Neighborhood filters

Prerequisites

Before starting this lesson, you should be familiar with:

Learning Objectives

After completing this lesson, learners should be able to:

  • Understand the basic principle of a neighborhood filter.

  • Apply basic neighborhood filters to an image.

Motivation

Images are quite often noisy or have other issues that make them hard to segment, e.g. by means of a simple intensity threshold. Neighborhood filters are often used to enhance the images in order to facilitate better performance of segmentation algorithms.

Concept map

graph TB P(Pixel coordinate) -->NBH(Neighborhood pixels) SE(Structuring element) --> NBH NBH -->A(Mathematical formula) A -->NP(New pixel value)



Figure


Image filtering with a pixel neighborhood. (a) Raw intensity image with pixel neighborhood (structuring element (SE), outer green box) and central pixel (inner orange box) on which the filtering operation will be performed. (b) Pixel values in the neighborhood. (c) X is the value that would be replaced after operation (indicated as op). Here, max, mean and variance operations are used. Note - One has to carefully look at the data type of the image as some operations can produce large/floating point values. (d) Different SEs (neighborhood in green and affected pixel in orange) top left - SE completely inside image boundaries; top right - SE at image boundaries (padding needed); bottom left - SE with different shape; bottom right - Line SE.



Neighborhood filters comprise two ingredients: a definition of the pixel neighborhood (size and shape) and a mathematical recipe what to compute on this neighborhood. The result of this computation will be used to replace the value of the central pixel in the neighborhood. This procedure can be applied to several (all) pixels of an image to obtain a filtered image. The animation shows a square neighborhood (3x3) applied to the inner pixels of the image.

There are tons of different neighborhood filters, and you can also invent one!

The neighborhoods

The neighborhood of a pixel is also called a structuring element (SE) and can have various sizes and shapes. Here, we use one of the simplest and most widely used neighborhoods, namely a circular neighborhood, which is defined by a certain radius. We will explore other shapes of SEs in more detail in a dedicated module.

Padding

Since the filtering operation takes into account all the directions of extent of SE, border pixels would be affected in a different way and one has to decide that which values they should assume. Padding is the operation of adding an additional layer around your data to get more accurate values after filtering process. It also gives you option to retain same dimensions for your data after filtering. Common padding methods are using zeros or to mirror/replicate the border pixel values.

The math

There are many ways how to cleverly compute on a pixel neighborhood. For example, one class of computations is called convolutional filters, another is called rank filters. Here, we focus on the relatively simple mean and variance filters.

Best practice

As usual, everything depends one the scientific question, but maybe one could say to use a filter that changes the image as little as possible.




Activities

Mean filter and structuring element exploration


Show activity for:  

ImageJ GUI

  • Mean filter
    • Open xy_8bit__nuclei_very_noisy.tif
    • It is helpful to first duplicate the image: [Image > Duplicate…] or [Ctrl-Shift-D]
      • Title = mean_1
    • Apply a mean filter to the image such that you can binarize the image into exactly three disjoint foreground regions (the three nuclei). The resulting mask should look like xy_8bit_binary__nuclei_very_noisy.tif
      • [Process > Filters > Mean…]
        • Radius = 1 pixels
      • [Image > Adjust > Threshold…]
        • Dark Background
        • Lower threshold level = 31
        • Higher threshold level = 255
        • Press Set
        • Press Apply
        • [Image > Lookup Tables > Invert LUT]
      • Repeat the procedure with radii = 2, 5, 7
    • What is the smallest size for the mean filter that does the job?
      • Radius = 2 pixels and Lower threshold level = 30 (other values can also be chosen) can remove the noise and find the appropriate nuclei size after manual thresholding
    • What happens if the filter size increases?
      • The boundaries of segmented nuclei are smoother
  • Structuring element inspection

skimage napari

# %% 
# Apply mean filters to an image to aid foreground background segmentation 

# %%
# Instantiate the napari viewer
import napari
from OpenIJTIFF import open_ij_tiff
viewer = napari.Viewer()

# %%
# Read the intensity image
image, *_ = open_ij_tiff('https://github.com/NEUBIAS/training-resources/raw/master/image_data/xy_8bit__nuclei_very_noisy.tif')

# %%
# View the image 
# - Appreciate that it is quite noisy
# - Inspect the pixel values to find a threshold that separates the nuclei from the background
viewer.add_image(image)

# %%
# Binarise the image
# - Appreciate that one cannot segment the nuclei by a simple intensity threshold
binary_image = image > 40
viewer.add_image(binary_image)

# %% 
# Prepare filtering the image by defining a circular structural element with a radius of 1 pixel
from skimage.morphology import disk
disk_radius_1 = disk(1)
print(disk_radius_1)

# %%
# Apply a mean filter to the image with the above structural element
from skimage.filters.rank import mean
mean_image_1 = mean(image, disk_radius_1)

# Add the filtered image to napari
# Napari:
# - Zoom in and inspect the pixel values to check that the filtered image indeed contains the local mean values of the raw image
viewer.add_image(mean_image_1)

# %%
# Binarise the filtered image
# - Appreciate that one still cannot readily segment the nuclei
binary_image_1 = mean_image_1 > 35
viewer.add_image(binary_image_1)

# %% 
# Apply mean filter with a disk of radius 3
mean_image_3 = mean(image, disk(3))
viewer.add_image(mean_image_3)

# %%
# Now the nuclei can be segmented by a simple threshold :)
binary_image_3 = mean_image_3 > 32
viewer.add_image(binary_image_3)



Variance filtering


Show activity for:  

ImageJ GUI

Exercise mean filter

  • Open image xy_8bit__noisy_two_nuclei_close.tif
  • The aim is to segment the image into foreground/background in order to get a similar result to xy_8bit_binary__noisy_two_nuclei_close.tif
  • To achive this aim, try thresholding the image after applying a mean filter with different radii.
    • Try thresholding
      • Without a mean filter
      • After applying a mean filter of radius: 1, 2, 3, or 5
    • For each filter setting find the lowest threshold that yields exactly four connected components (the nuclei).
      • Is this even possible without applying a mean filter?
      • For radius 5, what happens to the size of the segmented nuclei?

Solution

  • [ Image › Duplicate… ]
  • [ Process › Filters › Mean… ]
    • Radius = 1,2,3,5 pixel
  • [ Image > Adjust > Threshold… ]
    • Radius 1: threshold = 38
    • Radius 2: threshold = 43
    • Radius 3: threshold = 48
    • Radius 5: threshold = 57
  • Only segmenting the four nuclei in unfiltered image does not work, as it yields a large number of undesired connected components (noise).
  • Radius 1 yields a reasonable segmentation.
  • For radius 5 one has to choose a rather high threshold to still separate some of the nuclei and thus the segmented nuclei become very small.

Exercise variance filter

  • Open image xy_16bit__embryo_transmission.tif
  • Apply a variance filter [ Process > Filter > Variance… ] to segment the cell regions from the background
    • Hints:
      • Convert image to float, because the filter may yield high values [ Image > Type > 32-bit ]
  • What filter radius and threshold yield a good segmentation?

Solution

  • [ Image › Rename…]
    • Title = input
  • [ Image › Duplicate… ]
    • Title = variance_5
  • [ Image > Type > 32-bit ]
    • In variance calculation, pixel values can exceed 255 which is the maximum value that can be achieved in current bit depth (unsigned 8-bit) of input image.
  • [ Process › Filters › Variance… ]
    • Radius = 15 pixels
  • [ Image › Adjust › Threshold… ]
    • ([x] Dark Background)
    • Lower threshold level = 1.5
    • Higher threshold level = 1e30
    • Press Set
    • Press Apply
      • Press Convert to Mask

ImageJ Macro

// File > Close All run(“Close All”); // File > Open… open(“C:/Users/akhan/training-resources/image_data/xy_8bit__em_mitochondria_er.tif”);

// Image › Duplicate… run(“Duplicate…”, “title=variance_5”);

// Image > Type > 16-bit (since in variance calculation values can exceed 255 which is current bit depth of input image) run(“16-bit”);

// Process › Filters › Variance… run(“Variance…”, “radius=5”);

// Image › Adjust › Threshold… ([x] Dark Background) setAutoThreshold(“Default dark”);

//Press Set button (example upper and lower threshold values are (113, 255)) setThreshold(113, 255);

// Press Apply button run(“Convert to Mask”);

// Image › Lookup Tables › InvertLUT… (if highest values are darker and lowest brighter) run(“Invert LUT”);

skimage napari

# %% 
# Apply variance filters to an EM image to aid foreground background segmentation

# %%
# Import libraries and
# instantiate the napari viewer
import numpy as np
import napari
import matplotlib.pyplot as plt
from OpenIJTIFF import open_ij_tiff
viewer = napari.Viewer()

# %%
# Read the image and add it to napari
# - Try to find an intensity threshold that separates the intracellular region from the bright background
image, *_ = open_ij_tiff('https://github.com/NEUBIAS/training-resources/raw/master/image_data/xy_8bit__em_mitochondria_er.tif')
viewer.add_image(image)

# %% 
# Plot an intensity histogram as another way to find a threshold
plt.hist(image.flatten(), bins=np.arange(image.min(), image.max() + 1))

# %%
# Appreciate that a simple threshold does not suffice to distinguish the cytoplasm from the bright background outside the cell
binary_image = image < 230
viewer.add_image(binary_image)

# %% 
# Apply a variance filter to separate the two regions
# - The idea is to make use of the fact that the background had less local intensity variations than the cytoplasm
# - Note that a mean filter might also work here (please try), but we want to learn something new

# %%
# Convert to uint16 to be able to accommodate 
# the square of the unit8 image, which is needed to represent the variance
# - Appreciate that such datatype issues are common applying any kind of "image math"
print(image.dtype)
uint16_image = image.astype(np.uint16)
print(uint16_image.dtype)

# %% 
# View the uint16 image in napari and check whether the datatype conversion changed the pixel values
# - Going from a lower bit-depth to a higher bit-depth should normally not change the values
viewer.add_image(uint16_image)

# %%
# We could not find a function to directly compute the local variance of an image,
# thus we introduce a, very useful, generic approach to compute any kind of local neighborhood
# filter

# import the generic filter function
from scipy.ndimage import generic_filter

# apply the generic filter with a variance computation
var_image = generic_filter(uint16_image, function=np.var, size=11)

# %%
# View the variance filtered the image to napari
# Look for a threshold that separates the cell from the background
viewer.add_image(var_image)

# %%
# Also inspect the histogram to find a threshold
# - It is interesting to compare the histograms of the variance images computed
#   with different filter widths, to see how well there are two populations of intensity values
plt.hist(var_image.flatten(), bins=np.arange(var_image.min(), var_image.max() + 1))
plt.yscale('log')

# %%
# Segment the cellular region by applying a threshold to the variance filtered image
# - It can be interesting to also try some auto-threshold method here
binary_var_image = var_image > 100
viewer.add_image(binary_var_image)






Assessment

Fill in the blanks

Fill in the blanks, using these words: decrease, increase, size, structuring element, large

Solution

  • A synonym for neighborhood is structuring element (SE)
  • The filter radius characterize the filter size
  • large filter size can cause a loss of details in the filtered image
  • Filter can be used to decrease the noise in an image
  • The usage of filters can increase the quality of image segmentation/binarization




Follow-up material

Recommended follow-up modules:

Learn more: