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Distance and Similarity Measures

vectors color magnitude orientation

Definition: The notion of distance or similarity between two color vectors is of paramount importance for the development of the vector processing techniques such as noise removal filters, edge detectors and image zoomers.

Since each color vector is uniquely determined via its magnitude (length) and direction (orientation), the evaluation of the color vectors can be realized in the magnitude domain, the directional domain, or it can utilize both vectors’ magnitude and direction.

The most commonly used measure to quantify the distance between two color vectors x i = [ x i 1 , x i 2 , x i 3 ] and x j = [ x j 1 , x j 2 , x j 3 ], in the magnitude domain, is the generalized weighted Minkowski metric:

where c is the non-negative scaling parameter denoting the measure of the overall discrimination power and the exponent L , with L = 1 for the city-block distance, L = 2 for the Euclidean distance and L ? 8 for the chess-board distance, defines the nature of the distance metric. The parameter ? k , for S k ? k = 1, measures the proportion of attention allocated to the dimensional component k . Vectors having a range of values greater than a desirable threshold can be scaled down by the use of the weighting function ?. Alternative measures to the Minkowski metric include, but not limited to, the Canberra distance and the Czekanowski coefficient.

Apart form the distance measures, various similarity measures can be used in support of vector image processing operations. A similarity measure s (x i , x j ) is a symmetric function whose value is large when the vectorial inputs x i and x j are similar. Similarity in orientation is expressed through the normalized inner product s ( x i , x j ) = ( x i , x j /(| x i ? x j |)) which corresponds to the cosine of the angle between the two vectors x i and x j . Since similar colors have almost parallel orientations while significantly different colors point in different directions in a 3-D color space, such as the RGB space, the normalized inner product, or equivalently the angular distance d ( x i , x j ) = arccos( x i , x j /(| x i | x j |)) can be used to quantify orientation difference (angular distance) between the two vectors.

It is obvious that a generalized similarity measure model which can effectively quantify differences among color signals should take into consideration both the magnitude and orientation of the color vectors. Thus, a generalized measure based on both the magnitude and orientation of vectors should provide a robust solution to the problem of similarity quantification between two vectors. Such an idea is used in constructing the generalized content model family of measures s ( x i , x j ) = C ij / T ij which treat similarity between two vectors as the degree of common content, so-called commonality C ij , in relation to the total content, so-called totality T ij , of the two vectors x i and x j . Based on this general framework, different similarity measures can be obtained by utilizing different commonality and totality concepts.

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