Tensor scale: An analytic approach with efficient computation and applications

• Published in: Computer vision and image understanding Vol. 116; no. 10; pp. 1060 - 1075
• Main Authors: Xu, Ziyue, Saha, Punam K., Dasgupta, Soura
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01.10.2012; Elsevier
• Subjects: Algorithmics. Computability. Computer arithmetics; Applied sciences; Artificial intelligence; Computer science; control theory; systems; Earth sciences; Earth, ocean, space; Exact sciences and technology; Filtering; Filtration; Geomorphology, landform evolution; Image filtering; Image interpolation; Interpolation; Local scale; Mathematical analysis; Mathematical models; Morphometry; Pattern recognition. Digital image processing. Computational geometry; Scale-space; Surficial geology; Tensor scale; Tensors; Theoretical computing; Three dimensional; Local scale; Scale-space; Image filtering; Tensor scale; Morphometry; Image interpolation; Computer vision; Linear interpolation; Image resolution; Scale space; Image processing; Image interpretation; Differential geometry; Algorithmics; Computational complexity; Scale model; Geometrical model; Scaling law; Local structure; Anisotropy; Smoothing; Morphological analysis; Linear filtering; Ellipsoid
• ISSN: 1077-3142; 1090-235X
• DOI: 10.1016/j.cviu.2012.05.006

► Theoretical formulation of tensor scale – an anisotropic local scale model. ► Efficient algorithm for tensor scale computation in three-dimensions. ► Development and evaluation of tensor scale based anisotropic diffusive filtering. ► Development and evaluation of tensor scale based n-linear interpolation. Scale is a widely used notion in computer vision and image understanding that evolved in the form of scale-space theory where the key idea is to represent and analyze an image at various resolutions. Recently, we introduced a notion of local morphometric scale referred to as “tensor scale” using an ellipsoidal model that yields a unified representation of structure size, orientation and anisotropy. In the previous work, tensor scale was described using a 2-D algorithmic approach and a precise analytic definition was missing. Also, the application of tensor scale in 3-D using the previous framework is not practical due to high computational complexity. In this paper, an analytic definition of tensor scale is formulated for n-dimensional (n-D) images that captures local structure size, orientation and anisotropy. Also, an efficient computational solution in 2- and 3-D using several novel differential geometric approaches is presented and the accuracy of results is experimentally examined. Also, a matrix representation of tensor scale is derived facilitating several operations including tensor field smoothing to capture larger contextual knowledge. Finally, the applications of tensor scale in image filtering and n-linear interpolation are presented and the performance of their results is examined in comparison with respective state-of-art methods. Specifically, the performance of tensor scale based image filtering is compared with gradient and Weickert’s structure tensor based diffusive filtering algorithms. Also, the performance of tensor scale based n-linear interpolation is evaluated in comparison with standard n-linear and windowed-sinc interpolation methods.