fuzzy c means algorithm

docs/examples/image_segmentation/fuzzycmeans.jl

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+# ---
+# cover: assets/fuzzycmeans.gif
+# title: Fuzzy C-Means
+# description: This demo shows how to use the fuzzy c-means algorithm to segment an image.
+# author: Ashwani Rathee
+# date: 2021-09-22
+# ---
+
+# Fuzzy C-means clustering is widely used for unsupervised image segmentation. It is an
+# iterative algorithm which tries to minimize the cost function:
+
+# 
+# $J = \displaystyle\sum_{i=1}^{N} \sum_{j=1}^{C} \delta_{ij} \| x_{i} - c_{j} \|^2$
+# 
+
+# Unlike K-means, it allows pixels to belong to two or more clusters. It is widely used
+# for medical imaging like in the soft segmentation of brain tissue model.
+# Note that both Fuzzy C-means and K-means have an element of randomness, and it's possible
+# to get results that vary considerably from one run to the next.
+
+# Fuzzy C-Means clustering is a soft clustering approach, where each data point is assigned 
+# a likelihood or probability score to belong to that cluster. 
+
+# The step-wise approach of the Fuzzy c-means clustering algorithm is:
+# - Fix the value of c (number of clusters), and select a value of m (generally 1.25<m<2), 
+# and initialize partition matrix U.
+
+# ![](assets/p1.png)
+# - Calculate cluster centers (centroid).
+# ![](assets/p2.png)
+
+# Here,
+# µ: Fuzzy membership value
+# m: fuzziness parameter
+# 
+# - Update Partition Matrix
+# ![](assets/p3.png)
+# - Repeat the above steps until convergence.
+
+# **Time Complexity:** ``O(n \cdot C^2 \cdot \text{iter})`` where ``n`` is the number of pixels, ``C`` is
+# number of clusters and ``\text{iter}`` is the number of iterations.
+
+using Images
+using ImageSegmentation, TestImages
+using IndirectArrays
+
+img = testimage("blobs")
+
+
+# Briefly, `r` contains two fields of interest:
+
+# - `centers`, a `3×C` matrix of center positions for `C` clusters in RGB 
+# colorspace. You can extract it as a vector of colors using `centers = colorview(RGB, r.centers)`.
+# - `weights`, a `n×C` matrix such that `r.weights[10,2]` would be the weight
+#   of the 10th pixel in the green color channel (color channel 2).  
+
+# You can visualize this component as `centers[i]*reshape(r.weights[:,i], axes(img))`.
+# See the documentation in [Clustering.jl](https://github.com/JuliaStats/Clustering.jl) for further details.
+
+segments =  fuzzy_cmeans(img, 3, 2)
+
+# Let's segment it using fuzzy cmeans and show results
+
+centers = colorview(RGB, segments.centers);
+red = centers[1]*reshape(segments.weights[:,1], axes(img));
+green = centers[2]*reshape(segments.weights[:,2], axes(img));
+blue = centers[3]*reshape(segments.weights[:,3], axes(img));
+
+# Output with pixel intensity = cluster center intensity * membership of pixel in that class
+
+mosaic(img, red, green, blue; nrow=1)
+
+# Here we use `IndirectArray` to store the indexed image, for more explaination on it please
+# check the tutorial [Indexed image in 5 minutes](@ref demo_indexed_image).
+
+save("assets/fuzzycmeans.gif", cat(img, red, green, blue; dims=3); fps=1) #src