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.

中文翻译：

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张量尺度：一种具有有效计算和应用的分析方法。

比例尺是计算机视觉和图像理解中广泛使用的概念，它以比例空间理论的形式发展，其主要思想是代表和分析各种分辨率的图像。最近，我们使用椭球模型引入了局部形态计量尺度的概念，称为“张量尺度”，该模型可统一表示结构尺寸，方向和各向异性。在先前的工作中，使用2-D算法方法描述了张量标度，并且缺少精确的分析定义。而且，由于高计算复杂度，使用先前的框架在3D中应用张量标度是不实际的。在本文中，针对n维（nD）图像制定了张量尺度的解析定义，该图像捕获了局部结构的大小，方向和各向异性。也，提出了使用几种新颖的微分几何方法的二维和3-D高效计算解决方案，并通过实验检验了结果的准确性。而且，推导了张量尺度的矩阵表示，以促进包括张量场平滑以捕获更大的上下文知识在内的若干操作。最后，介绍了张量标度在图像滤波和n线性插值中的应用，并与相应的最新方法进行了比较，研究了其结果的性能。具体而言，将基于张量标度的图像滤波的性能与基于梯度和Weickert基于结构张量的扩散滤波算法进行了比较。也，