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526:“circles” in the parameter space that passing through the corresponding grid cell in the parameter space. The number is also called “voting number”. Initially, every element in the matrix is zeros. Then for each “edge” point in the original space, we can formulate a circle in the parameter space and increase the voting number of the grid cell which the circle passes through. This process is called “voting”.
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with radius r. An accumulator matrix is used for tracking the intersection point. In the parameter space, the voting number of points through which the circle passing would be increased by one. Then the local maxima point (the red point in the center in the right figure) can be found. The position (a, b) of the maxima would be the center of the original circle.
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J. Illingworth and J. Kittler introduced this method for implementing Hough
Transform efficiently. The AHT uses a small accumulator array and the idea of a flexible iterative "coarse to fine" accumulation and search strategy to identify significant peaks in the Hough parameter spaces. This method is
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If the radius is fixed, then the parameter space would be reduced to 2D (the position of the circle center). For each point (x, y) on the original circle, it can define a circle centered at (x, y) with radius R according to (1). The intersection point of all such circles in the parameter space would
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cone whose apex is at (x, y, 0). In the 3D space, the circle parameters can be identified by the intersection of many conic surfaces that are defined by points on the 2D circle. This process can be divided into two stages. The first stage is fixing radius then find the optimal center of circles in a
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Consider 4 points on a circle in the original image (left). The circle Hough transform is shown in the right. Note that the radius is assumed to be known. For each (x,y) of the four points (white points) in the original image, it can define a circle in the Hough parameter space centered at (x, y)
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For each A = 0; // fill with zeroes initially, instantiate 3D matrix For each cell(x,y) For each theta t = 0 to 360 // the possible theta 0 to 360 b = y – r * sin(t * PI / 180); //polar coordinate for center (convert to radians) a = x – r * cos(t * PI / 180); //polar
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However, choosing an appropriate grid size is difficult. Since too coarse a grid can lead to large values of the vote being obtained falsely because many quite different structures correspond to a single bucket. Too fine a grid can lead to structures not being found because votes resulting from
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Since the parameter space is 3D, the accumulator matrix would be 3D, too. We can iterate through possible radii; for each radius, we use the previous technique. Finally, find the local maxima in the 3D accumulator matrix. Accumulator array should be A in the 3D space. Voting should be for each
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In practice, an accumulator matrix is introduced to find the intersection point in the parameter space. First, we need to divide the parameter space into “buckets” using a grid and produce an accumulator matrix according to the grid. The element in the accumulator matrix denotes the number of
476:. If a 2D point (x,y) is fixed, then the parameters can be found according to (1). The parameter space would be three dimensional, (a, b, r). And all the parameters that satisfy (x, y) would lie on the surface of an inverted
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Mitra, Jubin et. el. "Peak trekking of hierarchy mountain for the detection of cerebral aneurysm using modified Hough circle transform." ELCVIA Electronic
Letters on Computer Vision and Image Analysis 12.1 (2013).
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for detecting circles in imperfect images. The circle candidates are produced by “voting” in the Hough parameter space and then selecting local maxima in an accumulator
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Since the parameter space of the CHT is three dimensional, it may require lots of storage and computation. Choosing a bigger grid size can ameliorate this problem.
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After voting, we can find local maxima in the accumulator matrix. The positions of the local maxima are corresponding to the circle centers in the original space.
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Since the head would be similar to a circle in an image, CHT can be used for detecting heads in a picture, so as to count the number of persons in the image.
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The original picture (right) is first turned into a binary image (left) using a threshold and
Gaussian filter. Then edges (mid) are found from it using
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Modified Hough Circle
Transform (MHCT) is used on the image extracted from Digital Subtraction Angiogram (DSA) to detect and classify aneurysms type.
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695:, By Harvey Rhody, Chester F. Carlson Center for Imaging Science, Rochester Institute of Technology (October 11, 2005)
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substantially superior to the standard Hough
Transform implementation in both storage and computational requirements.
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Process the filtering algorithm on image
Gaussian Blurring, convert the image to grayscale ( grayScaling), make
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Hong Liu, Yueliang Qian and
Shouxun Lin, “DETECTING PERSONS USING HOUGH CIRCLE TRANSFORM IN SURVEILLANCE VIDEO”
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597:. After this, all the edge points are used by the Circle Hough Transform to find underlying circle structure.
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2D parameter space. The second stage is to find the optimal radius in a one dimensional parameter space.
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J. Illingworth and J. Kittler, “The
Adaptive Hough Transform,” PAMI-9, Issue: 5, 1987, pp 690-698
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tokens that are not exactly aligned end up in different buckets, and no bucket has a large vote.
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Note that, in the accumulator matrix (right fig), there would be at least 3 local maxima points.
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coordinate for center (convert to radians) A +=1; //voting end end
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The local maximum voted circles of
Accumulator A gives the circle Hough space.
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462:{\displaystyle \left(x-a\right)^{2}+\left(y-b\right)^{2}=r^{2}\ \ \ \ \ (1)}
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Multiple circles with same radius can be found with the same technique.
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http://elcvia.cvc.uab.es/article/view/v12-n1-mitra-chandra-halder
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be corresponding to the center point of the original circle.
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667:, OpenCV-Python Tutorials (archived version on archive.org)
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In a two-dimensional space, a circle can be described by:
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The maximum voted circle of
Accumulator gives the circle.
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Circle finding technique used in digital image processing
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where (a,b) is the center of the circle, and r is the
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554:, The Canny operator gives the edges on image.
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659:Circle Detection via Standard Hough Transform
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557:Vote on all possible circles in accumulator.
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612:Also, the CHT is not very robust to noise.
568:The Incrementing for Best Candidate :
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533:Find circle parameter with unknown radius
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505:Multiple circles with known radius R
751:Feature detection (computer vision)
485:Find parameters with known radius R
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693:Lecture 10: Hough Circle Transform
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207:Affine invariant feature detection
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145:Maximally stable extremal regions
102:Hessian feature strength measures
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521:Accumulator matrix and voting
140:Determinant of Hessian (DoH)
135:Difference of Gaussians (DoG)
582:Find circles in a shoe-print
199:Generalized structure tensor
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683:Generalised Hough transform
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178:Generalized Hough transform
130:Laplacian of Gaussian (LoG)
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688:Randomized Hough transform
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643:Brain Aneurysm Detection
621:Adaptive Hough Transform
336:digital image processing
215:Affine shape adaptation
665:Hough Circle Transform
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324:circle Hough Transform
279:Implementation details
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97:Level curve curvature
595:canny edge detection
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49:Differential
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265:Scale space
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616:Extensions
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577:Examples
284:Pyramids
64:Robinson
59:Prewitt
44:Deriche
474:radius
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353:Theory
340:matrix
107:SUSAN
54:Sobel
39:Canny
322:The
251:GLOH
246:SURF
241:SIFT
150:PCBR
112:FAST
328:CHT
256:HOG
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