def test_trace_boundary(self):
# test moore neighbor algorithm
m_neighbor = np.array(
[[0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 1, 0],
[0, 0, 0, 0, 0, 0, 0, 1, 1, 0], [0, 0, 0, 1, 1, 1, 1, 0, 0, 0],
[0, 0, 1, 1, 1, 1, 1, 0, 0, 0], [0, 1, 1, 1, 1, 1, 1, 0, 0, 0],
[0, 0, 1, 1, 1, 1, 1, 0, 0, 0], [0, 0, 0, 1, 1, 1, 1, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 1, 1, 0], [0, 0, 0, 0, 0, 0, 0, 1, 1, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0]],
dtype=np.bool)
# refenece neighbors for isbf
rx_isbf = [
1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 8, 9, 9, 8, 7, 7, 7, 7, 6, 6, 5, 5,
5, 4, 4, 3, 3, 3, 3, 2, 1
]
ry_isbf = [
7, 8, 8, 7, 6, 6, 6, 6, 6, 7, 8, 8, 7, 7, 6, 5, 4, 3, 3, 2, 2, 1,
2, 2, 3, 3, 4, 5, 6, 7, 7
]
# refenece neighbors for moore
rx_moore = [
1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 8, 9, 9, 8, 7, 7, 7, 7, 6, 5, 4, 3,
3, 3, 3, 2, 1
]
ry_moore = [
7, 8, 8, 7, 6, 6, 6, 6, 6, 7, 8, 8, 7, 7, 6, 5, 4, 3, 2, 1, 2, 3,
4, 5, 6, 7, 7
]
output_isbf = trace_boundaries(m_neighbor)
np.testing.assert_allclose(rx_isbf, output_isbf[0][1])
np.testing.assert_allclose(ry_isbf, output_isbf[0][0])
output_moore = trace_boundaries(m_neighbor, 8)
np.testing.assert_allclose(rx_moore, output_moore[0][1])
np.testing.assert_allclose(ry_moore, output_moore[0][0])
def test_trace_boundary(self):
# test moore neighbor algorithm
m_neighbor = np.array(
[
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 1, 1, 0],
[0, 0, 0, 0, 0, 0, 0, 1, 1, 0],
[0, 0, 0, 1, 1, 1, 1, 0, 0, 0],
[0, 0, 1, 1, 1, 1, 1, 0, 0, 0],
[0, 1, 1, 1, 1, 1, 1, 0, 0, 0],
[0, 0, 1, 1, 1, 1, 1, 0, 0, 0],
[0, 0, 0, 1, 1, 1, 1, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 1, 1, 0],
[0, 0, 0, 0, 0, 0, 0, 1, 1, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
],
dtype=np.bool,
)
m_neighbor = np.ascontiguousarray(m_neighbor, dtype=np.int)
# refenece neighbors for isbf
rx_isbf = [1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 8, 9, 9, 8, 7, 7, 7, 7, 6, 6, 5, 5, 5, 4, 4, 3, 3, 3, 3, 2, 1]
ry_isbf = [7, 8, 8, 7, 6, 6, 6, 6, 6, 7, 8, 8, 7, 7, 6, 5, 4, 3, 3, 2, 2, 1, 2, 2, 3, 3, 4, 5, 6, 7, 7]
# refenece neighbors for moore
rx_moore = [1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 8, 9, 9, 8, 7, 7, 7, 7, 6, 5, 4, 3, 3, 3, 3, 2, 1]
ry_moore = [7, 8, 8, 7, 6, 6, 6, 6, 6, 7, 8, 8, 7, 7, 6, 5, 4, 3, 2, 1, 2, 3, 4, 5, 6, 7, 7]
output_isbf = trace_boundaries(m_neighbor)
np.testing.assert_allclose(rx_isbf, output_isbf[0][1])
np.testing.assert_allclose(ry_isbf, output_isbf[0][0])
output_moore = trace_boundaries(m_neighbor, 8)
np.testing.assert_allclose(rx_moore, output_moore[0][1])
np.testing.assert_allclose(ry_moore, output_moore[0][0])
def split_concavities(Label, MinDepth=4, MinConcavity=np.inf): # noqa: C901
"""Performs splitting of objects in a label image using geometric scoring
of concavities. Attempts to perform splits at narrow regions that are
perpendicular to the object's convex hull boundaries.
Parameters:
-----------
Label : array_like
A uint32 label image.
MinDepth : float
Minimum depth of concavities to consider during geometric splitting.
Default value = 2.
MinConcavity : float
Minimum concavity score to consider when performing for geometric
splitting. Default value = np.inf.
Notes:
------
Can produce a large number of thin "halo" objects surrouding the objects
with higher scores. These can be removed by filtering object width in the
resulting label image.
Returns:
--------
Label : array_like
A uint32 label image.
See Also:
---------
label_contours, min_model
References:
-----------
.. [1] S. Weinert et al "Detection and Segmentation of Cell Nuclei in
Virtual Microscopy Images: A Minimum-Model Approach" in Nature Scientific
Reports,vol.2,no.503, doi:10.1038/srep00503, 2012.
"""
# use shape profiles to split objects with concavities
# copy input label image
Convex = Label.copy()
# condense label image
if np.unique(Convex).size-1 != Convex.max():
Convex = label.condense(Convex)
# get locations of objects in initial image
Locations = ms.find_objects(Convex)
# initialize number of labeled objects and record initial count
Total = Label.max()
# initialize loop counter
i = 1
while i <= Total:
# get object window from label image
if i < len(Locations):
W = Convex[Locations[i-1]]
else:
Locations = ms.find_objects(Convex)
W = Convex[Locations[i-1]]
# embed masked object in padded boolean array
Mask = np.zeros((W.shape[0]+2, W.shape[1]+2), dtype=np.bool)
Mask[1:-1, 1:-1] = W == i
# generate convex hull of object
Hull = mo.convex_hull_image(Mask)
# generate boundary coordinates, trim duplicate point
X, Y = label.trace_boundaries(Mask, Connectivity=8)
X = np.array(X[:-1], dtype=np.uint32)
Y = np.array(Y[:-1], dtype=np.uint32)
# calculate distance transform of object boundary pixels to convex hull
Distance = mp.distance_transform_edt(Hull)
D = Distance[Y, X] - 1
# generate linear index of positions
Linear = np.arange(X.size)
# rotate boundary counter-clockwise until start position is on hull
while(D[0] != 0):
X = np.roll(X, -1)
Y = np.roll(Y, -1)
D = np.roll(D, -1)
Linear = np.roll(Linear, -1)
# find runs of concave pixels with length > 1
Concave = (D > 0).astype(np.int)
Start = np.where((Concave[1:] - Concave[0:-1]) == 1)[0]
Stop = np.where((Concave[1:] - Concave[0:-1]) == -1)[0] + 1
if(Stop.size == Start.size - 1):
Stop = np.append(Stop, 0)
# extract depth profiles, indices, distances for each run
iX = []
iY = []
Depths = []
Length = np.zeros((Start.size))
MaxDepth = np.zeros((Start.size))
for j in np.arange(Start.size):
if(Start[j] < Stop[j]):
iX.append(X[Start[j]:Stop[j]+1])
iY.append(Y[Start[j]:Stop[j]+1])
Depths.append(D[Start[j]:Stop[j]+1])
else: # run terminates at beginning of sequence
iX.append(np.append(X[Start[j]:], X[0]))
iY.append(np.append(Y[Start[j]:], Y[0]))
Depths.append(np.append(D[Start[j]:], D[0]))
Length[j] = iX[j].size
MaxDepth[j] = np.max(Depths[j])
# filter based on concave contour length and max depth
Keep = np.where((Length > 1) & (MaxDepth >= MinDepth))[0]
Start = Start[Keep]
Stop = Stop[Keep]
iX = [iX[Ind].astype(dtype=np.float) for Ind in Keep]
iY = [iY[Ind].astype(dtype=np.float) for Ind in Keep]
Depths = [Depths[Ind] for Ind in Keep]
# attempt cutting if more than 1 sequence is found
if Start.size > 1:
# initialize containers to hold cut scores, optimal cut locations
Scores = np.inf * np.ones((Start.size, Start.size))
Xcut1 = np.zeros((Start.size, Start.size), dtype=np.uint32)
Ycut1 = np.zeros((Start.size, Start.size), dtype=np.uint32)
Xcut2 = np.zeros((Start.size, Start.size), dtype=np.uint32)
Ycut2 = np.zeros((Start.size, Start.size), dtype=np.uint32)
# compare candidates pairwise between all runs and score
for j in np.arange(Start.size):
# get list of 'j' candidates that pass depth threshold
jCandidates = np.where(Depths[j] >= MinDepth)[0]
for k in np.arange(j+1, Start.size):
# get list of 'k' candidates that pass depth threshold
kCandidates = np.where(Depths[k] >= MinDepth)[0]
# initialize minimum score and cut locations
minScore = np.inf
minj = -1
mink = -1
# loop over each coordinate pair for concavities j,k
for a in np.arange(jCandidates.size):
for b in np.arange(kCandidates.size):
# calculate length score
Ls = length_score(iX[j][jCandidates[a]],
iY[j][jCandidates[a]],
iX[k][kCandidates[b]],
iY[k][kCandidates[b]],
Depths[j][jCandidates[a]],
Depths[k][kCandidates[b]])
# calculate angle score
As = angle_score(iX[j][0], iY[j][0],
iX[j][-1], iY[j][-1],
iX[k][0], iY[k][0],
iX[k][-1], iY[k][-1],
iX[j][jCandidates[a]],
iY[j][jCandidates[a]],
iX[k][kCandidates[b]],
iY[k][kCandidates[b]])
# combine scores
Score = (Ls + As) / 2
# replace if improvement
if Score < minScore:
minScore = Score
Scores[j, k] = minScore
minj = jCandidates[a]
mink = kCandidates[b]
# record best cut location
Xcut1[j, k] = iX[j][minj]
Ycut1[j, k] = iY[j][minj]
Xcut2[j, k] = iX[k][mink]
Ycut2[j, k] = iY[k][mink]
# pick the best scoring candidates and cut if needed
ArgMin = np.unravel_index(Scores.argmin(), Scores.shape)
if Scores[ArgMin[0], ArgMin[1]] <= MinConcavity:
# perform cut
SplitMask = cut(Mask,
Xcut1[ArgMin[0], ArgMin[1]].astype(np.float),
Ycut1[ArgMin[0], ArgMin[1]].astype(np.float),
Xcut2[ArgMin[0], ArgMin[1]].astype(np.float),
Ycut2[ArgMin[0], ArgMin[1]].astype(np.float))
# re-label cut image
SplitLabel = ms.label(SplitMask)[0]
# increment object count, and label new object at end
SplitLabel[SplitLabel > 1] = Total + 1
Total += 1
# label object '1' with current object value
SplitLabel[SplitLabel == 1] = i
# trim padding from corrected label image
Mask = Mask[1:-1, 1:-1]
SplitLabel = SplitLabel[1:-1, 1:-1]
# update label image
W[Mask] = SplitLabel[Mask]
else: # no cut made, move to next object
i = i + 1
else: # no cuts to attempt, move to next object
i = i + 1
return Convex
def trace_contours(I, X, Y, Min, Max, MaxLength=255):
"""Performs contour tracing of seed pixels in an intensity image using
gradient information.
Parameters:
-----------
I : array_like
An intensity image used for analyzing local minima/maxima and
gradients. Dimensions M x N.
X : array_like
A 1D array of horizontal coordinates of contour seed pixels for
tracing.
Y : array_like
A 1D array of the vertical coordinates of seed pixels for tracing.
Min : array_like
A 1D array of the corresponding minimum values for contour tracing of
seed point X, Y.
Max : array_like
A 1D array of the corresponding maximum values for contour tracing of
seed point X, Y.
MaxLength : int
Maximum allowable contour length. Default value = 255.
Notes:
------
Can be computationally expensive for large numbers of contours. Use
smoothing and delta thresholding when seeding contours to reduce burden.
Returns:
--------
cXs : list
A list of 1D numpy arrays defining the horizontal coordinates of object
boundaries.
cYs : list
A list of 1D numpy arrays defining the vertical coordinates of object
boundaries.
See Also:
---------
SeedContours, ScoreContours, MinimumModel
References:
-----------
.. [1] S. Weinert et al "Detection and Segmentation of Cell Nuclei in
Virtual Microscopy Images: A Minimum-Model Approach" in Nature Scientific
Reports,vol.2,no.503, doi:10.1038/srep00503, 2012.
"""
# initialize list of lists containing contours
cXs = []
cYs = []
# process each seed pixel sequentially
for i in np.arange(X.size):
# capture window surrounding (X[i], Y[i])
W = I[max(0, Y[i]-np.ceil(MaxLength/2.0)):
min(I.shape[0]+1, Y[i]+np.ceil(MaxLength/2.0)+1),
max(0, X[i]-np.ceil(MaxLength/2.0)):
min(I.shape[1]+1, X[i]+np.ceil(MaxLength/2.0))+1]
# binary threshold corresponding to seed pixel 'i'
W = (W <= Max[i]) & (W >= Min[i])
# embed with center pixel in middle of padded window
Embed = np.zeros((W.shape[0]+2, W.shape[1]+2), dtype=np.bool)
Embed[1:-1, 1:-1] = W
# calculate location of (X[i], Y[i]) in 'Embed'
pX = X[i] - max(0, X[i]-np.ceil(MaxLength/2.0)) + 1
pY = Y[i] - max(0, Y[i]-np.ceil(MaxLength/2.0)) + 1
# trace boundary, check stopping condition, append to list of contours
cX, cY = label.trace_boundaries(Embed, Connectivity=4,
XStart=pX, YStart=pY,
MaxLength=MaxLength)
if(cX[0] == cX[-1] and cY[0] == cY[-1] and len(cX) <= MaxLength):
# add window offset to contour coordinates
cX = [x + max(0, X[i]-np.ceil(MaxLength/2.0)) - 1 for x in cX]
cY = [y + max(0, Y[i]-np.ceil(MaxLength/2.0)) - 1 for y in cY]
# append to list of candidate contours
cXs.append(np.array(cX, dtype=np.uint32))
cYs.append(np.array(cY, dtype=np.uint32))
return cXs, cYs
def split_concavities(Label, MinDepth=4, MinConcavity=np.inf): # noqa: C901
"""Performs splitting of objects in a label image using geometric scoring
of concavities. Attempts to perform splits at narrow regions that are
perpendicular to the object's convex hull boundaries.
Parameters
----------
Label : array_like
A uint32 label image.
MinDepth : float
Minimum depth of concavities to consider during geometric splitting.
Default value = 2.
MinConcavity : float
Minimum concavity score to consider when performing for geometric
splitting. Default value = np.inf.
Notes
-----
Can produce a large number of thin "halo" objects surrouding the objects
with higher scores. These can be removed by filtering object width in the
resulting label image.
Returns
-------
Label : array_like
A uint32 label image.
See Also
--------
label_contours, min_model
References
----------
.. [1] S. Weinert et al "Detection and Segmentation of Cell Nuclei in
Virtual Microscopy Images: A Minimum-Model Approach" in Nature Scientific
Reports,vol.2,no.503, doi:10.1038/srep00503, 2012.
"""
# use shape profiles to split objects with concavities
# copy input label image
Convex = Label.copy()
# condense label image
if np.unique(Convex).size - 1 != Convex.max():
Convex = label.condense(Convex)
# get locations of objects in initial image
Locations = ms.find_objects(Convex)
# initialize number of labeled objects and record initial count
Total = Label.max()
# initialize loop counter
i = 1
while i <= Total:
# get object window from label image
if i < len(Locations):
W = Convex[Locations[i - 1]]
else:
Locations = ms.find_objects(Convex)
W = Convex[Locations[i - 1]]
# embed masked object in padded boolean array
Mask = np.zeros((W.shape[0] + 2, W.shape[1] + 2), dtype=np.bool)
Mask[1:-1, 1:-1] = W == i
# generate convex hull of object
Hull = mo.convex_hull_image(Mask)
# generate boundary coordinates, trim duplicate point
X, Y = label.trace_boundaries(Mask, conn=8)
X = np.array(X[:-1], dtype=np.uint32)
Y = np.array(Y[:-1], dtype=np.uint32)
# calculate distance transform of object boundary pixels to convex hull
Distance = mp.distance_transform_edt(Hull)
D = Distance[Y, X] - 1
# generate linear index of positions
Linear = np.arange(X.size)
# rotate boundary counter-clockwise until start position is on hull
while (D[0] != 0):
X = np.roll(X, -1)
Y = np.roll(Y, -1)
D = np.roll(D, -1)
Linear = np.roll(Linear, -1)
# find runs of concave pixels with length > 1
Concave = (D > 0).astype(np.int)
Start = np.where((Concave[1:] - Concave[0:-1]) == 1)[0]
Stop = np.where((Concave[1:] - Concave[0:-1]) == -1)[0] + 1
if (Stop.size == Start.size - 1):
Stop = np.append(Stop, 0)
# extract depth profiles, indices, distances for each run
iX = []
iY = []
Depths = []
Length = np.zeros((Start.size))
MaxDepth = np.zeros((Start.size))
for j in np.arange(Start.size):
if (Start[j] < Stop[j]):
iX.append(X[Start[j]:Stop[j] + 1])
iY.append(Y[Start[j]:Stop[j] + 1])
Depths.append(D[Start[j]:Stop[j] + 1])
else: # run terminates at beginning of sequence
iX.append(np.append(X[Start[j]:], X[0]))
iY.append(np.append(Y[Start[j]:], Y[0]))
Depths.append(np.append(D[Start[j]:], D[0]))
Length[j] = iX[j].size
MaxDepth[j] = np.max(Depths[j])
# filter based on concave contour length and max depth
Keep = np.where((Length > 1) & (MaxDepth >= MinDepth))[0]
Start = Start[Keep]
Stop = Stop[Keep]
iX = [iX[Ind].astype(dtype=np.float) for Ind in Keep]
iY = [iY[Ind].astype(dtype=np.float) for Ind in Keep]
Depths = [Depths[Ind] for Ind in Keep]
# attempt cutting if more than 1 sequence is found
if Start.size > 1:
# initialize containers to hold cut scores, optimal cut locations
Scores = np.inf * np.ones((Start.size, Start.size))
Xcut1 = np.zeros((Start.size, Start.size), dtype=np.uint32)
Ycut1 = np.zeros((Start.size, Start.size), dtype=np.uint32)
Xcut2 = np.zeros((Start.size, Start.size), dtype=np.uint32)
Ycut2 = np.zeros((Start.size, Start.size), dtype=np.uint32)
# compare candidates pairwise between all runs and score
for j in np.arange(Start.size):
# get list of 'j' candidates that pass depth threshold
jCandidates = np.where(Depths[j] >= MinDepth)[0]
for k in np.arange(j + 1, Start.size):
# get list of 'k' candidates that pass depth threshold
kCandidates = np.where(Depths[k] >= MinDepth)[0]
# initialize minimum score and cut locations
minScore = np.inf
minj = -1
mink = -1
# loop over each coordinate pair for concavities j,k
for a in np.arange(jCandidates.size):
for b in np.arange(kCandidates.size):
# calculate length score
Ls = length_score(iX[j][jCandidates[a]],
iY[j][jCandidates[a]],
iX[k][kCandidates[b]],
iY[k][kCandidates[b]],
Depths[j][jCandidates[a]],
Depths[k][kCandidates[b]])
# calculate angle score
As = angle_score(
iX[j][0], iY[j][0], iX[j][-1], iY[j][-1],
iX[k][0], iY[k][0], iX[k][-1], iY[k][-1],
iX[j][jCandidates[a]], iY[j][jCandidates[a]],
iX[k][kCandidates[b]], iY[k][kCandidates[b]])
# combine scores
Score = (Ls + As) / 2
# replace if improvement
if Score < minScore:
minScore = Score
Scores[j, k] = minScore
minj = jCandidates[a]
mink = kCandidates[b]
# record best cut location
Xcut1[j, k] = iX[j][minj]
Ycut1[j, k] = iY[j][minj]
Xcut2[j, k] = iX[k][mink]
Ycut2[j, k] = iY[k][mink]
# pick the best scoring candidates and cut if needed
ArgMin = np.unravel_index(Scores.argmin(), Scores.shape)
if Scores[ArgMin[0], ArgMin[1]] <= MinConcavity:
# perform cut
SplitMask = cut(Mask, Xcut1[ArgMin[0],
ArgMin[1]].astype(np.float),
Ycut1[ArgMin[0], ArgMin[1]].astype(np.float),
Xcut2[ArgMin[0], ArgMin[1]].astype(np.float),
Ycut2[ArgMin[0], ArgMin[1]].astype(np.float))
# re-label cut image
SplitLabel = ms.label(SplitMask)[0]
# increment object count, and label new object at end
SplitLabel[SplitLabel > 1] = Total + 1
Total += 1
# label object '1' with current object value
SplitLabel[SplitLabel == 1] = i
# trim padding from corrected label image
Mask = Mask[1:-1, 1:-1]
SplitLabel = SplitLabel[1:-1, 1:-1]
# update label image
W[Mask] = SplitLabel[Mask]
else: # no cut made, move to next object
i = i + 1
else: # no cuts to attempt, move to next object
i = i + 1
return Convex
def trace_contours(I, X, Y, Min, Max, MaxLength=255):
"""Performs contour tracing of seed pixels in an intensity image using
gradient information.
Parameters
----------
I : array_like
An intensity image used for analyzing local minima/maxima and
gradients. Dimensions M x N.
X : array_like
A 1D array of horizontal coordinates of contour seed pixels for
tracing.
Y : array_like
A 1D array of the vertical coordinates of seed pixels for tracing.
Min : array_like
A 1D array of the corresponding minimum values for contour tracing of
seed point X, Y.
Max : array_like
A 1D array of the corresponding maximum values for contour tracing of
seed point X, Y.
MaxLength : int
Maximum allowable contour length. Default value = 255.
Notes
-----
Can be computationally expensive for large numbers of contours. Use
smoothing and delta thresholding when seeding contours to reduce burden.
Returns
-------
cXs : list
A list of 1D numpy arrays defining the horizontal coordinates of object
boundaries.
cYs : list
A list of 1D numpy arrays defining the vertical coordinates of object
boundaries.
See Also
--------
SeedContours, ScoreContours, MinimumModel
References
----------
.. [1] S. Weinert et al "Detection and Segmentation of Cell Nuclei in
Virtual Microscopy Images: A Minimum-Model Approach" in Nature Scientific
Reports,vol.2,no.503, doi:10.1038/srep00503, 2012.
"""
# initialize list of lists containing contours
cXs = []
cYs = []
# process each seed pixel sequentially
for i in np.arange(X.size):
# capture window surrounding (X[i], Y[i])
W = I[max(0, Y[i] -
np.ceil(MaxLength /
2.0)):min(I.shape[0] + 1, Y[i] +
np.ceil(MaxLength / 2.0) + 1),
max(0, X[i] - np.ceil(MaxLength / 2.0)
):min(I.shape[1] + 1, X[i] + np.ceil(MaxLength / 2.0)) + 1]
# binary threshold corresponding to seed pixel 'i'
W = (W <= Max[i]) & (W >= Min[i])
# embed with center pixel in middle of padded window
Embed = np.zeros((W.shape[0] + 2, W.shape[1] + 2), dtype=np.bool)
Embed[1:-1, 1:-1] = W
# calculate location of (X[i], Y[i]) in 'Embed'
pX = X[i] - max(0, X[i] - np.ceil(MaxLength / 2.0)) + 1
pY = Y[i] - max(0, Y[i] - np.ceil(MaxLength / 2.0)) + 1
# trace boundary, check stopping condition, append to list of contours
cX, cY = label.trace_boundaries(Embed,
conn=4,
x_start=pX,
y_start=pY,
MaxLength=MaxLength)
if (cX[0] == cX[-1] and cY[0] == cY[-1] and len(cX) <= MaxLength):
# add window offset to contour coordinates
cX = [x + max(0, X[i] - np.ceil(MaxLength / 2.0)) - 1 for x in cX]
cY = [y + max(0, Y[i] - np.ceil(MaxLength / 2.0)) - 1 for y in cY]
# append to list of candidate contours
cXs.append(np.array(cX, dtype=np.uint32))
cYs.append(np.array(cY, dtype=np.uint32))
return cXs, cYs
