2D Shape Measures for Computer Vision

Nayak, Amiya

doi:10.1002/9780470175668.ch12

Cited by 10 publications

(6 citation statements)

References 65 publications

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“…In addition, there are some general approaches to defining shape descriptors that could be adopted to defining squareness [22]. For example, first, fit (somehow) a square K f it (S) to a measured shape S, and then estimate the squareness of the shape S by comparing K f it (S) and S. One straightforward possibility for a fitted square K f it (S) is the square whose centroid coincides with the centroid of S, whose area is equal to the area of S, and finally K f it (S) is rotated such that the area of K f it (S)∩S is maximised.…”

Section: Introductionmentioning

confidence: 99%

Measuring Squareness and Orientation of Shapes

Rosin

Žunić

2010

J Math Imaging Vis

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In this paper we propose a measure which defines the degree to which a shape differs from a square. The new measure is easy to compute and being area based, is robust-e.g., with respect to noise or narrow intrusions. Also, it satisfies the following desirable properties:-it ranges over (0, 1] and gives the measured squareness equal to 1 if and only if the measured shape is a square;-it is invariant with respect to translations, rotations and scaling. In addition, we propose a generalisation of the new measure so that shape squareness can be computed while controlling the impact of the relative position of points inside the shape. Such a generalisation enables a tuning of the behaviour of the squareness measure and makes it applicable to a range of applications. A second generalisation produces a measure, parameterised by δ, that ranges in the interval (0, 1] and equals 1 if and only if the measured shape is a rhombus whose diagonals are in the proportion 1 : δ. The new measures (the initial measure and the generalised ones) are naturally defined and theoretically well founded-consequently, their behaviour can be well understood. As a by-product of the approach we obtain a new method for the orienting of shapes, which is demonstrated to be superior with respect to the standard method in several situations. The usefulness of the methods described in the manuscript is illustrated on three large shape databases: diatoms (ADIAC), MPEG-7 CE-1, and trademarks.

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Section: Introductionmentioning

confidence: 99%

Measuring Squareness and Orientation of Shapes

Rosin

Žunić

2010

J Math Imaging Vis

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“…Обычно авторы рассматривают задачу в случае, если области из нулей имеют регулярную структуру или их не так много. К примеру, заранее известны позиции K нулевых точек [8,9] или нулевые области имеют форму отрезков [10]. В таких случаях удается построить эффективные алгоритмы.…”

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Quantum algorithm for search of largest square on two dimensional map

Khadiev¹,

Remidovsky²

2022

Proceedings of Academician O.B. Lupanov 14th International Scientific Seminar "Discrete Mathematics and Its Applications"

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In the framework of this work, we consider the problem of finding the largest square in a 0-1 matrix consisting of ones. The task is considered with point of view of quantum algorithms. For the problem of finding a square of the largest size on a two-dimensional map, there is a quantum algorithm with running time O(n<sup>1.5</sup>log n) and error probability at most 0.1. At the same time, any classical algorithm (probabilistic or deterministic) has a lower bound on the running time Omega(n<sup>2</sup>).

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“…The main contribution of [4] is to replace the diagonal constraint matrix C by a matrix which corresponds to the condition 4ac-b 2 =1, which then always produces an ellipse. Such a matrix C satisfies a T Ca = 1 and has all zeros except C 13…”

Section: Direct Least Square Fittingmentioning

confidence: 99%

“…The problem now is to minimize a 1 T S 1 a 1 subject to a 1 T C'a 1 =1, where C' is the top 3x3 sub-matrix from C (nonzero elements are C' 13…”

Section: Then G = a 1 T S 11 A 1 + A 1 T S 12 A 2 =A 1 T (S 1 1 -S mentioning

confidence: 99%

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Direct Ellipse Fitting and Measuring Based on Shape Boundaries

Stojmenović

Nayak

Advances in Image and Video Technology

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Measuring ellipticity is an important area of computer vision systems. Most existing ellipticity measures are area based and cannot be easily applied to point sets such as extracted edges from real world images. We are interested in ellipse fitting and ellipticity measures which rely exclusively on shape boundary points which are practical in computer vision. They should also be calculated very quickly, be invariant to rotation, scaling and translation. Direct ellipse fitting methods are guaranteed to specifically return an ellipse as the fit rather than any conic. We argue that the only existing direct ellipse fit method does not work properly and propose a new simple scheme. It will determine the optimal location of the foci of the fitted ellipse along the orientation line (symmetrically with respect to the shape center) such that it minimizes the variance of sums of distances of points to the foci. We next propose a novel way of measuring the accuracy of ellipse fits against the original point set. The evaluation of fits proceeds by our novel ellipticity measure which transforms the point data into polar representation where the radius is equal to the sum of distances from the point to both foci, and the polar angle is equal to the one the original point makes with the center relative to the x-axis. The linearity of the polar representation will correspond to the quality of the ellipse fit for the original data. We also propose an ellipticity measure based on the average ratio of distances to the ellipse and to its center. The choice of center for each shape impacts the overall ellipticity measure. We discuss two ways of determining the center of the shape. The measures are tested on a set of shapes. The proposed algorithms work well on both open and closed curves.

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2D Shape Measures for Computer Vision

Cited by 10 publications

References 65 publications

Measuring Squareness and Orientation of Shapes

Measuring Squareness and Orientation of Shapes

Quantum algorithm for search of largest square on two dimensional map

Direct Ellipse Fitting and Measuring Based on Shape Boundaries

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