Scaling Theorems for Zero Crossings

By Alan L. Yuille, Tomaso A. Poggio. 1987

We characterize some properties of the zero crossings of the Laplacian of signals¿in particular images¿filtered with linear filters, as a function of the scale of the filter (extending recent work by Witkin [16]). We prove that in any dimension the only filter that does not create generic zero crossings as the scale increases is the Gaussian. This result can be generalized to apply to level crossings of any linear differential operator: it applies in particular to ridges and ravines in the image intensity. In the case of the second derivative along the gradient, there is no filter that avoids creation of zero crossings, unless the filtering is performed after the derivative is applied.

Subfields: Signal Processing

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Citations in the corpus

Holistically-Nested Edge Detection (2015, neutral): “This integrated learning of hierarchical features is in distinction to previous multi-scale approaches [38, 39, 28] in which scale-space edge ﬁelds are neither automatically learned nor connected hierarchically.”
Gaussian-based edge-detection methods - a survey (2002, background): “Yuille and Poggio [39] extended this work to two-dimensional (2-”
A Theory for Multiresolution Signal Decomposition: The Wavelet Representation (1989, background): “Because of (2) it is sufficient to express the above principle for j = 0 .”
A multiscale dynamic programming procedure for boundary detection in ultrasonic artery images (2000, methodology): “In order to facilitate an automated detection without an initial setting, we performed multiscale DP based on the properties of the scale space [26]‐[ 28 ].”
Detecting salient blob-like image structures and their scales with a scale-space primal sketch: A method for focus-of-attention (1993, background): “The idea of scale-space representation of images, suggested by Witkin (1983) has, in particular, been developed by Koenderink and van Doorn (1984, 1986, 1992), Babaud et al. (1986), Yuille and Poggio (1986), Hummel (1987), Lindeberg (1990, 1993) and Florack et al. (1992).”
Clustering by Scale-Space Filtering (2000, methodology): “For example, the scale-space theory is developed in image and signal processing [22], [23], [24], [25], [ 26 ], [27], [28].”
Edge Detection Techniques-An Overview (1998, unknown): “Nice scaling behavior means that edges are not created as the scale increases; in general this requirement does not hold for non-linear operators [121, 122].”
Linear scale-space (1994, background): “A similar result was given by Yuille and Poggio ( 1985 , 1986 ) concerning the zero - crossings of the Laplacian of the Gaussian .”
Edge and line oriented contour detection: State of the art (2011, background): “The best known mother wavelet ψs r ð Þ is the 2D Gaussian function, due to several mathematical properties (such as isotropy, separability, entropy minimization, no zero-crossing creation [230], solution of the linear diffusion equation ∂I ∂s = ∇ 2I, monotonicity of local maxima”