Convolutional neural networks (CNNs) are increasingly adopted in medical imaging, e.g., to reconstruct high-quality images from undersampled magnetic resonance imaging (MRI) acquisitions or estimate subject motion during an examination. MRI is naturally acquired in the complex domain $\mathbb{C}$, obtaining magnitude and phase information in k-space. However, CNNs in complex regression tasks are almost exclusively trained to minimize the L2 loss or maximizing the magnitude structural similarity (SSIM), which are possibly not optimal as they do not take full advantage of the magnitude and phase information present in the complex domain. This work identifies that minimizing the L2 loss in the complex field has an asymmetry in the magnitude/phase loss landscape and is biased, underestimating the reconstructed magnitude. To resolve this, we propose a new loss function for regression in the complex domain called $\perp$-loss, which adds a novel phase term to established magnitude loss functions, e.g., L2 or SSIM. We show $\perp$-loss is symmetric in the magnitude/phase domain and has favourable properties when applied to regression in the complex domain. Specifically, we evaluate the $\perp$+${\ell }^2$-loss and $\perp$+SSIM-loss for complex undersampled MR image reconstruction tasks and MR image registration tasks. We show that training a model to minimize the $\perp$+${\ell }^2$-loss outperforms models trained to minimize the L2 loss and results in similar performance compared to models trained to maximize the magnitude SSIM while offering high-quality phase reconstruction. Moreover, $\perp$-loss is defined in ${\mathbb{R}}^n$, and we apply the loss function to the ${\mathbb{R}}^2$ domain by learning 2D deformation vector fields for image registration. We show that a model trained to minimize the $\perp$+${\ell }^2$-loss outperforms models trained to minimize the end-point error loss.

卷积神经网络 (CNN) 越来越多地用于医学成像，例如，从欠采样的磁共振成像 (MRI) 采集中重建高质量图像或估计检查期间的对象运动。MRI 是在复杂领域自然获得的$\mathbb{C}$，获得k空间中的幅度和相位信息。然而，复杂回归任务中的 CNN 几乎完全被训练来最小化 L2 损失或最大化幅度结构相似性 (SSIM)，这可能不是最优的，因为它们没有充分利用复杂域中存在的幅度和相位信息。这项工作确定了最小化复杂场中的 L2 损失在幅度/相位损失情况中具有不对称性，并且是有偏差的，低估了重建的幅度。为了解决这个问题，我们提出了一种新的损失函数，用于复杂域中的回归，称为$\perp$-loss，为已建立的幅度损失函数添加一个新的相位项，例如 L2 或 SSIM。我们展示$\perp$-损失在幅度/相位域中是对称的，并且在应用于复域中的回归时具有良好的特性。具体来说，我们评估$\perp$+${\ell }^2$-损失和$\perp$+SSIM 损失用于复杂的欠采样 MR 图像重建任务和 MR 图像配准任务。我们展示了训练模型以最小化$\perp$+${\ell }^2$-loss 优于经过训练以最小化 L2 损失的模型，并且与经过训练以最大化 SSIM 幅度同时提供高质量相位重建的模型相比，其性能相似。而且，$\perp$-损失定义在${\mathbb{R}}^n$，我们将损失函数应用于${\mathbb{R}}^2$域通过学习用于图像配准的 2D 变形向量场。我们证明了一个经过训练的模型可以最小化$\perp$+${\ell }^2$-loss 优于经过训练以最小化端点误差损失的模型。