# Hierarchical Merging of Region Adjacency Graphs

Region Adjacency Graphs model regions in an image as nodes of a graph with edges between adjacent regions. Superpixel methods tend to over segment images, ie, divide into more regions than necessary. Performing a Normalized Cut and Thresholding Edge Weights are two ways of extracting a better segmentation out of this. What if we could combine two small regions into a bigger one ? If we keep combining small similar regions into bigger ones, we will end up with bigger regions which are significantly different from its adjacent ones. Hierarchical Merging explores this possibility. The current working code can be found at this Pull Request

## Code Example

The merge_hierarchical function performs hierarchical merging on a RAG. It picks up the smallest weighing edge and combines the regions connected by it. The new region is adjacent to all previous neighbors of the two combined regions. The weights are updated accordingly. It continues doing so till the minimum edge weight in the graph in more than the supplied thresh value. The function takes a RAG as input where smaller edge weight imply similar regions. Therefore, we use the rag_mean_color function with the default "distance" mode for RAG construction. Here is a minimal code snippet.

from skimage import graph, data, io, segmentation, color

img = data.coffee()
labels = segmentation.slic(img, compactness=30, n_segments=400)
g = graph.rag_mean_color(img, labels)
labels2 = graph.merge_hierarchical(labels, g, 40)
g2 = graph.rag_mean_color(img, labels2)

out = color.label2rgb(labels2, img, kind='avg')
out = segmentation.mark_boundaries(out, labels2, (0, 0, 0))
io.imsave('out.png',out)


I arrived at the threshold 40 after some trial and error. Here is the output.

The drawback here is that the thresh argument can vary significantly depending on image to image.

## Comparison with Normalized Cut

Loosely speaking the normalized cut follows a top-down approach where as the hierarchical merging follow a bottom-up approach. Normalized Cut starts with the graph as a whole and breaks it down into smaller parts. On the other hand hierarchical merging, starts with individual regions and merges them into bigger ones till a criteria is reached. The Normalized Cut however, is much more robust and requires little tuning of its parameters as images change. Hierarchical merging is a lot faster, even though most of its computation logic is written in Python.

## Effect of change in threshold

Setting a very low threshold, will not merge any regions and will give us back the original image. A very large threshold on the other hand would merge all regions and give return the image as just one big blob. The effect is illustrated below.

## Hierarchical Merging in Action

With this modification the following code can output the effect of all the intermediate segmentation during each iteration.

from skimage import graph, data, io, segmentation, color
import time
from matplotlib import pyplot as plt

img = data.coffee()
labels = segmentation.slic(img, compactness=30, n_segments=400)
g = graph.rag_mean_color(img, labels)
labels2 = graph.merge_hierarchical(labels, g, 60)

c = 0

out = color.label2rgb(graph.graph_merge.seg_list[-10], img, kind='avg')
for label in graph.graph_merge.seg_list:
out = color.label2rgb(label, img, kind='avg')
out = segmentation.mark_boundaries(out, label, (0, 0, 0))
io.imsave('/home/vighnesh/Desktop/agg/' + str(c) + '.png', out)
c += 1


I then used avconv -f image2 -r 3 -i %d.png -r 20 car.mp4 to output a video. Below are a few examples.

In each of these videos, at every frame, a boundary dissapears. This means that the two regions separated by that boundary are merged. The frame rate is 5 FPS, so more than one region might be merged at a time.

# A closert look at Normalized Cut

## Variation with number of regions

In this post I explained how the Normalized Cut works and demonstrated some examples of it. This post aims to take a closer look at the code. I ran the following code to monitor the time taken by NCut with respect to initial number of regions.

from __future__ import print_function
from skimage import graph, data, io, segmentation, color
import time
from matplotlib import pyplot as plt

image = data.coffee()
segment_list = range(50, 801, 50)

for nseg in segment_list:
labels = segmentation.slic(image, compactness=30, n_segments=nseg)
rag = graph.rag_mean_color(image, labels, mode='similarity')
T = time.time()
new_labels = graph.ncut(labels, rag)
time_taken = time.time() - T
out = color.label2rgb(new_labels, image, kind='avg')
io.imsave('/home/vighnesh/Desktop/ncut/' + str(nseg) + '.png', out)
print(nseg, time_taken)


Here is the output sequentially.

By a little guess-work, I figured that the curve approximately varies as x^2.2. For 800 nodes, the time taken is around 35 seconds.

## Line Profile

I used line profiler to examine the time taken by each line of code in threshold_normalized. Here are the results.

   218                                           @profile
219                                           def _ncut_relabel(rag, thresh, num_cuts):
220                                               """Perform Normalized Graph cut on the Region Adjacency Graph.
221
222                                               Recursively partition the graph into 2, until further subdivision
223                                               yields a cut greather than thresh or such a cut cannot be computed.
224                                               For such a subgraph, indices to labels of all its nodes map to a single
225                                               unique value.
226
227                                               Parameters
228                                               ----------
229                                               labels : ndarray
230                                                   The array of labels.
231                                               rag : RAG
233                                               thresh : float
234                                                   The threshold. A subgraph won't be further subdivided if the
235                                                   value of the N-cut exceeds thresh.
236                                               num_cuts : int
237                                                   The number or N-cuts to perform before determining the optimal one.
238                                               map_array : array
239                                                   The array which maps old labels to new ones. This is modified inside
240                                                   the function.
241                                               """
242        59       218937   3710.8      3.2      d, w = _ncut.DW_matrices(rag)
243        59          151      2.6      0.0      m = w.shape[0]
244
245        59           61      1.0      0.0      if m > 2:
246        44         3905     88.8      0.1          d2 = d.copy()
247                                                   # Since d is diagonal, we can directly operate on its data
248                                                   # the inverse of the square root
249        44          471     10.7      0.0          d2.data = np.reciprocal(np.sqrt(d2.data, out=d2.data), out=d2.data)
250
251                                                   # Refer Shi & Malik 2001, Equation 7, Page 891
252        44        26997    613.6      0.4          vals, vectors = linalg.eigsh(d2 * (d - w) * d2, which='SM',
253        44      6577542 149489.6     94.9                                       k=min(100, m - 2))
254
255                                                   # Pick second smallest eigenvector.
256                                                   # Refer Shi & Malik 2001, Section 3.2.3, Page 893
257        44          618     14.0      0.0          vals, vectors = np.real(vals), np.real(vectors)
258        44          833     18.9      0.0          index2 = _ncut_cy.argmin2(vals)
259        44         2408     54.7      0.0          ev = _ncut.normalize(vectors[:, index2])
260
261        44        22737    516.8      0.3          cut_mask, mcut = get_min_ncut(ev, d, w, num_cuts)
262        44           78      1.8      0.0          if (mcut < thresh):
263                                                       # Sub divide and perform N-cut again
264                                                       # Refer Shi & Malik 2001, Section 3.2.5, Page 893
265        29        78228   2697.5      1.1              sub1, sub2 = partition_by_cut(cut_mask, rag)
266
267        29          175      6.0      0.0              _ncut_relabel(sub1, thresh, num_cuts)
268        29           92      3.2      0.0              _ncut_relabel(sub2, thresh, num_cuts)
269        29           32      1.1      0.0              return
270
271                                               # The N-cut wasn't small enough, or could not be computed.
272                                               # The remaining graph is a region.
273                                               # Assign ncut label by picking any label from the existing nodes, since
274                                               # labels are unique, new_label is also unique.
275        30          685     22.8      0.0      _label_all(rag, 'ncut label')


As you can see above 95% of the time is taken by the call to eigsh.

To take a closer look at it, I plotted time while ensuring only one iteration. This commit here takes care of it. Also, I changed the eigsh call to look for the largest eigenvectors instead of the smallest ones, with this commit here. Here are the results.

A single eignenvalue computation is bounded by O(n^1.5) as mentioned in the original paper. The recursive NCuts are pushing the time required towards more than O(n^2).eigsh solves the eigenvalue problem for a symmetric hermitian matrix. It in turn relies on a library called ARPack. As documented here ARPack isn’t very good at finding the smallest eigenvectors. If the value supplied as the argument k is too small, we get the ArpackNoConvergence Exception. As seen from the above plot, finding the largest eigenvectors is much more efficient using the eigsh function.

Since the problem is specific to ARPack, using other libraries might lead to faster computation. slepc4py is one such BSD licensed library. The possibility of optionally importing slec4py should be certainly explored in the near future.

Also, we can optionally ask the user for a function to solve the eigenvalue problem, so that he can use a matrix library of his choice if he/she so desires.

## Final Thoughts

Although the current Normalized Cut implementation takes more than quadratic time, the preceding over segmentation method does most of the heavy lifting. With something like SLIC, we can be sure of the number of nodes irrespective of the input image size. Although, a better eigenvalue finding technique for smallest eigenvectors would immensely improve its performance.

A lot of Image Processing algorithms are based on intuition from visual cues. Region Adjacency Graphs would also benefit if they were somehow drawn back on the images they represent. If we are able to see the nodes, edges, and the edges weights, we can fine tune our parameters and algorithms to suit our needs. I had written a small hack in this blog post to help better visualize the results. Later, Juan suggested I port if for scikit-image. It will indeed be a very helpful tool for anyone who wants to explore RAGs in scikit-image.

## Getting Started

You will need to pull for this Pull Request to be able to execute the code below. I’ll start by defining a custom show_image function to aid displaying in IPython notebooks.

from skimage import graph, data, io, segmentation, color
from matplotlib import pyplot as plt
from skimage.measure import regionprops
import numpy as np
from matplotlib import colors

def show_image(img):
width = img.shape[1] / 50.0
height = img.shape[0] * width/img.shape[1]
f = plt.figure(figsize=(width, height))
plt.imshow(img)


We will start by loading a demo image just containing 3 bold colors to help us see how the draw_rag function works.

image = io.imread('/home/vighnesh/Desktop/images/colors.png')
show_image(image)


We will now use the SLIC algorithm to give us an over-segmentation, on which we will build our RAG.

labels = segmentation.slic(image, compactness=30, n_segments=400)


Here’s what the over-segmentation looks like.

border_image = segmentation.mark_boundaries(image, labels, (0, 0, 0))
show_image(border_image)


## Drawing the RAGs

We can now form out RAG and see how it looks.

rag = graph.rag_mean_color(image, labels)
out = graph.draw_rag(labels, rag, border_image)
show_image(out)


In the above image, nodes are shown in yellow whereas edges are shown in green. Each region is represented by its centroid. As Juan pointed out, many edges will be difficult to see because of low contrast between them and the image, as seen above. To counter this we support the desaturate option. When set to True the image is converted to grayscale before displaying. Hence all the image pixels are a shade of gray, while the edges and nodes stand out.

out = graph.draw_rag(labels, rag, border_image, desaturate=True)
show_image(out)


Although the above image does very well to show us individual regions and their adjacency relationships, it does nothing to show us the magnitude of edges. To give us more information about the magnitude of edges, we have the colormap option. It colors edges between the first and the second color depending on their weight.

blue_red = colors.ListedColormap(['blue', 'red'])
out = graph.draw_rag(labels, rag, border_image, desaturate=True,
colormap=blue_red)
show_image(out)


As you can see, the edges between similar regions are blue, whereas edges between dissimilar regions are red. draw_rag also accepts a thresh option. All edges above thresh are not considered for drawing.

out = graph.draw_rag(labels, rag, border_image, desaturate=True,
colormap=blue_red, thresh=10)
show_image(out)


Another clever trick is to supply a blank image, this way, we can see the RAG unobstructed.

cyan_red = colors.ListedColormap(['cyan', 'red'])
out = graph.draw_rag(labels, rag, np.zeros_like(image), desaturate=True,
colormap=cyan_red)
show_image(out)


Ahhh, magnificent.

Here is a small piece of code which produces a typical desaturated color-distance RAG.

image = data.coffee()
labels = segmentation.slic(image, compactness=30, n_segments=400)
rag = graph.rag_mean_color(image, labels)
cmap = colors.ListedColormap(['blue', 'red'])
out = graph.draw_rag(labels, rag, image, border_color=(0,0,0), desaturate=True,
colormap=cmap)
show_image(out)


If you notice the above image, you will find some edges crossing over each other. This is because, some regions are convex. Hence their centroid lies outside their boundary and edges emanating from it can cross other edges.

## Examples

I will go over some examples of RAG drawings, since most of it is similar, I won’t repeat the code here. The Ncut technique, wherever used, was with its default parameters.

### Color distance RAG of Coffee after applying NCut

Notice how the centroid of the white rim of the cup is placed at its centre. It is the one adjacent to the centroid of the gray region of the upper part of the spoon, connected to it via a blue edge. Notice how this edge crosses others.

## Further Improvements

• A point that was brought up in the PR as well is that thick lines would immensely enhance the visual
appeal of the output. As and when they are implemented, rag_draw should be modified to support drawing
thick edges.
• As centroids don’t always lie in within an objects boundary, we can represent regions by a point other than their centroid, something which always lies within the boundary. This would allow for better visualization of the actual RAG from its drawing.

# Normalized Cuts on Region Adjacency Graphs

In my last post I demonstrated how removing edges with high weights can leave us with a set of disconnected graphs, each of which represents a region in the image. The main drawback however was that the user had to supply a threshold. This value varied significantly depending on the context of the image. For a fully automated approach, we need an algorithm that can remove edges automatically.

The first thing that I can think of which does something useful in the above mention situation is the Minimum Cut Algorithm. It divides a graph into two parts, A and B such that the weight of the edges going from nodes in Set A to the nodes in Set B is minimum.

For the Minimum Cut algorithm to work, we need to define the weights of our Region Adjacency Graph (RAG) in such a way that similar regions have more weight. This way, removing lesser edges would leave us with the similar regions.

## Getting Started

For all the examples below to work, you will need to pull from this Pull Request. The tests fail due to outdated NumPy and SciPy versions on Travis. I have also submitted a Pull Request to fix that. Just like the last post, I have a show_img function.

from skimage import graph, data, io, segmentation, color
from matplotlib import pyplot as plt
from skimage.measure import regionprops
from skimage import draw
import numpy as np

def show_img(img):

width = img.shape[1]/75.0
height = img.shape[0]*width/img.shape[1]
f = plt.figure(figsize=(width, height))
plt.imshow(img)


I have modified the display_edges function for this demo. It draws nodes in yellow. Edges with low edge weights are greener and edges with high edge weight are more red.

def display_edges(image, g):
"""Draw edges of a RAG on its image

Returns a modified image with the edges drawn. Edges with high weight are
drawn in red and edges with a low weight are drawn in green. Nodes are drawn
in yellow.

Parameters
----------
image : ndarray
The image to be drawn on.
g : RAG
threshold : float
Only edges in g below threshold are drawn.

Returns:
out: ndarray
Image with the edges drawn.
"""

image = image.copy()
max_weight = max([d['weight'] for x, y, d in g.edges_iter(data=True)])
min_weight = min([d['weight'] for x, y, d in g.edges_iter(data=True)])

for edge in g.edges_iter():
n1, n2 = edge

r1, c1 = map(int, rag.node[n1]['centroid'])
r2, c2 = map(int, rag.node[n2]['centroid'])

green = 0,1,0
red = 1,0,0

line  = draw.line(r1, c1, r2, c2)
circle = draw.circle(r1,c1,2)
norm_weight = ( g[n1][n2]['weight'] - min_weight ) / ( max_weight - min_weight )

image[line] = norm_weight*red + (1 - norm_weight)*green
image[circle] = 1,1,0

return image


To see demonstrate the display_edges function, I will load an image, which just has two regions of black and white.

demo_image = io.imread('bw.png')
show_img(demo_image)


Let’s compute the pre-segmenetation using the SLIC method. In addition to that, we will also use regionprops to give us the centroid of each region to aid the display_edges function.

labels = segmentation.slic(demo_image, compactness=30, n_segments=100)
labels = labels + 1  # So that no labelled region is 0 and ignored by regionprops
regions = regionprops(labels)


We will use label2rgb to replace each region with its average color. Since the image is so monotonous, the difference is hardly noticeable.

label_rgb = color.label2rgb(labels, demo_image, kind='avg')
show_img(label_rgb)


We can use mark_boundaries to display region boundaries.

label_rgb = segmentation.mark_boundaries(label_rgb, labels, (0, 1, 1))
show_img(label_rgb)


As mentioned earlier we need to construct a graph with similar regions having more weights between them. For this we supply the "similarity" option to rag_mean_color.

rag = graph.rag_mean_color(demo_image, labels, mode="similarity")

for region in regions:
rag.node[region['label']]['centroid'] = region['centroid']

label_rgb = display_edges(label_rgb, rag)
show_img(label_rgb)


If you notice above the black and white regions have red edges between them, i.e. they are very similar. However the edges between the black and white regions are green, indicating they are less similar.

## Problems with the min cut

Consider the following graph

The minimum cut approach would partition the graph as {A, B, C, D} and {E}. It has a tendency to separate out small isolated regions of the graph. This is undesirable for image segmentation as this would separate out small, relatively disconnected regions of the image. A more reasonable partition would be {A, C} and {B, D, E}. To counter this aspect of the minimum cut, we used the Normalized Cut.

## The Normalized Cut

It is defined as follows
Let $V$ be the set of all nodes and $w(u,v)$ for $u, v \in V$ be the edge weight between $u$ and $v$

$NCut(A,B) = \frac{cut(A,B)}{Assoc(A,V)} + \frac{cut(A,B)}{Assoc(B,V)}$
where
$cut(A,B) = \sum_{a \in A ,b \in B}{w(a,b)}$

$Assoc(X,V) = cut(X,V) = \sum_{x \in X ,v \in V}{w(x,v)}$

With the above equation, NCut won’t be low is any of A or B is not well-connected with the rest of the graph. Consider the same graph as the last one.

We can see that minimizing the NCut gives us the expected partition, that is, {A, C} and {B, D, E}.

## Normalized Cuts for Image Segmentation

The idea of using Normalized Cut for segmenting images was first suggested by Jianbo Shi and Jitendra Malik in their paper Normalized Cuts and Image Segmentation. Instead of pixels, we are considering RAGs as nodes.

The problem of finding NCut is NP-Complete. Appendix A of the paper has a proof for it. It is made tractable by an approximation explained in Section 2.1 of the paper. The function _ncut_relabel is responsible for actually carrying out the NCut. It divides the graph into two parts, such that the NCut is minimized. Then for each of the two parts, it recursively carries out the same procedure until the NCut is unstable, i.e. it evaluates to a value greater than the specified threshold. Here is a small snippet to illustrate.

img = data.coffee()

labels1 = segmentation.slic(img, compactness=30, n_segments=400)
out1 = color.label2rgb(labels1, img, kind='avg')

g = graph.rag_mean_color(img, labels1, mode='similarity')
labels2 = graph.cut_normalized(labels1, g)
out2 = color.label2rgb(labels2, img, kind='avg')

show_img(out2)


## NCut in Action

To observe how the NCut works, I wrote a small hack. This shows us the regions as divides by the method at every stage of recursion. The code relies on a modification in the original code, which can be seen here.

from skimage import graph, data, io, segmentation, color
from matplotlib import pyplot as plt
import os

#img = data.coffee()
os.system('rm *.png')
img = data.coffee()
#img = color.gray2rgb(img)

labels1 = segmentation.slic(img, compactness=30, n_segments=400)
out1 = color.label2rgb(labels1, img, kind='avg')

g = graph.rag_mean_color(img, labels1, mode='similarity')
labels2 = graph.cut_normalized(labels1, g)

offset = 1000
count = 1
tmp_labels = labels1.copy()
for g1,g2 in graph.graph_cut.sub_graph_list:
for n,d in g1.nodes_iter(data=True):
for l in d['labels']:
tmp_labels[labels1 == l] = offset
offset += 1
for n,d in g2.nodes_iter(data=True):
for l in d['labels']:
tmp_labels[labels1 == l] = offset
offset += 1
tmp_img = color.label2rgb(tmp_labels, img, kind='avg')
io.imsave(str(count) + '.png',tmp_img)
count += 1


The two components at each stage are stored in the form of tuples in sub_graph_list. Let’s say, the Graph was divided into A and B initially, and later A was divided into A1 and A2. The first iteration of the loop will label A and B. The second iteration will label A1, A2 and B, and so on. I used the PNGs saved and converted them into a video with avconv using the command avconv -f image2 -r 1 -i %d.png -r 20 demo.webm. GIFs would result in a loss of color, so I made webm videos. Below are a few images and their respective successive NCuts. Use Full Screen for better viewing.

Note that although there is a user supplied threshold, it does not have to vary significantly. For all the demos below, the default value is used.

### Colors Image

During each iteration, one region (area of the image with the same color) is split into two. A region is represented by its average color. Here’s what happens in the video

• The image is divided into red, and the rest of the regions (gray at this point)
• The grey is divided into a dark pink region (pink, maroon and yellow) and a
dark green ( cyan, green and blue region ).
• The dark green region is divided into light blue ( cyan and blue ) and the
green region.
• The light blue region is divided into cyan and blue
• The dark pink region is divided into yellow and a darker pink (pink and marron
region.
• The darker pink region is divided into pink and maroon regions.

# Graph based Image Segmentation

My GSoC project this year is Graph based segmentation algorithms using region adjacency graphs. Community binding period is coming to an end. I have experimented a bit with Region Adjacency Graphs (RAGs) and Minimum Spanning Trees (MSTs) with this ugly piece of Python code.  I will try to describe in brief what I plan to do during this GSoC period.