On a Non-monotonicity Effect of Similarity Measures

Moser, Bernhard; Stübl, Gernot; Bouchot, Jean-Luc

doi:10.1007/978-3-642-24471-1_4

Cited by 7 publications

(3 citation statements)

References 13 publications

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“…The interesting point about this is that based on Weyl's discrepancy concept distance measures can be constructed that guarantee the desirable registration properties: (R1) the measure vanishes if and only if the lag vanishes, (R2) the measure increases monotonically with an increasing lag, and (R3) the measure obeys a Lipschitz condition that guarantees smooth changes also for patterns with high frequencies. As proven in [26], properties (R1)-(R3) are not satisfied simultaneously by commonly used measures in this context like mutual information, Kullback-Leibler distance or the Jensen-Rényi divergence measure which are special variants of f -divergence and f -information measures, see, e.g., [4,20,29,30,37], nor do the standard measures based on p-norms or the widely used correlation measures due to Pearson or Spearman, see [8,17,34].…”

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confidence: 77%

“…Such problems encounter as autocorrelation in signal processing, see [5]. In computer vision such problems are particularly encountered in stereo matching as point correspondence problem, see, e.g., [32] and [26], in template matching, e.g., for the purpose of print inspection, see, e.g., [7] and [23], in superpixel matching [16] or in defect detection in textured images, see [6,25,35]. In these cases, for high-frequency patterns, the discrepancy norm leads to cost functions with less local extrema and a more distinctive region of convergence in the neighborhood of the global minimum compared to commonly used (dis-)similarity measures.…”

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confidence: 99%

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Geometric Characterization of Weyl’s Discrepancy Norm in Terms of Its n-Dimensional Unit Balls

Moser

2012

Discrete Comput Geom

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Weyl's discrepancy measure induces a norm on R n which shows a monotonicity and a Lipschitz property when applied to differences of index-shifted sequences. It turns out that its n-dimensional unit ball is a zonotope that results from a multiple sheared projection from the (n + 1)-dimensional hypercube which can be interpreted as a discrete differentiation. This characterization reveals that this norm is the canonical metric between sequences of differences of values from the unit interval in the sense that the n-dimensional unit ball of the discrepancy norm equals the space of such sequences. MotivationIn the mathematical literature discrepancy theory is devoted to problems related to irregularities of distributions. In this context the term discrepancy refers to a measure that evaluates to which extent a given distribution deviates from total uniformity in measure-theoretic, combinatorial and geometric settings. This theory goes back to Weyl [39] and is still an active field of research, see, e.g., [3,12,19]. Applications can be found in the field of numerical integration, especially for Monte Carlo methods in high dimensions, see, e.g., [28,36,40], or in computational geometry, see, e.g., [1,9,21]. For applications to data storage problems on parallel disks, see [10,13] and for halftoning images, see [31].This paper is motivated by [24] which applies Weyl's discrepancy concept in order to derive an ordering-dependent norm for measuring the (dis-)similarity between patterns. In this context the focus lies on evaluating the auto-misalignment that measures

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Geometric Characterization of Weyl’s Discrepancy Norm in Terms of Its n-Dimensional Unit Balls

Moser

2012

Discrete Comput Geom

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“…Furthermore it is more robust to Gaussian noise than traditional measures, for examples see Moser et al 21 .…”

Section: Discrepancy Norm For Robust Periodicity Estimationmentioning

confidence: 98%

Periodicity estimation of nearly regular textures based on discrepancy norm

et al. 2013

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This paper proposes a novel approach to determine the texture periodicity, the texture element size and further characteristics like the area of the basin of attraction in the case of computing the similarity of a test image patch with a reference. The presented method utilizes the properties of a novel metric, the so-called discrepancy norm. Due to the Lipschitz and the monotonicity property the discrepancy norm distinguishes itself from other metrics by well-formed and stable convergence regions. Both the periodicity and the convergence regions are closely related and have an immediate impact on the performance of a subsequent template matching and evaluation step.The general form of the proposed approach relies on the generation of discrepancy norm induced similarity maps at random positions in the image. By applying standard image processing operations like watershed and blob analysis on the similarity maps a robust estimation of the characteristic periodicity can be computed. From the general approach a tailored version for orthogonal aligned textures is derived which shows robustness to noise disturbed images and is suitable for estimation on near regular textures. In an experimental set-up the estimation performance is tested on samples of standardized image databases and is compared with state-of-theart methods. Results show that the proposed method is applicable to a wide range of nearly regular textures and estimation results keeps up with current methods. When adding a hypothesis generation/selection mechanism it even outperforms the current state-or-the-art.

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