title: Transforms and Operators for Directional Bioimage Analysis: A Survey.
abstract: We give a methodology-oriented perspective on directional image analysis and rotation-invariant processing. We review the state of the art in the field and make connections with recent mathematical developments in functional analysis and wavelet theory. We unify our perspective within a common framework using operators. The intent is to provide image-processing methods that can be deployed in algorithms that analyze biomedical images with improved rotation invariance and high directional sensitivity. We start our survey with classical methods such as directional-gradient and the structure tensor. Then, we discuss how these methods can be improved with respect to robustness, invariance to geometric transformations (with a particular interest in scaling), and computation cost. To address robustness against noise, we move forward to higher degrees of directional selectivity and discuss Hessian-based detection schemes. To present multiscale approaches, we explain the differences between Fourier filters, directional wavelets, curvelets, and shearlets. To reduce the computational cost, we address the problem of matching directional patterns by proposing steerable filters, where one might perform arbitrary rotations and optimizations without discretizing the orientation. We define the property of steerability and give an introduction to the design of steerable filters. We cover the spectrum from simple steerable filters through pyramid schemes up to steerable wavelets. We also present illustrations on the design of steerable wavelets and their application to pattern recognition.
authors.affiliation: Biomedical Imaging Group, École polytechnique fédérale de Lausanne (EPFL), Station 17, 1015, Lausanne VD, Switzerland. zsuzsanna.puspoki@epfl.ch.
authors.forename: Zsuzsanna
authors.initials: Z
authors.lastname: Püspöki
authors.affiliation: Biomedical Imaging Group, École polytechnique fédérale de Lausanne (EPFL), Station 17, 1015, Lausanne VD, Switzerland.
authors.forename: Martin
authors.initials: M
authors.lastname: Storath
authors.affiliation: Biomedical Imaging Group, École polytechnique fédérale de Lausanne (EPFL), Station 17, 1015, Lausanne VD, Switzerland.
authors.forename: Daniel
authors.initials: D
authors.lastname: Sage
authors.affiliation: Biomedical Imaging Group, École polytechnique fédérale de Lausanne (EPFL), Station 17, 1015, Lausanne VD, Switzerland.
authors.forename: Michael
authors.initials: M
authors.lastname: Unser
mesh_terms: D000818:Animals; D001185:Artificial Intelligence; D006801:Humans; D007091:Image Processing, Computer-Assisted / Q000379:methods* / Q000706:statistics & numerical data; D008432:Mathematical Computing; D008856:Microscopy, Fluorescence / Q000295:instrumentation / Q000379:methods*; D055620:Optical Phenomena; D010363:Pattern Recognition, Automated / Q000379:methods / Q000706:statistics & numerical data*; D059629:Signal-To-Noise Ratio; D058067:Wavelet Analysis