1 Introduction
Skin cancer is the most common type of cancer in the world. Early detection of skin cancer can increase the five year survival rate of patients from 18% to 98% [1]. While skin cancer can be detected by visual examination, distinguishing malignant from nonmalignant lesions is a challenging task. In recent years, computer aided diagnosis has been widely leveraged in automated assessment of dermoscopy and clinical images to assist dermatologists evaluation. Semantic segmentation, the task of labeling each image pixel with the class label of its surrounding object, is generally the first step toward the automatic understanding of images. Remarkable variations in the appearance of healthy and unhealthy skin, including color, texture, lesion shape and size originating from image acquisition and inter and intraclass variation, complicates the skin lesion segmentation problem.
For decades, since the seminal work of Kass et al. [13], energy functional minimization techniques were the most popular approaches to solve image segmentation problems [15]. Imaging artifacts and variability in the appearance of image regions make the data fidelity term insufficient to achieve robust segmentation results. Therein, the segmentation that minimizes a weighted sum of unary (data) and regularization energy functional terms is sought. Incorporating prior knowledge about the structure of target object in the objective function to regularize plausible solutions with anatomically meaningful constraints have been widely leveraged to obtain more reliable delineations [10, 17]. Active shape models (ASM) was one of the pioneering works to incorporate shape priors into deformable models [9]
. To effectuate the shape prior, ASM and many other shapeencoding segmentation methods required an estimate of the object pose (i.e., the orientation, scale, and location of the target object in the image)
[11, 22]. Some examples of priors which have been utilized in energy optimization based segmentation methods are shape models, topology preservation, moment constraints and geometrical and distance interaction between image regions.
Recently deep fully convolutional networks have achieved significant success in the task of semantic segmentation. Hierarchical extraction of features followed by skip connections and upsampling operations was first introduced by Long et al. in an endtoend trainable framework [14]
. Despite the success of FCNs, they have indicated clear limitations in the dense perpixel prediction task. Consecutive spatial pooling and striding convolutions in FCNs reduce the initial image resolution and lead to loss of the image fine structures. Some techniques have been proposed to address these limitations of FCNs. Learning multiple deconvolutional layers and concatenating lowlevel fine features with highlevel coarse features through skip connections are commonly used to retrieve lowlevel visual features
[20]. Dilated convolutional has also been introduced to aggregate multiscale contextual information without losing image resolutions [24]. Although pixelwise prediction benefits from these resolution enlarging techniques, they are only capable to partially recover detailed spatial information.In the context of fully convolutional networks, leveraging prior information about the target object structure in the segmentation model has not been widely studied. By optimizing individual pixel level class predictions in the FCNs loss function, independent class labels are assigned to image pixels without considering highlevel label dependencies. There have been some efforts towards structured prediction and leveraging meaningful priors into deep learning frameworks. DeeplabCRF and CRFRNN employ probabilistic graphical modeling either as a post processing step or by implementing recurrent layers in FCNs to enforce assigning similar labels to pixels with similar color and position and further improve the object boundaries [6]. Recently BenTaieb et al. proposed a new loss function to encode the geometrical and topological priors of containment and detachment in an endtoend FCN framework [2, 27]
. To leverage the shape prior in segmentation models, Chen et al. learn a shape constraint by a deep Boltzmann machine and then employ the learned prior in a variational segmentation method
[5]. In addition, training convolutional autoencoder networks to learn anatomical shape variations has demonstrated improvements in the robustness of FCN segmentation models [18, 19].To the best of our knowledge, none of the existing works incorporates a star shape prior as a regularization term in the loss function of FCNs trained in an endtoend fashion. The star shape prior was first introduced in the context of image segmentation by Veksler, where it was encoded as a regularization term into the cost function formulation of a graphbased (discrete) image segmentation approach [21]. Later, Chittajallu et al. incorporated three types of shape constraints including star shape prior into a Markov random field based segmentation model and applied their method to noncontrast cardiac computed tomography scans [7]. Yuan et al. extended the star shape prior to 3D objects and applied it to prostate magnetic resonance images [25]. Nosrati et al. derived a star shape prior in a continuous variational formulation and applied it to segmenting overlapping cervical cells [16]. Although the star shape prior clearly improved results for a variety of target objects, one limiting requirement of Veksler’s approach and its variants, however, is the assumption that the center of foreground objects is known (e.g. provided by user interaction).
We aim to harness the powerful proven capabilities of deep learning in automatically extracting learnt (i.e., not handcrafted) pixeldriven image features (i.e., likelihood) and augment it with demonstrably useful shape priors without requiring the knowledge of the target object pose. We propose to encode the star shape prior into the training of fully convolutional networks to improve segmentation of skin lesions from their surrounding healthy skin. Our idea is to formulate the star shape prior in the loss function of FCN frameworks to penalize nonstar shape segments in prediction maps and preserve global structures in the output space. Integration of the star shape prior in the loss function makes it possible to train the whole FCN framework in an endtoend manner. In contrast to Veksler’s work and its variants, our approach to star shape prior in a deep learning setting not only eliminates the need for manually setting object centers, but also alleviates, at inference time, the computationally intensive optimization associated with the energy minimizing approaches. Our experimental results illustrate how imposing the shape prior constraint in deep networks refines skin lesion segmentation in comparison to using a single pixel level loss in FCNs.
2 Methodology
Our goal is to leverage the star shape prior into the learning process of an FCN to generate plausible segmentation maps (e.g. skin lesions) from their surrounding background without requiring additional training, user interaction, pre or postprocessing.
FCN’s pixelwise loss In FCNs, given a set of training images and their corresponding ground truth segmentations,
, the deep network learns to take unseen image samples and generate a segmentation probability map, the same size as the input images that assigns a semantic label to each pixel. Learning the deep network parameters
, is performed by maximizing the a posteriori probability of giving the true label to each image pixel given the input image. Maximizing the a posteriori probability is usually replaced by minimizing its negative loglikelihood function as a cost function :(1) 
For binary dense class prediction, a binary cross entropy loss is generally deployed:
(2) 
where is the pixel space, is the ground truth label of pixel in image and
is the FCN sigmoid function output indicating the predicted probability of the
pixel of the image being a skin lesion. The pixelwise binary logistic loss penalizes the deviation of the predicted label for each pixel from its true label.Star shape regularized loss Assuming is the center of object , object is a star shape object if, for any point interior to the object, all the pixels lying on the straight line segment connecting to the object center are inside the object (Fig. 1(a)). This definition of star shape prior holds for a large group of object shapes including convex ones. To incorporate the star shape prior as a new regularization term, we augment the loss function in (2) with a new loss term to penalize line segments that violate the prior (e.g. Fig. 1(b)) in the prediction maps:
(3) 
where and are hyperparameters setting the contribution of each term in the optimization function, is the binary cross entropy loss and is our star shape prior:
(4) 
(5) 
where is the line segment connecting pixel to the object center and is any pixel incident on line . is trained to assign to all such pixels a label identical to the label of pixel as long as (i) and have the same ground truth labels () and (ii) the difference between the ground truth label and the predicted labels for is nonzero (). The 3rd term of (4) determines how labels of pixels internal to the lesion are penalized to ensure star shapes, whereas the first two terms of (4) are designed to allow discontinuities of pixel labels across the ground truth boundary of the lesion and ignore the star shape term when the given label is true. In Fig. 1(c), and are examples where the value of is positive while their assigned labels should not be penalized. Condition (i) chooses a set of pixels on and allows discontinuities between the background () and foreground assigned labels and, condition (ii) enforces the loss function not to penalize the label assigned to .
In our implementation of 4, instead of penalizing the difference between the predicted probabilities and ground truth labels for all the points on the straight line , we only examine the closest pixels to on and compute the loss value per pixel based on those predicted probabilities. We also quantize, to a set of directions, the possible angles of all lines passing through .
In the training of our deep network, we automatically find the star object center from binary ground truth maps. At inference time, we do not need to supply the center of star objects as prediction maps are achieved by a forward pass through the network whose parameters are already trained to generate segmentations.
3 Experiments
Data description We validated our proposed segmentation approach on dermoscopy data provided by the International Skin Imaging Collaboration (ISIC) at ISBI Skin Lesion Analysis Towards Melanoma Detection Challenge [8]. The data set contains training, validation, and test images. We first rescaled all images to
pixels and normalized each RGB channel by the mean and standard deviation of the training data. To confirm the suitability of adopting the starshape prior for this task, we calculated the percentage of segmentation mask pixels that violate the star shape definition to be only 0.14% over the whole dataset (0.05% of training, 0.3% of validation, and 0.38% of test image pixels). Fig.
2 shows examples of rare pixels where the star shape constraint is violated.Network architecture We exploited two stateoftheart fully convolutional network architectures to evaluate our proposed new loss: 1)UNet[20]
2)ResNetDUC. ResNetDUC deploys the FCN version of ResNet152, pretrained on ImageNet as an encoder
[12]. Instead of using multiple deconvolutional layers to decode low resolution feature maps into the original image size prediction maps, single Dense Upsampling Convolution (DUC) layer is used to reconstruct finedetailed information from coarse feature maps [23]. Furthermore, dilated convolutions are used in the encoder to benefit from multiscale contextual information from previous layers activations [24].Implementation
We trained deep networks implemented with the PyTorch library, over minibatches of size 12. We tuned all hyperparameters on the validation set. Loss functions are optimized using the stochastic gradient descent algorithm with an initial learning rate of
. The learning rate was divided by when the performance of model on validation data set stopped improving. Momentum and weight decay were set to and , respectively. For the implementation of the star shape regularized loss function, , , and . We first trained the deep network with binary cross entropy function for epochs and then finetuned the network parameters with the proposed loss function. Training takes 2 days and test takes 1 sec/image on our 12 GB GPU.Results We evaluated the performance of UNet and ResNetDUC trained with and without the star shape prior. As shown in Table 1
, using our shape regularized loss function in the training of UNet and ResNetDUC, the Jaccard index is improved by more than
(row A vs. B and row C vs. D). We measured the statistical significance of our results by exploring the Jaccard index over the test data. We used the nonparametric Wilcoxon signed rank sum test and found that the results of UNet and ResNetDUC with and without incorporation of star shape prior are statistically significantly different at .We compared our proposed method with competing methods participating in the challenge. The ResNetDUC architecture trained with our star shape regularized loss achieved the first rank based on the challenge ranking metric, Jaccard index. Table 1, rows E, F and G, show results of the first three ranked teams. Although all top three teams used FCNs to perform image segmentation, in contrast to our work, they employed various additional steps like averaging over multiple model results, multiscale image input as well as pre and postprocessing approaches like inclusion of different color spaces in the input and multithresholding. Qualitative results of our proposed approach are presented in Fig. 3. Encoding star shape prior into the loss function results in smoother prediction maps with a single connected component as lesion for most cases.
Method  Jaccard  Dice  Accuracy  Specificity  Sensitivity  

A  UNet [20]  70.5  79.7  91.8  97.8  77.0 
B  UNet + Star Shape  73.3  82.4  92.4  95.3  85.4 
C  ResNetDUC [23]  74.0  83.3  93.00  98.2  80.0 
D  ResNetDUC + Star Shape  77.3  85.7  93.8  97.3  85.5 
E  Yuan et al. [26]  76.5  84.9  93.4  97.5  82.5 
F  Berseth et al. [3]  76.2  84.7  93.2  97.8  82.0 
G  Bi et al. [4]  76.0  84.4  93.4  98.5  80.2 
4 Conclusion
We encoded the star shape prior in the loss function of an endtoend trainable fully convolutional network to generate more accurate and plausible skin lesion segmentations. In contrast to energy minimization approaches, our proposed framework does not require computationally expensive optimization at inference time nor a userdefined object centre. Our experiments indicated that leveraging the prior knowledge in fully convolutional networks yield convergence to an improved output space. In future works, we will extend to other prior information including but not limited to anatomically meaningful priors in fully convolutional networks trained for other 2D and 3D medical imaging applications.
Acknowledgments. We gratefully thank NVIDIA Corporation for the donation of the Titan X GPU used for this research.
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