Methods for uncertainty estimation.

CNNs have some shortcomings despite their progress in a wide range of applications. One of the concerning drawbacks among them is its inability to provide a notion of uncertainty in its prediction, which is crucial in the medical domain [48]. For example, in a case where the CNN model was trained on a range of car data in order to predict the category to which the given car belongs, the hypothetical model should return a prediction with a high level of confidence. But what if the model is validated with a bike image and asked to choose a car category? This is a situation where the test data is far from the distribution as the model is trained on distinguishing among various car classes and have never seen the image of a bike. In such cases, the model is expected to return a prediction and some additional details communicating the high degree of uncertainty with these kinds of data. Uncertainty estimate can be used to assess samples which are difficult to identify, thereby requiring a further expert review, and to detect samples that deviate from the data used to train the model.

There are mainly two types of uncertainty viz., Aleatoric and Epistemic uncertainty [49]. Aleatoric uncertainty captures noise inherent in the data and cannot be abated by collecting more data [50]. Epistemic uncertainty, also known as model uncertainty, accounts for variability in the parameters of the model and analyzes what the model is not aware owing to the lack of training data [42]. Epistemic uncertainty is helpful to understand examples that vary from training data especially in situations where we have small and imbalanced datasets, which is common in CAD systems [50].

Uncertainties are formulated as probability distributions over the model parameters (for epistemic uncertainty) or model inputs (for aleatoric uncertainty) [42]. Bayesian statistics have largely inspired most of the work done till now on uncertainty estimation techniques. Bayesian Neural Network (BNN) [51] is the probabilistic variant of the traditional neural networks and provides a mathematical framework for uncertainty estimation. Most of the earlier works on epistemic uncertainty estimation are based on Bayesian inference. However, in practice, Bayesian inference is computationally expensive; therefore, extensive research has been done in developing various techniques to approximate Bayesian deep networks although they are not scalable for larger convolutional networks [52–55]. Research has also been carried out to develop alternative strategies, which are suitable for approximating the uncertainty [42, 56]. The work proposed by [42] demonstrated how dropout [57] applied on a neural network with an arbitrary number of layers is mathematically equivalent to estimating variational inference in Gaussian process model [58]. This was later extended to CNNs in [59] explaining that dropout can be used to enforce a Bernoulli distribution over the weights of the CNN without any additional model parameters. This method is known as Monte Carlo (MC) Dropout and is successfully employed in some of the applications in the medical imaging domain [26, 60].

The dropout layers are generally added in many deep neural networks to reduce overfitting by randomly dropping weights with a fixed probability. Inspired by the capability of the MC-Dropout technique in estimating uncertainty, we employ the same to build our proposed model. Given a test sample s^*, we sample the network B times over its parameters and thereby giving an estimate of the predictive posterior distribution. This sampling is known as Monte Carlo sampling and the mean μ_e over these iterations is considered as the final result on a given test sample. μ_e is computed as shown in the equation below [42] (1) where denotes the weights of the network with dropouts in B^th MC iteration and B is the total number of sampled sets of weights. Among several classes y*, the one with μ_max is selected as the outcome for each test sample s*.

Aleatoric uncertainty captures noise inherent in the data and cannot be abated by collecting more data [50]. Aleatoric uncertainty can be estimated either by learning a mapping directly from the input data [50] or by test-time data augmentation [41, 61, 62]. However, the former technique suffers from the drawback, as is it involves adapting the network architecture and loss function, which restricts the application to trained models. Therefore, we employ test-time data augmentation technique in our pipeline. In this approach, a test sample s* is augmented to form V different versions of the image and is forwarded to the network. The mean μ_a over these iterations is considered the final result of a given test sample. μ_a is computed as shown in the equation below [50] (2) where denotes the v^th augmented image, denotes the weights of the network and V is the total number of image augmentations. Among several classes y*, the one with μ_max is selected as the outcome for each test sample s*.

These two approaches are then combined to calculate the overall uncertainty where a test sample s* is augmented to form M different versions of the image and is forwarded to the network with the dropout activated during the test time. The mean μ over these iterations is considered as the final result on a given test sample. μ is computed as shown in the equation below [41, 61] (3) where denotes the augmented image passed and denotes the weights of the network with dropouts during the m^th iteration and M is the total number of iterations. Among several classes y*, the one with μ_max is selected as the outcome for each test sample s*.

In order to estimate the model uncertainty φ, we calculate the entropy of the averaged probability vector across the N classes using the equation as given below [41, 61, 62] (4) here p_n is the probability of nth class.

Methods for explainability.

CNNs lack interpretability, which is an essential requirement in the medical domain due to the possibility of life-threatening consequences. A medical practitioner needs to understand the key features in the image used by the given model to make predictions to verify if it is consistent with medical knowledge and build trust in the model’s capability. While interpretability is desirable in all domains, since medical practitioners have to deal with medico-legal, ethical, and strict regulations it becomes all the more essential in the medical domain. Recently, there has been considerable research on saliency methods that relate CNNs prediction to the inputs that have maximum influence on the prediction. These techniques may be useful in a variety of ways, including tracking a model’s assessment, ensuring that the model does not learn false correlations, and evaluating the model for issues related to fairness [63, 64].

Saliency based methods can broadly be classified into two categories. One collection of methods modifies the input and computes the effect of this change on the output by making a forward pass through the network using these altered inputs [65, 66]. The other set of approaches calculate attributions by returning the prediction score back to the input features through each layer of the network. In general, second category methods are faster than the initial set of methods, as they usually require a single or constant number of neural network queries [67]. Guided Backprop [68], Grad CAM [64], Guided GradCAM [64] and XRAI [67] are some of the promising approaches in this category. Therefore we explore these techniques in our approach to bring model interpretability in the context of skin lesion detection.

Guided Backpropagation is a technique for visualizing CNN by slightly modifying the backpropagation algorithm wherein the negative gradients are set to zero in each layer, allowing only positive gradients to flow backwards through the network. Guided Backpropagation is a combination of Backpropagation and deconvolution. During forward pass, due to the presence of ReLU activations, all the negative input values passed through neurons are set to zero. Therefore, during the backward pass of Backpropagation, the gradients don’t flow back through these neurons. In deconvolution, during the backwards pass all the negative gradients are suppressed to zero. In guided Backpropagation, both the negative gradients and the gradients with negative input are suppressed to zero. The rationale behind this modification is that all the positive gradients of higher magnitude imply key pixels, while negative gradients denote the pixels the model wants to suppress.

Grad-CAM provides a visual explanation by leveraging the gradient information coming into the final convolutional layer. The last layer is chosen as it provides the best tradeoff between detailed spatial information and high-level semantics [64]. It considers the convolutional layer since the convolutional features generally possess spatial information. The key pixels responsible for categorising a particular class are determined by forward propagation through the network by obtaining gradients for each class. The gradients during backpropagation are average-pooled to obtain the weights that are important for the target class prediction. The weights obtained are combined with activations maps using ReLU operation to compute the Grad-CAM heatmap. To generate a Grad-CAM heatmap , of width w and height h for a class of interest e, the gradient of the score of class e, z^e is calculated with respect to the feature maps F^a i.e. Then these gradients are global average pooled to get the neuron importance weights using the equation given below [64]: (5) correspond to the global average pooling. The represents a partial linearization [64] of the network downstream from F for a class of interest e.Finally to obtain the heatmap, a weighted combination of forward activation maps is computed followed by a ReLU as given below [64]. (6)

Guided Grad-CAM overcomes the drawback of Grad-CAM, which is the inability to show fine-grained importance like Guided Backpropagation, a pixel-space gradient visualization method [64]. Guided Grad-CAM is the blend of Guided Backpropagation and Grad-CAM algorithms via pointwise multiplication to incorporate the advantages of both methods. First, the heatmap of an input image is obtained via Grad-CAM. Then, this heatmap is upsampled to the input image resolution. Finally, it is pointwise multiplied with Guided Backpropagation to get Guided Grad-CAM visualization. The resultant visualization has high resolution and is class discriminative.

XRAI is the most recently proposed saliency method based on Integrated Gradients [69] that decides the key inputs by changing the network input from baseline to the original input and consolidating these gradients. It begins by segmenting the image using Felzenswalb’s graph-based segmentation [70] technique, followed by repeated testing of the significance of each segment using attributions. Integrated gradients are used as attribution with black and white baselines to resolve their setback as they are insensitive to pixels similar to or equal to the baseline image. Thus, every pixel gets an equal chance to contribute to the attributions regardless of the distance from the baseline. Finally, it merges regions with a higher positive value of the sum of all the attributions of that region until it has the complete image as the mask or runs out of regions to add [67].

Methods for segmentation.

In medical image analysis, some pixels in the image contain vital information that might play a crucial role in decision-making, thereby providing a rationale for the treatment. Segmentation would help in augmenting the classification model performance in most cases and, furthermore, would reduce the computation time [71]. In the latter part of the last decade, CNN based segmentation algorithms performed well in biomedical image segmentation tasks. More importantly, U-Net [72] has emerged as one of the most promising architecture in this domain and has been applied to various image segmentation tasks [73–75].

U-Net defined the state of the art in the medical image segmentation tasks [76], however it is not robust enough to analyze objects in the image present at different scales. One of the novel ideas of U-Net architecture has been the implementation of shortcut links between the corresponding layers before the max-pooling and after the deconvolution operations, to relay the spatial information that gets lost from encoder to decoder during the pooling process. The dispelled spatial features though retained, still suffers from shortcomings in the skip connections i.e., there is a plausible semantic gap between the two sets of features being merged. The features from the encoder are supposed to be lower-level features, and on the contrary, the decoder features are of much higher level because they come from deeper layers after fairly complex computation [77].

In order to tackle the shortcomings discussed above, [77] proposed few structural changes in the form of ‘MultiRes block’ and ‘Res path’ to the U-Net architecture drawing inspiration from [76, 78, 79]. Inspired by the sucessful working of MutiRes block and Res path structures, we employ it in our segmentation architecture known as Bayesian MultiResUNet. Similar to the Inception blocks [80], where convolutional layers of different kernel sizes are adopted to inspect the points of interest in images from different scales, MultiRes blocks employs 3 × 3, 5 × 5 and 7 × 7 filters in parallel with the larger and computationally expensive 5 × 5 and 7 × 7 blocks factorized as a succession of 3 × 3 without affecting the objective function [78]. Additionally, MultiRes blocks contain 1 × 1 convolutional layers, for better comprehension of spatial information as shown in Fig 2. Rather than just concatenating the feature maps from the encoder stages to the decoder stages as in the shortcut connection of U-Net, Res paths transfers them through a chain of convolution layers with residual connections and then concatenates them with the decoder features to mitigate the gap between encoder and decoder features. Res path is represented in Fig 3 below.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. MultiRes block: The rounded rectangle represents a concatenation operation where the black block represents a 3 × 3 convolution, the green block represents a 5 × 5 convolution and the red one represents a 7 × 7 convolution.

Finally a skip connection is added along with 1×1 filter.

https://doi.org/10.1371/journal.pone.0276836.g002

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. Res path: The encoder features are passed through a series of convolutions instead of linearly connecting them to the decoder features.

https://doi.org/10.1371/journal.pone.0276836.g003

As depicted in Fig 4, Bayesian MultiResUNet has symmetric architecture where the encoder is responsible for extracting spatial features from the input image while the decoder produces the segmentation map using the encoded features. In the encoder, the weights obtained from the MultiRes block are passed to a pooling block where a dropout layer is appended after the pooling operation and these acquired weights are used as an input to the next MultiRes block. The fifth MultiRes block acts as a bridge between encoder and decoder with three 3 × 3 convolution operations followed by one 1 × 1 convolutional operation. On the other hand, the decoder begins at the upsampling block, which incorporates 2 × 2 transposed convolution operation [81] to perform upsampling thereby reducing the feature channels by half. These weights are then passed on to the MultiRes block, similar to the encoder. This succession of upsampling and MultiRes operations is repeated four times, reducing the number of filters by two at each stage. Finally, a 1 × 1 convolution operation is performed to generate the segmentation map. As we step towards the inner shortcut routes, the intensity of the semantic gap between the encoder and the decoder function maps would possibly decrease; thus we reduce the number of convolutional blocks, i.e., we employ 4, 3, 2, 1 convolutional blocks respectively along the four Res paths. We use 32, 64, 128, 256 filters in the blocks of the four Res paths respectively to compensate for the number of feature maps in encoder-decoder similar to [77]. ReLU [82] activation function and batch-normalization [83] are employed by all convolutional layers in this architecture, except for the final one which uses a Sigmoid activation function.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 4. Bayesian MultiResUNet comprises an encoder and a decoder pathway, with skip connections and dropout layers between the corresponding layers in pooling and upsampling blocks.

https://doi.org/10.1371/journal.pone.0276836.g004

Methods for classification.

In our experimentation, we initially trained various off-the-self Classification architectures like Inception [80], Xception [84], VGG-19 [85], DenseNet-169 [86] and ResNet-50 [87]. Furthermore, we have selected the top two models and obtained the Bayesian version of these networks by adding dropouts. The dropout can even degrade the performance of the model; therefore we empirically evaluate the performance of several Bayesian models with various configurations, which include the positioning of dropout layers as well as the dropout rate, to identify those with the best performance of prediction for the skin lesion classification task. Moreover, all the Bayesian networks employed in our analysis are approximate Bayesian models, as the exact Bayesian inference for neural networks is computationally intractable.

Metrics for classification.

For classification, we employ accuracy metric and F1-score as given in the equations below. (8) Here T_p represents True Positives, i.e, the number of samples correctly predicted as belonging to a given class. True negatives is given by T_n, which denotes the number of samples correctly identified as not belonging to a given class. F_p and F_n denote the false positive and false negative sample predictions respectively.

F1 score is the harmonic mean of precision and recall [89]. Recall is defined as the number of true positives T_p over the number of true positives T_p plus the number of false negatives F_n [90] while Precision is given by the number of true positives T_p divided by the number of true positives T_p plus the number of false positives F_p [90]. F1−score is a more robust metric to evaluate the classification performance as it takes into consideration the class imbalance problem by giving equal importance to precision and recall, thus involving both false positives and false negatives. For classification tasks where both precision and recall are of high significance, F1-score should be maximized. F1 score ranges between 0 and 1, reaches the best value of 1 when the balance between precision and recall is perfect. F1−score is calculated as given below. (9)

Metrics for segmentation.

We have employed commonly used metrics such as Dice coefficient (DI) and Jaccard index (JI) to quantify image segmentation efficiency. Both these metrics essentially measure the similarity between the ground truth and the predicted segmented image in terms of the extent of overlap between the two images. The Dice coefficient (DI) is given by (10) The Jaccard index is given by: (11) where M represents the ground truth of segmentation, which is normally a manually-identified salient region, and C represent a mask.

Metrics for uncertainty.

Uncertainty is measured using monte carlo dropout and test-time data augmentation. As mentioned in the above section, we calculate uncertainty φ but the range of these values would vary depending on the number of Monte Carlo samples. Hence we calculate normalised uncertainty φ_norm where φ_norm ∈ [0, 1] [43]. (12)

To split the predictions into certain and uncertain categories, we set a threshold φ_T ∈ [0, 1] where a prediction is deemed to be certain if φ_norm < φ_T and uncertain if φ_norm > φ_T.

When it comes to classification, we usually end up with 4 kinds of predictions i.e incorrect-uncertain (iu), correct-uncertain (cu), correct-certain (cc), and incorrect-certain (ic) predictions, where incorrect-uncertain(iu) refers to a prediction that was incorrect and the model was uncertain about it. Correct-uncertain(cu) refers to one where the model prediction is correct but the model is uncertain about it. The remaining correct-certain(cc) and incorrect-certain(ic) refer to predictions that were correct or incorrect but the mode is certain. The overall accuracy of the uncertainty estimation could be expressed as a ratio of all the desirable cases i.e correct-certain (cc) and incorrect-uncertain (iu), and all the possible cases. This diagnostic accuracy(A) can be represented in the form [43] (13) where L represents the count for each possible combination.

Metrics for explainability.

Different explainability techniques like Grad-Cam, Guided Backprop, Guided Grad-Cam, and XRAI have been discussed in the previous section. To compare the performance of these techniques, we have used the bokeh effect and measured the accuracies as mentioned in [67]. The basic intuition behind this analysis is that if the above explainability techniques identify important pixels to the model’s prediction, then the model’s output of the original image and reconstructed image must go hand in hand [67]. Therefore, the bokeh effect is used to reconstruct the image, in which initially the original image is blurred and the important pixels given by the explainability techniques are added. This is done for the entire test set. Later the resultant images are passed through the classification model. The explainability techniques used are thus compared using the prediction accuracy of the classification algorithm on these reconstructed images.