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Deep learning methods have contributed substantially to the rapid advancement of medical image segmentation, the quality of which relies on the suitable design of loss functions. Popular loss functions, including the cross-entropy and dice losses, often fall short of boundary detection, thereby limiting high-resolution downstream applications such as automated diagnoses and procedures. We developed a novel loss function that is tailored to reflect the boundary information to enhance the boundary detection. As the contrast between segmentation and background regions along the classification boundary naturally induces heterogeneity over the pixels, we propose the piece-wise two-sample t-test augmented (PTA) loss that is infused with the statistical test for such heterogeneity. We demonstrate the improved boundary detection power of the PTA loss compared to benchmark losses without a t-test component.

Solving semiparametric models can be computationally challenging because the dimension of parameter space may grow large with increasing sample size. Classical Newton's method becomes quite slow and unstable with intensive calculation of the large Hessian matrix and its inverse. Iterative methods separately update parameters for finite dimensional component and infinite dimensional component have been developed to speed up single iteration, but they often take more steps until convergence or even sometimes sacrifice estimation precision due to sub-optimal update direction. We propose a computationally efficient implicit profiling algorithm that achieves simultaneously the fast iteration step in iterative methods and the optimal update direction in the Newton's method by profiling out the infinite dimensional component as the function of the finite dimensional component. We devise a first order approximation when the profiling function has no explicit analytical form. We show that our implicit profiling method always solve any local quadratic programming problem in two steps. In two numerical experiments under semiparametric transformation models and GARCH-M models, we demonstrated the computational efficiency and statistical precision of our implicit profiling method.