The enhancement and detection of elongated structures in noisy image data are relevant for many biomedical imaging applications. To handle complex crossing structures in 2D images, 2D orientation scores \(U: {\mathbb {R}} ^ 2\times S ^ 1 \rightarrow {\mathbb {C}}\) were introduced, which already showed their use in a variety of applications. Here we extend this work to 3D orientation scores \(U: {\mathbb {R}} ^ 3 \times S ^ 2\rightarrow {\mathbb {C}}\). First, we construct the orientation score from a given dataset, which is achieved by an invertible coherent state type of transform. For this transformation we introduce 3D versions of the 2D cake wavelets, which are complex wavelets that can simultaneously detect oriented structures and oriented edges. Here we introduce two types of cake wavelets: the first uses a discrete Fourier transform, and the second is designed in the 3D generalized Zernike basis, allowing us to calculate analytical expressions for the spatial filters. Second, we propose a nonlinear diffusion flow on the 3D roto-translation group: crossing-preserving coherence-enhancing diffusion via orientation scores (CEDOS). Finally, we show two applications of the orientation score transformation. In the first application we apply our CEDOS algorithm to real medical image data. In the second one we develop a new tubularity measure using 3D orientation scores and apply the tubularity measure to both artificial and real medical data.

中文翻译：

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3D 图像可逆方向分数的设计和处理。

噪声图像数据中细长结构的增强和检测与许多生物医学成像应用相关。为了处理 2D 图像中的复杂交叉结构，引入了2D 方向分数\(U: {\mathbb {R}} ^ 2\times S ^ 1 \rightarrow {\mathbb {C}}\)，它已经在各种应用。在这里，我们将这项工作扩展到 3D 方向分数\(U: {\mathbb {R}} ^ 3 \times S ^ 2\rightarrow {\mathbb {C}}\)。首先，我们根据给定的数据集构建方向得分，这是通过可逆相干态类型的变换来实现的。对于这种变换，我们引入了 2D cake 小波的 3D 版本，它们是可以同时检测定向结构和定向边缘的复杂小波。这里我们介绍两种类型的蛋糕小波：第一种使用离散傅立叶变换，第二种是在 3D 广义 Zernike 基础上设计的，允许我们计算空间滤波器的解析表达式。其次，我们提出了 3D 旋转平移组上的非线性扩散流：通过方向分数保持交叉保持相干性增强扩散（CEDOS）。最后，我们展示了方向得分变换的两种应用。在第一个应用中，我们将 CEDOS 算法应用于真实的医学图像数据。在第二个中，我们使用 3D 方向得分开发了一种新的管状度量，并将管状度量应用于人工和真实的医学数据。