There is no universally mathematical model for texture feature extraction. Because the gray-level co-occurrence matrix (GLCM) model method is not restricted by the analysis object, it can well reflect the spatial gray distribution of the image and the texture features of the image, and has been widely used [34]. GLCM which describes the grayscale of adjacent pixels (or within a certain distance) is a statistical matrix, and reflects the comprehensive information which consists of the image gray change in the direction, interval, and amplitude. Assume the gray level of a digital image is N, p (i, j) represents the possibility (or frequency) of the appearance of grayscale j under the condition that the starting grayscale is i, where it is assumed that j is along the direction $\theta$ of i and the space distance is d. GLCM shows statistical information, and can be calculated using the following equations.

(1) Mean:

(4)$$\text{Mean}=\overline{x}=\sum _{i=0}^N\sum _{j=0}^{N}p(i,j)\times i$$

The mean reflects the regularity of texture. The smaller the mean is, the more disorganized the texture is.

(2) Variance:

(5)$$\text{Variance}=\sum _{i=0}^N\sum _{j=0}^{N}p(i,j)\times {(i-\overline{x})}^2$$

The variance measures the deviation of the pixel value from the mean. The larger the variance is, the more the change of gray scale is.

(3) Entropy:

(6)$$\text{Entropy}=-\sum _{i=0}^N\sum _{j=0}^{N}p(i,j)\times \ln p(i,j)$$

Entropy is the measurement of information contained in an image. The greater the value of entropy is, the more complex the texture is.

(4) Contrast:

(7)$$\text{Contrast}=\sum _{i=0}^N\sum _{j=0}^{N}p(i,j)\times {(i-j)}^2$$

Contrast reflects the total amount of local gray scale changes in an image. The greater the contrast of an image is, the clearer the visual effect of this image is.

(5) Correlation:

(8)$$\text{Correlation}=\sum _{\mathrm{i}=0}^{\mathrm{N}}\sum _{\mathrm{j}=0}^{\mathrm{N}}\frac{(\mathrm{i}-\text{Mean})\times (\mathrm{j}-\text{Mean})\times \mathrm{p}{(\mathrm{i},\mathrm{j})}^2}{\text{Variance}}$$

The correlation is a measure of the linear relationship of the gray scale. The longer the extension of the gray value in a certain direction is, the greater the correlation is.

(6) Homogeneity:

(9)$$\text{Homogeneity}=\sum _{i=0}^N\sum _{j=0}^{N}p(i,j)\times \frac{1}{1+{(i-j)}^2}$$

Homogeneity is used to measure the uniformity of the local gray level of an image. The more homogeneous the local gray scale is, the greater the value is.

(7) Energy:

(10)$$\text{Energy}=\sum _{i=0}^N\sum _{j=0}^{N}p{(i,j)}^2$$

Energy is the measurement of uniformity of gray distribution in the image.

Using the above 7 kinds of texture features, a texture vector representing an image can be obtained.

The feature vectors need to be normalized so that any feature does not dominate among all of these features. All the feature vectors are normalized to contain zero mean and unit variance. The normalization is done by using the equation:

(11)$$x_i=\frac{x_i-\mu }{{\sigma }_x}$$

where $\mu$ and are respectively the mean and standard deviation of all the features $x_i$.

In summary, for each of these three weighted and averaged gray images (Img1, Img2, Img3), the above 7 statistics are calculated and normalized. Thus every subject can be represented using a vector containing 21 features.

Machine learning algorithm usually learns a task every time, and decomposes the complex problem into the theoretically independent sub-problems. Then it learns each sub-problem separately. Finally, it constructs the mathematical model of complex problems by combining the learning results of the sub-problems, namely single task learning [35, 36]. Multi-task learning is a machine learning method relative to single task learning. It uses information shared by multiple tasks to learn multiple tasks simultaneously, and solves multiple problems simultaneously. The obtained results interact with each other. Sharing information between tasks is the prerequisite for multi-task learning. On this basis, the training of multiple tasks can improve the overall generalization performance of the model. The main difference between multi-task learning and single task learning is that the training process of the model is different [31, 32]. In the training process of single task learning, each task is independent and does not affect each other. Its disadvantage is that it ignores the information contained in other tasks during the process of single task training. To some extent, the loss of relevant training information is caused, and this part of the lost information may be very useful for the training process. Nevertheless, the training of multi-task learning takes into account the correlation and useful information shared between tasks. At the same time it learns multiple tasks in parallel. The difference between the two training model is shown in the Fig. 2 below.

The most important issue of multi-task learning is how to build the model of relationship between tasks, and to make the relevant tasks share information. Finally, the goal of using the correlation between different tasks to improve the learning performance of the algorithm is achieved. We expect that the model fits the training data as much as possible, and is not too complicated at the same time. Therefore, the support vector machine algorithm with regularized multi-task is adopted to solve MRI image classification problems of mental illness in multi-site data centers.

Assuming that there are t supervised learning tasks in a multi-task learning problem, for each task i, the learning function is assumed to be $f_t:R^d\to R$. Training set is $X_i=[x_1,x_2,\mathrm{\dots },x_n]\in R^{d\times n}(i=1,2,\mathrm{\dots },t)$, where n is the number of input sample, and d is the dimension of the sample feature vector. $Y_i=[y_1,y_2,\mathrm{\dots },y_n]\in R^n(i=1,2,\mathrm{\dots },t)$, and $y_i\in \{+1,-1\}$ is the label of each sample in the i-th task. The weight coefficient matrix of t-th supervised learning tasks is $W=[w_1,w_2,\mathrm{\dots },w_t]\in R^{d\times t}$. The goal of multi-task learning is to get t related tasks’ regression or classification function $f_t(x)$ by learning the training data. In order to accurately find out the $f_t(x)$ function, the multi-task objective function should be determined first. Assuming that $f_t(w_i^T{x}_i,Y_i)$ is the loss function of the t-th task, in the classification problem, the classical loss function includes log-likelihood function, exponential function and hinge function. Support vector machine model with regularized multi-task learning and a least empirical error can be expressed by [28].

(12)$$\underset{W}{\min }\frac{1}{n}\sum _{i=1}^{n}f(w_i^T{X}_i,Y_i)+\lambda \mathrm{\Omega }(W)$$

where the first item is the empirical loss function on training data. $f(w_i^T{X}_i,Y_i)$ uses hinge loss function. The second one is the regularized term which can encode correlation between tasks. $\lambda$ is the parameter of the regularized term, and $\lambda$ $>$ 0.

The optimal solution for a single t task is equivalent to the global problem of solving the target function of the joint t task, and is described by

(13)$$\underset{W}{\min }\sum _{t=1}^t\frac{1}{n}\sum _{i=1}^{n}f(w_i^T{X}_i,Y_i)+\lambda \sum _{t=1}^t\mathrm{\Omega }(W)$$

The norm of the model parameter vector is usually used as regularized term in machine learning. The regularization order needs to be set in advance. $l_0$, $l_1$ and $l_2$ norms are commonly used. In our experiments, we found that different regularization order can improve the classification accuracy of different data. So SVM classification algorithm with p-norm regularized multi-task learning is proposed in this paper. p-norm is not only effective during processing image data, but also easy to optimize. And it can reduce the computational complexity of the model. The formula is described by [29]

(14)$$h(w)={\parallel w\parallel }_p={\left(\sum _{i=1}^n{\left|w_i\right|}^p\right)}^{\frac{1}{p}}$$

where $x=\{x_1,\mathrm{\dots },x_n\}$ is a vector. p-norm is a measure of the sparsity of the vector. The desirable range of the order p is 0 p $\le$ 2, and the choice of p depends on the related degree between the tasks. The more correlation and shared information between the tasks is, the larger the p value is. Let $k(w)=h^p(w)$ , when 0 p $\le$ 2. Its derivative equation is shown by

(15)$$\frac{\partial k(w)}{\partial w_i}=p{\left|w_i\right|}^{p-1}\times \mathrm{sgn}\left(w_i\right)$$

where $\mathrm{sgn}\left(w_i\right)=\frac{w_i}{\left|w_i\right|}$. So the Eqn. 15 can be written by

(16)$$\frac{\partial k(w)}{\partial w_i}=p{\left|w_i\right|}^{p-2}\times w_i$$

Finally, the objective function of multi-task learning SVM with p-norm regularization is shown by

(17)$$\mathrm{L}=\underset{\mathrm{w}}{\min }\sum _{\mathrm{t}=1}^{\mathrm{t}}\frac{1}{\mathrm{n}}\sum _{\mathrm{i}=1}^{\mathrm{n}}\max (0,1-{\mathrm{y}}_{\mathrm{i}}{\mathrm{w}}_{\mathrm{i}}^{\mathrm{T}}{\mathrm{x}}_{\mathrm{i}})+\lambda \sum _{\mathrm{t}=1}^{\mathrm{t}}{\parallel {\mathrm{w}}_{\mathrm{i}}\parallel }_{\mathrm{p}}$$

According to the different situation, the derivation of the Eqn. 17 is as follows.

If , then

(18)$$\frac{\partial L}{\partial w_{t,i}}=0$$

If $1-y_i{w}_{t,i}^T{x}_i>0$, then

(19)$$\frac{\partial L}{\partial w_{t,i}}=-y_{t,i}{x}_{t,i}+p{\left|w_{t,i}\right|}^{p-2}\times w_{t,i}$$

The gradient descent method is used to update the weight coefficient matrix $W_{t,i}=\{{\omega }_{t,1},{\omega }_{t,2},\mathrm{\dots },{\omega }_{t,k}\}$ of t-th task, i.e., the following equation

(20)$$W_{t,i}=W_{t-1,i}+r\cdot \nabla L\left(W_{t-1,i}\right)$$