## 1 Introduction  
Deep convolutional neural networks have driven significant advancements in medical image analysis [1, 2], substantially improving segmentation accuracy [3, 4]. Numerous studies have demonstrated substantial enhancements and achieved cutting-edge performance such as cardiac [5, 6, 7] and abdominal [8, 9] imaging. However, most deep learning systems rely on large labeled datasets to mitigate overfitting and ensure reliable generalization to unseen test data. Acquiring such extensive medical annotations poses significant challenges, as it requires domain expertise met only by well-trained specialists. Manual segmentation by radiologists is often inconsistent, resource-intensive, and time-consuming. Progress remains reliant on accessing abundant high-quality ground truths that encapsulate anatomical intricacies. Mitigating the dependence on exhaustive annotations for training medical imaging models has garnered significant interest recently [10, 11, 12]. Our work targets semi-supervised learning to optimize segmentation using limited expert-labeled data and abundant unlabeled data. This approach facilitates economizing human effort while maximizing the utilization of all available samples [13]. Recent semi-supervised advances for medical segmentation focus on leveraging unlabeled data with consistency regularization. Specifically, mean teacher architectures promote consensus between student and teacher models for perturbed inputs [14]. Mean teacher paradigms utilize student-teacher frameworks to promote consensus between two networks. Alternatively, auxiliary tasks predicting boundary maps can provide additional constraints [15; 16]. However, current approaches overlook inter-sample correlations and treat labeled and unlabeled data separately. Our core innovation lies in concurrently exploiting both annotated and unannotated images by deeply coupling their representations to share learned semantic patterns. We operationalize this via transformers within a mutually connected multi-task network. Motivated by multi-task learning's proven effectiveness in many applications of computer vision [12; 17], we propose concurrently learning complementary shadow regions and shadow edge predictions to improve shadow detection using both global and local information [18]. In order to leverage multi-task learning principles, we synergistically combine a coarse global shadow region task with a fine-grained localized shadow edge task, allowing the model to integrate signals across multiple granularities and leverage both region-level and boundary-level cues to boost overall shadow segmentation accuracy through a complementary multi-task setup. Specifically, detecting shadow edges provides detailed constraints on shadow region boundaries. To boost medical image segmentation, we develop a multi-task framework based on mean teacher called transYLmtMT. We first design a multi-task transformer-based YOLOv5 model, mt-transYL that jointly trains on two tasks: shadow region and edge detection. The mt-transYL incorporates transformers within YOLOv5 for encoding sequential inter-dependencies through self-attentions [19; 20]. To retain fine-grained spatial details, we incorporate a hybrid architecture blending high-resolution decoders with contextual transformer encoders. This forms the base network operationalizing region and edge detection objectives. We employ the identical mt-transYL variant in a student-teacher arrangement with additional unsupervised regularizations. For labeled data, supervised losses monitor both tasks. Further, prediction consistency is enforced between student and teacher outputs on unlabeled samples. By synergizing segmentation across granularities while concurrently leveraging labeled and unlabeled data, our framework targets advancing challenges constrained by limited annotations. The semi-supervised multi-task formulation combines benefits from transformer-based context modeling, pixel-level decoding, and transductive regularizations for optimizing utilization.