The structure in Figure 1 labels the main procedure of our experimental approach. First of all, we have a database of various images of minerals. These images will be preprocessed with a certain number of operations (cutting, resizing, etc.). Then divided into a training and validation data set, the images are used to train a convolutional neural network via AlexNet. Finally, the membership classification of each mineral image is calculated and the precision rate is recorded serving as the performance of the chosen optimizers.

For the needs of the project and in order to give it a particular meaning, the creation of the database began with the collection of 15 samples of magmatic rocks in the field. These samples were used to prepare 15 thin sections for each sample. Each slide was admitted under homogeneous polarized light from an MD500 microscope (using Amscope 3.7 software) on which a camera is mounted and transmitted to a computer (Intel(R) Core(TM) i5-8265U CPU @ 1.60GHz 1.80 GHz, RAM size 16.0 GB (15.8 GB usable)) to collect slide images.

For our research, we needed a fairly large number of images. By capture, the microscope scene images are acquired following the rotation of its stage ranging from 0 to 315 degrees in increments of 45 degrees. Images are observed at magnification (×40). The flowchart for this step is illustrated in Figure 1 . All images (RGB) were stored at the size of 2592 × 1944 pixels with a resolution of 120 dpi, and saved in “JPG” format. The increment made is due to the fact that different minerals have different quenching properties.

The images acquired include one or more minerals at a time with different hues. The pretreatment began with the isolation of each mineral from its matrix, resulting in a resizing. The operation was done by screenshot with the XnView

software. The images occupy on average a storage space of between 3.5 and 8 KB with each a resolution of 144 ppi (pixels per inch). The minerals dataset in this experiment is a new dataset. A total of 700 images (due to 100 images per class) of single species minerals with different hues constituted our database ( Figure 2 ). The classes were chosen because of their abundance in igneous rocks and whose proportions are used to name the rocks based on the Streckeisen diagram. We distinguish: amphibole, biotite, alkali feldspar, muscovite, plagioclase, pyroxene and quartz. Feldspathoids were not taken into account in this study because they are very rare in rock formations in Ivory Coast. To reduce the model parameters, the original images were compressed to 224 × 224 pixels. However, each image was labeled according to its class.

The exact distribution of data by classes and types is described in Table 1 . The number of images in the test folder has no impact on that of the training or learning divided into two (training images or train, and validation images) with the respective proportion (70% - 30%) [49] . We just need a few different images to test the prediction once processing is complete.

After acquiring and organizing the data, it’s time for training which consists of different processes.

1) Choice of model

In this work, mineral classification is implemented using a convolutional neural network with the AlexNet model. The choice of AlexNet is both a function of the quantity of our images (700 in total for the 7 classes or 100/class) and for its convolutional and fully connected layers. The latter mentioned are adjusted so as to maximize the extraction of descriptor details at the intermediate level of features. The network was developed by Alex Krizhevsky in 2012, hence its name with his collaborators Ilya Sutskever and Geoffrey Hinton [50] . Winner of the ILSRC2012 ImageNet challenge, AlexNet is a CNN model with more depth and width allowing it to adapt to graphics processing units (GPUs) with their great potential for parallel calculations. The reasons for our choice of AlexNet are as follows:

Unlike the all-connected layers of CNNs, its convolutional layers maintain spatial coherence of information. This is the first work that propelled convolutional neural networks (CNN). It takes as input a 224 × 224 pixels’ image with 3 color channels. Its architecture includes 8 layers including 5 convolutional and 3 fully connected ( Figure 3 ).

The block of the first convolutional layers filters the input image (224 × 224 × 3) with 96 kernels of size (11 × 11 × 3) with a step of 4 pixels. The second layer, which has the output (normalized and grouped) of the first as input, filters with 256 kernels of size (5 × 5 × 48). The next 3 layers are connected to each other without any intermediate normalization or pooling layers. However, the third layer has 384 cores of size (3 × 3 × 256) connected to the outputs (normalized and grouped) of the second. The fourth also has 384 cores but of size (3 × 3 × 192). The fifth has 256 cores of size (3 × 3 × 192). The fully connected layers have 4096 neurons each and perform global classification [55] .

AlexNet promotes a fast GPU implementation of CNNs for image recognition with the “sigmoid” or softmax function as activation function [56] . This function acts as an output layer and calculates the output probability.

2) Optimization algorithms

Typically, in machine learning, the goal of a model is to create a prediction function $f\left(x\right)$ from a data set D, also known as a set of training. In the context of supervised learning, D consists of pairs of examples (x, y), where x represents an input vector to the model and y is a target vector indicating what we are trying to predict. Regardless of how the data was acquired, it is then put through a learning algorithm that aims to model the relationship between the inputs and the targets. The inputs to a network are generally denoted $x_1,x_2,\cdots ,x_p$, acting as the explanatory variables of the model. The weights associated with these inputs are represented by the parameters α and also β which must be estimated during the learning procedure. The output of the model y, represents the variable to be explained or the target of the model, formulated as:

$y=f\left(x_1,x_2,\cdots ,x_p;\alpha \right)$. (1)

Learning therefore consists of estimating these parameters (α, β) by minimizing the prediction error. To measure the prediction error, we use a function called the loss function which defines the learning rate and gradient descent is a technique to do this. It updates the parameters for each example of data x and labels y ; we speak of an iterative algorithm. The lower its value, the more robust the model. If the model has correctly learned on all the data, the value of the cost function becomes zero.

a) Stochastic gradient descent momentum (sgdm) optimizers

In order to improve the efficiency of stochastic gradient descent (SGD) [57] , the concept of momentum [58] (combination of the current direction of the gradient and from the previous direction) which in the case of optimization gives a certain “inertia” to the updating of the model parameters. Therefore, rather than moving only in the direction of the instantaneous gradient at each iteration, the momentum allows a sort of “memory” of previous directions to be retained. Thus, the algorithm keeps track of the direction in which the model parameters move. This helps accelerate the convergence of the optimization by attenuating unwanted oscillations or noisy variations in the gradient descent ( Figure 4 ).

For a loss function $J\left(\theta \right)$, the network escapes from traps using the relation using the momentum coefficient [58] defined by:

$v_{t-1}=\gamma -\eta {\nabla }_{\theta }J\left({\theta }_t\right)$ (2)

where:

$\gamma \in \left[0,1\right]$ represents the momentum coefficient,

v_t, the momentum vector at iteration t,

η, the learning rate,

${\nabla }_{\theta }J\left({\theta }_t\right)$, the gradient of the cost function with respect to the parameters at iteration t

The complete formula for updating the parameters with SGDM (stochastic gradient momentum descent) is then:

${\theta }_{t+1}={\theta }_t-v_{t+1}$ (3)

where : θ represents the model parameters. Using the momentum coefficient can help speed up the convergence of the optimization algorithm and smooth out oscillations when training the model.

b) Optimizers Root Mean Square Propagation (RMSprop) Algorithm

This optimizer was designed to address some issues related to updating Learning rate in gradient descent. Instead of accumulating all the squares of the previous gradients, we restrict the window of accumulated gradients to a fixed size. Therefore, we apply an exponential moving average of the squares of the gradients to automatically adapt the learning rate for each parameter of the model rather than storing the squares of the previous gradients. The parameter update (θ) with RMSprop is calculated as follows at each iteration t.

$E{\left[g^2\right]}_t=\beta E{\left[g^2\right]}_{t-1}+\left(1-\beta \right)g_t^2$ (4)

${\theta }_{t+1}={\theta }_t-\frac{\alpha }{\sqrt{E{\left[g^2\right]}_t+\epsilon }}\cdot g_t$ (5)

where:

g_t, the gradient with respect to the parameters at iteration t;

[g²]_t, the exponential moving average of the squares of the gradients at iteration t;

α, the learning rate;

β, attenuation coefficient, generally close to 1 (for example, 0.9);

ε, small constant added to avoid division by zero.

[59] , propose to set β at 0.9 while a good default value for a better learning step α is 0.001.

c) Optimizers Adaptive moment estimation (Adam)

Adam, is a method which calculates a learning step qualified as adaptive for each parameter [60] . During training, in addition to storing an exponentially decreasing average of the squares of the previous gradients v_t as with RMSprop, it also keeps an exponentially decreasing average of the previous gradients m_t following the equations:

$m_t={\beta }_1{m}_{t-1}+\left(1-{\beta }_1\right)g_t$ (6)

$v_t={\beta }_2{v}_{t-1}+\left(1-{\beta }_2\right)g_t^2$ (7)

It should be noted that m_t makes it possible to update the exponential average moments of the gradient and is qualified as an order 1 estimation while v_t updates the exponential average moments of the squares of the gradient and is qualified as an order estimation 2. According to these authors, m_t and v_t are initialized as vectors of 0 and are biased around this point during the first steps, specifically when the decay coefficients (β₁, β₂) are small (close to 1). They propose a bias correction for the estimates with the relationships:

${\hat{m}}_t=\frac{m_t}{1-{\beta }_1^t}$ (8)

${\hat{v}}_t=\frac{v_t}{1-{\beta }_2^t}$ (9)

Then comes the settings update (θ):

${\theta }_{t+1}={\theta }_t-\frac{\eta }{\sqrt{{\hat{v}}_t}+\epsilon }\cdot g_t$ (10)

where:

η, the learning rate;

β₁ et β₂, attenuation coefficients (typically close to 1, for example, 0.9 and 0.999);

ε, small constant added to avoid division by zero.

For the parameters β₁, β₂ and ε, the authors propose default values respectively 0.9, 0.999 and 10⁻⁸.

3) AlexNet and Transfer Learning

Lack of knowledge of open source reference images of minerals is a disadvantage in having a sufficient quantity of images. This leaves us unable to train the AlexNet network from scratch. Our solution to this deficit takes us to work through Transfer Learning [61] [62] [63] . This method refers to the ability of a system to recognize and use knowledge and skills acquired in previous tasks, to apply them to new tasks, often in different domains. Thus, the influence of transfer learning follows the principle of Figure 5 .

This change consists of modifying the classification of the 1000 classes of ImageNet into a classification of seven (7) classes. This will involve determining as outputs the presence of images of quartz, biotite, amphibole, plagioclase, feldspar, muscovite, pyroxene. The new network to obtain follows the following steps: This principle works as follows:

In each configuration, the parameters of the new model are optimized using the grid search method. This method uses two hyper-parameters: the number of epochs and the “Learning rate”. The grid method consists of defining a certain number of values (or grid of values) for each hyper-parameter [64] . The metric chosen to evaluate the performance of the said model is precision. However, the “loss” curve can be used ( Figure 6 (curve in orange)). The combination that provides the best performance is then selected to certify the quality of the images.

Trained to classify images into 1000 ImageNet categories, we need to adjust the output layer of the AlexNet pre-trained network to match the number of classes in our work. This leads to replacing the last dense layer with a new dense layer with the appropriate number of neurons for our case. Through transfer learning and data augmentation operations, the AlexNet pre-trained network will optimize its parameters to solve the classification of rock mineral images in the database. In this work, to prevent the size of the images from affecting the classification results [65] , each image is compressed to [224 × 224 pixels] in order to fully assess the accuracy. The ReLu activation function is used to introduce non-linearity into the model, facilitating the learning of complex relationships between rock minerals. At the output of the network, Softmax is used specifically to convert scores into class probabilities.

Deep learning with its networks with complex architectures as well as its important training parameters, which by using transfer learning freezes some of its layers to a certain point requires evaluation. The evaluation phase will therefore determine what is correct in the model. A way to quantify which predictions is correct. The performance of a model results from the confusion matrix ( Table 2 ) presented by the model including the metrics: precision, recall, accuracy and trade-off between precision and recall (F1-score).

For a confusion matrix the metrics raised are determined as follows:

$Précision=\frac{TP}{TP+FP}$ (11)

$Rappel=\frac{TP}{TP+FN}$ (12)

$Accuracy=\frac{TP+TN}{TP+TN+PF+FN}$ (13)

$F1-mesure=\frac{2\ast \left(Précision\ast Rappel\right)}{Précision+Rappel}$ (14)

With TP, TN, FP, FN respectively true positive, true negative, false positive and false negative.

This section of this article details the results after training our AlexNet model on image of rock minerals. Three optimizers were selected in order to understand and know their mode of operation by varying the Learning rate for a series of experiments and then doing the same for different epochs on our dataset. For a method, the learning step remains a very sensitive parameter for its convergence [66] . Figure 6 shows an example of the curve obtained after training. At this step, the learning step is executed for N iterations, thus updating the parameters N times using N examples from the dataset. However, an epoch constitutes T updates, that is to say the equivalent of the entire dataset intended for this phase.

The results of each variation linked to each optimizer are recorded in Table 3(a) , Table 3(b) , Table 4(a) , Table 4(b) , Table 5(a) and Table 5(b) . The proportions of data for each training and validation set are respectively 70% - 30% and remain the same in each case.

Table 3(a) and Table 3(b) describe the results obtained with the “sgdm” optimizer with the AlexNet architecture using images of rock minerals using Transfer Learning. Part (3a) describes the accuracy scores for different Learning rate values. The highest score is 96.2% at the learning rate of value 10⁻³. As for Table 3(b) , we record scores in two phases. The first phase has the same evolution as the values of the grid epochs. The step is set to 10⁻³ during the experiments. The highest score recorded is 96.7%. Subsequently, the second phase is positioned with a drop in precision values.

The stochastic gradient descent with momentum (sgdm) optimizer recorded its lowest score value (82.4%) for an estimated learning rate of 10⁻¹ and 84.8% when the algorithm went through the entire training data set 50 times. During the different types of training to evaluate sgdm, with the different Learning Rate values. The curve of the loss function or “Loss” (curve shown in orange) completely begins to converge around 100 epochs, or around 400 iterations.

Table 4(a) and Table 4(b) show the results obtained with the “Adam” optimizer, still using Transfer Learning. Part (4a), however, describes the accuracy scores for different learning rate values. The highest score value is 95.2% at the learning rate value 10⁻¹.

As for Table 4(b) , we note the scores also corresponding to different epochs with a learning rate, this time maintained at 10⁻¹ during the experimental phase. The highest score is 93.8% after 200 epochs.

The Adam optimizer recorded its lowest score values, which are 86.7% with the lowest learning rate value set to 10⁻⁶ and 76.2% when the algorithm performed 50 epochs on the entire training data.

Table 5(a) and Table 5(b) show the results obtained with the “RMSprop” optimizer. Part (5a) shows the model accuracy scores for different learning step values. The highest score with RMSprop is 89.5% for 10⁻⁴ as the optimal value of the learning step. As for Table 5(b) , the learning rate was maintained at 10⁻⁴ and we record the scores corresponding to the epochs proposed for the experiment. The highest score is 84.3% at 350 epochs.

This optimizer presents its lowest score value which is 75.2% with the high value of the Learning rate estimated at 10⁻³. Unlike the other epochs, the logic is respected, i.e. a low score of 41.9% at the start of training and which increases as the epochs also increase before making a learning jump after 350.