Biomedical Imaging Group, École Polytechnique Fédérale de Lausanne, Lausanne CH-1015, Switzerland.
IEEE Trans Image Process. 2012 Nov;21(11):4522-33. doi: 10.1109/TIP.2012.2206044. Epub 2012 Jun 26.
We introduce a systematic and practical design for steerable wavelet frames in 3D. Our steerable wavelets are obtained by applying a 3D version of the generalized Riesz transform to a primary isotropic wavelet frame. The novel transform is self-reversible (tight frame) and its elementary constituents (Riesz wavelets) can be efficiently rotated in any 3D direction by forming appropriate linear combinations. Moreover, the basis functions at a given location can be linearly combined to design custom (and adaptive) steerable wavelets. The features of the proposed method are illustrated with the processing and analysis of 3D biomedical data. In particular, we show how those wavelets can be used to characterize directional patterns and to detect edges by means of a 3D monogenic analysis. We also propose a new inverse-problem formalism along with an optimization algorithm for reconstructing 3D images from a sparse set of wavelet-domain edges. The scheme results in high-quality image reconstructions which demonstrate the feature-reduction ability of the steerable wavelets as well as their potential for solving inverse problems.
我们提出了一种系统而实用的 3D 可转向小波框架设计方法。我们的可转向小波是通过将广义里兹变换的 3D 版本应用于基本各向同性小波框架而获得的。该新型变换是自可逆的（紧框架），其基本组成部分（里兹小波）可以通过形成适当的线性组合，在任何 3D 方向上有效地旋转。此外，可以在线性组合给定位置的基函数，以设计定制（和自适应）可转向小波。所提出的方法的特点通过 3D 生物医学数据的处理和分析得到说明。特别地，我们展示了这些小波如何用于通过 3D 单形分析来描述方向模式和检测边缘。我们还提出了一种新的反问题形式和一种用于从稀疏的小波域边缘集重建 3D 图像的优化算法。该方案产生高质量的图像重建，展示了可转向小波的特征减少能力及其解决反问题的潜力。
