Although a number of normalized edit distances presented so far may offer good performance in some applications, none of them can be regarded as a genuine metric between strings because they do not satisfy the triangle inequality. Given two strings X and Y over a finite alphabet, this paper defines a new normalized edit distance between X and Y as a simple function of their lengths (|X| and |Y|) and the Generalized Levenshtein Distance (GLD) between them. The new distance can be easily computed through GLD with a complexity of O(|X| \cdot |Y|) and it is a metric valued in [0, 1] under the condition that the weight function is a metric over the set of elementary edit operations with all costs of insertions/deletions having the same weight. Experiments using the AESA algorithm in handwritten digit recognition show that the new distance can generally provide similar results to some other normalized edit distances and may perform slightly better if the triangle inequality is violated in a particular data set.

Roads are important mode of transportation, which are very convenient for people’s daily work and life. However, it is challenging to accuratly extract road information from a high-resolution remote sensing image. This paper presents a road extraction method for remote sensing images with a complement UNet (C-UNet). C-UNet contains four modules. Firstly, the standard UNet is used to roughly extract road information from remote sensing images, getting the first segmentation result; secondly, a fixed threshold is utilized to erase partial extracted information; thirdly, a multi-scale dense dilated convolution UNet (MD-UNet) is introduced to discover the complement road areas in the erased masks, obtaining the second segmentation result; and, finally, we fuse the extraction results of the first and the third modules, getting the final segmentation results. Experimental results on the Massachusetts Road dataset indicate that our C-UNet gets the higher results than the state-of-the-art methods, demonstrating its effectiveness.

Based on the “convexly separable” concept, we present a solid geometric theory and a new general framework to design piecewise linear classifiers for two arbitrarily complicated nonintersecting classes by using a “multiconlitron,” which is a union of multiple conlitrons that comprise a set of hyperplanes or linear functions surrounding a convex region for separating two convexly separable datasets. We propose a new iterative algorithm called the cross distance minimization algorithm (CDMA) to compute hard margin non-kernel support vector machines (SVMs) via the nearest point pair between two convex polytopes. Using CDMA, we derive two new algorithms, i.e., the support conlitron algorithm (SCA) and the support multiconlitron algorithm (SMA) to construct support conlitrons and support multiconlitrons, respectively, which are unique and can separate two classes by a maximum margin as in an SVM. Comparative experiments show that SMA can outperform linear SVM on many of the selected databases and provide similar results to radial basis function SVM on some of them, while SCA performs better than linear SVM on three out of four applicable databases. Other experiments show that SMA and SCA may be further improved to draw more potential in the new research direction of piecewise linear learning.