CN117764917A - Yolov 8-based lung nodule image detection method - Google Patents

Abstract

The invention discloses a lung nodule image detection method based on YOLOv8, which comprises the following steps: step1: acquiring a lung image data set, and preprocessing the lung image data set; step2: carrying out lung parenchyma segmentation on the lung image preprocessed in the step S1; step3: adding an NR-SCA module after the C2f convolution module in the backbone network of the YOLOv8 model; step4: weighting a size similarity formula of the NWD loss function and an IOU similarity formula of the CIOU loss function to serve as a loss function of the YOLOv8 model; step5: and (3) training the constructed YOLOv8 model through the lung image obtained in the step (2), and detecting a lung nodule image through the trained YOLOv8 model. The invention provides a noise reduction space coordinate attention (NR-SCA) mechanism, which can more effectively detect lung nodules with different positions and sizes by combining a Gaussian filtering noise reduction technology, a space attention and a coordinate attention into one module, and improves the accuracy and the robustness of detection.

Description

A pulmonary nodule image detection method based on YOLOv8

Technical field

The present invention relates to the field of image detection technology, and specifically relates to a YOLOv8-based pulmonary nodule image detection method.

Background technique

Lung cancer is a malignant tumor that seriously endangers human health. Early diagnosis and timely intervention are the keys to improving its clinical efficacy. Currently, the diagnostic techniques for pulmonary nodules include CT, chest X-ray and MRI. Among them, CT is widely used in clinical practice due to its high resolution, high sensitivity and strong specificity. Traditional pulmonary nodule detection methods mainly rely on doctors’ experience and visual observation, which have problems such as long diagnosis time, low efficiency, and error-prone. With the advancement of computer and image processing technology, pulmonary nodules can be automatically identified and located through automated or semi-automated analysis of CT images, thus greatly improving the efficiency and accuracy of early screening for lung cancer.

In recent years, medical image analysis methods based on deep learning have been widely used. Among these studies, the use of convolutional neural networks for pulmonary nodule identification is currently the most important method. Traditional object detection methods based on convolutional neural networks mainly include Faster R-CNN, YOLO, SSD, etc. These algorithms can automatically detect and locate pulmonary nodules through training models, greatly improving the accuracy and reliability of early screening and diagnosis of lung cancer. Among them, YOLO technology is an efficient, fast and accurate detection method, but it still faces many problems and challenges. For example, it is difficult to detect small pulmonary nodules; the impact of pulmonary arteries, lymph nodes, and bones can easily lead to missed diagnosis or misdiagnosis; while the detection speed is increased, its accuracy has also declined. Therefore, how to improve its modeling and algorithm to improve its accuracy and robustness is an issue that needs to be solved urgently.

Contents of the invention

The present invention provides a YOLOv8-based pulmonary nodule image detection method to solve the technical problems in the prior art that smaller pulmonary nodules are easily missed, have low accuracy, and have slow detection speed.

The present invention provides a YOLOv8-based pulmonary nodule image detection method, which includes the following steps:

Step 1: Obtain the lung image data set and preprocess the lung image data set;

Step 2: Perform lung parenchyma segmentation on the lung image preprocessed in step S1;

Step 3: Add the NR-SCA module after the C2f convolution module in the backbone network of the YOLOv8 model. The NR-SCA module includes: Gaussian filter, coordinate attention mechanism, and spatial attention mechanism; input to the NR-SCA module After the data is passed through the Gaussian filter, it enters the coordinate attention mechanism and the spatial attention mechanism respectively. The outputs of the coordinate attention mechanism and the spatial attention mechanism are element-wise multiplied with the output of the Gaussian filter. The product results are spliced and fused as the NR-SCA module. Output;

Step 4: Weight the size similarity formula of the NWD loss function and the IOU similarity formula of the CIOU loss function as the loss function of the YOLOv8 model;

Step 5: Use the lung images obtained in step 2 to train the constructed YOLOv8 model, and use the trained Yolov8 model to detect pulmonary nodule images.

Further, in step 3, the output of the NR-SCA module is:

In the formula, is the feature map after noise reduction by Gaussian filtering; y _c is the feature map output by the coordinate attention mechanism; y _s is the feature map output by the spatial attention mechanism; ⊙ is the element-wise product.

Further, in step 3, the output of the Gaussian filter is:

In the formula, is the pixel at position (i, j) in the output feature map after filtering; X _{u, v, k} is the pixel value at position (u, v) in the input feature map; k is the number of feature map channels; (u, v ) is the position offset of the filter, (i,j) is the position offset of the output feature map; σ is the standard deviation of Gaussian filtering;

Further, in step 3, the output of the coordinate attention mechanism is:

In the formula, X _c (i, j) is the value of the input feature map on channel c, vertical position i and horizontal position j; is the channel attention weight in the vertical direction;/> is the channel attention weight in the horizontal direction;

Further, in step 3, the output of the spatial attention mechanism is:

In the formula, It is the feature map obtained by splicing the average pooling feature map and the maximum pooling feature map according to the channel dimension; f ^7*7 is a 7×7 convolution operation on the spliced feature map; σ is the activation function.

Further, step 3 also includes: the Neck part of the YOLOv8 model is a BiFPN structure.

Further, in step 4, the loss function of the constructed YOLOv8 model is:

In the formula, S _IOU (N _{a ,} N _b ) represents the IOU similarity between Na and N _b ; W _θ (N _a , N _b ) represents the size similarity between Na _and N _b ; C is the nonlinear transformation function. Parameters; α and β are adjustment parameters, the value of α is (0,1), and the value of β is (1, +∞).

Further, step 4 also includes optimizing the size similarity formula of the NWD loss function. The optimized size similarity formula is:

In the formula, h _i and w _i are the height and width of the i-th feature scale respectively; θ is a positive constant, which is the attenuation degree of size similarity; C is the baseline of size similarity, which is the parameter of the nonlinear transformation function; λ _i is the weight of the feature scale.

Further, step 4 also includes optimizing the IOU similarity formula of the CIOU loss function. The optimized IOU similarity formula is:

In the formula, Area(N _a ∩N _b ) is the intersection area of the two bounding boxes; Area(N _a ∪N _b ) is the union area of the two bounding boxes.

Further, step 4 also includes optimizing the parameters of the nonlinear transformation function. The formula of the parameters of the optimized nonlinear transformation function is:

In the formula, N is the number of feature scales; h _i is the height of the i-th bounding box; w _i is the width of the i-th bounding box.

Beneficial effects of the present invention:

The present invention proposes a noise reduction spatial coordinate attention (NR-SCA) mechanism. By combining the three mechanisms of Gaussian filter noise reduction technology, spatial attention and coordinate attention into one module, noise reduction, noise reduction, and coordinate attention can be considered at the same time. spatial location and nodule coordinate information simultaneously to achieve better performance. The above combination enables the model to more effectively detect pulmonary nodules in different locations and sizes, improving detection accuracy and robustness.

The present invention uses a bidirectional weighted feature pyramid network to simplify network parameters and make feature fusion more accurate.

The present invention combines the SNWD loss function constructed by size similarity and IOU similarity to more comprehensively consider the size and position information of objects in target detection. Dimensional similarity helps to handle the detection of objects of different sizes, while IOU similarity helps to accurately locate the location of objects. Combining them can better balance detection accuracy and location accuracy, and improve the detection efficiency of small target nodules.

Description of the drawings

The features and advantages of the present invention will be more clearly understood by referring to the accompanying drawings, which are schematic and should not be construed as limiting the invention in any way, in which:

Figure 1 is a schematic structural diagram of the YOLOv8 model of the present invention;

Figure 2 is a flow chart of a specific embodiment of the present invention;

Figure 3 is a schematic structural diagram of the NR-SCA module in a specific embodiment of the present invention;

Figure 4 is a schematic structural diagram of a bidirectional weighted feature pyramid (BiFPN) in a specific embodiment of the present invention.

Detailed ways

In order to make the purpose, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments These are some embodiments of the present invention, rather than all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without making creative efforts fall within the scope of protection of the present invention.

The embodiment of the present invention provides a YOLOv8-based pulmonary nodule image detection method. The specific process is shown in Figure 2, including the following steps:

Step 1: Obtain the lung image data set and preprocess the lung image data set;

The lung image data set used is LIDC-IDRI. This data set is a public pulmonary medical imaging data set, containing 1018 patients' CT scan images and corresponding pulmonary nodule lesion annotation results. The annotation results were analyzed by four experienced medical experts in two stages. Finish. Based on the linear attenuation coefficients of air, water and X-rays, the lung CT scan data is formatted and converted into JPG format.

Step 2: Perform lung parenchyma segmentation on the lung image preprocessed in step S1;

Since the U-Net segmentation network has a simple structure and lacks the extraction of local detailed features, it will lose fine target boundary information and lead to a decrease in segmentation accuracy. Therefore, the present invention uses a convolutional neural network (AttentionU-Net) combined with an attention mechanism to perform lung parenchyma analysis. Segmentation enables network learning to focus more on the area of interest, enhances the sensitivity and accuracy of the model, and allows the model to better identify target boundaries in different scenarios. The segmentation result can be expressed as:

S(X)＝Conv(Attention(X,C(E(X)))+D(C(E(X))),W)

In the formula, S(X) represents the final output of the segmentation network; Attention(X,C(E(X))) represents the output of the attention mechanism; Conv represents the convolution of the final segmentation layer; D(C(E(X)) ) represents the fast output of the decoder; W represents the convolution kernel.

Step 3: Add the NR-SCA module after the C2f convolution module in the backbone network of the YOLOv8 model. The Neck part of the YOLOv8 model is a BiFPN structure. Among them, the NR-SCA module includes: Gaussian filter, coordinate attention mechanism, spatial attention Force mechanism; the data input to the NR-SCA module enters the coordinate attention mechanism and the spatial attention mechanism respectively after passing through the Gaussian filter. The outputs of the coordinate attention mechanism and the spatial attention mechanism are element-wise multiplied with the output of the Gaussian filter respectively. The product The results are spliced and fused as the output of the NR-SCA module;

The output of the NR-SCA module is:

In the formula, is the feature map after noise reduction by Gaussian filtering; y _c is the feature map output by the coordinate attention mechanism; y _s is the feature map output by the spatial attention mechanism; ⊙ is the element-wise product.

Among them, the output of the Gaussian filter is:

In the formula, is the pixel at position (i, j) in the output feature map after filtering; X _{u, v, k} is the pixel value at position (u, v) in the input feature map; k is the number of feature map channels; (u, v ) is the position offset of the filter, (i,j) is the position offset of the output feature map; σ is the standard deviation of Gaussian filtering;

Among them, the output of the coordinate attention mechanism is:

In the formula, X _c (i, j) is the value of the input feature map on channel c, vertical position i and horizontal position j; is the channel attention weight in the vertical direction;/> is the channel attention weight in the horizontal direction;

Among them, the output of the spatial attention mechanism is:

In the formula,/> It is the feature map obtained by splicing the average pooling feature map and the maximum pooling feature map according to the channel dimension;

f ^7*7 is a 7×7 convolution operation on the spliced feature map; σ is the activation function.

Adding key features in the NR-SCA attention focused feature map to the backbone network can handle lung nodule detection more comprehensively by jointly considering noise reduction, spatial and coordinate information compared to single coordinate attention or spatial attention. tasks, improve detection performance, perception capabilities, and simplify the training and optimization process, helping to achieve more accurate and reliable pulmonary nodule detection. The specific structure is shown in Figure 3, including:

step1: Perform Gaussian filtering and denoising on the input feature map. There are impurities such as blood vessels and bones in lung CT images. Using Gaussian filter can help reduce noise in lung CT images, smooth the image and highlight pulmonary nodules. The specific formula is as follows:

In the formula, X _{i, j, k} represents the pixel at position (i, j) in the output feature map after filtering; X _{u, v, k} represents the pixel value at position (u, v) in the input feature map; k represents The number of feature map channels; (u, v) represents the position offset of the filter; (i, j) represents the position offset of the output feature map; σ is the standard deviation of Gaussian filtering. When Gaussian filtering is performed on the feature map, the kernel moves around each pixel of the image and performs a weighted average of surrounding pixels.

Step2: Input the feature map processed by Gaussian filtering into the coordinate attention branch. In order to improve the accuracy of nodule positioning and capture long-range spatial interaction through precise positioning information to avoid all spatial information being compressed into the channel, press the following The formula to decompose global average pooling is as follows:

In the formula, X _c (h, i) and X _c (j, w) represent the pixel value at the corresponding position of the c-th channel of the feature map; W represents the width of the feature map; The input feature map is average pooled in the W and H directions to obtain feature maps with sizes C×H×1 and C×1×W respectively. Then the generated C×1×W feature map is transformed, and then the concat operation is performed. The formula looks like this:

f＝δ(F ₁ ([z ^h ,z ^w ]))

In the formula, [z ^h , z ^w ] means connecting the average z ^h in the vertical direction and the average z ^w in the horizontal direction to form a new feature vector; F ₁ represents the forward direction of the neural network Propagation; δ represents the activation function, which can increase the expression ability of the neural network; f represents the final feature vector obtained after calculation by the neural network.

Step3: Coordinate attention can obtain spatial information in both horizontal and vertical directions, and integrate position information into the attention mechanism so that the model can enhance or suppress features based on the location of the lung nodule. Along the spatial dimension, f is split and separated, and then divided into f ^h ∈R ^c/r×H×1 and f ^w ∈R ^c/r×1×W , and then 1×1 convolution is used to perform dimension-raising operations respectively. Combined with the sigmoid activation function, the final attention vectors g ^h ∈R ^c×H×1 and g ^w ∈R ^c×1×W are obtained. The formula is as follows:

g ^h =σ(F _h (f ^h ))

g ^w =σ(F _w f ^w ))

In the formula, f ^h and f ^w represent the input feature vectors in the vertical and horizontal directions respectively, F represents the forward propagation of the neural network, σ represents the activation function, g ^h and g ^w represent the final vertical and horizontal directions after calculation by the neural network. Feature vector. Then g ^h and g ^w are expanded, and the final output of the coordinate attention branch can be expressed as follows:

In the formula, Represents the value of the input feature map on channel c, vertical position i and horizontal position j, /> Represents the channel attention weight in the vertical direction, /> Represents the channel attention weight in the horizontal direction.

Step 4: Spatial attention can learn the weight distribution of each channel, thereby enhancing the perception of the morphology and characteristics of pulmonary nodules and improving the feature expression ability of the model. The feature map processed by Gaussian filtering is input into the spatial attention branch, and two 1×H×W feature maps are obtained through maximum pooling and average pooling. The two feature maps are then spliced through the Concat operation, and 7× The 7-channel convolution becomes a 1-channel feature map, and then a sigmoid is used to obtain the spatial attention feature map. The specific formula is as follows:

In the formula, It means splicing the average pooling feature map and the maximum pooling feature map according to the channel dimension to obtain a richer feature map; f ^7*7 means performing a 7×7 convolution operation on the spliced feature map; σ means activation function.

Step5: Perform element-wise multiplication of the feature maps output by the coordinate attention branch and spatial attention branch with the original Gaussian denoised feature map to obtain the product feature map and then perform splicing and fusion in the channel dimension. The fused NR-SCA attention The force can play a synergistic effect among the three modules, reduce the information loss of multiple forward propagations between modules, retain global information, and obtain the final output feature map, that is:

In the formula, represents the feature map after Gaussian filtering and denoising, y _c represents the feature map output by the coordinate attention branch, y _s represents the feature map output by the spatial attention branch, and ⊙ represents the element-wise product.

Replacing the original PANet module with the BiFPN module can perform weighted fusion of different contributions of input feature maps of different resolutions. The BiFPN module removes nodes with only one input edge and adds an extra edge between the original input node and the output node of the same layer to fuse more features; processes each top-down and bottom-up path as a Feature network layer, and repeat the same layer multiple times to achieve higher-level feature fusion. The structure is shown in Figure 4.

The weighted feature fusion method is used in BiFPN to scale the range to [0,1], making the training fast and efficient. The expression is as follows:

In the formula, w _i represents the weight parameter, I _i represents the input feature map, O represents the fused feature map, and ∈ is a small constant used to prevent the denominator from being zero.

Step 4: Weight the size similarity formula of the NWD loss function and the IOU similarity formula of the CIOU loss function as the loss function of the YOLOv8 model;

The loss function of the constructed YOLOv8 model is:

Among them, the IOU similarity formula is:

In the formula, Area(N _a ∩N _b ) is the intersection area of two bounding boxes; Area(N _a ∪N _b ) is the union area of two bounding boxes;

The size similarity formula is:

In the formula, h _i and w _i are the height and width of the i-th feature scale respectively; C and θ are positive constants, which are the baseline and attenuation degree of size similarity respectively; λ _i is the weight of the feature scale;

The formula for the parameters of the nonlinear transformation function is:

In the formula, N is the number of feature scales; h _i is the height of the i-th bounding box; w _i is the width of the i-th bounding box.

SNWD models the bounding box as a 2D Gaussian distribution, and then uses Wasserstein distance to measure the similarity between the two Gaussian distributions. In order to balance the robustness and applicability of detecting nodules of different sizes, the similarity is divided into The two parts of IOU similarity and size similarity are input into the nonlinear adjustment function according to different proportions, and the average absolute size is used to adjust the exponential nonlinear transformation function to improve the detection efficiency of small target nodules. Specifically:

In step 1, for small targets, the foreground pixels within the bounding box are often distributed in the center, while the background pixels are distributed at the edges. In order to better weight each pixel within the bounding box, the bounding box can be modeled as a two-dimensional Gaussian distribution, the formula is as follows:

In the formula, C _x , C _y , w, h represent the center coordinates, width and height of the horizontal bounding box respectively.

Step 2 uses the Wasserstein distance in optimal transmission theory to calculate the distance between the two distributions, and uses the normalized index to obtain a new metric. Its second-order Wasserstein distance and normalized Wasserstein distance are respectively defined as:

In the formula, μ ₁ and μ ₂ represent two Gaussian distributions, m ₁ and m ₂ represent the means of the two distributions, and/> represents the square root of the covariance matrix of the two distributions, ||·|| ₂ represents the Euclidean distance, NWD(N _a , N _b ) represents the normalized Wasserstein distance, and C represents the positive constant related to the data set.

Step 3 In the NWD method, only the center position and width and height of the bounding box are considered. However, in fact, for lung nodule images, the bounding box may also contain other features, such as texture, color, shape, etc. Therefore, we can consider calculating the similarity between bounding boxes at different feature scales to describe the relationship between them more comprehensively. The IOU similarity expression and size similarity expression are respectively:

In the formula, Area(N _a ∩N _b ) represents the intersection area of two bounding boxes, Area(N _a ∪N _b ) represents the union area of two bounding boxes, h _i and w _i represent the i-th feature scale respectively. The height and width of , C and θ are positive constants, indicating the baseline and attenuation degree of size similarity respectively, λ _i is the weight of the feature scale, indicating the contribution of this position to the calculation of size similarity.

Step4 The size relationship of the target is crucial for correct positioning and recognition. In order to better consider the average size of the target in the data set and make the size similarity calculation more reasonable to improve the generalization ability of the model, by calculating the height of all bounding boxes The parameter C of the new exponential nonlinear transformation function is constructed by taking the average of the sum of the width and the width, and the formula is as follows:

In the formula, N represents the number of feature scales, h _i represents the height of the i-th bounding box, w _i represents the width of the i-th bounding box, and the expression of SNWD is obtained through weighted calculation:

In the formula, S _IOU (N _a , N _b ) represents the IOU similarity between Na _and N _b , W _θ (N _a , N _b ) represents the size similarity between _Na and N _b , α and β are adjustment parameters, And the value of α is (0,1), and the value of β is (1, +∞), which is used to balance the impact between IOU similarity and size similarity.

Step 5: Use the lung images obtained in step 2 to train the constructed YOLOv8 model, and use the trained Yolov8 model to detect and identify pulmonary nodules on the pulmonary nodule images.

By improving the yolov8 model, the structure is shown in Figure 4, it can reduce noise interference, enhance coordinate attention to local features, simplify the fusion network parameters, effectively improve the detection efficiency of small pulmonary nodules, and reduce the missed detection rate, thereby achieving Rapid and accurate detection of pulmonary nodules.

Although the embodiments of the present invention have been described in conjunction with the accompanying drawings, those skilled in the art can make various modifications and variations without departing from the spirit and scope of the invention. Such modifications and variations are covered by the appended claims. within the limited scope.

Claims (10)

1. A YOLOv8-based pulmonary nodule image detection method, which is characterized by including the following steps: Step 1: Obtain the lung image data set and preprocess the lung image data set; Step 2: Perform lung parenchyma segmentation on the lung image preprocessed in step S1; Step 3: Add the NR-SCA module after the C2f convolution module in the backbone network of the YOLOv8 model. The NR-SCA module includes: Gaussian filter, coordinate attention mechanism, and spatial attention mechanism; input to the NR-SCA module After the data is passed through the Gaussian filter, it enters the coordinate attention mechanism and the spatial attention mechanism respectively. The outputs of the coordinate attention mechanism and the spatial attention mechanism are element-wise multiplied with the output of the Gaussian filter. The product results are spliced and fused as the NR-SCA module. Output; Step 4: Weight the size similarity formula of the NWD loss function and the IOU similarity formula of the CIOU loss function as the loss function of the YOLOv8 model; Step 5: Use the lung images obtained in step 2 to train the constructed YOLOv8 model, and use the trained Yolov8 model to detect pulmonary nodule images. 2. The YOLOv8-based pulmonary nodule image detection method as claimed in claim 1, characterized in that in step 3, the output of the NR-SCA module is: In the formula, is the feature map after noise reduction by Gaussian filtering; y _c is the feature map output by the coordinate attention mechanism; y _s is the feature map output by the spatial attention mechanism; ⊙ is the element-wise product. 3. The YOLOv8-based pulmonary nodule image detection method as claimed in claim 1 or 2, characterized in that, in step 3, the output of the Gaussian filter is: In the formula, is the pixel at position (i, j) in the output feature map after filtering; X _{u, v, k} is the pixel value at position (u, v) in the input feature map; k is the number of feature map channels; (u, v ) is the position offset of the filter, (i, j) is the position offset of the output feature map; σ is the standard deviation of Gaussian filtering. 4. The YOLOv8-based pulmonary nodule image detection method as claimed in claim 1 or 2, characterized in that in step 3, the output of the coordinate attention mechanism is: In the formula, X _c (i, j) is the value of the input feature map on channel c, vertical position i and horizontal position j; is the channel attention weight in the vertical direction;/> is the channel attention weight in the horizontal direction. 5. The YOLOv8-based pulmonary nodule image detection method as claimed in claim 1, characterized in that in step 3, the output of the spatial attention mechanism is: In the formula, It is the feature map obtained by splicing the average pooling feature map and the maximum pooling feature map according to the channel dimension; f ^7*7 is a 7×7 convolution operation on the spliced feature map; σ is the activation function. 6. The YOLOv8-based pulmonary nodule image detection method according to claim 1, wherein the step 3 further includes: the Neck part of the YOLOv8 model is a BiFPN structure. 7. The YOLOv8-based pulmonary nodule image detection method as claimed in claim 1, characterized in that in step 4, the loss function of the constructed YOLOv8 model is: In the formula, S _IOU (N _{a ,} N _b ) represents the IOU similarity between Na and N _b ; W _θ (N _a , N _b ) represents the size similarity between Na _and N _b ; C is the nonlinear transformation function. Parameters; α and β are adjustment parameters, the value of α is (0,1), and the value of β is (1, +∞). 8. The YOLOv8-based pulmonary nodule image detection method as claimed in claim 7, wherein the step 4 further includes optimizing the size similarity formula of the NWD loss function, and the optimized size similarity formula is : In the formula, h _i and w _i are the height and width of the i-th feature scale respectively; θ is a positive constant, which is the attenuation degree of size similarity; C is the baseline of size similarity, which is the parameter of the nonlinear transformation function; λ _i is the weight of the feature scale. 9. The YOLOv8-based pulmonary nodule image detection method as claimed in claim 7, wherein the step 4 further includes optimizing the IOU similarity formula of the CIOU loss function, and the optimized IOU similarity formula is : In the formula, Area(N _a ∩N _b ) is the intersection area of the two bounding boxes; Area(N _a ∪N _b ) is the union area of the two bounding boxes. 10. The pulmonary nodule image detection method based on YOLOv8 as claimed in claim 7, characterized in that, the step 4 also includes optimizing the parameters of the nonlinear transformation function, and the formula of the parameters of the optimized nonlinear transformation function for: In the formula, N is the number of feature scales; h _i is the height of the i-th bounding box; w _i is the width of the i-th bounding box.