On a Non-monotonicity Effect of Similarity Measures

The effect of non-monotonicity of similarity measures is addressed which can be observed when measuring the similarity between patterns with increasing displacement. This effect becomes the more apparent the less smooth the pattern is. It is proven that commonly used similarity measures like f -divergence measures or kernel functions show this non-monotonicity effect which results from neglecting any ordering in the underlying construction principles. As an alternative approach Weyl’s discrepancy measure is examined by which this non-monotonicity effect can be avoided even for patterns with high-frequency or chaotic characteristics. The impact of the non-monotonicity effect to applications is discussed by means of examples from the field of stereo matching, texture analysis and tracking.