Tianyi Shi

Tianyi Shi, Xiaohuan Ding, Wei Zhou et al.

Vessel segmentation is crucial in many medical image applications, such as detecting coronary stenoses, retinal vessel diseases and brain aneurysms. However, achieving high pixel-wise accuracy, complete topology structure and robustness to various contrast variations are critical and challenging, and most existing methods focus only on achieving one or two of these aspects. In this paper, we present a novel approach, the affinity feature strengthening network (AFN), which jointly models geometry and refines pixel-wise segmentation features using a contrast-insensitive, multiscale affinity approach. Specifically, we compute a multiscale affinity field for each pixel, capturing its semantic relationships with neighboring pixels in the predicted mask image. This field represents the local geometry of vessel segments of different sizes, allowing us to learn spatial- and scale-aware adaptive weights to strengthen vessel features. We evaluate our AFN on four different types of vascular datasets: X-ray angiography coronary vessel dataset (XCAD), portal vein dataset (PV), digital subtraction angiography cerebrovascular vessel dataset (DSA) and retinal vessel dataset (DRIVE). Extensive experimental results demonstrate that our AFN outperforms the state-of-the-art methods in terms of both higher accuracy and topological metrics, while also being more robust to various contrast changes. The source code of this work is available at https://github.com/TY-Shi/AFN.

Tianyi Shi, Xiaohuan Ding, Liang Zhang et al.

Curvilinear object segmentation is critical for many applications. However, manually annotating curvilinear objects is very time-consuming and error-prone, yielding insufficiently available annotated datasets for existing supervised methods and domain adaptation methods. This paper proposes a self-supervised curvilinear object segmentation method that learns robust and distinctive features from fractals and unlabeled images (FreeCOS). The key contributions include a novel Fractal-FDA synthesis (FFS) module and a geometric information alignment (GIA) approach. FFS generates curvilinear structures based on the parametric Fractal L-system and integrates the generated structures into unlabeled images to obtain synthetic training images via Fourier Domain Adaptation. GIA reduces the intensity differences between the synthetic and unlabeled images by comparing the intensity order of a given pixel to the values of its nearby neighbors. Such image alignment can explicitly remove the dependency on absolute intensity values and enhance the inherent geometric characteristics which are common in both synthetic and real images. In addition, GIA aligns features of synthetic and real images via the prediction space adaptation loss (PSAL) and the curvilinear mask contrastive loss (CMCL). Extensive experimental results on four public datasets, i.e., XCAD, DRIVE, STARE and CrackTree demonstrate that our method outperforms the state-of-the-art unsupervised methods, self-supervised methods and traditional methods by a large margin. The source code of this work is available at https://github.com/TY-Shi/FreeCOS.

Nicolas Boullé, Seick Kim, Tianyi Shi et al.

Neural operators are a popular technique in scientific machine learning to learn a mathematical model of the behavior of unknown physical systems from data. Neural operators are especially useful to learn solution operators associated with partial differential equations (PDEs) from pairs of forcing functions and solutions when numerical solvers are not available or the underlying physics is poorly understood. In this work, we attempt to provide theoretical foundations to understand the amount of training data needed to learn time-dependent PDEs. Given input-output pairs from a parabolic PDE in any spatial dimension $n\geq 1$, we derive the first theoretically rigorous scheme for learning the associated solution operator, which takes the form of a convolution with a Green's function $G$. Until now, rigorously learning Green's functions associated with time-dependent PDEs has been a major challenge in the field of scientific machine learning because $G$ may not be square-integrable when $n>1$, and time-dependent PDEs have transient dynamics. By combining the hierarchical low-rank structure of $G$ together with randomized numerical linear algebra, we construct an approximant to $G$ that achieves a relative error of $\smash{\mathcal{O}(Γ_ε^{-1/2}ε)}$ in the $L^1$-norm with high probability by using at most $\smash{\mathcal{O}(ε^{-\frac{n+2}{2}}\log(1/ε))}$ input-output training pairs, where $Γ_ε$ is a measure of the quality of the training dataset for learning $G$, and $ε>0$ is sufficiently small.