fitting a circle to a binary image
Question
I have been using skim age's thresholding algorithms to get some binary mask. For example, I obtain binary images like this:
What I am trying to figure out is how can I fit a circle to this binary mask. The constraint is the circle should cover as much of the white areas as possible and the whole circumference of the circle should lie entirely on the white parts.
I have been wrecking my head on how I can do this efficiently but have come up with no solution that works.
One approach I thought that might be something is:
- Find some optimal center of the image/circle (I am not sure how to do this yet. Perhaps some raster scan like approach).
- Compute circle for increasing radiii and figure out when it starts getting out of the white area or outside the image.
- Then the centroid and radius will describe the circle.
5 answers
solution1
7 ACCPTED 2015-02-02 22:09:40
Here is a solution that tries to make an optimal circle fit via minimization. It soon becomes apparent that the bubble isn't a circle :) Note the use of "regionprops" for easily determining area, centroid, etc. of regions.
from skimage import io, color, measure, draw, img_as_bool
import numpy as np
from scipy import optimize
import matplotlib.pyplot as plt
image = img_as_bool(color.rgb2gray(io.imread('bubble.jpg')))
regions = measure.regionprops(image)
bubble = regions[0]
y0, x0 = bubble.centroid
r = bubble.major_axis_length / 2.
def cost(params):
x0, y0, r = params
coords = draw.circle(y0, x0, r, shape=image.shape)
template = np.zeros_like(image)
template[coords] = 1
return -np.sum(template == image)
x0, y0, r = optimize.fmin(cost, (x0, y0, r))
import matplotlib.pyplot as plt
f, ax = plt.subplots()
circle = plt.Circle((x0, y0), r)
ax.imshow(image, cmap='gray', interpolation='nearest')
ax.add_artist(circle)
plt.show()
solution2
6 2015-02-03 00:06:29
This should in general give very good and robust results:
import numpy as np
from skimage import measure, feature, io, color, draw
img = color.rgb2gray(io.imread("circle.jpg"))
img = feature.canny(img).astype(np.uint8)
img[img > 0] = 255
coords = np.column_stack(np.nonzero(img))
model, inliers = measure.ransac(coords, measure.CircleModel,
min_samples=3, residual_threshold=1,
max_trials=500)
print model.params
rr, cc = draw.circle(model.params[0], model.params[1], model.params[2],
shape=img.shape)
img[rr, cc] = 128
solution3
2 2015-02-02 16:21:35
This is actually a mostly solved problem in image processing. Looks like what you want is a Hough Transform , specifically the circular or elliptical kind. I believe the circular one is a bit less computationally intensive in general.
Here are some code examples for scikit-image that show pretty much exactly what you're trying to do. And here is a link to the documentation .
solution4
1 2015-02-02 17:59:51
Updated Answer
Actually, if you use Connected Components Analysis , aka Blob Analysis , you can do it with ImageMagick much more succinctly and accurately like this:
convert 3J3qz.jpg \
-define connected-components:verbose=true \
-define connected-components:area-threshold=100 \
-connected-components 8 null:
Output:
Objects (id: bounding-box centroid area mean-color):
0: 720x576+0+0 370.6,322.1 213779 srgb(0,0,0)
13: 488x513+104+0 347.7,250.7 200941 srgb(255,255,255) <-- answer
which shows your largest blob (the speech bubble) has its centroid at coordinates 347,250 from the top-left corner, and also gives you the bounding box which measures 488x513 pixels and its top-left corner is at 104,0 from which you can derive a radius.
I can mark these with ImageMagick like this:
convert 3J3qz.jpg \
-fill red -draw "rectangle 342,245 352,255"
-stroke red -fill none -draw "rectangle 104,0 592,513"
out.png
Original Answer
As you are curious... you can do what I was suggesting with ImageMagick in two lines:
convert 3J3qz.jpg -resize 1x! -colorspace gray txt:
# ImageMagick pixel enumeration: 1,576,255,gray
0,0: (66,66,66) #424242 gray(66)
0,1: (70,70,70) #464646 gray(70)
0,2: (72,72,72) #484848 gray(72)
0,3: (76,76,76) #4C4C4C gray(76)
...
0,152: (176,176,176) #B0B0B0 gray(176)
0,153: (176,176,176) #B0B0B0 gray(176)
0,154: (177,177,177) #B1B1B1 gray(177)
0,155: (177,177,177) #B1B1B1 gray(177)
0,156: (177,177,177) #B1B1B1 gray(177)
0,157: (177,177,177) #B1B1B1 gray(177)
0,158: (178,178,178) #B2B2B2 gray(178)
0,159: (178,178,178) #B2B2B2 gray(178)
0,160: (179,179,179) #B3B3B3 gray(179)
0,161: (179,179,179) #B3B3B3 gray(179)
0,162: (179,179,179) #B3B3B3 gray(179)
0,163: (179,179,179) #B3B3B3 gray(179)
0,164: (179,179,179) #B3B3B3 gray(179)
0,165: (179,179,179) #B3B3B3 gray(179)
0,166: (179,179,179) #B3B3B3 gray(179)
0,167: (179,179,179) #B3B3B3 gray(179)
0,168: (180,180,180) #B4B4B4 gray(180)
0,169: (180,180,180) #B4B4B4 gray(180)
0,170: (180,180,180) #B4B4B4 gray(180)
0,171: (180,180,180) #B4B4B4 gray(180)
0,172: (180,180,180) #B4B4B4 gray(180)
0,173: (180,180,180) #B4B4B4 gray(180)
0,174: (180,180,180) #B4B4B4 gray(180)
0,175: (180,180,180) #B4B4B4 gray(180)
0,176: (181,181,181) #B5B5B5 gray(181)
0,177: (181,181,181) #B5B5B5 gray(181)
0,178: (182,182,182) #B6B6B6 gray(182)
0,179: (182,182,182) #B6B6B6 gray(182)
0,180: (182,182,182) #B6B6B6 gray(182)
0,181: (182,182,182) #B6B6B6 gray(182)
0,182: (182,182,182) #B6B6B6 gray(182)
0,183: (182,182,182) #B6B6B6 gray(182)
0,184: (183,183,183) #B7B7B7 gray(183)
0,185: (183,183,183) #B7B7B7 gray(183)
0,186: (183,183,183) #B7B7B7 gray(183)
0,187: (183,183,183) #B7B7B7 gray(183)
0,188: (183,183,183) #B7B7B7 gray(183)
0,189: (183,183,183) #B7B7B7 gray(183)
0,190: (183,183,183) #B7B7B7 gray(183)
0,191: (183,183,183) #B7B7B7 gray(183)
0,192: (184,184,184) #B8B8B8 gray(184)
0,193: (184,184,184) #B8B8B8 gray(184)
0,194: (184,184,184) #B8B8B8 gray(184)
0,195: (184,184,184) #B8B8B8 gray(184)
0,196: (184,184,184) #B8B8B8 gray(184)
0,197: (184,184,184) #B8B8B8 gray(184)
0,198: (184,184,184) #B8B8B8 gray(184)
0,199: (184,184,184) #B8B8B8 gray(184)
0,200: (185,185,185) #B9B9B9 gray(185)
0,201: (185,185,185) #B9B9B9 gray(185)
0,202: (185,185,185) #B9B9B9 gray(185)
0,203: (185,185,185) #B9B9B9 gray(185)
0,204: (185,185,185) #B9B9B9 gray(185)
0,205: (185,185,185) #B9B9B9 gray(185)
0,206: (185,185,185) #B9B9B9 gray(185)
0,207: (185,185,185) #B9B9B9 gray(185)
0,208: (186,186,186) #BABABA gray(186)
0,209: (186,186,186) #BABABA gray(186)
0,210: (186,186,186) #BABABA gray(186)
0,211: (186,186,186) #BABABA gray(186)
0,212: (185,185,185) #B9B9B9 gray(185)
0,213: (186,186,186) #BABABA gray(186)
0,214: (186,186,186) #BABABA gray(186)
0,215: (186,186,186) #BABABA gray(186)
0,216: (186,186,186) #BABABA gray(186)
0,217: (186,186,186) #BABABA gray(186)
0,218: (186,186,186) #BABABA gray(186)
0,219: (186,186,186) #BABABA gray(186)
0,220: (186,186,186) #BABABA gray(186)
0,221: (186,186,186) #BABABA gray(186)
0,222: (186,186,186) #BABABA gray(186)
0,223: (186,186,186) #BABABA gray(186)
0,224: (186,186,186) #BABABA gray(186)
0,225: (186,186,186) #BABABA gray(186)
0,226: (186,186,186) #BABABA gray(186)
0,227: (186,186,186) #BABABA gray(186)
0,228: (187,187,187) #BBBBBB gray(187)
0,229: (187,187,187) #BBBBBB gray(187)
0,230: (187,187,187) #BBBBBB gray(187)
0,231: (187,187,187) #BBBBBB gray(187)
0,232: (187,187,187) #BBBBBB gray(187)
0,233: (187,187,187) #BBBBBB gray(187)
0,234: (187,187,187) #BBBBBB gray(187) <---- max=234
0,235: (187,187,187) #BBBBBB gray(187)
0,236: (187,187,187) #BBBBBB gray(187)
0,237: (187,187,187) #BBBBBB gray(187)
0,238: (187,187,187) #BBBBBB gray(187)
0,239: (187,187,187) #BBBBBB gray(187)
0,240: (187,187,187) #BBBBBB gray(187)
0,241: (187,187,187) #BBBBBB gray(187)
0,242: (187,187,187) #BBBBBB gray(187)
0,243: (187,187,187) #BBBBBB gray(187)
0,244: (187,187,187) #BBBBBB gray(187)
0,245: (187,187,187) #BBBBBB gray(187)
0,246: (187,187,187) #BBBBBB gray(187)
0,247: (187,187,187) #BBBBBB gray(187)
0,248: (187,187,187) #BBBBBB gray(187)
0,249: (187,187,187) #BBBBBB gray(187)
0,250: (187,187,187) #BBBBBB gray(187)
...
0,573: (0,0,0) #000000 gray(0)
0,574: (0,0,0) #000000 gray(0)
0,575: (0,0,0) #000000 gray(0)
And the other side
convert 3J3qz.jpg -resize x1! -colorspace gray txt:
# ImageMagick pixel enumeration: 720,1,255,gray
0,0: (0,0,0) #000000 gray(0)
1,0: (0,0,0) #000000 gray(0)
2,0: (0,0,0) #000000 gray(0)
3,0: (0,0,0) #000000 gray(0)
4,0: (0,0,0) #000000 gray(0)
...
241,0: (219,219,219) #DBDBDB gray(219)
242,0: (220,220,220) #DCDCDC gray(220)
243,0: (220,220,220) #DCDCDC gray(220)
244,0: (221,221,221) #DDDDDD gray(221)
245,0: (222,222,222) #DEDEDE gray(222)
246,0: (223,223,223) #DFDFDF gray(223)
247,0: (223,223,223) #DFDFDF gray(223)
248,0: (224,224,224) #E0E0E0 gray(224)
249,0: (224,224,224) #E0E0E0 gray(224)
250,0: (225,225,225) #E1E1E1 gray(225)
251,0: (227,227,227) #E3E3E3 gray(227)
252,0: (229,229,229) #E5E5E5 gray(229)
253,0: (230,230,230) #E6E6E6 gray(230)
254,0: (231,231,231) #E7E7E7 gray(231)
255,0: (232,232,232) #E8E8E8 gray(232) <--- max=255
256,0: (231,231,231) #E7E7E7 gray(231)
257,0: (231,231,231) #E7E7E7 gray(231)
258,0: (231,231,231) #E7E7E7 gray(231)
259,0: (231,231,231) #E7E7E7 gray(231)
260,0: (230,230,230) #E6E6E6 gray(230)
261,0: (230,230,230) #E6E6E6 gray(230)
262,0: (230,230,230) #E6E6E6 gray(230)
263,0: (230,230,230) #E6E6E6 gray(230)
264,0: (230,230,230) #E6E6E6 gray(230)
265,0: (230,230,230) #E6E6E6 gray(230)
266,0: (230,230,230) #E6E6E6 gray(230)
267,0: (230,230,230) #E6E6E6 gray(230)
268,0: (229,229,229) #E5E5E5 gray(229)
269,0: (230,230,230) #E6E6E6 gray(230)
270,0: (229,229,229) #E5E5E5 gray(229)
271,0: (229,229,229) #E5E5E5 gray(229)
272,0: (229,229,229) #E5E5E5 gray(229)
273,0: (229,229,229) #E5E5E5 gray(229)
274,0: (229,229,229) #E5E5E5 gray(229)
275,0: (229,229,229) #E5E5E5 gray(229)
276,0: (229,229,229) #E5E5E5 gray(229)
277,0: (229,229,229) #E5E5E5 gray(229)
278,0: (229,229,229) #E5E5E5 gray(229)
279,0: (229,229,229) #E5E5E5 gray(229)
280,0: (229,229,229) #E5E5E5 gray(229)
281,0: (229,229,229) #E5E5E5 gray(229)
282,0: (229,229,229) #E5E5E5 gray(229)
283,0: (229,229,229) #E5E5E5 gray(229)
284,0: (229,229,229) #E5E5E5 gray(229)
285,0: (229,229,229) #E5E5E5 gray(229)
286,0: (229,229,229) #E5E5E5 gray(229)
287,0: (230,230,230) #E6E6E6 gray(230)
288,0: (230,230,230) #E6E6E6 gray(230)
289,0: (230,230,230) #E6E6E6 gray(230)
290,0: (230,230,230) #E6E6E6 gray(230)
291,0: (230,230,230) #E6E6E6 gray(230)
292,0: (230,230,230) #E6E6E6 gray(230)
293,0: (230,230,230) #E6E6E6 gray(230)
294,0: (230,230,230) #E6E6E6 gray(230)
295,0: (231,231,231) #E7E7E7 gray(231)
296,0: (231,231,231) #E7E7E7 gray(231)
297,0: (231,231,231) #E7E7E7 gray(231)
298,0: (231,231,231) #E7E7E7 gray(231)
299,0: (231,231,231) #E7E7E7 gray(231)
300,0: (231,231,231) #E7E7E7 gray(231)
301,0: (231,231,231) #E7E7E7 gray(231)
302,0: (231,231,231) #E7E7E7 gray(231)
303,0: (231,231,231) #E7E7E7 gray(231)
304,0: (232,232,232) #E8E8E8 gray(232)
305,0: (231,231,231) #E7E7E7 gray(231)
306,0: (231,231,231) #E7E7E7 gray(231)
307,0: (231,231,231) #E7E7E7 gray(231)
308,0: (231,231,231) #E7E7E7 gray(231)
309,0: (232,232,232) #E8E8E8 gray(232)
310,0: (232,232,232) #E8E8E8 gray(232)
311,0: (232,232,232) #E8E8E8 gray(232)
312,0: (233,233,233) #E9E9E9 gray(233)
313,0: (232,232,232) #E8E8E8 gray(232)
314,0: (232,232,232) #E8E8E8 gray(232)
315,0: (232,232,232) #E8E8E8 gray(232)
316,0: (232,232,232) #E8E8E8 gray(232)
317,0: (232,232,232) #E8E8E8 gray(232)
318,0: (232,232,232) #E8E8E8 gray(232)
319,0: (232,232,232) #E8E8E8 gray(232)
320,0: (232,232,232) #E8E8E8 gray(232)
321,0: (233,233,233) #E9E9E9 gray(233)
322,0: (233,233,233) #E9E9E9 gray(233)
323,0: (233,233,233) #E9E9E9 gray(233)
324,0: (233,233,233) #E9E9E9 gray(233)
325,0: (233,233,233) #E9E9E9 gray(233)
326,0: (233,233,233) #E9E9E9 gray(233)
327,0: (233,233,233) #E9E9E9 gray(233)
328,0: (233,233,233) #E9E9E9 gray(233)
329,0: (233,233,233) #E9E9E9 gray(233)
330,0: (233,233,233) #E9E9E9 gray(233)
331,0: (233,233,233) #E9E9E9 gray(233)
332,0: (233,233,233) #E9E9E9 gray(233)
333,0: (233,233,233) #E9E9E9 gray(233)
334,0: (233,233,233) #E9E9E9 gray(233)
335,0: (233,233,233) #E9E9E9 gray(233)
336,0: (233,233,233) #E9E9E9 gray(233)
337,0: (233,233,233) #E9E9E9 gray(233)
338,0: (233,233,233) #E9E9E9 gray(233)
339,0: (233,233,233) #E9E9E9 gray(233)
340,0: (233,233,233) #E9E9E9 gray(233)
341,0: (233,233,233) #E9E9E9 gray(233)
342,0: (233,233,233) #E9E9E9 gray(233)
343,0: (233,233,233) #E9E9E9 gray(233)
344,0: (233,233,233) #E9E9E9 gray(233)
345,0: (233,233,233) #E9E9E9 gray(233)
346,0: (233,233,233) #E9E9E9 gray(233)
347,0: (233,233,233) #E9E9E9 gray(233)
348,0: (233,233,233) #E9E9E9 gray(233)
349,0: (233,233,233) #E9E9E9 gray(233)
350,0: (233,233,233) #E9E9E9 gray(233)
351,0: (233,233,233) #E9E9E9 gray(233)
352,0: (233,233,233) #E9E9E9 gray(233)
353,0: (233,233,233) #E9E9E9 gray(233)
354,0: (233,233,233) #E9E9E9 gray(233)
...
717,0: (0,0,0) #000000 gray(0)
718,0: (0,0,0) #000000 gray(0)
719,0: (0,0,0) #000000 gray(0)
solution5
0 2015-02-02 20:23:25
For someone looking to code Mark's suggestion in python, it is quite easy.
collapsed = np.sum(binary_array, axis=0)
# These indices will be already sorted
indices = np.where(collapsed == collapsed.max())[0]
c = indices[int(round((len(indices) - 1) / 2))]
# Same for rows
collapsed = np.sum(binary_array, axis=1)
# These indices will be already sorted
indices = np.where(collapsed == collapsed.max())[0]
r = indices[int(round((len(indices) - 1) / 2))]
# circle center is (r, c)
This code takes care when your shape is not spherical and the collapsing along an axes can have multiple maxima. In that case, it takes the middle one (the one that can give you the largest radius when you fit the circle).
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