174 lines
6.0 KiB
Python
174 lines
6.0 KiB
Python
"""
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==============
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Edge operators
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==============
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Edge operators are used in image processing within edge detection algorithms.
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They are discrete differentiation operators, computing an approximation of the
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gradient of the image intensity function.
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"""
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import numpy as np
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import matplotlib.pyplot as plt
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from skimage import filters
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from skimage.data import camera
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from skimage.util import compare_images
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image = camera()
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edge_roberts = filters.roberts(image)
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edge_sobel = filters.sobel(image)
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fig, axes = plt.subplots(ncols=2, sharex=True, sharey=True,
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figsize=(8, 4))
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axes[0].imshow(edge_roberts, cmap=plt.cm.gray)
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axes[0].set_title('Roberts Edge Detection')
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axes[1].imshow(edge_sobel, cmap=plt.cm.gray)
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axes[1].set_title('Sobel Edge Detection')
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for ax in axes:
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ax.axis('off')
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plt.tight_layout()
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plt.show()
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######################################################################
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# Different operators compute different finite-difference approximations of
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# the gradient. For example, the Scharr filter results in a less rotational
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# variance than the Sobel filter that is in turn better than the Prewitt
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# filter [1]_ [2]_ [3]_. The difference between the Prewitt and Sobel filters
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# and the Scharr filter is illustrated below with an image that is the
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# discretization of a rotation- invariant continuous function. The
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# discrepancy between the Prewitt and Sobel filters, and the Scharr filter is
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# stronger for regions of the image where the direction of the gradient is
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# close to diagonal, and for regions with high spatial frequencies. For the
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# example image the differences between the filter results are very small and
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# the filter results are visually almost indistinguishable.
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#
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# .. [1] https://en.wikipedia.org/wiki/Sobel_operator#Alternative_operators
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#
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# .. [2] B. Jaehne, H. Scharr, and S. Koerkel. Principles of filter design.
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# In Handbook of Computer Vision and Applications. Academic Press,
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# 1999.
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#
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# .. [3] https://en.wikipedia.org/wiki/Prewitt_operator
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x, y = np.ogrid[:100, :100]
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# Creating a rotation-invariant image with different spatial frequencies.
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image_rot = np.exp(1j * np.hypot(x, y) ** 1.3 / 20.).real
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edge_sobel = filters.sobel(image_rot)
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edge_scharr = filters.scharr(image_rot)
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edge_prewitt = filters.prewitt(image_rot)
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diff_scharr_prewitt = compare_images(edge_scharr, edge_prewitt)
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diff_scharr_sobel = compare_images(edge_scharr, edge_sobel)
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max_diff = np.max(np.maximum(diff_scharr_prewitt, diff_scharr_sobel))
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fig, axes = plt.subplots(nrows=2, ncols=2, sharex=True, sharey=True,
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figsize=(8, 8))
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axes = axes.ravel()
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axes[0].imshow(image_rot, cmap=plt.cm.gray)
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axes[0].set_title('Original image')
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axes[1].imshow(edge_scharr, cmap=plt.cm.gray)
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axes[1].set_title('Scharr Edge Detection')
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axes[2].imshow(diff_scharr_prewitt, cmap=plt.cm.gray, vmax=max_diff)
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axes[2].set_title('Scharr - Prewitt')
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axes[3].imshow(diff_scharr_sobel, cmap=plt.cm.gray, vmax=max_diff)
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axes[3].set_title('Scharr - Sobel')
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for ax in axes:
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ax.axis('off')
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plt.tight_layout()
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plt.show()
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######################################################################
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# As in the previous example, here we illustrate the rotational invariance of
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# the filters. The top row shows a rotationally invariant image along with the
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# angle of its analytical gradient. The other two rows contain the difference
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# between the different gradient approximations (Sobel, Prewitt, Scharr &
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# Farid) and analytical gradient.
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#
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# The Farid & Simoncelli derivative filters [4]_, [5]_ are the most
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# rotationally invariant, but require a 5x5 kernel, which is computationally
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# more intensive than a 3x3 kernel.
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#
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# .. [4] Farid, H. and Simoncelli, E. P., "Differentiation of discrete
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# multidimensional signals", IEEE Transactions on Image Processing
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# 13(4): 496-508, 2004. :DOI:`10.1109/TIP.2004.823819`
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#
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# .. [5] Wikipedia, "Farid and Simoncelli Derivatives." Available at:
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# <https://en.wikipedia.org/wiki/Image_derivatives#Farid_and_Simoncelli_Derivatives>
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x, y = np.mgrid[-10:10:255j, -10:10:255j]
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image_rotinv = np.sin(x ** 2 + y ** 2)
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image_x = 2 * x * np.cos(x ** 2 + y ** 2)
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image_y = 2 * y * np.cos(x ** 2 + y ** 2)
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def angle(dx, dy):
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"""Calculate the angles between horizontal and vertical operators."""
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return np.mod(np.arctan2(dy, dx), np.pi)
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true_angle = angle(image_x, image_y)
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angle_farid = angle(filters.farid_h(image_rotinv),
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filters.farid_v(image_rotinv))
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angle_sobel = angle(filters.sobel_h(image_rotinv),
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filters.sobel_v(image_rotinv))
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angle_scharr = angle(filters.scharr_h(image_rotinv),
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filters.scharr_v(image_rotinv))
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angle_prewitt = angle(filters.prewitt_h(image_rotinv),
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filters.prewitt_v(image_rotinv))
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def diff_angle(angle_1, angle_2):
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"""Calculate the differences between two angles."""
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return np.minimum(np.pi - np.abs(angle_1 - angle_2),
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np.abs(angle_1 - angle_2))
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diff_farid = diff_angle(true_angle, angle_farid)
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diff_sobel = diff_angle(true_angle, angle_sobel)
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diff_scharr = diff_angle(true_angle, angle_scharr)
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diff_prewitt = diff_angle(true_angle, angle_prewitt)
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fig, axes = plt.subplots(nrows=3, ncols=2, sharex=True, sharey=True,
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figsize=(8, 8))
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axes = axes.ravel()
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axes[0].imshow(image_rotinv, cmap=plt.cm.gray)
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axes[0].set_title('Original image')
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axes[1].imshow(true_angle, cmap=plt.cm.hsv)
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axes[1].set_title('Analytical gradient angle')
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axes[2].imshow(diff_sobel, cmap=plt.cm.inferno, vmin=0, vmax=0.02)
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axes[2].set_title('Sobel error')
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axes[3].imshow(diff_prewitt, cmap=plt.cm.inferno, vmin=0, vmax=0.02)
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axes[3].set_title('Prewitt error')
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axes[4].imshow(diff_scharr, cmap=plt.cm.inferno, vmin=0, vmax=0.02)
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axes[4].set_title('Scharr error')
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color_ax = axes[5].imshow(diff_farid, cmap=plt.cm.inferno, vmin=0, vmax=0.02)
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axes[5].set_title('Farid error')
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fig.subplots_adjust(right=0.8)
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colorbar_ax = fig.add_axes([0.90, 0.10, 0.02, 0.50])
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fig.colorbar(color_ax, cax=colorbar_ax, ticks=[0, 0.01, 0.02])
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for ax in axes:
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ax.axis('off')
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plt.show()
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