On the translation-invariance of image distance metric

Abstract

An appropriate choice of the distance metric is a fundamental problem in pattern recognition, machine learning and cluster analysis. Some methods that based on the distance of samples, e.g, the k-means clustering algorithm and the k-nearest neighbor classifier, are crucially relied on the performance of the distance metric. In this paper, the property of translation invariance for the distance metric of images is especially emphasized. The consideration is twofold. Firstly, some of the commonly used distance metrics, such as the Euclidean and Minkowski distance, are independent of the training set and/or the domain-specific knowledge. Secondly, the translation invariance is a necessary property for any intuitively reasonable image metric. The image Euclidean distance (IMED) and generalized Euclidean distance (GED) are image metrics that take the spatial relationship between pixels into consideration. Sun et al.(IEEE Conference on Computer Vision and Pattern Recognition, pp 1398–1405, 2009) showed that IMED is equivalent to a translation-invariant transform and proposed a metric learning algorithm based on the equivalency. In this paper, we provide a complete treatment on this topic and extend the equivalency to the discrete frequency domain. Based on the connection, we show that GED and IMED can be implemented as low-pass filters, which reduce the space and time complexities significantly. The transform domain metric learning proposed in (Sun et al. 2009) is also resembled as a translation-invariant counterpart of LDA. Experimental results demonstrate improvements in algorithm efficiency and performance boosts on the small sample size problems.