Automated thresholding suggestions

episodes/07-thresholding.md

@@ -347,14 +347,16 @@ plt.xlim(0, 1.0)
 
 ![](fig/maize-root-cluster-histogram.png){alt='Grayscale histogram of the maize root image'}
 
-The histogram has a significant peak around 0.2, and a second,
-smaller peak very near 1.0.
+The histogram has a significant peak around 0.2 and then a broader "hill" around 0.6 followed by a smaller peak near 1.0. Looking at the grayscale image, we can identify the lower peak at 0.2 as the background and the larger pixel values as the foreground, but it is not so obvious what the threshold should be.
 Thus, this image is a good candidate for thresholding with Otsu's method.
 The mathematical details of how this works are complicated (see
 [the scikit-image documentation](https://scikit-image.org/docs/dev/api/skimage.filters.html#threshold-otsu)
 if you are interested),
 but the outcome is that Otsu's method finds a threshold value between
 the two peaks of a grayscale histogram.
+:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: instructor
+The histogram of the maize root image may prompt questions from learners about the interpretation of the peaks and the broader region around 0.6. The focus here is on the separation of background and foreground pixel values. We note that Otsu's method does not work well for the image with the shapes used earlier in this episode, as the foreground pixel values are more distributed. These examples could be augmented by a discussion of unimodal, bimodal, and multimodal histograms. While these points can lead to fruitful discussions, the text in this episode attempts to reduce cognitive load and deliberately simplifies the discussion.
+::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
 
 The `ski.filters.threshold_otsu()` function can be used to determine
 the threshold automatically via Otsu's method.
@@ -462,10 +464,10 @@ def measure_root_mass(filename, sigma=1.0):
     binary_mask = blurred_image > t
 
     # determine root mass ratio
-    rootPixels = np.count_nonzero(binary_mask)
+    root_pixels = np.count_nonzero(binary_mask)
     w = binary_mask.shape[1]
     h = binary_mask.shape[0]
-    density = rootPixels / (w * h)
+    density = root_pixels / (w * h)
 
     return density
 ```
@@ -612,9 +614,8 @@ and label from the image before applying Otsu's method.
 ## Solution
 
 We can apply a simple binary thresholding with a threshold
-`t=0.95` to remove the label and circle from the image. We use the
-binary mask to set the pixels in the blurred image to zero
-(black).
+`t=0.95` to remove the label and circle from the image. We can then use the
+binary mask to calculate the Otsu threshold without the pixels from the label and circle.
 
 ```python
 def enhanced_root_mass(filename, sigma):
@@ -627,21 +628,22 @@ def enhanced_root_mass(filename, sigma):
 
     # perform binary thresholding to mask the white label and circle
     binary_mask = blurred_image < 0.95
-    # use the mask to remove the circle and label from the blurred image
-    blurred_image[~binary_mask] = 0
-
-    # perform automatic thresholding to produce a binary image
-    t = ski.filters.threshold_otsu(blurred_image)
-    binary_mask = blurred_image > t
+
+    # perform automatic thresholding using only the pixels with value True in the binary mask
+    t = ski.filters.threshold_otsu(blurred_image[binary_mask])
+
+    # update binary mask to identify pixels which are both less than 0.95 and greater than t
+    binary_mask = np.logical_and(binary_mask, blurred_image > t)
 
     # determine root mass ratio
-    rootPixels = np.count_nonzero(binary_mask)
+    root_pixels = np.count_nonzero(binary_mask)
     w = binary_mask.shape[1]
     h = binary_mask.shape[0]
-    density = rootPixels / (w * h)
+    density = root_pixels / (w * h)
 
     return density
 
+
 all_files = glob.glob("data/trial-*.jpg")
 for filename in all_files:
     density = enhanced_root_mass(filename=filename, sigma=1.5)
@@ -653,12 +655,26 @@ The output of the improved program does illustrate that the white circles
 and labels were skewing our root mass ratios:
 
 ```output
-data/trial-016.jpg,0.045935837765957444
-data/trial-020.jpg,0.058800033244680854
-data/trial-216.jpg,0.13705003324468085
-data/trial-293.jpg,0.13164461436170213
+data/trial-016.jpg,0.046250166223404256
+data/trial-020.jpg,0.05886968085106383
+data/trial-216.jpg,0.13712117686170214
+data/trial-293.jpg,0.13190342420212767
 ```
+:::::::::::::::::::::::::::::::::::::::::: spoiler
 
+### What is `&` doing in the example above?
+
+The `&` operator above means that we have defined a logical AND statement. This combines the two tests of pixel intensities in the blurred image such that both must be true for a pixel's position to be set to `True` in the resulting mask.
+
+| Result of `t < blurred_image` | Result of `blurred_image < 0.95 | Resulting value in `binary_mask` |
+|----------|---------|---------|
+| False | True | False |
+| True | False | False |
+| True | True | True |
+
+Knowing how to construct this kind of logical operation can be very helpful in image processing. The University of Minnesota Library's [guide to Boolean operators](https://libguides.umn.edu/BooleanOperators) is a good place to start if you want to learn more.
+
+::::::::::::::::::::::::::::::::::::::::::::::::::
 Here are the binary images produced by the additional thresholding.
 Note that we have not completely removed the offending white pixels.
 Outlines still remain.