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Multi-task learning proved to be an effective strategy to increase the performance of a dense prediction network on a segmentation task, by defining auxiliary tasks to reflect different aspects of the problem and concurrently learning them with the main task of segmentation. Up to now, previous studies defined their auxiliary tasks in the Euclidean space. However, for some segmentation tasks, the complexity and high variation in the texture of a region of interest may not follow the smoothness constraint in the Euclidean geometry. This paper addresses this issue by introducing a new multi-task network, MTFD-Net , which utilizes the fractal geometry to quantify texture complexity through self-similar patterns in an image. To this end, we propose to transform an image into a map of fractal dimensions and define its learning as an auxiliary task, which will provide auxiliary supervision to the main segmentation task, towards betterment of left atrium (LA) segmentation in computed tomography (CT) images. To the best of our knowledge, this is the first proposal of a dense prediction network that employs the fractal geometry to define an auxiliary task and learns it in parallel to the segmentation task in a multi-task learning framework. Our experiments revealed that the proposed MTFD-Net model led to more accurate LA segmentations compared to its counterparts. • A segmentation network quantifying texture complexity through self-similar patterns. • The model learns fractal dimension map estimation in a multi-task learning framework. • The model led to improved performance for left atrium segmentation in CT images. [ABSTRACT FROM AUTHOR]