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Li thresholding#
In 1993, Li and Lee proposed a new criterion for finding the “optimal” threshold to distinguish between the background and foreground of an image [1]. They proposed that minimizing the cross-entropy between the foreground and the foreground mean, and the background and the background mean, would give the best threshold in most situations.
Until 1998, though, the way to find this threshold was by trying all possible
thresholds and then choosing the one with the smallest cross-entropy. At that
point, Li and Tam implemented a new, iterative method to more quickly find the
optimum point by using the slope of the cross-entropy [2]. This is the method
implemented in scikit-image’s skimage.filters.threshold_li().
Here, we demonstrate the cross-entropy and its optimization by Li’s iterative
method. Note that we are using the private function _cross_entropy, which
should not be used in production code, as it could change.
import numpy as np
import matplotlib.pyplot as plt
from skimage import data
from skimage import filters
from skimage.filters.thresholding import _cross_entropy
cell = data.cell()
camera = data.camera()
First, we let’s plot the cross entropy for the skimage.data.camera()
image at all possible thresholds.
thresholds = np.arange(np.min(camera) + 1.5, np.max(camera) - 1.5)
entropies = [_cross_entropy(camera, t) for t in thresholds]
optimal_camera_threshold = thresholds[np.argmin(entropies)]
fig, ax = plt.subplots(1, 3, figsize=(8, 3))
ax[0].imshow(camera, cmap='gray')
ax[0].set_title('image')
ax[0].set_axis_off()
ax[1].imshow(camera > optimal_camera_threshold, cmap='gray')
ax[1].set_title('thresholded')
ax[1].set_axis_off()
ax[2].plot(thresholds, entropies)
ax[2].set_xlabel('thresholds')
ax[2].set_ylabel('cross-entropy')
ax[2].vlines(
optimal_camera_threshold,
ymin=np.min(entropies) - 0.05 * np.ptp(entropies),
ymax=np.max(entropies) - 0.05 * np.ptp(entropies),
)
ax[2].set_title('optimal threshold')
fig.tight_layout()
print('The brute force optimal threshold is:', optimal_camera_threshold)
print('The computed optimal threshold is:', filters.threshold_li(camera))
plt.show()
The brute force optimal threshold is: 78.5
The computed optimal threshold is: 78.91288426606151
Next, let’s use the iter_callback feature of threshold_li to examine
the optimization process as it happens.
iter_thresholds = []
optimal_threshold = filters.threshold_li(camera, iter_callback=iter_thresholds.append)
iter_entropies = [_cross_entropy(camera, t) for t in iter_thresholds]
print('Only', len(iter_thresholds), 'thresholds examined.')
fig, ax = plt.subplots()
ax.plot(thresholds, entropies, label='all threshold entropies')
ax.plot(iter_thresholds, iter_entropies, label='optimization path')
ax.scatter(iter_thresholds, iter_entropies, c='C1')
ax.legend(loc='upper right')
plt.show()
Only 5 thresholds examined.
This is clearly much more efficient than the brute force approach. However, in some images, the cross-entropy is not convex, meaning having a single optimum. In this case, gradient descent could yield a threshold that is not optimal. In this example, we see how a bad initial guess for the optimization results in a poor threshold selection.
iter_thresholds2 = []
opt_threshold2 = filters.threshold_li(
cell, initial_guess=64, iter_callback=iter_thresholds2.append
)
thresholds2 = np.arange(np.min(cell) + 1.5, np.max(cell) - 1.5)
entropies2 = [_cross_entropy(cell, t) for t in thresholds]
iter_entropies2 = [_cross_entropy(cell, t) for t in iter_thresholds2]
fig, ax = plt.subplots(1, 3, figsize=(8, 3))
ax[0].imshow(cell, cmap='magma')
ax[0].set_title('image')
ax[0].set_axis_off()
ax[1].imshow(cell > opt_threshold2, cmap='gray')
ax[1].set_title('thresholded')
ax[1].set_axis_off()
ax[2].plot(thresholds2, entropies2, label='all threshold entropies')
ax[2].plot(iter_thresholds2, iter_entropies2, label='optimization path')
ax[2].scatter(iter_thresholds2, iter_entropies2, c='C1')
ax[2].legend(loc='upper right')
plt.show()
In this image, amazingly, the default initial guess, the mean image value, actually lies right on top of the peak between the two “valleys” of the objective function. Without supplying an initial guess, Li’s thresholding method does nothing at all!
iter_thresholds3 = []
opt_threshold3 = filters.threshold_li(cell, iter_callback=iter_thresholds3.append)
iter_entropies3 = [_cross_entropy(cell, t) for t in iter_thresholds3]
fig, ax = plt.subplots(1, 3, figsize=(8, 3))
ax[0].imshow(cell, cmap='magma')
ax[0].set_title('image')
ax[0].set_axis_off()
ax[1].imshow(cell > opt_threshold3, cmap='gray')
ax[1].set_title('thresholded')
ax[1].set_axis_off()
ax[2].plot(thresholds2, entropies2, label='all threshold entropies')
ax[2].plot(iter_thresholds3, iter_entropies3, label='optimization path')
ax[2].scatter(iter_thresholds3, iter_entropies3, c='C1')
ax[2].legend(loc='upper right')
plt.show()
To see what is going on, let’s define a function, li_gradient, that
replicates the inner loop of the Li method and returns the change from the
current threshold value to the next one. When this gradient is 0, we are at
a so-called stationary point and Li returns this value. When it is
negative, the next threshold guess will be lower, and when it is positive,
the next guess will be higher.
In the plot below, we show the cross-entropy and the Li update path when the
initial guess is on the right side of that entropy peak. We overlay the
threshold update gradient, marking the 0 gradient line and the default
initial guess by threshold_li.
def li_gradient(image, t):
"""Find the threshold update at a given threshold."""
foreground = image > t
mean_fore = np.mean(image[foreground])
mean_back = np.mean(image[~foreground])
t_next = (mean_back - mean_fore) / (np.log(mean_back) - np.log(mean_fore))
dt = t_next - t
return dt
iter_thresholds4 = []
opt_threshold4 = filters.threshold_li(
cell, initial_guess=68, iter_callback=iter_thresholds4.append
)
iter_entropies4 = [_cross_entropy(cell, t) for t in iter_thresholds4]
print(len(iter_thresholds4), 'examined, optimum:', opt_threshold4)
gradients = [li_gradient(cell, t) for t in thresholds2]
fig, ax1 = plt.subplots()
ax1.plot(thresholds2, entropies2)
ax1.plot(iter_thresholds4, iter_entropies4)
ax1.scatter(iter_thresholds4, iter_entropies4, c='C1')
ax1.set_xlabel('threshold')
ax1.set_ylabel('cross entropy', color='C0')
ax1.tick_params(axis='y', labelcolor='C0')
ax2 = ax1.twinx()
ax2.plot(thresholds2, gradients, c='C3')
ax2.hlines(
[0], xmin=thresholds2[0], xmax=thresholds2[-1], colors='gray', linestyles='dashed'
)
ax2.vlines(
np.mean(cell),
ymin=np.min(gradients),
ymax=np.max(gradients),
colors='gray',
linestyles='dashed',
)
ax2.set_ylabel(r'$\Delta$(threshold)', color='C3')
ax2.tick_params(axis='y', labelcolor='C3')
fig.tight_layout()
plt.show()
8 examined, optimum: 111.68876119648344
In addition to allowing users to provide a number as an initial guess,
skimage.filters.threshold_li() can receive a function that makes a
guess from the image intensities, just like numpy.mean() does by
default. This might be a good option when many images with different ranges
need to be processed.
def quantile_95(image):
# you can use np.quantile(image, 0.95) if you have NumPy>=1.15
return np.percentile(image, 95)
iter_thresholds5 = []
opt_threshold5 = filters.threshold_li(
cell, initial_guess=quantile_95, iter_callback=iter_thresholds5.append
)
iter_entropies5 = [_cross_entropy(cell, t) for t in iter_thresholds5]
print(len(iter_thresholds5), 'examined, optimum:', opt_threshold5)
fig, ax1 = plt.subplots()
ax1.plot(thresholds2, entropies2)
ax1.plot(iter_thresholds5, iter_entropies5)
ax1.scatter(iter_thresholds5, iter_entropies5, c='C1')
ax1.set_xlabel('threshold')
ax1.set_ylabel('cross entropy', color='C0')
ax1.tick_params(axis='y', labelcolor='C0')
plt.show()
5 examined, optimum: 111.68876119648344
Total running time of the script: (0 minutes 5.120 seconds)
