Signatures versus histograms: Definitions, distances and algorithms

Abstract

Abstract. The aim of this paper is to present a new method to compare histograms. The main advantage is that there is an important time-complexity reduction respect the methods presented before. This reduction is statistically and analytically demonstrated in the paper. The distances between histograms that we present are defined on a structure called signature, which is a lossless representation of histograms. Moreover, the type of the elements of the sets that the histograms represent are ordinal, nominal and modulo. We show that the computational cost of these distances is O(z') for the ordinal and nominal types and O(z'2 ) for the modulo one, being z' the number of non-empty bins of the histograms. The computational cost of the algorithms presented in the literature depends on the number of bins of the histograms. In most of the applications, the obtained histograms are sparse, then considering only the non-empty bins makes the time consuming of the comparison drastically decrease. The distances and algorithms presented in this paper are experimentally validated on the comparison of images obtained from public databases and positioning of mobile robots through the recognition of indoor scenes (captured in a learning stage).

Analysis of Hydrologic Alteration Due to River Diversion

“…Serratosa & Sanfeliu (2006) observed that most distance measures considers the overlap or intersection between two histograms as a function of the distance value, but they do not take accurately into account the similarity of the non-overlapping parts of the two histograms. The effects of these drawbacks are examined through the test case reported below.…”

“…It is an analysis of the frequency modification of each hydrologic variable from pre-to post-impact condition based on the Earth Mover's Distance (EMD) algorithm (Rubner et al, 1998(Rubner et al, , 2000 applied to "extended signatures" (Serratosa & Sanfeliu, 2006).…”

“…If the ordering of the elements in the set is unimportant, a histogram is a lossless representation of the set itself (Serratosa & Sanfeliu, 2006). Thus, the "distance" between two sets can be computed in an efficient way by computing the distance between their histograms.…”

“…These include: the quadratic-form distance proposed by Niblack et al (1993), the B-distance, the Matusita approach and the "K-L distance" (Cha & Srihari, 2002). As noted above, many methods do not accurately take into account the similarity of the non-overlapping parts of the two histograms (Serratosa & Sanfeliu, 2006). This problem was overcome by Rubner et al (1998Rubner et al ( , 2000, who presented a new definition of the distance measure between histograms called Earth Mover's Distance (EMD), that is more robust than histogram matching techniques.…”

“…As subsequent step, the histograms can be represented as "extended signatures" (Serratosa & Sanfeliu, 2006), in which the minimum number of empty bins is added to assure that the number of bins for both pre-and post-impact data are the same. To this aim, the vector m' is obtained adding equal size bins to m a ' in order to include all values of b'.…”