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emmanuelle
55f5103dd8
Modified travis_script.sh to account for the new structure of the gallery Added README.txt files in directories of gallery examples Fixed references to gallery images in user guide pages Fixed broken links
134 lines
4.5 KiB
Python
134 lines
4.5 KiB
Python
r"""
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=============================
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Straight line Hough transform
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=============================
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The Hough transform in its simplest form is a `method to detect straight lines
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<http://en.wikipedia.org/wiki/Hough_transform>`__.
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In the following example, we construct an image with a line intersection. We
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then use the Hough transform to explore a parameter space for straight lines
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that may run through the image.
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Algorithm overview
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------------------
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Usually, lines are parameterised as :math:`y = mx + c`, with a gradient
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:math:`m` and y-intercept `c`. However, this would mean that :math:`m` goes to
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infinity for vertical lines. Instead, we therefore construct a segment
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perpendicular to the line, leading to the origin. The line is represented by
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the length of that segment, :math:`r`, and the angle it makes with the x-axis,
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:math:`\theta`.
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The Hough transform constructs a histogram array representing the parameter
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space (i.e., an :math:`M \times N` matrix, for :math:`M` different values of
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the radius and :math:`N` different values of :math:`\theta`). For each
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parameter combination, :math:`r` and :math:`\theta`, we then find the number of
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non-zero pixels in the input image that would fall close to the corresponding
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line, and increment the array at position :math:`(r, \theta)` appropriately.
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We can think of each non-zero pixel "voting" for potential line candidates. The
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local maxima in the resulting histogram indicates the parameters of the most
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probably lines. In our example, the maxima occur at 45 and 135 degrees,
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corresponding to the normal vector angles of each line.
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Another approach is the Progressive Probabilistic Hough Transform [1]_. It is
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based on the assumption that using a random subset of voting points give a good
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approximation to the actual result, and that lines can be extracted during the
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voting process by walking along connected components. This returns the
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beginning and end of each line segment, which is useful.
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The function `probabilistic_hough` has three parameters: a general threshold
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that is applied to the Hough accumulator, a minimum line length and the line
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gap that influences line merging. In the example below, we find lines longer
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than 10 with a gap less than 3 pixels.
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References
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----------
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.. [1] C. Galamhos, J. Matas and J. Kittler,"Progressive probabilistic
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Hough transform for line detection", in IEEE Computer Society
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Conference on Computer Vision and Pattern Recognition, 1999.
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.. [2] Duda, R. O. and P. E. Hart, "Use of the Hough Transformation to
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Detect Lines and Curves in Pictures," Comm. ACM, Vol. 15,
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pp. 11-15 (January, 1972)
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"""
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from skimage.transform import (hough_line, hough_line_peaks,
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probabilistic_hough_line)
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from skimage.feature import canny
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from skimage import data
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import numpy as np
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import matplotlib.pyplot as plt
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# Construct test image
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image = np.zeros((100, 100))
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# Classic straight-line Hough transform
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idx = np.arange(25, 75)
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image[idx[::-1], idx] = 255
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image[idx, idx] = 255
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h, theta, d = hough_line(image)
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fig, (ax1, ax2, ax3) = plt.subplots(1, 3, figsize=(8,4))
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ax1.imshow(image, cmap=plt.cm.gray)
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ax1.set_title('Input image')
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ax1.set_axis_off()
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ax2.imshow(np.log(1 + h),
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extent=[np.rad2deg(theta[-1]), np.rad2deg(theta[0]),
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d[-1], d[0]],
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cmap=plt.cm.gray, aspect=1/1.5)
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ax2.set_title('Hough transform')
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ax2.set_xlabel('Angles (degrees)')
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ax2.set_ylabel('Distance (pixels)')
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ax2.axis('image')
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ax3.imshow(image, cmap=plt.cm.gray)
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rows, cols = image.shape
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for _, angle, dist in zip(*hough_line_peaks(h, theta, d)):
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y0 = (dist - 0 * np.cos(angle)) / np.sin(angle)
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y1 = (dist - cols * np.cos(angle)) / np.sin(angle)
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ax3.plot((0, cols), (y0, y1), '-r')
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ax3.axis((0, cols, rows, 0))
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ax3.set_title('Detected lines')
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ax3.set_axis_off()
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# Line finding, using the Probabilistic Hough Transform
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image = data.camera()
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edges = canny(image, 2, 1, 25)
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lines = probabilistic_hough_line(edges, threshold=10, line_length=5,
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line_gap=3)
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fig, (ax1, ax2, ax3) = plt.subplots(1, 3, figsize=(8,4), sharex=True, sharey=True)
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ax1.imshow(image, cmap=plt.cm.gray)
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ax1.set_title('Input image')
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ax1.set_axis_off()
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ax1.set_adjustable('box-forced')
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ax2.imshow(edges, cmap=plt.cm.gray)
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ax2.set_title('Canny edges')
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ax2.set_axis_off()
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ax2.set_adjustable('box-forced')
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ax3.imshow(edges * 0)
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for line in lines:
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p0, p1 = line
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ax3.plot((p0[0], p1[0]), (p0[1], p1[1]))
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ax3.set_title('Probabilistic Hough')
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ax3.set_axis_off()
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ax3.set_adjustable('box-forced')
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plt.show()
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