Method for estimating atmospheric turbulence wavefront phase factor in long distance imaging

Abstract

The application provides a method for estimating atmospheric turbulence wavefront phase factors in long-distance imaging, directly constructs a mapping relationship model between an optical system detection image and a wavefront phase factor, and utilizes a trained deep neural network to learn the mapping relationship, realizes end-to-end output of a real part and an imaginary part of a wavefront phase factor corresponding to each frame of detection image, and thus completes estimation of the atmospheric degradation wavefront phase factor on a transmission path. The application expands the application category and research ideas in the field of turbulence medium light propagation, guarantees that the output of the double network model is not only the best fitting in the data space distribution, but also guarantees that the predicted phase factor is consistent with Kolmogorov turbulence statistics, namely, the scheme is data driven combined with physical constraints.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of photoelectric detection and imaging technology, and in particular to a wavefront phase factor estimation method. BACKGROUND

[0002] The performance of an optical system in a long-range imaging scene such as an airborne oblique range or a shipborne horizontal range is usually described by parameters such as signal-to-noise ratio, resolution, contrast, or is characterized by the effective range of the system. Among them, the imaging resolution is one of the main parameters indicating the performance of the system. After the detector model is determined, the atmospheric turbulence often becomes the key factor affecting the system resolution, and the imaging quality is largely limited by the distortion wavefront phase introduced by atmospheric turbulence. Fried (Fried D L.Optical resolution through a randomly inhomogeneous medium for very long and very short exposures[J].JOSA, 1966, 56(10): 1372-1379.) and others have made many in-depth studies on the optical system resolution under long exposure and short exposure atmospheric turbulence, established the basic imaging physical model, explored the statistical properties of the turbulence phase screen, and pointed out the basic research ideas and research directions for removing the influence of atmospheric turbulence after obtaining the long-range detection image.

[0003] Traditional atmospheric turbulence degradation image correction techniques mainly include phase recovery, phase extraction, blind deconvolution, wavefront deconvolution, phase difference, point spread function estimation, etc. Based on the imaging model and Fourier optical principles, the image de-turbulence technology is modeled as an inverse problem of the imaging process, and is iteratively optimized by adding physical constraints, or the clear image or the diffraction-limited point spread function is directly solved, and then non-blind deconvolution processing is performed. Or first estimate the atmospheric distortion wavefront phase from the detected degraded image, and then combine the corresponding relationship between the point spread function and the wavefront phase to perform non-blind deconvolution processing. It can be seen that the distortion wavefront phase, which is an intermediate variable, is a key physical quantity in the process of improving the imaging quality and resolution in a long-range scene. Therefore, it can also be said that the accuracy of the wavefront phase estimation directly affects the improvement of the effective range of the long-range imaging optical system.

[0004] The adverse factors or specific manifestations that affect the ability of atmospheric degradation wavefront phase estimation at present can include: arbitrariness of Zernike mode order selection, limited phase peak-valley value estimation range, manual intervention required for iteration stop, poor adaptability to different turbulence intensities, poor adaptability to differences in imaging background, etc. In recent years, with the development of deep learning, high-performance computing and other technologies, methods based on artificial intelligence have rapidly promoted the estimation of distorted wavefront phase. Deep neural networks can directly output the Zernike mode coefficients of the phase or the two-dimensional distribution of the wavefront phase. This type of method generally does not require a forward analytical model of the wavefront phase to the detected image, and can to some extent get rid of the constraints of the physical imaging model, but at the same time, this type of technology is generally based on the Zernike mode representation of the wavefront phase. The research focus is on how to improve the prediction accuracy of the Zernike coefficients. For example, the patent with the patent number CN202010075311.X and the name of "a Shack-Hartmann wavefront detector based on deep learning" uses a deep neural network to directly predict the Zernike polynomial coefficients of the degraded wavefront and then reconstructs the turbulence wavefront.

[0005] For actual shipborne horizontal or airborne oblique long-distance photoelectric detection and imaging scenes, there are many factors that limit the specific imaging environment such as small field of view, low light flux or changing turbulence. Direct estimation of the wavefront phase by a deep neural network will inevitably introduce two errors: one is the model error, which is caused by the modeling approximation ability of the selected network architecture for the mapping relationship between the phase and the observed image; the other is the truncation error, which is caused by the number of Zernike mode orders selected to represent the phase. In addition, to improve the scene adaptability of the estimated phase coefficient, an additional optical defocusing component or a preset device is generally required for the current deep learning method. In view of the defects of the above methods or the adverse factors of the dynamic changing scene, it is urgent to further expand the research methods and research ideas, or to improve the traditional wavefront phase estimation technology, or to construct a new deep neural network mapping model, and to further improve the prediction accuracy of the wavefront phase in long-distance imaging. SUMMARY

[0006] In order to overcome the shortcomings of the prior art, the present application provides a method for estimating atmospheric turbulence wavefront phase factor in long-distance imaging. The wavefront phase is a key physical quantity in the process of light propagation in a turbulent medium, however, more directly used in theoretical and applied research is the wavefront phase factor rather than the optical phase itself. Based on this, the present application considers converting the research on the extraction, prediction, estimation and other technologies of the wavefront phase in the traditional method into the research on the wavefront phase factor The study of estimation and statistical properties explores solutions for directly estimating wavefront phase factors from detection images acquired from optical systems.

[0007] To overcome the distortion of optical wavefront phase in existing long-distance imaging scenarios To address the shortcomings in the predictive power of this random variable, this invention proposes an atmospheric turbulence wavefront phase factor for long-range imaging. Estimation Method. This method directly constructs a mapping model between the optical system's detection image and the wavefront phase factor, and uses a trained deep neural network to learn this mapping relationship. It achieves end-to-end output of the real and imaginary parts of the wavefront phase factor corresponding to each frame of the detection image, thus completing the estimation of the atmospheric degradation wavefront phase factor on the transmission path.

[0008] The technical solution adopted by this invention to solve its technical problem specifically includes the following steps:

[0009] Step 1: Construct a pair of generative adversarial networks (PGANs). The network model represents the mapping relationship between the probe image and the real and imaginary parts of the wavefront phase factor, respectively. The pair of generative adversarial networks P... R and P I They are used to estimate the real part of the wavefront phase factor, respectively. and the virtual part During training, the parameters of the two networks are passed to each other alternately, while during prediction, they are independent of each other.

[0010] Step 2: Construct a paired dataset of "wavefront phase factor-degraded observation images". The clear images in the paired dataset are measured data collected in clear weather, and the degraded observation images in the paired dataset are phase screens based on the power spectrum inversion method. Simulation construction; Power spectral density function PSD n (k) Employing structural parameters that include atmospheric refractive index The expression, during horizontal scene simulation Typical values ​​are taken from the ground plane, and the slant path is simulated. Using an atmospheric structure parameter model that varies with altitude Wavefront phase factor in paired datasets The real and imaginary parts are directly derived from Build;

[0011] Step 3: Iteratively train and test the dual generative adversarial network designed in Step 1 on the paired dataset. Generator network GR of the domain and Domain Generator Network (GI) and its Corresponding Domain Discriminator Network (DR) and The discriminator networks (DI) of each domain are jointly iteratively optimized, with the following specific steps:

[0012] The dual generative adversarial network P R and P I is trained simultaneously, and the loss function L adds a wavefront phase factor constraint relationship, and domain and domain and the corresponding degraded observation image pair are synthesized as a training set input, a neural network method is used for training, until the training set and the test set converge, and a trained dual generative adversarial network model is obtained.

[0013] Step 4, verify the trained dual generative adversarial network model in step 3, first verify it by using a deep neural network, and secondly verify the generated wavefront phase factor by using the atmospheric turbulence statistical characteristics, and refer to the phase structure function of the turbulence phase screen simulation test method, using the phase factor structure function consistent with Kolmogorov statistics as a theoretical value, verify the consistency of the wavefront phase factor curve estimated by the model with the theoretical value, as shown in Figure 3 , if the consistency is poor, re-optimize the network structure in step 1, and the judgment of the consistency can be based on the subjective judgment of the human eye on the closeness of the two curves.

[0014] At this point, the estimation of the atmospheric turbulence wavefront phase factor in long-distance imaging is completed, and when used, the collected single-frame observation image is input into the trained domain generator network GR and domain generator network GI, and the output is the real part and the imaginary part of the wavefront phase factor corresponding to the image.

[0015] In the process of constructing the dual generative adversarial network PGAN model in step 1, the generator networks GR and GI both use Unet or ResNet architecture, and the specific design of the loss function L is as follows:

[0016]

[0017] In the formula: respectively represent the adversarial loss function of the model PGAN discriminant network DR, DI;

[0018] respectively represent the content loss function of the generation network GR, GI; the design of the third term represents the constraint relationship of the real part and the imaginary part determined by the definition of the wavefront phase factor; γ1 and γ2 are regularization parameters of the balance weight.

[0019] In the process of constructing the "wavefront phase factor-degraded observation image" pair data set in step 2, the phase screen simulation is performed by using a power spectral density function PSD n (k) using a modified Von Karman spectrum model containing atmospheric refractive index structure parameters ,

[0020]

[0021] where k is the spatial frequency, the turbulence inner scale k m = 5.92 / l0; the turbulence outer scale k0 = 2π / L0; l0 is the turbulence inner scale, L0 is the turbulence outer scale, for a far distance terrestrial imaging scene, the atmospheric refractive index structure parameters take the typical value of the ground C0 = 1.7 × 10 -14 m -2 / 3 ; for an oblique range imaging scene, using the parameter model varying with height proposed by the International Telecommunication Union

[0022]

[0023] where h is the altitude (m); v RMS is the vertical path root mean square wind speed, e is the natural constant.

[0024] In step 3, when the dual generative adversarial network PGAN is jointly trained, the two networks P R and P I are alternately performed, and the estimated value of the network output output by each iteration is transmitted. and When the network P R is alternately trained, the loss function of the network P R degenerates into the following expression:

[0025]

[0026] Similarly, the loss function of the network P I degenerates into the following expression:

[0027]

[0028] In step 4, when the wavefront phase factor generated by the dual network model is verified for atmospheric turbulence statistical characteristics, the expression of the theoretical value of the phase factor structure function conforming to the Kolmogorov statistics is:

[0029]

[0030] where: To conform to the theoretical phase structure function of Kolmogorov statistics, r is the spatial interval vector.

[0031] An electronic device includes: one or more processors; a memory; and one or more programs, wherein the one or more programs are stored in the memory and configured to be executed by the one or more processors, and the one or more programs are configured to perform the methods described above.

[0032] A computer-readable storage medium storing program code that can be invoked by a processor to perform the method described above.

[0033] The beneficial effects of this invention are as follows:

[0034] (1) An alternative scheme is designed to extract wavefront information of atmospheric turbulence distortion carried in deteriorated observation images obtained from long-distance imaging. This scheme estimates the real and imaginary parts of the wavefront phase factor from the observation image, rather than the wavefront phase itself. This is because the wavefront phase factor is more directly used in theoretical and applied research. right Direct predictions are made, and theoretical analysis and simulations are performed on its statistical characteristics, which expands the application scope and research ideas in the field of light propagation in turbulent media.

[0035] (2) Due to wavefront phase factor The real and imaginary parts of the t_t are bounded in the range [-1, 1], making them well-suited for being considered as two-dimensional spatially distributed data. This allows them to be input into deep neural networks for modeling and prediction, with the model applicable to turbulent environments of varying intensities. For distorted wavefront phases... Due to the influence of the intensity of turbulence, its value range is dynamically and randomly changing, making it unsuitable as a direct input to deep networks and unable to be directly predicted. Indirect prediction is required using methods such as Zernike model representation, but this does not avoid the error in the selection of the model order.

[0036] (3) The statistical characteristic verification method based on the theoretical value of the wavefront phase factor structure function ensures that the output of the dual network model is not only the best fit in the data space distribution, but also that the predicted phase factor conforms to Kolmogorov turbulence statistics. That is, the scheme is data-driven combined with physical constraints. Attached Figure Description

[0037] Figure 1 This invention relates to the phase factor real part prediction network P. R right An estimated single-frame illustration, Figure 1 (a) is a schematic diagram of the predicted real part. Figure 1 (b) is a schematic diagram of the real part reference value.

[0038] Figure 2 is a real part prediction network P I to a schematic diagram of an estimated frame, Figure 2 (a) is a schematic diagram of a predicted value of the imaginary part, Figure 2 (b) is a schematic diagram of a reference value of the imaginary part.

[0039] Figure 3 a schematic diagram of the degree of coincidence of the output of the phase factor structure function with the theoretical value DETAILED DESCRIPTION

[0040] The application will be further described below in conjunction with the accompanying drawings and examples.

[0041] The hardware environment used to implement the scheme in the example: CPU, Core i5-13600KF, 14 cores, 20 threads, 3.5GHz, 24M cache; memory, 16G, 3200DDR4; GPU, XFX RTX4070ti S, 16G graphics card. The software environment used to implement the scheme in the example: integrated development environment, PyCharm 2024.2.3; operating system, Microsoft Windows 11.

[0042] A method for estimating atmospheric turbulence wavefront phase factors in long-distance imaging, characterized in that it comprises the following steps:

[0043] Step 1, constructing a dual generative adversarial network (PGAN) model for mapping the relationship between the probe image and the real and imaginary parts of the wavefront phase factor, the dual network P R ,P I are used to simultaneously extract the real and imaginary parts of the wavefront phase factor from the observed degraded image, respectively and During training, the parameters of the dual networks are alternately transmitted and predicted independently during prediction.

[0044] In the dual generative adversarial network PGAN model, P R and P IThe generator network GR and GI both adopt the ResNet101 network architecture. The first layer is a 7*7 convolutional layer; then there are four stages, each of which contains several residual blocks; followed by a global average pooling layer and a fully connected layer, which is used to expand the output of the pooling layer into a vector and map it to the dimension of the number of categories. Each residual block of ResNet101 consists of two 3*3 convolutional layers, each followed by batch normalization (Batch Normalization) and a ReLU activation function, and there is also batch normalization and a ReLU activation function between the residual blocks, but no convolutional layer. The first residual block of each stage uses a 1*1 convolutional layer to convert the number of input channels to the number of output channels, so as to be added to the subsequent residual blocks.

[0045] P R and P I The discriminator network DR and DI adopts the conventional structure design of the discriminator in the generative adversarial network.

[0046] The loss function of the dual generative adversarial network PGAN model is specifically designed as follows:

[0047]

[0048] In the formula: respectively represent the adversarial loss function of the model PGAN discriminator network DR and DI; respectively represent the content loss function of the generator network GR and GI; the design of the third term indicates that the constraint relationship between the real part and the imaginary part determined by the wavefront phase factor is added; γ1 and γ2 are regularization parameters for balancing the weights of the three parts.

[0049] Take P R , the adversarial loss of the discriminator network DR is defined using the Wasserstein distance with gradient penalty,

[0050]

[0051] In the formula, represents the estimation of the real part of the atmospheric degradation wavefront phase factor output by the generator network each time, and i(x) is the degraded observation image input into the network. The content loss of the generator network GR only uses the pixel content loss

[0052]

[0053] In the formula, o(x) represents the clear real scene image corresponding to the atmospheric degradation observation image. P IThe loss function of the network is defined as P R The network is similar. Two networks are trained simultaneously, which improves efficiency and fully utilizes mathematical constraint relationships.

[0054] Step 2, construct the "wavefront phase factor-degraded observation image" paired data set. The clear image of the scene is approximated by the measured data collected when the atmospheric influence is weak. In the embodiment, 10 scenes including road, grassland, sandy land, forest and typical targets are collected. The degraded observation image in the data set is based on the phase screen Simulation construction, the turbulence degradation imaging model is i(x) = o(x) x h(x) + n(x), h(x) is the degradation point spread function, and the corresponding relationship between h(x) and the phase screen is n(x) is random noise; here x represents two-dimensional convolution operation.

[0055] The power spectral density function PSD n (k) adopts a modified Von Karman spectrum model containing atmospheric refractive index structure parameters ,

[0056]

[0057] wherein: k m = 5.92 / l0; k0 = 2π / L0; l0 and L0 are the inner and outer scales of turbulence respectively. For a long-range horizontal imaging scene, the atmospheric refractive index structure parameter takes the typical value C0 = 1.7*10 -14 m -2 / 3 near the ground; for a slant range imaging scene, the parameter model varying with height proposed by the International Telecommunication Union is adopted

[0058]

[0059] wherein: h is the height above the ground; v RMS is the vertical path root mean square wind speed, which is related to the near-ground wind speed v g , and here v

[0060] According to the above simulation method, 2K phase screen data are simulated for horizontal and slant range scenes respectively, which are simulated to be added to 10 clear images of the scene to construct a paired data set of 40,000 degraded images and 10 clear images, and the paired data set is divided into a training set, a test set and a verification set according to the conventional processing method.

[0061] Step 3, on the data set constructed in step 2, the double generative adversarial network PR and P I Iterative alternating training and testing are performed until convergence.

[0062] The predicted domain and The generator network GR, GI and the discriminator network DR, DI of the domain data are jointly iteratively optimized, which is equivalent to training two networks P R and P I are trained simultaneously and alternately, and the loss function adds a constraint relationship of the wavefront phase factor , so that the and The domain and The degraded observation image pairs corresponding to the domain are synthesized as a training set input until the training set and the test set converge.

[0063] The loss function of the network P R degenerates to the following expression during alternating training:

[0064]

[0065] Similarly, the loss function of the network P I degenerates to the following expression:

[0066]

[0067] In the embodiment, the parameter γ1 is 300, the parameter γ2 is 0.05, and the three loss functions have all converged after about 600 times of alternating iteration training.

[0068] Step 4, verifying the trained double generative adversarial network P R and P I of step 3. In addition to the verification method commonly used for generative adversarial network models, the embodiment mainly verifies the atmospheric turbulence statistical characteristics of the generated wavefront phase factor. Referring to the phase structure function inspection method of turbulence phase screen simulation, the phase factor structure function consistent with Kolmogorov statistics is used as a theoretical value to verify the degree of conformity of the wavefront phase factor curve estimated by the model, and if the conformity is poor, the network structure in step 1 needs to be re-optimized and designed. The embodiment assumes that the atmospheric turbulence wavefront phase factor is homogeneous in a statistical sense, and thus defines the expression of the phase factor structure function theoretical value consistent with Kolmogorov statistics:

[0069]

[0070] In the formula, is the theoretical phase structure function according to Kolmogorov statistics, r = x' - x is a spatial interval vector, and r0 is the atmospheric coherence length defined by Fried constant. The phase factor structure function is specifically derived as follows:

[0071] First, the phase factor covariance function is derived. Assuming that the wavefront phase factor is statistically homogeneous, the covariance of random variables is defined as where <·> represents ensemble average, is the phase spatial interval change. If p(α) is the probability density function of α(r), based on the statistical characteristics of , α(r) is a zero-mean Gaussian random variable, that is

[0072]

[0073] In the formula, σ 2 is the variance of α(r),

[0074] Thus, according to the definition of , there is Substitute the expression of p(α) into the definition of , combine the duality of p(α) and Euler formula, and calculate the integral to obtain the expression of the phase factor covariance function as

[0075]

[0076] At this point, according to the relationship between the random variable structure function and the covariance function , the theoretical expression of the phase factor structure function according to Kolmogorov statistics can be obtained,

[0077]

[0078] The phase factor structure function predicted by the double network model needs to be checked for compliance with .

[0079] At this point, the embodiment completes the estimation of the atmospheric turbulence wavefront phase factor in long-distance imaging. The above description is only part of some embodiments of the present technology, not all embodiments. Based on the embodiments in the present application, all other embodiments obtained by those of ordinary skill in the art without creative labor should be within the scope of protection of the present application.

Claims

1. A method for estimation of atmospheric turbulence wavefront phase factor in long distance imaging, characterized in that The method comprises the following steps: Step 1, constructing a double generative adversarial network model, the network model is the mapping relationship of the probe image with the real part of the wavefront phase factor and the imaginary part of the wavefront phase factor, double generative adversarial network P R and P I are respectively used to estimate the real part and the imaginary part of the wavefront phase factor The double network parameters are alternately transmitted to each other during training, and are independent of each other during prediction. Step 2: Construct a paired dataset of "wavefront phase factor-deterioration observation images". The clear images in the paired dataset are measured data collected in clear weather, and the deterioration observation images in the paired dataset are phase screens based on the power spectrum inversion method. Simulation construction; Power spectral density function PSD n (k) Employing structural parameters that include atmospheric refractive index The expression, during horizontal scene simulation Typical values ​​are taken from the ground plane, and the slant path is simulated. Using an atmospheric structure parameter model that varies with altitude Wavefront phase factor in paired datasets The real and imaginary parts are directly derived from Build; Step 3, iteratively alternating training and testing the dual generative adversarial network designed in step 1 on the paired dataset, and obtaining the generator network GR of the source domain and the generator network GI of the target domain and the corresponding the discriminator network DR of the source domain and the discriminator network DI of the target domain are jointly iteratively optimized, and the specific steps are as follows: The dual generative adversarial network P R and P I are trained simultaneously, and a loss function L is added with a wavefront phase factor A constraint relationship is added to the loss function L, and the domain and domain and the corresponding degraded observation image pairs are synthesized as a training set input, a neural network method is used for training, until the training set and the test set converge, and a trained dual generative adversarial network model is obtained. Step 4, the trained dual generative adversarial network model of step 3 is verified, first, the deep neural network is verified, and then the generated wavefront phase factor is verified for atmospheric turbulence statistical characteristics, referring to the phase structure function of turbulence phase screen simulation Verification method, the phase factor structure function conforming to Kolmogorov statistics is used As a theoretical value, the wavefront phase factor estimated by the model is verified The degree of coincidence of the curve, if the coincidence is poor, the network structure in step 1 is re-optimized and designed, and the degree of coincidence can be judged according to the subjective judgment of the human eye on the closeness of the two curves; This completes the estimation of the atmospheric turbulence wavefront phase factor in long-range imaging. When using it, the acquired single-frame observation images are input into the trained... Domain Generator Network GR and The generator network GI of the domain outputs the real and imaginary parts of the wavefront phase factor corresponding to the frame image.

2. The atmospheric turbulence wavefront phase factor estimation method in long-distance imaging according to claim 1, characterized in that: In the process of constructing the dual generative adversarial network PGAN model in step 1, the generator networks GR and GI both adopt Unet or ResNet architecture, and the specific design of the loss function L is as follows: In the formula: respectively represent the adversarial loss functions of the model PGAN discriminant network DR and DI. respectively denote the content loss functions of the generator networks GR, GI; the design of the third term indicates that the constraint relationship between the real part and the imaginary part determined by the wavefront phase factor definition is added; γ1 and γ2 are the regularization parameters of the balance weight.

3. The atmospheric turbulence wavefront phase factor estimation method in long-distance imaging according to claim 1, characterized in that: In the process of constructing the "wavefront phase factor-degraded observation image" pair dataset in step 2, the phase screen simulation is performed by using a power spectral density function PSD n (k) using a modified Von Karman spectrum model containing atmospheric refractive index structure parameters , where k is the spatial frequency, k0is the inner scale of turbulence, and L0is the outer scale of turbulence m = 5.92 / 10; k0= 2π / L0; l0is the inner scale of turbulence, L0is the outer scale of turbulence, for far distance terrestrial imaging scene, the atmospheric refractive index structure parameter Take the typical value of the ground plane C0= 1.7 x 10 -14 m -2 / 3 ; for oblique range imaging scene, The model of the parameter changing with height proposed by the International Telecommunication Union is adopted where: h is the altitude; v RMS is the root mean square wind speed for the vertical path, and e is the natural constant.

4. The atmospheric turbulence wavefront phase factor estimation method in long-distance imaging according to claim 2, characterized in that: The step 3 is to jointly train the two generative adversarial networks PGAN, wherein the two networks P R and P I are alternately performed, and the estimated values of the network outputs outputted by respective iterations are alternately transmitted. and When the two networks are alternately trained, the loss function of the network P R degenerates into the following expression: Similarly, the P I The loss function degenerates to the following expression:

5. The atmospheric turbulence wavefront phase factor estimation method in long-distance imaging according to claim 1, characterized in that: The expression of the theoretical value of the phase factor structure function in accordance with the Kolmogorov statistics when verifying the atmospheric turbulence statistical characteristics of the wavefront phase factor generated by the double-network model is: The expression is: In the formula: is the theoretical phase structure function according to the Kolmogorov statistics, and r is a spatial interval vector.

6. An electronic device, comprising: comprising: one or more processors; a memory; one or more programs, wherein the one or more programs are stored in the memory and configured to be executed by the one or more processors, and the one or more programs are configured to execute the method according to any one of claims 1-5.

7. A computer readable storage medium characterized in that, The computer readable storage medium stores program code, and the program code can be called and executed by the processor to execute the method according to any one of claims 1-5.

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