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The encoder-decoder model is a commonly used Deep Neural Network (DNN) model for medical image segmentation. Conventional encoder-decoder models make pixel-wise predictions focusing heavily on local patterns around the pixel. This makes it challenging to give segmentation that preserves the object's shape and topology, which often requires an understanding of the global context of the object. In this work, we propose a Fourier Coefficient Segmentation Network~(FCSN) -- a novel DNN-based model that segments an object by learning the complex Fourier coefficients of the object's masks. The Fourier coefficients are calculated by integrating over the whole contour. Therefore, for our model to make a precise estimation of the coefficients, the model is motivated to incorporate the global context of the object, leading to a more accurate segmentation of the object's shape. This global context awareness also makes our model robust to unseen local perturbations during inference, such as additive noise or motion blur that are prevalent in medical images. When FCSN is compared with other state-of-the-art models (UNet+, DeepLabV3+, UNETR) on 3 medical image segmentation tasks (ISIC\_2018, RIM\_CUP, RIM\_DISC), FCSN attains significantly lower Hausdorff scores of 19.14 (6\%), 17.42 (6\%), and 9.16 (14\%) on the 3 tasks, respectively. Moreover, FCSN is lightweight by discarding the decoder module, which incurs significant computational overhead. FCSN only requires 22.2M parameters, 82M and 10M fewer parameters than UNETR and DeepLabV3+. FCSN attains inference and training speeds of 1.6ms/img and 6.3ms/img, that is 8$\times$ and 3$\times$ faster than UNet and UNETR.