Merge pull request #250 from ahojnnes/approx-polygon

ENH: Add polygon approximation and subdivision.

CONTRIBUTORS.txt

@@ -105,11 +105,8 @@
   Shape views: ``util.shape.view_as_windows`` and ``util.shape.view_as_blocks``
 
 - Johannes Schönberger
-  Polygon, circle and ellipse drawing functions
-  Adaptive thresholding
-  Implementation of Matlab's `regionprops`
-  Estimation of geometric transformation parameters
-  Local binary pattern texture classification
+  Drawing functions, adaptive thresholding, regionprops, geometric
+  transformations, LBPs, polygon approximations, and more.
 
 - Pavel Campr
   Fixes and tests for Histograms of Oriented Gradients.

doc/examples/plot_polygon.py

@@ -0,0 +1,77 @@
+"""
+====================
+Approximate Polygons
+====================
+
+This example shows how to approximate polygonal chains with the Douglas-Peucker
+algorithm.
+"""
+
+import numpy as np
+import matplotlib.pyplot as plt
+from skimage.draw import ellipse
+from skimage.measure import find_contours, approximate_polygon, \
+    subdivide_polygon
+
+
+hand = np.array([[1.64516129, 1.16145833],
+                 [1.64516129, 1.59375   ],
+                 [1.35080645, 1.921875  ],
+                 [1.375     , 2.18229167],
+                 [1.68548387, 1.9375    ],
+                 [1.60887097, 2.55208333],
+                 [1.68548387, 2.69791667],
+                 [1.76209677, 2.56770833],
+                 [1.83064516, 1.97395833],
+                 [1.89516129, 2.75      ],
+                 [1.9516129 , 2.84895833],
+                 [2.01209677, 2.76041667],
+                 [1.99193548, 1.99479167],
+                 [2.11290323, 2.63020833],
+                 [2.2016129 , 2.734375  ],
+                 [2.25403226, 2.60416667],
+                 [2.14919355, 1.953125  ],
+                 [2.30645161, 2.36979167],
+                 [2.39112903, 2.36979167],
+                 [2.41532258, 2.1875    ],
+                 [2.1733871 , 1.703125  ],
+                 [2.07782258, 1.16666667]])
+
+# subdivide polygon using 2nd degree B-Splines
+new_hand = hand.copy()
+for _ in range(5):
+    new_hand = subdivide_polygon(new_hand, degree=2, preserve_ends=True)
+
+# approximate subdivided polygon with Douglas-Peucker algorithm
+appr_hand = approximate_polygon(new_hand, tolerance=0.02)
+
+print "Number of coordinates:", len(hand), len(new_hand), len(appr_hand)
+
+fig, (ax1, ax2) = plt.subplots(ncols=2, figsize=(9, 4))
+
+ax1.plot(hand[:, 0], hand[:, 1])
+ax1.plot(new_hand[:, 0], new_hand[:, 1])
+ax1.plot(appr_hand[:, 0], appr_hand[:, 1])
+
+
+# create two ellipses in image
+img = np.zeros((800, 800), 'int32')
+rr, cc = ellipse(250, 250, 180, 230, img.shape)
+img[rr, cc] = 1
+rr, cc = ellipse(600, 600, 150, 90, img.shape)
+img[rr, cc] = 1
+
+plt.gray()
+ax2.imshow(img)
+
+# approximate / simplify coordinates of the two ellipses
+for contour in find_contours(img, 0):
+    coords = approximate_polygon(contour, tolerance=2.5)
+    ax2.plot(coords[:, 1], coords[:, 0], '-r', linewidth=2)
+    coords2 = approximate_polygon(contour, tolerance=39.5)
+    ax2.plot(coords2[:, 1], coords2[:, 0], '-g', linewidth=2)
+    print "Number of coordinates:", len(contour), len(coords), len(coords2)
+
+ax2.axis((0, 800, 0, 800))
+
+plt.show()

doc/examples/plot_shapes.py

@@ -19,11 +19,11 @@ import numpy as np
 
 img = np.zeros((500, 500, 3), 'uint8')
 
-#: draw line
+# draw line
 rr, cc = line(120, 123, 20, 400)
 img[rr,cc,0] = 255
 
-#: fill polygon
+# fill polygon
 poly = np.array((
     (300, 300),
     (480, 320),
@@ -34,11 +34,11 @@ poly = np.array((
 rr, cc = polygon(poly[:,0], poly[:,1], img.shape)
 img[rr,cc,1] = 255
 
-#: fill circle
+# fill circle
 rr, cc = circle(200, 200, 100, img.shape)
 img[rr,cc,:] = (255, 255, 0)
 
-#: fill ellipse
+# fill ellipse
 rr, cc = ellipse(300, 300, 100, 200, img.shape)
 img[rr,cc,2] = 255
 

skimage/measure/__init__.py

@@ -1,3 +1,4 @@
 from .find_contours import find_contours
 from ._regionprops import regionprops, perimeter
 from ._structural_similarity import structural_similarity
+from ._polygon import approximate_polygon, subdivide_polygon

skimage/measure/_polygon.py

@@ -0,0 +1,169 @@
+import numpy as np
+from scipy import signal
+
+
+def approximate_polygon(coords, tolerance):
+    """Approximate a polygonal chain with the specified tolerance.
+
+    It is based on the Douglas-Peucker algorithm.
+
+    Note that the approximated polygon is always within the convex hull of the
+    original polygon.
+
+    Parameters
+    ----------
+    coords : (N, 2) array
+        Coordinate array.
+    tolerance : float
+        Maximum distance from original points of polygon to approximated
+        polygonal chain. If tolerance is 0, the original coordinate array
+        is returned.
+
+    Returns
+    -------
+    coords : (M, 2) array
+        Approximated polygonal chain where M <= N.
+
+    References
+    ----------
+    .. [1] http://en.wikipedia.org/wiki/Ramer-Douglas-Peucker_algorithm
+    """
+    if tolerance == 0:
+        return coords
+
+    chain = np.zeros(coords.shape[0], 'bool')
+    # pre-allocate distance array for all points
+    dists = np.zeros(coords.shape[0])
+    chain[0] = True
+    chain[-1] = True
+    pos_stack = [(0, chain.shape[0] - 1)]
+    end_of_chain = False
+
+    while not end_of_chain:
+        start, end = pos_stack.pop()
+        # determine properties of current line segment
+        r0, c0 = coords[start, :]
+        r1, c1 = coords[end, :]
+        dr = r1 - r0
+        dc = c1 - c0
+        segment_angle = - np.arctan2(dr, dc)
+        segment_dist = c0 * np.sin(segment_angle) + r0 * np.cos(segment_angle)
+
+        # select points in-between line segment
+        segment_coords = coords[start + 1:end, :]
+        segment_dists = dists[start + 1:end]
+
+
+        # check whether to take perpendicular or euclidean distance with
+        # inner product of vectors
+
+        # vectors from points -> start and end
+        dr0 = segment_coords[:, 0] - r0
+        dc0 = segment_coords[:, 1] - c0
+        dr1 = segment_coords[:, 0] - r1
+        dc1 = segment_coords[:, 1] - c1
+        # vectors points -> start and end projected on start -> end vector
+        projected_lengths0 = dr0 * dr + dc0 * dc
+        projected_lengths1 = - dr1 * dr - dc1 * dc
+        perp = np.logical_and(projected_lengths0 > 0,
+                              projected_lengths1 > 0)
+        eucl = np.logical_not(perp)
+        segment_dists[perp] = np.abs(
+            segment_coords[perp, 0] * np.cos(segment_angle)
+            + segment_coords[perp, 1] * np.sin(segment_angle)
+            - segment_dist
+        )
+        segment_dists[eucl] = np.minimum(
+            # distance to start point
+            np.sqrt(dc0[eucl] ** 2 + dr0[eucl] ** 2),
+            # distance to end point
+            np.sqrt(dc1[eucl] ** 2 + dr1[eucl] ** 2)
+        )
+
+        if np.any(segment_dists > tolerance):
+            # select point with maximum distance to line
+            new_end = start + np.argmax(segment_dists) + 1
+            pos_stack.append((new_end, end))
+            pos_stack.append((start, new_end))
+            chain[new_end] = True
+
+        if len(pos_stack) == 0:
+            end_of_chain = True
+
+    return coords[chain, :]
+
+
+# B-Spline subdivision
+_SUBDIVISION_MASKS = {
+    # degree: (mask_even, mask_odd)
+    #         extracted from (degree + 2)th row of Pascal's triangle
+    1: ([1, 1], [1, 1]),
+    2: ([3, 1], [1, 3]),
+    3: ([1, 6, 1], [0, 4, 4]),
+    4: ([5, 10, 1], [1, 10, 5]),
+    5: ([1, 15, 15, 1], [0, 6, 20, 6]),
+    6: ([7, 35, 21, 1], [1, 21, 35, 7]),
+    7: ([1, 28, 70, 28, 1], [0, 8, 56, 56, 8]),
+}
+
+
+def subdivide_polygon(coords, degree=2, preserve_ends=False):
+    """Subdivision of polygonal curves using B-Splines.
+
+    Note that the resulting curve is always within the convex hull of the
+    original polygon. Circular polygons stay closed after subdivision.
+
+    Parameters
+    ----------
+    coords : (N, 2) array
+        Coordinate array.
+    degree : {1, 2, 3, 4, 5, 6, 7}, optional
+        Degree of B-Spline. Default is 2.
+    preserve_ends : bool, optional
+        Preserve first and last coordinate of non-circular polygon. Default is
+        False.
+
+    Returns
+    -------
+    coords : (M, 2) array
+        Subdivided coordinate array.
+
+    References
+    ----------
+    .. [1] http://mrl.nyu.edu/publications/subdiv-course2000/coursenotes00.pdf
+    """
+    if degree not in _SUBDIVISION_MASKS:
+        raise ValueError("Invalid B-Spline degree. Only degree 1 - 7 is "
+                         "supported.")
+
+    circular = np.all(coords[0, :] == coords[-1, :])
+
+    method = 'valid'
+    if circular:
+        # remove last coordinate because of wrapping
+        coords = coords[:-1, :]
+        # circular convolution by wrapping boundaries
+        method = 'same'
+
+    mask_even, mask_odd = _SUBDIVISION_MASKS[degree]
+    # divide by total weight
+    mask_even = np.array(mask_even, np.float) / (2 ** degree)
+    mask_odd = np.array(mask_odd, np.float) / (2 ** degree)
+
+    even = signal.convolve2d(coords.T, np.atleast_2d(mask_even), mode=method,
+                             boundary='wrap')
+    odd = signal.convolve2d(coords.T, np.atleast_2d(mask_odd), mode=method,
+                            boundary='wrap')
+
+    out = np.zeros((even.shape[1] + odd.shape[1], 2))
+    out[1::2] = even.T
+    out[::2] = odd.T
+
+    if circular:
+        # close polygon
+        out = np.vstack([out, out[0, :]])
+
+    if preserve_ends and not circular:
+        out = np.vstack([coords[0, :], out, coords[-1, :]])
+
+    return out

skimage/measure/tests/test_polygon.py

@@ -0,0 +1,56 @@
+import numpy as np
+from skimage.measure import approximate_polygon, subdivide_polygon
+from skimage.measure._polygon import _SUBDIVISION_MASKS
+
+square = np.array([
+    [0, 0], [0, 1], [0, 2], [0, 3],
+    [1, 3], [2, 3], [3, 3],
+    [3, 2], [3, 1], [3, 0],
+    [2, 0], [1, 0], [0, 0]
+])
+
+
+def test_approximate_polygon():
+    out = approximate_polygon(square, 0.1)
+    np.testing.assert_array_equal(out, square[(0, 3, 6, 9, 12), :])
+
+    out = approximate_polygon(square, 2.2)
+    np.testing.assert_array_equal(out, square[(0, 6, 12), :])
+
+    out = approximate_polygon(square[(0, 1, 3, 4, 5, 6, 7, 9, 11, 12), :], 0.1)
+    np.testing.assert_array_equal(out, square[(0, 3, 6, 9, 12), :])
+
+    out = approximate_polygon(square, -1)
+    np.testing.assert_array_equal(out, square)
+
+
+def test_subdivide_polygon():
+    new_square1 = square
+    new_square2 = square[:-1]
+    new_square3 = square[:-1]
+    # test iterative subdvision
+    for _ in range(10):
+        square1, square2, square3 = new_square1, new_square2, new_square3
+        # test different B-Spline degrees
+        for degree in range(1, 7):
+            mask_len = len(_SUBDIVISION_MASKS[degree][0])
+            # test circular
+            new_square1 = subdivide_polygon(square1, degree)
+            np.testing.assert_array_equal(new_square1[-1], new_square1[0])
+            np.testing.assert_equal(new_square1.shape[0],
+                                    2 * square1.shape[0] - 1)
+            # test non-circular
+            new_square2 = subdivide_polygon(square2, degree)
+            np.testing.assert_equal(new_square2.shape[0],
+                                    2 * (square2.shape[0] - mask_len + 1))
+            # test non-circular, preserve_ends
+            new_square3 = subdivide_polygon(square3, degree, True)
+            np.testing.assert_equal(new_square3[0], square3[0])
+            np.testing.assert_equal(new_square3[-1], square3[-1])
+            print mask_len
+            np.testing.assert_equal(new_square3.shape[0],
+                                    2 * (square3.shape[0] - mask_len + 2))
+
+
+if __name__ == "__main__":
+    np.testing.run_module_suite()