3 The images show the preprocessed results of the original stripe pattern. The left image is a typical original stripe pattern, and the right image is a stripe pattern after preprocessing.

[0035] Figure 4 This is a schematic diagram of the generator network architecture;

[0036] Figure 5 This is a diagram of the residual module architecture;

[0037] Figure 6 This is a diagram of the self-attention module architecture.

[0038] Figure 7 This is a schematic diagram of the discriminator network architecture;

[0039] Figure 8 Original underwater target stripe pattern;

[0040] Figure 9 This is the striped pattern after image restoration and enhancement;

[0041] Figure 10 To detect satellite images of the target;

[0042] Figure 11 This is the original signal point cloud image;

[0043] Figure 12 The point cloud image is after signal enhancement and restoration.

[0044] Figure 13 The image shows a point cloud map of an underwater target. The left image is the original point cloud map, and the right image is the enhanced and restored point cloud map.

[0045] Figure 14 This is the original signal point cloud density distribution map;

[0046] Figure 15 This is a point cloud density distribution map after signal restoration and enhancement. Detailed Implementation

[0047] The technical solution of the present invention will be further described below with reference to the accompanying drawings, but it is not limited thereto. Any modifications or equivalent substitutions to the technical solution of the present invention that do not depart from the spirit and scope of the technical solution of the present invention should be covered within the protection scope of the present invention.

[0048] This invention provides a method for enhancing and restoring stripe echo signals from a stripe tube lidar based on SAGAN. The method utilizes SAGAN to restore and enhance the echo signals of low-quality target signals from an airborne stripe tube lidar. The overall process is as follows: Figure 2 As shown, SAGAN is similar to the original GAN, consisting of a generator network G and a discriminator network D.

[0049] During training, the generator network G takes a noisy signal as input and outputs a false, sharp signal generated by G. The discriminator network D's main function is to distinguish whether the input signal is a true, sharp signal or a false signal generated by the generator network. The ultimate training goal is to make the discriminator network D unable to distinguish between a true, sharp signal and a false signal generated by G. In other words, the generator network G should be able to generate a realistically restored target signal from a noisy signal. The trained generator network G can then be used for the restoration and enhancement of low-quality target signals. Using the original target stripe signal as input to the generator network G, the output after computation is the result of image enhancement and restoration of the low-quality target signal.

[0050] I. Training Data Preprocessing

[0051] Before network training, the training data needs to be preprocessed to simulate the characteristics of low-quality signals using clear ground target signals. This requires adding noise to the clear ground target signals to simulate the echo signals of low-quality targets, which are then used as input to the generator during training. The original size of the echo signal stripe pattern of the striped laser lidar is 512×1024.

[0052] According to the literature, laser light in water is mainly affected by the following two factors: firstly, the laser undergoes multiple reflections and refractions at the interface of the medium when it enters and leaves the water surface; secondly, the laser is absorbed and scattered by the water body during its propagation.

[0053] Considering the effects of multiple reflections and refractions of the laser at the interface and the scattering effect of water, reverberant noise and random Gaussian noise are added to the clear ground target signal; random attenuation is added to the signal to simulate the absorption of the laser by water. A comparison is made between the original ground target signal and the ground target signal with added noise. Figure 3 As shown.

[0054] Meanwhile, since the echo signal (i.e., the stripe pattern structure) of stripe tube LiDAR is usually relatively simple, to reduce the computational load of subsequent model training, the original 512×1024 stripe pattern is proportionally reduced to a 256×512 image. This processing introduces an error of about one pixel in the original stripe pattern, meaning that the point position in the subsequently generated point cloud will have an error of less than 0.3m, which is within an acceptable error range. At the same time, the total number of parameters in the generator network is reduced from about 160M to about 41.9M, and the total number of parameters in the discriminator network is reduced from about 8.5M to about 1.93M, meaning that the network training parameters are reduced to about one-quarter of the original network. This operation can greatly reduce the computational load and time required for model training, while the error introduced by this optimization operation is also controlled within a reasonable range.

[0055] II. SAGAN Model Network Architecture

[0056] The overall architecture of the network mainly consists of a generator network G and a discriminator network D. The main function of the generator network G is to enhance and restore the noisy signal; the main function of the discriminator network D is to evaluate the similarity between the signal fringe pattern generated by the generator network G and the original clear signal fringe pattern, and to guide the training process of the generator network G accordingly.

[0057] To improve the training stability of the network, SAGAN makes the following modifications to the original Generative Adversarial Network: First, a self-attention module is introduced into the discriminator and generator networks. The effective integration of this module enables the network to realistically capture and associate long-range spatial information, allowing the model to simultaneously capture global semantic information in the image and improve its ability to recognize details and pattern structures in the image. Second, spectral normalization is used in the discriminator and generator networks. This method can effectively constrain the parameter update speed of each convolutional layer during training, preventing gradient explosion or gradient vanishing during training and improving the convergence of model training. At the same time, the TTUR method is also used, in which the generator and discriminator use different learning rates, which can balance the training speed of the generator and discriminator networks and improve the stability of model training.

[0058] To improve network training speed, a mixed-precision floating-point method is employed during training, in addition to reducing image size. This method leverages different precision numerical formats to enhance training efficiency. Since half-precision floating-point (FP16) reduces memory usage by 50% and increases computation speed by 2-8 times compared to single-precision floating-point (FP32), FP32 is used in normalization layers and model saving, while FP16 is used in other layers and during training. This strategy significantly improves training speed (estimated speedup of 30-50%) while maintaining model performance, and reduces GPU memory consumption by approximately 30%. Furthermore, compared to directly using 16-bit floating-point numbers, mixed-precision training does not affect the storage format of model parameters, and the saved or loaded model files are fully compatible with the original FP32 version.

[0059] 1. Generator Network Architecture

[0060] The architecture of the generator network G is as follows Figure 4 As shown, its main architecture is a U-Net model, incorporating a residual block and a self-attention block. The generator network G takes a noisy signal as input and outputs a false, clear signal. The training objective of the generator network G is to generate clear stripe signals from noisy stripe signals. After training, the generator network G can output clear underwater target stripe signals by inputting real underwater target stripe signals.

[0061] In the generator network G, all normalization layers employ Instance Normalization Layers, which normalize each channel of each sample during computation. The specific computation method of the Instance Normalization Layer is as follows:

[0062] (1)

[0063] in, It is the input tensor In the The sample, the first The first channel, the first line, number Column elements; It is the first The sample, the first The average of each channel; It is the first The sample, the first The variance of each channel; It is the normalized scaling factor; It is the normalized offset; and These are learnable parameters; It is a small constant to prevent division by zero.

[0064] Compared to commonly used Batch Normalization Layer and Layer Normalization Layer, which normalize each channel of each batch and all channels of each sample respectively, Instance Normalization Layer can preserve the differences between samples, effectively improving model performance and stability in image generation tasks.

[0065] The specific architecture of the residual module in the network architecture is as follows: Figure 5 As shown, the residual module consists of two convolutional layers, two normalization layers, and one ReLU layer. The output size of the residual module is the same as the input size, and the output of the residual module is the sum of the original input of the module and the computation results of each layer. Therefore, introducing the residual module can effectively alleviate the problems of gradient vanishing and deep network degradation, while preserving the features of the previous layer and improving the network's performance and generalization ability.

[0066] The specific architecture of the self-attention module in the network architecture is as follows: Figure 6As shown, the introduction of a self-attention module enhances the model's ability to capture long-range dependencies and its parallelism, thereby improving the model's expressive power and highlighting important features in the image. The self-attention module used here is largely similar to the original self-attention module, but a step similar to the residual module is added to the module's output, directly adding the original input to the self-attention module's output. This further enhances the model's resistance to gradient vanishing and deep network degradation, improving the network's performance and generalization ability.

[0067] 2. Discriminator Network Architecture

[0068] The architecture of the discriminator network D is as follows: Figure 7 As shown, its main architecture is a convolutional neural network model with a self-attention module and spectral normalization applied to the convolutional layers. The discriminator network D is used to determine whether the input image is a real, clear signal or a signal generated by the generator network G. After training, the accuracy of the discriminator network D should fluctuate around 50%, indicating that at this point, the discriminator network D can no longer distinguish whether the input data was generated by the generator network G.

[0069] The self-attention module in the discriminator network D is the same as that in the generator network G, and will not be described again here.

[0070] The discriminator network D employs LeakyReLU (Leaky Rectified Linear Unit), an improved ReLU activation function widely used in deep learning, particularly excelling in Generative Adversarial Networks (GANs). Its functional form is shown below:

[0071]

[0072] in, The slope of the negative half-axis is set to 0.2, which, compared to a regular ReLU layer that completely discards gradients in the negative interval, LeakyReLU retains small gradients in the negative interval, maintaining the ability to update parameters, preventing the discriminator from converging prematurely, and balancing the training dynamics of the generator and discriminator, thus significantly improving the training stability of deep models.

[0073] The goal of spectral normalization is to make the spectral norm (maximum singular value) of the weight matrix equal to 1. This is achieved by iteratively calculating the left and right singular vectors of the weight matrix to approximate the singular vector corresponding to the maximum singular value, thus realizing spectral normalization of the weight matrix. Spectral normalization constrains the numerical update range of the weight matrix, preventing gradient explosion.

[0074] III. Based on the design plan, the specific implementation steps are as follows:

[0075] Step 1: Save the ground signal stripe pattern to be processed and the original underwater signal stripe pattern to the relevant folders respectively;

[0076] Step 2: Add the designed noise signal to the original ground signal stripe pattern in Step 1 to simulate an underwater signal stripe pattern;

[0077] Step 3: Input the simulated underwater signal stripe pattern obtained in Step 2 as a noise signal into the generator network, and input the generator output and the ground stripe pattern in Step 1 into the discriminator network;

[0078] Step 4: Train the network as shown in Step 3 until the discriminator network can no longer distinguish between the generator's output and the original ground stripe pattern;

[0079] Step 5: Obtain the parameters of the generator network trained in Step 4, input the original underwater signal stripe pattern into the trained generator network, and you can get the restored and enhanced underwater signal stripe pattern.

[0080] IV. Technical Effects:

[0081] Table 1 compares the training performance of the original GAN ​​and the improved SAGAN. After training both networks 300 times, the evaluation results of the restored images compared to the original images are shown. Compared to GAN, after training with SAGAN, the peak signal-to-noise ratio (PSNR) increased by 3.857, the structural similarity index (SSIM) increased by 0.09, and the mean squared error (MSE) decreased to 30.5% of the GAN result. This indicates that SAGAN significantly improves the image enhancement and restoration capabilities and enhances the quality of the restored images compared to the original GAN.

[0082] Table 1 Comparison of training results

[0083]

[0084] by Figure 8 Taking a single underwater target signal stripe pattern as an example, using a trained generator model G, the original signal stripe pattern is enhanced and restored, and the result is as follows. Figure 9 As shown, the signal-to-noise ratio of the striped image is significantly improved.

[0085] Real satellite images of the target area, such as Figure 10 As shown, after plotting the original fringe pattern obtained from the detection and the fringe pattern obtained after model processing into point cloud maps, the results are as follows. Figure 11 and Figure 12 As shown. Figure 13As shown, the point cloud map of the water area was cropped. According to statistics, the original underwater target echo stripe map generated a total of 51,894 points, while after model processing, the underwater target stripe map generated a total of 94,916 points, which increased the number of point clouds by 82.9%.

[0086] The point cloud density distributions of the original signal and the image after enhancement and restoration are as follows: Figure 14 and Figure 15 As shown, after SAGAN processing, the point cloud density in the water area is significantly improved. In some areas with better results, the point cloud density of the underwater signal after enhancement and restoration can approach that of the point cloud density of the ground signal.

Claims

1. A SAGAN-based streak tube lidar streak echo signal enhancement restoration method, characterized by The method comprises the following steps: Step 1: save the ground signal stripe pattern to be processed and the original underwater signal stripe pattern to the relevant folders respectively; Step 2: add noise signal to the original ground signal stripe pattern in step 1 to simulate the underwater signal stripe pattern; Step 3: improve the original GAN, the generator network uses the U-Net architecture, and a residual module is introduced in the generator to construct the adversarial generation network with the self-attention mechanism; Step 4: the size of the simulated underwater signal stripe pattern obtained in step 2 is reduced to 256x512 in proportion from 512x1024, which is input into the generator network as noise signal, and the output result of the generator and the ground stripe pattern in step 1 are input into the discriminator network; Step 5: train the adversarial generation network by using the mixed precision floating point method until the discriminator network cannot distinguish the output result of the generator from the original ground stripe pattern, wherein the single precision floating point method is used in the normalization layer and the model saving process, and the half precision floating point method is used in other layers and the training process; Step 6: obtain the generator network parameters trained in step 5, input the original underwater signal stripe pattern into the trained generator network, and the restored and enhanced underwater signal stripe pattern can be obtained.

2. The SAGAN-based streak tube lidar streak echo signal enhancement restoration method according to claim 1, characterized in that The specific steps of step 2 are as follows: Step 2-1: add reverberation noise and random Gaussian noise to the original ground signal stripe pattern; Step 2-2: add random attenuation to the original ground signal stripe pattern.

3. The SAGAN-based streak tube lidar streak echo signal enhancement restoration method of claim 1, wherein The main architecture of the generator network is a U-Net model, and a residual module and a self-attention module are introduced therein, the input of the generator network is a signal containing noise, the output is a false clear signal, the training target of the generator network is to generate a clear stripe signal from a stripe signal containing noise, and after the training of the generator network is completed, the real underwater target stripe signal is input to output a clear underwater target stripe signal.

4. The SAGAN-based streak tube lidar streak echo signal enhancement restoration method according to claim 3, characterized in that In the generator network, all normalization layers are instance normalization layers, the instance normalization layer normalizes each channel of each sample in the calculation process, and the specific calculation method of the instance normalization layer is as follows: (1) in, It is the input tensor In the The sample, the first The first channel, the first line, number Column elements; It is the first The sample, the first The average of each channel; It is the first The sample, the first The variance of each channel; It is the normalized scaling factor; It is the normalized offset; and It is a learnable parameter; It is a constant to prevent division by zero.

5. The SAGAN-based streak tube lidar streak echo signal enhancement restoration method according to claim 1 or 3, characterized in that The residual module is composed of two convolution layers, two normalization layers and a ReLU layer, the output size of the residual module is the same as the input size, and the output of the residual module is composed of the original input of the module and the calculation results of each layer network.

6. The SAGAN-based streak tube lidar streak echo signal enhancement restoration method of claim 3, wherein The main architecture of the discriminator network is a convolutional neural network model, a self-attention module is added therein, and spectral normalization is performed on the convolution layer.

7. The SAGAN-based streak tube lidar streak echo signal enhancement restoration method of claim 6, wherein LeakyReLU is used in the discriminator network, and its function form is as follows: wherein, is the negative half-axis slope.

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