Tree Internal Defected Imaging Using Model-Driven Deep Learning Network

Abstract

The health of trees has become an important issue in forestry. How to detect the health of trees quickly and accurately has become a key area of research for scholars in the world. In this paper, a living tree internal defect detection model is established and analyzed using model-driven theory, where the theoretical fundamentals and implementations of the algorithm are clarified. The location information of the defects inside the trees is obtained by setting a relative permittivity matrix. The data-driven inversion algorithm is realized using a model-driven algorithm that is used to optimize the deep convolutional neural network, which combines the advantages of model-driven algorithms and data-driven algorithms. The results of the comparison inversion algorithms, the BP neural network inversion algorithm, and the model-driven deep learning network inversion algorithm, are analyzed through simulations. The results shown that the model-driven deep learning network inversion algorithm maintains a detection accuracy of more than 90% for single defects or homogeneous double defects, while it can still have a detection accuracy of 78.3% for heterogeneous multiple defects. In the simulations, the single defect detection time of the model-driven deep learning network inversion algorithm is kept within 0.1 s. Additionally, the proposed method overcomes the high nonlinearity and ill-posedness electromagnetic inverse scattering and reduces the time cost and computational complexity of detecting internal defects in trees. The results show that resolution and accuracy are improved in the inversion image for detecting the internal defects of trees.

1. Introduction

As a renewable resource, wood is widely used in construction, decoration, energy, and other fields [1]. When defects, such as voids and decay, occur in the trunk due to various natural factors, and its characteristics, not only will the quality of the wood products not meet standards, but the tree may even collapse in severe cases [2]. The detection of living trees can prevent the influence of various unfavorable factors in time, reduce unnecessary waste and make full use of forest resources [3,4,5].

For the detection of internal defects of living trees, the current mainstream methods include the stress wave method, ultrasound method, and computer tomography (CT) scan [6,7,8]. However, most methods have their corresponding shortcomings [9,10]. For example, the stress wave method must drive nails into each measurement point on the trunk due to its detection characteristics; tree needle detection also requires probes to be drilled into the trunk [11]. Both detection methods will cause damage to the tree and cannot be defined as non-destructive testing. The ultrasonic detection process is susceptible to interference from the external environment, and the use of coupling agents may cause environmental pollution [12]. The cost of CT equipment is relatively expensive, and it is easy to cause radiation hazards to researchers in terms of safety [13,14]. Compared with other non-destructive testing technologies applied in the forestry field, electromagnetic waves have received great attention because of their fast, high-efficiency, easy-to-operate, non-susceptible external interference, and the ability to achieve non-intrusive and non-destructive testing [15,16,17]. With the substantial improvement of computer performance, some researchers have developed progressive algorithms to identify defects in common wood by means of a BP neural network and a convolution neural network, which improves the detection accuracy and efficiency [18].

In this article, we analyzed the contrast source inversion (CSI) and the neural network algorithm and proposed a model-driven deep learning network inversion algorithm to conduct simulation experiments on the detection of internal defects in trees. The CSI, BP neural network algorithm and the model-driven deep learning network inversion algorithm are compared and analyzed. The results show that the model-driven deep learning network inversion algorithm improves the defect inversion imaging rate and image quality. The main work of this paper is as follows:

• The objective function of the comparison source inversion is obtained by using the Lippmann–Schwinger equation and the equivalent current source radiation process of the scattering field. The models of comparison; source inversion algorithm, BP neural network inversion algorithm, and model-driven inversion algorithm based on deep learning networks, are established;
• Determine the simulated imaging evaluation metrics; on the basis of building simulation environment and training database, the inversion imaging of single defect, homogeneous double defect and heterogeneous multi-defect is realized, and the algorithm iterative stability is analyzed.

The main physical variables involved in this article and their meanings are shown in

Table 1. The main physical variables and meanings involved in this paper.

Physical Variable	Interpretation	Physical Variable	Interpretation
$E^t$	Total field of electric field	$E^i$	Incident field of electric field
$E^s$	Scattering field of electric field	$\omega$	Angular frequency
${\mu }_0$	Air-permeability	${\epsilon }_0$	Air-permittivity
${\epsilon }_r$	Relative permittivity	$k_0$	Wavenumber
$J$	Contrast source	$\chi$	Contrast factor
$\eta$	Normalization parameter

.

2. Method

Trees are composed of materials with different dielectric properties. The dielectric properties of trunks and defects are obviously different. Electromagnetic waves will produce reflections when they pass through trunks and defects. By setting the dielectric properties of different tree tissues, the tree model can be simulated when the detection system is faced with real trees; therefore, the simulation results will be expected. Data-driven algorithms have been widely applied in the field of artificial intelligence, but their applications in tree defect detection are less. The combination of a model-driven algorithm based on contrast source and a data-driven algorithm are the main work and innovation of this paper.

In this paper, aiming at the forward process of electromagnetic wave propagation, the theoretical equations of scattering and the basic model of scatterers are constructed through Maxwell’s equations and then the parameter optimization for solving integral equations are obtained. Based on the research and analysis of model-driven and data-driven inversion algorithms, the model-driven deep learning network inversion algorithm is established [19,20]. The workflow chart is shown as Figure 1.

2.1. Scattering Problem

The inverse scattering imaging model in this paper is shown in Figure 2, where the incident wave is mainly a two-dimensional plane wave, and the forward process of the electromagnetic wave includes the following two equations. The first is that the interaction form of the wave scatterer satisfies the Lippmann–Schwinger equation [21]:

$$E^t\left(r\right)=E^i\left(r\right)+i\omega {\mu }_0{\int }_{D}g\left(r,r^{\prime }\right)\left[-i\omega {\epsilon }_0\left({\epsilon }_r\left(r^{\prime }\right)-1\right)E^t\left(r^{\prime }\right)\right]d{r}^{\prime },r\in D$$

(1)

The second is the equivalent current source radiation process of the scattered field:

$$E^s\left(r\right)=i\omega {\mu }_0{\int }_{D}g\left(r,r^{\prime }\right)\left[-i\omega {\epsilon }_0\left({\epsilon }_r\left(r^{\prime }\right)-1\right)E^t\left(r^{\prime }\right)\right]d{r}^{\prime },r\in S$$

(2)

where $\mathit{\omega }=2\mathit{\pi }f$ is the angular frequency, ${\mu }_0$ is air-permeability, $g\left(r,r^{\prime }\right)$ is two-dimensional Green’s function, ${\epsilon }_0$ is air-permittivity, and ${\epsilon }_r\left(r^{\prime }\right)$ is relative permittivity in the electromagnetic detection model. In Equation (2), $-i\omega {\epsilon }_0\left({\epsilon }_r\left(r\right)-1\right)E^t\left(r\right)$ represents the induced contrast current density in a physical sense, where we define $J\left(r\right)=\left({\epsilon }_r\left(r\right)-1\right)E^t\left(r\right)$ as the standardized contrast current density. $k_0=\omega \sqrt{{\mu }_0{\epsilon }_0}$ is the wavenumber of the homogeneous background medium, and${\epsilon }_r\left(r\right)-1$ reflects the contrast between xylem and defects $\chi \left(r\right)$.

$$k_0^2{\int }_{D}g\left(r,r^{\prime }\right)J\left(r^{\prime }\right)d{r}^{\prime }=\{\begin{array}{c}|G_S\left(J\right),r\in S\\ |G_D\left(J\right),r\in D\end{array}$$

(3)

Then the governing equation can be written in two incompatible forms. The first type is called field equations, where the electric field involves two equations:

$$E^t\left(r\right)=E^i\left(r\right)+G_D\left(\chi E^t\right),r\in D$$

(4)

$$E^s\left(r\right)=G_S\left(\chi E^t\right),r\in S$$

(5)

The second type is called the source equation, where the current source involves two equations:

$$J\left(r\right)=\chi \left(r\right)\left[E^i\left(r\right)+G_D\left(J\right)\right],r\in D$$

(6)

$$E^s\left(r\right)=G_S\left(J\right),r\in S$$

(7)

The core of the CSI is to process the source integral Equations (6) and (7). The inverse problem is interpreted as the minimization of the objective function, which is a linear combination of the normalized data equation and the normalized state equation. The objective function formula is as follows:

$$F\left(J_j,\chi ,\beta \right)=\beta \frac{{\sum }_j\Vert \chi E_j^i-J_j+\chi G_D{J}_j{\Vert }_D^2}{{\sum }_j\Vert \chi E_j^i{\Vert }_D^2}+\frac{{\sum }_j\Vert E_j^s-G_S{J}_j{\Vert }_S^2}{{\sum }_j\Vert E_j^s{\Vert }_S^2}$$

(8)

where $\Vert \chi E_j^i-J_j+\chi G_D{J}_j{\Vert }_D^2$ is the $\Vert ·{\Vert }_D$ measure of the right side of the equation minus the left end of the equation in Equation (7), and $\Vert E_j^s-G_S{J}_j{\Vert }_S^2$ is the $\Vert ·{\Vert }_S$ measure of the right side of the equation minus the left residual of the Equation (6).

The CSI uses the conjugate gradient descent method to control the iterative process. The contrast source J_j and the contrast $\chi$ are updated alternately, and the output value of the objective function is gradually reduced and minimized to reach the exact solution.

2.2. Contrast Source Inversion (CSI) Method

The CSI method is a model-driven algorithm which combines contrast and total field to create a new contrast source and uses the conjugate gradient method to update the contrast and contrast source to minimize the objective function and complete the electrical parameters of the target medium to achieve inversion imaging. The core point of the CSI is to use the contrast and the total field $E_j^t$ to form the contrast induction current $J_j=\chi E_j^t$. In the solution process, $E_j^t$ is not solved directly, and the solution is solved iteratively by alternately updating the CSI $J_j$ and $\chi$ to continuously approximate the solution of the integral equation.

The CSI approach mainly utilizes the source-type Lippmann–Schwinger equation:

$$J\left(r\right)=\chi \left(r\right)\left[E^i\left(r\right)+G_D\left(J\right)\right],r\in D$$

(9)

$$E^s\left(r\right)=G_S\left(J\right),r\in S$$

(10)

Equation (10) is a pathological equation and requires a priori information to recover its uniqueness in the practical solution.

The main idea of the iterative solution is to construct the objective function. The objective function is a normalized measure of the error value on both sides of the equal sign of the source-type equation. The CSI transforms the solution of the problem into a minimization objective function that approximates the exact solution [22]. Again, the objective function of the CSI is defined as:

$$F\left(J_j,\chi ,\beta \right)=\beta \frac{{\sum }_j\Vert \chi E_j^i-J_j+\chi G_D{J}_j{\Vert }_D^2}{{\sum }_j\Vert \chi E_j^i{\Vert }_D^2}+\frac{{\sum }_j\Vert E_j^s-G_S{J}_j{\Vert }_S^2}{{\sum }_j\Vert E_j^s{\Vert }_S^2}$$

(11)

The update scheme for contrast ${\chi }_n$ is:

$${\chi }_n={\chi }_{n-1}+{\alpha }_n^{\chi }{g}_n^{\chi },n\ge 1$$

(12)

$$g_n^{\chi }=-{\eta }_{D,n-1}\frac{{\sum }_j\left({\chi }_{n-1}{E}_{j,n}^t-{\omega }_{j,n}\right)\overline{E_{j,n}^t}}{{\sum }_j{\left|E_{j,n}^t\right|}^2}$$

(13)

where ${\eta }_{D,n-1}$ is the normalization parameter. To keep the contrast consistent with the CSI updating, Equation (12) is rewritten as:

$${\chi }_n={\chi }_{n-1}+{\alpha }_n^{\chi }{d}_n,n\ge 1$$

(14)

$$d_n=g_n^{\chi }+{\gamma }_n^{\chi }{d}_{n-1},d_0=0,n\ge 1$$

(15)

where ${\gamma }_n^{\chi }$ is:

$${\gamma }_n^{\chi }=\frac{\mathit{Re}\langle g_n^{\chi },g_n^{\chi }-g_{n-1}^{\chi }{\rangle }_D}{\langle g_{n-1}^{\chi },g_{n-1}^{\chi }{\rangle }_D}$$

(16)

Replacing ${\chi }_n$ in the second term of the objective function, we have:

$$F_{D,n}=\frac{{\sum }_j\Vert {\chi }_n{E}_{j,n}^t-J_{j,n}{\Vert }_D^2}{{\sum }_j\Vert {\chi }_n{E}_{j,n}^t{\Vert }_D^2}=\frac{{\sum }_j\Vert \left({\chi }_{n-1}+{\alpha }_n^{\chi }{d}_n\right)E_{j,n}^t-J_{j,n}{\Vert }_D^2}{{\sum }_j\Vert \left({\chi }_{n-1}+{\alpha }_n^{\chi }{d}_n\right)E_{j,n}^t{\Vert }_D^2}$$

(17)

In the minimizing Equation (17), we have:

$${\alpha }_n^{\chi }=\frac{-\left(aC-Ac+\sqrt{{(aC-Ac)}^2-4\left(aB-Ab\right)(bC-Bc}\right)}{2\left(aB-Ab\right)}$$

(18)

2.3. BP Neural Network Inversion Algorithm

The core purpose of the neural network inversion method is to gain accurate detection results by training a neural network model and gradually fitting the relationship between the input and output data during the training iterations [23,24].

$b_j^l$ denotes the bias of the $j^{th}$ neuron in the $l^{th}$ layer, and $a_j^l$ denotes the activation value of the $j^{th}$ neuron in the $l^{th}$ layer. We have:

$$a_j^l=\sigma \left({\sum }_k{\omega }_{jk}^l{a}_k^{l-1}+b_j^l\right),$$

(19)

where the summation is performed over all k neurons in the ${(l-1)}^{th}$-layer and ${\omega }^l$ is the weight matrix on the l-layer, where $\sigma$ is expressed as:

$$\sigma \left(\omega \cdot x+b\right)\equiv \frac{1}{1+\mathit{exp}\left(-\omega x-b\right)}$$

(20)

Thus, Equation (19) can be rewritten in matrix form as:

$$a_j^l=\sigma \left({\omega }^l{a}^{l-1}+b^l\right)$$

(21)

The intermediate quantity z^l is often used in the calculation process to simplify the form of the calculation:

$$z^l={\omega }^l{a}^{l-1}+b^l$$

(22)

$z^l$ is the weighted input of the l-layer neuron.

The intermediate quantity ${\delta }_j^l$ is the error of $j^{th}$ neuron on the $l^{th}$ layer:

$${\delta }_j^l\equiv \frac{\partial C}{\partial z_j^l}$$

(23)

The backward propagation operation from the output layer is called backpropagation, and ${\delta }^L$ denotes the output layer error, which is obtained using Equations (19)–(23):

$${\delta }^L\equiv \frac{\partial C}{\partial a_j^L}\sigma \prime \left(z_j^L\right)$$

(24)

Rewriting Equation (24) in matrix form, we achieve:

$${\delta }^L={\mathit{𝛻}}_{a}C\odot \sigma \prime \left(z^L\right)$$

(25)

$${\delta }^L=\left(a^L-y\right)\odot \sigma \prime \left(z^L\right)$$

(26)

The rate of change of bias and weights in the substitution function is:

$$\frac{\partial C}{\partial b_j^l}={\delta }_j^l$$

(27)

$$\frac{\partial C}{\partial {\omega }_{jk}^l}=a_k^{l-1}{\delta }_j^l$$

(28)

2.4. Model-Driven Inversion Algorithm Based on Deep Learning Networks

CSI methods, including model-driven algorithms, are highly dependent on accurate mathematical models. However, due to the highly ill-posed nature of the electromagnetic wave inverse scattering problem, if the detection environment changes greatly, it is very likely that the detection results of the CSI will be inaccurate [25]. A new parameter setting and modeling for the changed environment is required. This feature increases the time cost and computational complexity of the CSI and reduces the scope of application. The field standing wood inspection environment is complex and variable, and many parameters fixed in the simulation experiment are changed in the actual measurement process. For example, the relative dielectric constants of wood with different moisture contents are different, and the relative dielectric constants of different types of defects are also different. Therefore, the traditional CSI algorithm cannot adapt to the requirements of modern defect detection with fast, accurate, and strong generalization ability.

To overcome the drawbacks of model-driven algorithms represented by comparison sources, some data-driven algorithms are introduced [26]. Data-driven algorithms are represented by neural networks, among which deep convolutional neural networks have the strongest ability to extract and classify data features and have also been a popular research topic in recent years [27,28]. Neural networks can better fit nonlinear problems by learning from a large amount of data, overcoming discomfort, improving the detection speed and generalization ability of the detection system, and enabling fast and accurate defect detection, which cannot be achieved via traditional model-driven algorithms. However, the selection of the neural network structure and the parameter optimization process is dependent on the researcher’s experience [29,30]. Herein, the CSI is introduced to combine the advantages of model-driven and data-driven algorithms to form the model-driven deep learning network.

The structure of the deep convolutional neural network used for constructing new algorithms is shown in Figure 3.

The sensory field size in the network is 5 × 5, and the feature data are pooled in the pooling layer to perform a 2 × 2 pooling operation, and finally, a 100 × 100 regression value is obtained for imaging through the fully connected layer. The cost function is a quadratic cost function with following form:

$$C\left(\omega ,b\right)\equiv \frac{1}{2n}{\sum }_x\Vert y\left(x\right)-a\Vert .$$

(29)

To improve the defect detection speed of the algorithm and the network learning rate, cross-entropy cost function is used and expressed as [31]:

$$C=-\frac{1}{n}{\sum }_x\left[y\ln a+\left(1-y\right)\ln (1-a\right)].$$

(30)

Bringing $a=\sigma \left(z\right)$ into Equation (30), we achieve:

$$\frac{\partial C}{\partial {\omega }_j}=-\frac{1}{n}{\sum }_x\left(\frac{y}{\sigma \left(z\right)}-\frac{1-y}{1-\sigma \left(z\right)}\right)\frac{\partial \sigma }{\partial {\omega }_j}=-\frac{1}{n}{\sum }_x\left(\frac{y}{\sigma \left(z\right)}-\frac{1-y}{1-\sigma \left(z\right)}\right){\sigma }^{\prime }\left(z\right)x_{j }$$

(31)

Taking $\sigma \prime \left(z\right)=\sigma \left(z\right)\left(1-\sigma \left(z\right)\right)$ into Equation (31) gives:

$$\frac{\partial C}{\partial {\omega }_j}=\frac{1}{n}{\sum }_x{x}_j\left(\sigma \left(z\right)-y\right).$$

(32)

Similarly, the bias derivative is obtained as:

$$\frac{\partial C}{\partial b}=\frac{1}{n}{\sum }_x\left(\sigma \left(z\right)-y\right).$$

(33)

The cross-entropy cost function eliminates $\sigma \prime \left(z\right)$ from the bias derivatives of weights and biases using intermediate quantities so that it can avoid the slow learning process associated with too small $\sigma \prime \left(z\right)$ values [32].

In this paper, to speed up the training process of deep convolutional neural networks as well as to optimize the hyperparameter selection, the CSI combines with the deep convolutional neural network to obtain a new cost function. The core equation of CSI minimizes the objective function, which takes the form of:

$$F\left(J_j,\chi ,\beta \right)=\beta \frac{{\sum }_j\Vert \chi E_j^i-J_j+\chi G_D{J}_j{\Vert }_D^2}{{\sum }_j\Vert \chi E_j^i{\Vert }_D^2}+\frac{{\sum }_j\Vert E_j^s-G_S{J}_j{\Vert }_S^2}{{\sum }_j\Vert E_j^s{\Vert }_S^2}$$

(34)

The objective function of the CSI is normalized and minimized, and the DCNN is obtained by minimizing the cost function to obtain the appropriate weights and biases, the normalization process of the CSI is introduced into the DCNN, and the value of the hyperparameter $\lambda$ is determined by the information of the CSI than the validation set with the researcher′s experience. Formally the value of $\lambda$ is changed from a constant value to a dynamic real value determined by the CSI, forming the model-driven deep learning network. Thus, the model-driven deep learning network cost function takes the form of:

$$C=-\frac{1}{n}{\sum }_{x_j}\frac{\left[y_j\ln a_j^L+\left(1-y_j\right)\ln (1-a_j^L\right)]}{{\sum }_j{\sum }_k\Vert E_k^s{\Vert }_s^2}+\frac{{\lambda }_0}{2n{\sum }_j{\sum }_k\Vert E_k^s{\Vert }_s^2}{\sum }_{\omega }{\omega }^2$$

(35)

$\sum _j\sum _k\Vert E_k^s{\Vert }_s^2$ in Equation (35) is two-parametric sum of the scattered field data $E_k^s$ obtained by k receivers in each set of j sets of data contained in the small batch. Because $E_k^s$ is known and is not equal in each set of training data, $\sum _j\sum _k\Vert E_k^s{\Vert }_s^2$ is a dynamic real value. Although $\sum _j\sum _k\Vert E_k^s{\Vert }_s^2$ introduces more information about the model mechanism into the training process of the DCNN, it does not affect the stochastic descent process of the cost function. ${\lambda }_0$ is a weighting factor, which is a constant, and ${\lambda }_0$ has the function of preventing the DCNN from overfitting.

3. Experimental Results and Analysis

The imaging data of the standing wood defect model in this paper is calculated from a scattering equation. Before imaging, the scattering process should be modeled. The specific method is to set the relative dielectric constant matrix according to the distribution of common defects in the living tree to express the defect data information in the trunk of the tree. In this process, parameters, such as frequency, position and wave source, need to be set to calculate the scattering field data and prepare for the inversion of subsequent data.

3.1. Simulated Imaging Evaluation Metrics

3.1.1. Intersection over Union

In the process of detecting internal defects in trees, the accuracy of the inversion results during a single inspection is determined by Equation (36) [33]:

$$IOU=\frac{S_i{\cap }^{ }{S}_f}{S_i{{\displaystyle \cup }}^{ }{S}_f}$$

(36)

In this paper, IOU essentially represents the degree of overlap between inverse map defects and model defects [34]. In the single detection of internal defects in trees, it is judged to be accurate when IOU ≥ 0.87. The most ideal case is that the inverse defect is the same as the defect set in the model when IOU = 1, the rest of the cases are judged to be inaccurate for the single detection.

3.1.2. Algorithm Detection Accuracy

To detect the generalization ability of the algorithm, this paper sets up several sets of test data to verify the performance of the proposed algorithm and find the detection accuracy of the devised algorithm under different detection environments. The calculation accuracy is shown in Equation (37):

$$Acc=\frac{N_{tp}}{N_t}\times 100\%$$

(37)

Bulleted lists look like this: Acc in Equation (37) indicates the detection accuracy of the proposed algorithm for all test data, N_tp is the number of all test results that are judged to be detected accurately, and N_t is the total number of test data.

The scattered field data are used to reconstruct the target medium and the model-driven deep learning network and BP neural network, and the CSI are tested for inverse imaging, where the scattered field data are obtained by the forward process [35]. The 18,000 data sets in the training database are divided into three groups: 14,800 data sets as the training set of the model-driven deep learning network and the BP neural network; 2000 data sets as the validation set of the BP neural network; finally, 1200 data sets as the test set, including 300 sets each of single defect, homogeneous double defect, and heterogeneous multiple defects. All inversion imaging algorithms were tested, the number of test data that each algorithm could accurately invert was counted, and the final algorithm detection accuracy was obtained according to Equation (37). All algorithms were selected for detailed analysis and the presentation of typical defect inversion images, in which the IOU values of inversion maps were obtained according to Equation (36), and the imaging evaluation indexes of all algorithms, were compared.

3.2. Model Settings

3.2.1. Build Simulation Environment

The simulated imaging experiments were performed with the help of a simulation environment built in Matlab R2017b. The BP neural network simulation imaging and model-driven deep learning network-based inversion algorithms were mainly performed under the TensorFlow framework in Python 3.6. The computer has a Win10 operating system, Intel(R) Core(TM) i7-8700 CPU @ 3.20 GHz, 16 GB of RAM, and NVIDIA GeForce RTX 2060 display adapter. In the actual standing wood defect model, parameters that affect electromagnetic wave scattering are the dielectric constant of the material, the conductivity information, and the model size. Thus, in the simulation test, the relative permittivity of the model is set to one, and the rest of the relative permittivity is set according to the actual medium information, as shown in

Table 2. Simulation model parameter settings.

Parameter Name	Value	Parameter Name	Value
Domain	0.32 m × 0.32 m	Relative permittivity of internal defects	20/40/60
Radius of trunk	0.1 m	Impedance of air	120π
Radius of internal defects	0.01 m/0.02 m	Number of electromagnetic wave transmitters	32
Frequency	200 MHz~700 MHz	Number of electromagnetic wave receivers	32
Resolution	100 × 100	Relative permittivity of xylem	5
Relative permittivity of air	1	-	-

. In the process of tree growth, the relative permittivity information is mainly related to the xylem water content, and not much related to the tree species. Therefore, this paper only considers the relative permittivity distribution inside the xylem affected by the tree water content.

In the simulation model, white represents air whose relative permittivity is set to 1, and the air impedance value is set to 120$\pi$. The yellow color is the two-dimensional cross-section of the xylem, which is a circle of 10 cm radius, and its relative permittivity is set to five. The gray part inside the xylem is a circular defect of 2 cm radius. The imaging resolution is 100 × 100. To obtain the scattered field data of the solution domain, electromagnetic wave transceivers with the same transmitting and receiving frequencies are selected for the detection, and the imaging frequencies are from 200 MHz to 700 MHz. A total of 32 groups of electromagnetic wave transmitters and 32 groups of electromagnetic wave receivers are evenly distributed on the circular arc around the solution domain.

3.2.2. Build Training Database

The training database consists of two parts. The first part includes 1500 sets of existing laboratory data and 1500 sets of electromagnetic field scattering data obtained by Matlab R2017b simulation. The second part is to use Matlab R2017b to simulate the forward process of electromagnetic wave propagation and obtain 15,000 sets of electromagnetic field scattering data through simulation. The two parts total of 18,000 sets of data form a training database, which is used to train model-driven deep learning networks and BP neural networks. Since 32 transmitters are used to transmit electromagnetic waves and 32 receivers are used to measure the scattered field data in the live wood defect scattering model, all training data are in the form of a 32 × 32 matrix. The training database reflects a negative factor in the testing process as much as possible, thus enabling adequate training of the model-driven deep learning network and improving the generalization capability of the algorithm [36,37].

3.3. Single Defect Inverse Imaging

In the 320 mm × 320 mm square region, white represents air, the relative dielectric constant is set to 1 and the air impedance is set to 120, with 100 × 100 imaging resolution. The yellow region represents the two-dimensional cross section of xylem, which is shaped as a circle with a radius of 10 m. The relative dielectric constant is set to five. The gray area represents the defect, which is the circle of the radius 20 mm, and the relative dielectric constant of the defect is set to 40.

Figure 4 shows the inversion imaging effect of each algorithm when the defect is in the center of the xylem. The boundary transition phenomenon in the inverse map of the contrast source is serious, which makes the inverse defect area larger than the defect area set in the model, and the IOU value is 0.885. The BP neural network has a better inversion of the defect size and location, and the IOU value reaches 0.964, but the boundary between wood and air in the inverse map is not clear enough. The model-driven deep learning inverse map has the highest IOU value of 0.977, which not only inverts the defect size and location but also clearly inverts the media boundary between wood and air.

In order to express the relationship between the output data of the model result and the true value, the mean square error is introduced:

$$C\left(\omega ,b\right)\equiv \frac{1}{2n}{{\displaystyle \sum }}_x\Vert y\left(x\right)-a{\Vert }^2$$

(38)

The weight set of network connections is expressed as $\omega$, offset to b. The total amount of input data is n. When the input is x, the output vector of the last layer is $a$. $C\left(\omega ,b\right)$ is the average generation value obtained from the total input data.

Table 3. Mean square error and average single detection time for each algorithm.

	Contrast Source Inversion	bp Neural Network	Model-Driven Deep Learning Networks
Mean Square Error	0.2826	0.1732	0.0825
Single Detection Time	116s	0.059s	0.063s

shows the average detection time of each algorithm and the maximum mean square error of the relative permittivity in the inverse map of each algorithm versus the relative permittivity in the model. Among the algorithms, the CSI has the largest mean squared error and the model-driven deep learning network has the smallest mean squared error. In terms of time cost, the CSI takes the longest time for a single detection, reaching more than 100 s; the BP neural network and the model-driven deep learning network require the shortest detection time, both around 0.06 s.

3.4. Homogeneous Double Defect Inversion Imaging

The homogeneous double-defect model means that the parameters of the two defects in the xylem are identical, and the coaxial means that the circle centers of the two defects are at the same coordinate coaxial. Figure 5 shows the inversion results of the homogeneous coaxial double-defect model for each algorithm.

As shown in Figure 5, the boundary transition phenomenon in the inverse map of the contrast source is serious, which makes it impossible to identify the media boundary between the two defects in the inverse map, and the IOU value is only 0.872, reducing the inverse accuracy of the defect size and location. The BP neural network can reflect the defect size and location better, and the IOU value of the inverse map is 0.963. However, the media boundary between the wood and air in the inverse map is not clear enough. The model-driven deep learning inversion not only has less noise but also clearly inverts the defect size and location, as well as the media boundary between wood and air, with an IOU value of 0.975.

Table 4. Mean square error and average single detection time for each algorithm.

	Contrast Source Inversion	BP Neural Network	Model-Driven Deep Learning Networks
Mean Square Error	0.3526	0.1932	0.0937
Single Detection Time	119 s	0.078 s	0.066 s

shows the average single detection time and mean square error for each algorithm. When the number of defects increases to two, the CSI is unable to invert the defect and xylem media boundaries, where the relative permittivity in the inverse map is between 5 and 40. This mean square error is 0.3526 for the CSI, while the model-driven deep learning network inversion is the most accurate with a mean square error of 0.0937. In terms of time cost, the detection time required by the model-driven deep learning network is 0.066 s; the average detection time of the CSI increases by 3 s compared to the single defect detection time.

3.5. Heterogeneous Multi-Defect Inversion Imaging

To further expand the application for real tree internal defect detection, three internal defects are set up for an inverse imaging test, and the electrical properties of each defect are different to consider the existence of three different defects in the same two-dimensional section of the wood. The relative dielectric constants of the three defects are 20, 40, and 60, respectively, and the live wood defect model is set up as shown in Figure 6a, where the relative dielectric constant of the defect on the right side of the xylem is 20, the relative dielectric constant of the defect above the xylem is 40, and the relative dielectric constant of the defect below the xylem is 60. The effect of each algorithm for defect inversion is shown in Figure 6.

As shown in Figure 6, for the detection of heterogeneous multi-defects inside the trees, the CSI cannot locate the defect location. The BP neural network better inverts the defect size and location, while the boundary between wood and air in the result is not clear enough, and the IOU values for BP are 0.928 and 0.941, indicating that this algorithm is not accurate enough for feature extraction of the training data. The model-driven deep learning inversion has less noise, accurately reflecting the defect size and location, and also clearly reflecting the media boundary between wood, defect and air, and the IOU value reaches 0.961.

As shown in

Table 5. Mean square error and average single detection time for each algorithm.

	Contrast Source Inversion	BP Neural Network	Model-Driven Deep Learning Networks
Mean Square Error	None	0.2679	0.1345
Single Detection Time	None	0.077 s	0.065 s

, under the standard of mean square error, the result of the model-driven depth neural network is significantly better than that of the BP neural network. The consumption of the two methods is roughly the same.

3.6. Algorithm Iterative Stability Analysis

BP neural networks and the model-driven deep learning networks need to be trained to achieve good generalization capabilities, but the iterative process is not required to solve the problem after the algorithm is trained, and iterations only exist in the algorithm training process so that it can reduce the defect detection time [38,39]. The iterative process of CSI, the BP neural network and the model-driven deep learning network are compared to analyze the stability in the iterative process. The iterative process of each algorithm is shown in Figure 7 and Figure 8, where the output value of the objective function and cost function are processed in absolute value.

Figure 7 shows the change in the output value of the objective function during the solution process for CSI. The iteration process is the detection process of the CSI for the homogeneous double defects of radius 2 cm, which can be seen as the output value of the CSI objective function can converge to a stable range after about 90 iterations. If more iterations are used in the solution process, the detection time cost of the CSI will be increased. As can be seen from Figure 8, the BP neural network converges to the stable range only after 150 training iterations, while the model-driven deep learning network converges to the stable range only after 80 training iterations. Thus, the model-driven deep learning network reduces the number of training iterations by improving the cost function, and it can reasonably control the number of iterations in the network training process to reduce the time cost and improve network training efficiency.

Figure 9 shows a group of abnormal training processes of BP neural networks. Although the method converges to a stable value after 130 iterations, the “spike” phenomenon can be observed obviously on the iterative curve. These “spikes” are caused by abnormal training data, which leads to a sudden change in the output value of the cost function. These mutations seriously affect the performance of the algorithm. The improvement of cost function is an effective way to avoid this phenomenon. The results of Figure 8b show the effectiveness of this change. It can be observed that the stability of the iterative process reflected in Figure 8b is significantly better than that in Figure 9 when using the same data.

4. Conclusions

In this paper, a live-standing wood defect detection model is established, and a model-driven deep learning inversion algorithm based on the advantages of model-driven and data-driven algorithms is proposed by learning the propagation process of electromagnetic waves in the scatterer. By establishing a deep convolutional neural network, the simulation imaging test is conducted using the CSI and the model-driven deep learning algorithm, and the test results show that the model-driven deep learning network detection accuracy can reach 91.6% for single defects within trees; for heterogeneous double defects and complex multi-media defects, the model-driven deep learning network detection accuracy is 86.3% and 78.3%, respectively. At the same time, the algorithm proposed in this paper reduces the single detection time to less than 0.1 s. The model-driven deep learning algorithm provides the theoretical possibility for large-scale, real-time, all-tree internal defect detection. Most of the training data in this paper are obtained from simulation experiments. In future work, more laboratory data will be added to ensure the accuracy of internal defect detection in trees. At the same time, it is necessary to further consider the idealized setting of xylem and defects of the simulation model in this paper, which does not conform to the irregular shape of trees.