Abstract Segmentation, a useful/powerful technique in pattern recognition, is the process of identifying object outlines within images. There are a number of efficient algorithms for segmentation in Euclidean space that depend on the variational approach and partial differential equation modelling. Wavelets have been used successfully in various problems in image processing, including segmentation, inpainting, noise removal, super-resolution image restoration, and many others. Wavelets on the sphere have been developed to solve such problems for data defined on the sphere, which arise in numerous fields such as cosmology and geophysics. In this work, we propose a wavelet-based method to segment images on the sphere, accounting for the underlying geometry of spherical data. Our method is a direct extension of the tight-frame based segmentation method used to automatically identify tube-like structures such as blood vessels in medical imaging. It is compatible with any arbitrary type of wavelet frame defined on the sphere, such as axisymmetric wavelets, directional wavelets, curvelets, and hybrid wavelet constructions. Such an approach allows the desirable properties of wavelets to be naturally inherited in the segmentation process. In particular, directional wavelets and curvelets, which were designed to efficiently capture directional signal content, provide additional advantages in segmenting images containing prominent directional and curvilinear features. We present several numerical experiments, applying our wavelet-based segmentation method, as well as the common K-means method, on real-world spherical images, including an Earth topographic map, a light probe image, solar data-sets, and spherical retina images. These experiments demonstrate the superiority of our method and show that it is capable of segmenting different kinds of spherical images, including those with prominent directional features. Moreover, our algorithm is efficient with convergence usually within a few iterations.

中文翻译：

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基于小波的球体分割

摘要 分割是模式识别中一种有用/强大的技术，是识别图像中对象轮廓的过程。有许多有效的欧几里德空间分割算法，它们依赖于变分方法和偏微分方程建模。小波已成功用于图像处理中的各种问题，包括分割、修复、噪声去除、超分辨率图像恢复等。球体上的小波已被开发用于解决定义在球体上的数据的此类问题，这些问题出现在许多领域，例如宇宙学和地球物理学。在这项工作中，我们提出了一种基于小波的方法来分割球体上的图像，解释球面数据的基本几何形状。我们的方法是基于紧框架的分割方法的直接扩展，用于自动识别医学成像中的血管等管状结构。它与定义在球面上的任意类型的小波框架兼容，例如轴对称小波、方向小波、曲波和混合小波结构。这种方法允许在分割过程中自然地继承小波的理想特性。特别是定向小波和曲波，旨在有效地捕获定向信号内容，在分割包含突出方向和曲线特征的图像方面提供了额外的优势。我们提出了几个数值实验，将我们基于小波的分割方法以及常见的 K 均值方法应用于真实世界的球面图像，包括地球地形图、光探测图像、太阳数据集和球形视网膜图像。这些实验证明了我们方法的优越性，并表明它能够分割不同种类的球面图像，包括具有突出方向特征的球面图像。此外，我们的算法通常在几次迭代内收敛是有效的。