Near-Optimal Detection of Geometric Objects by Fast Multiscale Methods

Suppose n points are scattered uniformly at random in the unit square [0, 1] 2 . Question: How many of these points can possibly lie on some curve of length λ? Answer, proved here:We consider a general class of such questions; in each case, we are given a class Γ of curves in the square, and we ask: in a cloud of n uniform random points, how many can lie on some curve γ ∈ Γ? Classes of interest include (in addition to the rectifiable curves mentioned above): Lipschitz graphs, monotone graphs, twice-differentiable curves, graphs of smooth functions with m-bounded derivatives. In each case we get order-of-magnitude estimates; for example, there are twice-differentiable curves containing as many as O P (n 1/3 ) uniform random points, but not essentially more than this.We also consider generalizations to higher dimensions and to hypersurfaces of various codimensions. Thus, twice-differentiable k-dimensional hypersurfaces in R d may contain as many as O P (n k/(2d−k) ) uniform random points. We also consider other notions of 'passing through' such as passing through given space/direction pairs. Thus, twice-differentiable curves in R 2 may pass through at most O P (n 1/4 ) uniform random location/direction pairs.We give both concrete approaches to our results, based on geometric multiscale analysis, and abstract approaches, based on ε-entropy.Stylized applications in image processing and perceptual psychophysics are described.

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“…Then, Lemma 7.2 implies that the Hausdorff distance between ξ a and γ does not exceed c 3 κε 2 1 , for a constant c 3 > 0.…”

Section: Lemma 74 There Is a Constant Cmentioning

confidence: 96%

“…Our beamlets amount to a variation on the notion of Beamlets [10,2,11]. They provide an efficient, dyadically organized, multiscale, multi-orientation organizational structure.…”

Section: Upper Boundmentioning

confidence: 99%

Connect the dots: how many random points can a regular curve pass through?

et al. 2005

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“…Arias-Castro et al (2005) describe a general theory for detecting two dimensional structures in digital images. The pixels outside the structure take white noise values.…”

Section: Detection Of Ellipses and Other Structuresmentioning

confidence: 99%

“…Optimal detection methods are described for a range of structures including line segments, rectangles, disks and objects bounded by an ellipse. In their analysis of structure detection, Arias-Castro et al (2005) define a distance between any two given structures. The distance depends on the degree of overlap of the structures in the image.…”

Section: Detection Of Ellipses and Other Structuresmentioning

confidence: 99%

“…A second criterion is to detect an ellipse if the set of pixel values inside the ellipse is different in some well defined way from the set of pixel values outside the ellipse. For example, in Arias-Castro et al (2005) an ellipse is detected if the set of values of the pixels within it has an elevated mean value. The choice of a particular criterion depends on the application, and especially on any prior knowledge about the ellipses which are to be detected.…”

Section: Ellipse Detectionmentioning

confidence: 99%

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Application of the Fisher-Rao Metric to Ellipse Detection

Maybank

2006

Int J Comput Vision

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Abstract. The parameter space for the ellipses in a two dimensional image is a five dimensional manifold, where each point of the manifold corresponds to an ellipse in the image. The parameter space becomes a Riemannian manifold under a Fisher-Rao metric, which is derived from a Gaussian model for the blurring of ellipses in the image. Two points in the parameter space are close together under the Fisher-Rao metric if the corresponding ellipses are close together in the image. The Fisher-Rao metric is accurately approximated by a simpler metric under the assumption that the blurring is small compared with the sizes of the ellipses under consideration. It is shown that the parameter space for the ellipses in the image has a finite volume under the approximation to the Fisher-Rao metric. As a consequence the parameter space can be replaced, for the purpose of ellipse detection, by a finite set of points sampled from it. An efficient algorithm for sampling the parameter space is described. The algorithm uses the fact that the approximating metric is flat, and therefore locally Euclidean, on each three dimensional family of ellipses with a fixed orientation and a fixed eccentricity. Once the sample points have been obtained, ellipses are detected in a given image by checking each sample point in turn to see if the corresponding ellipse is supported by the nearby image pixel values. The resulting algorithm for ellipse detection is implemented. A multiresolution version of the algorithm is also implemented. The experimental results suggest that ellipses can be reliably detected in a given low resolution image and that the number of false detections can be reduced using the multiresolution algorithm.

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Template Matching Techniques in Computer Vision

Brunelli¹

2009

573

339

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Near-Optimal Detection of Geometric Objects by Fast Multiscale Methods

Cited by 148 publications

References 49 publications

Connect the dots: how many random points can a regular curve pass through?

Connect the dots: how many random points can a regular curve pass through?

Application of the Fisher-Rao Metric to Ellipse Detection

Template Matching Techniques in Computer Vision

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