Procedia Technology 10 ( 2013 ) 271 – 278
2212-0173 © 2013 The Authors. Published by Elsevier Ltd.
Selection and peer-review under responsibility of the University of Kalyani, Department of Computer Science & Engineering doi: 10.1016/j.protcy.2013.12.361
ScienceDirect
1
stInternational Conference on Computational Intelligence: Modelling, Techniques and Applications
(CIMTA- 2013)
Random Walks Based Image Segmentation using Color Space Graphs
Sonu Kumar Jha
a,*, Purnendu Bannerjee
b, Subhadeep Banik
aaApplied Statistics Unit, Indian Statistical Institute,Kolkata-700108 bSociety for Natural Language Technology Research, Kolkata
Abstract
Image segmentation is one of the most involved topics of research in the area of Computer Vision and has attracted a lot of attention from the researchers. The goal of image segmentation is to change the representation of the image into something that is more meaningful and easier to analyze. In this process an image is partitioned into multiple segments by assigning label to pixels of the image so that the pixels sharing the same label can be characterized as a distinct object present in that image. Several automatic and semi-automatic approaches have been designed in order to segment an image. In this paper we will focus on a semi-automatic approach called the Random Walker approach which was first introduced by Leo Grady [1] and we will contribute several improvements which leads to better segmented images compared to the image segmentations achieved by the ideas involved in the above said paper. We will use graph theoretical approaches to partition the color space and give a new definition of the distance between two pixels that results in an improved segmentation.
© 2013 The Authors. Published by Elsevier Ltd.
Selection and peer-review under responsibility of the University of Kalyani, Department of Computer Science & Engineering.
Keywords: Image segmentation; Random Walk; Combinatorial Dirichlet problem; Color Space Graph; LUV space; Laplacian Matrix; Laplacian Equation.
1.Introduction
Semi-automatic or interactive segmentation have been a vast area of research in Image Processing and has gained a lot of interest from the researchers in computer vision. The objective is to segment an image into different regions in which each region corresponds to a particular object of interest with the help of user interaction. User input is required to initialize the segmentation algorithm (semi-automatic) in order to segment an image into different regions. User input can be provided in many ways of which the simplest is the region of interest being outlined by the user with the mouse clicks.
Semi-automatic segmentation has gain importance by its vast applications on image editing, medical imaging, machine vision, object detection, recognition tasks, traffic control systems, video surveillance and many other fields. Several algorithms have been developed so far for general purpose semi-automatic segmentation and can be made useful by combining it with the domain
knowledge to solve the domain’s segmentation problem. The other aspect of the importance of semi-automatic segmentation is this problem is a challenging one because the task is to initialize the process by a small set of user data and then to propagate the information throughout the image.
* Sonu Kumar Jha. Tel.:+91-33-2575-2821.
E-mail address: jhasonu1987@yahoo.com
© 2013 The Authors. Published by Elsevier Ltd.
Selection and peer-review under responsibility of the University of Kalyani, Department of Computer Science & Engineering
Open access under CC BY-NC-ND license.
There are many methods proposed for semi-automatic image segmentation. Our interest is to focus on the Random Walker method proposed by L. Grady [1, 13, 14] and try to contribute some improvements in the algorithm which gives better segmented images. The Random Walker method has several advantages. Generally any segmentation scheme focuses on dividing the image into two regions, namely the background and the object. On the other hand the Random Walker method provides us with the flexibility of segmenting an image into ݇ number of objects where ݇ is the arbitrary number of objects the user wants to segment in the image. Random walker method assumes the image as a graph where the nodes or vertices of the graph are the set of pixels. The edge weights represent the transition probabilities from one node to another. So in the given
scenario, it’s way too flexible and easier for the information given by the user to propagate through all over the image.
1.1.Paper Structure
The structure of the paper is as follows. In section 2 we explain the related works done in the area of semi-automatic segmentation. Section 3 gives the detailed idea of Grady’s Random Walker method, a brief explanation of our new introduced improvements. Section 4 provides the algorithm and results of our technique when applied for image segmentation. Section 5 shows the qualitative improvements achieved as compared to the original results of Random Walker algorithm by L. Grady. Finally in section 6 we conclude our paper.
2.Related works
As mentioned earlier, several works have been done and several algorithms have been designed for semi-automatic image segmentation till now and we discuss here some of the related works done in the area. To start with, we can look at an approach introduced by Boykov and Jolly in which they used graph cuts for interactive segmentation [6, 7]. In this approach the user provides seeds by marking the background and the object. An energy function is formulated which is composed of two different models namely color and coherence. The energy function labels the background and the foreground with 0 and 1 and at last minimization is done using a standard Max-flow Min-cut problem.
Further works on graph cut methodology have been done. A method presented by Li et al [8] was a combination of watershed segmentation and graph cut minimization. Nagahashi et al [3] presented an image segmentation scheme in which they used iterated graph cuts based on multi scale smoothing. The idea was to stepwise segment the regions of an object from global to local segmentation using iterations of graph cut and Gaussian smoothing. Shi et al [4] introduced a novel approach of segmentation based on normalized cuts in which the image segmentation was treated as a graph partitioning problem and normalized cut technique was used to segment the image. Rother et al [9] introduced the grab cut technique which came with a better color modeling and an extra layer of local minimization to graph cuts. See [9, 15-20] for more.
Mortensen and Barrett introduced the technique of intelligent scissors [10-12]. Intelligent scissors allows the objects within a digital image to be extracted quickly and accurately. The user needs to draw a curve around the object of interest using simple gestures with a mouse and when the gestured mouse position comes in proximity with the object edge, the live wire boundary wraps around the object.
The Random Walker approach introduced by Grady [1, 13] which will be the main focus of this paper is more flexible than the above mentioned techniques because it has an inherent advantage of handling arbitrary number of objects easily. In this
process input is given by the user namely the “seeds” and the information gained from the seeds is propagated throughout the image. The initial works in this approach includes the computation of probability of a random walker reaching the seeded pixels starting from a given pixel. The transition probability is inversely proportional to the contrast between the pixels which does not allow the walks to cross the edges.
There have been many extensional works on the Random walker method. These references [14, 21-27] provide the details of these works.
3.Basic idea of the Random Walker method
This section describes the details of the Random Walker approach which was introduced by Grady [1]. An undirected graph can be presented asܩ ൌ ሺܸǡ ܧሻ, where ܸand ܧ are the set of vertex and the set of the edges, respectively. The set of the vertex ܸ
is the set of pixels present in the image. The vertex set can be partitioned into two set of vertices. One set is “labelled vertices”ܸ, which are marked by user as belonging to the several objects, i.e. the seeds; and the rest of the image pixels is the
other setܸ௨ or the “unlabelled vertices”. The set of the edges ܧ consists of the pairs of pixels which are neighbours in the image,
using either of standard4 or 8-neigbourhood. Depending on the difference between the intensities of grayscale or colour scale of the two pixels the weight of an edge݁ ൌ ሺݒǡ ݒሻ can be represented either as ߱ሺ݁ሻ or ߱ሺݒǡ ݒሻ. Let݀൫ݒǡ ݒ൯ ൌ ฮ݃ሺݒሻ െ
݃൫ݒ൯ฮ, where ݃ is either intensity of grayscale (a scalar) or colour (a 3-dimensional vector); then the weight can be express as:
߱൫ݒǡ ݒ൯ ൌ ݁ିௗ൫௩ǡ௩ೕ൯ మ Ȁఙమ
or
߱൫ݒǡ ݒ൯ ൌ ଵ ଵାௗሺ௩ǡ௩ೕሻȀఙwhere the value of the parameterߪ can be tuned accordingly. The weights of the edge lie in the range ሺͲǡͳሻ; for similar pixels, we will have a weight close to 1, whereas for very different pixels the weight is close to 0.
Having the above graph structure in hand, the idea of the Random Walker method is as follows. It is assumed that image consists of ݇ possible objects and each labelled vertices of ܸ belongs to one of these ݇ objects. Now consider a weighted edge ݁
whose endpoints are ݒ and ݒ i.e. ݁ ൌ ሺݒǡ ݒሻ. The weight of the edge ߱ሺ݁ሻ lies within the range ሺͲǡͳሻ and it can be interpreted
as the measurement of transition probability of a random walk from one vertex to another vertex. Depending on the weight of the edge, the random walk is likely to transition form ݒ to ݒ if ݒ and ݒ are very similar in color or intensity, and is unlikely to
make transition if they are very dissimilar.
Given the above transition probabilities we can consider a specific random walk on the graph. In particular, for each unlabelled vertex ݒא ܸ௨; the probability that a random walk beginning at that vertex reaches any one of the labelled vertices
belonging to a particular object݇ is computed; this probability is denoted by the quantity . Then the image segmentation is
done according to these probabilities. More specifically, for any vertex ݒ, we classify it as belonging to segment ݇ if ᇱ
for all ݇ᇱ് ݇. Note that edges in the image (as opposed to edges in the graph) correspond to low transition probabilities, as they
involve a rapid change in colour or intensity. Thus, this algorithm will tend to respect image edges in performing the segmentation.
It turns out that the probabilities may be computed through the solution of a large, sparse linear system. The combinatorial
Dirichlet problem has the same solution as the Random Walker probabilities [1].The Dirichlet integralmay be defined as
ܦሾݑሿ ൌͳ ʹන ȁݑȁ
ଶ݀ȳ ஐ
for a field ݑ and region ȳ [1]. The next step now is to find a harmonic function. A harmonic function is a function which satisfies the Laplace equation
ଶݑ ൌ ͲǤ
The problem of finding a harmonic function subject to its boundary values is called the Dirichlet problem. The harmonic function that satisfies the boundary conditions minimizes the Dirichlet integral, since the Laplace equation is the Euler-Lagrange equation for the Dirichlet integral. Define the combinatorial Laplacian matrix as
ܮൌ ൝
߲݂݅݅ ൌ ݆
െ݂݆߱݅݅ܽ݊݀ܽݎ݆݁ܽ݀ܽܿ݁݊ݐ
Ͳݐ݄݁ݎݓ݅ݏ݁
where ܮ is indexed by vertices ݅ and ݆.
With these definitions in place, we can determine how to solve for the harmonic function that finds probabilities/potentials on unseeded nodes, while keeping the seed nodes fixed. A combinatorial formulation of the Dirichlet integral is
ܦሾݔሿ ൌͳ ʹݔ ்ܮݔ ൌͳ ʹ ߱ሺݔെ ݔሻ ଶ ೕאா
and a combinatorial harmonic is a function ݔthat minimizes the integral. Since ܮis positive semi-definite, the only critical points of ܦሾݔሿwill be minima. Partition the vertices into two sets, ܸ(marked/seed nodes) and ܸ௨(unseeded nodes) such that
ܸ ܸ௨ൌ ܸandܸת ܸ௨ൌ . Note that ܸcontains all seed points, regardless of their label. We may assume without loss of
generality that the nodes in ܮand ݔare ordered such that seed nodes are first and unseeded nodes are second. Therefore, we may decompose the above equation into
ܦሾݔ௨ሿ ൌ ͳ ʹሾݔ ்ݔ ௨்ሿ ܮ ܤ ܤ் ܮ ௨൨ቂ ݔ ݔ௨ቃ
where ݔ and ݔ௨ correspond to the potentials of the seeded and unseeded nodes respectively. Differentiating ܦሾݔ௨ሿwith respect
to ݔ௨ and finding the critical point yields
ܮ௨ݔ௨ൌ െܤ்ݔ
which is a system of linear equations with ȁܸ௨ȁunknowns. If the graph is connected, or if every connected component contains a
seed, then this equation will be non-singular. Denote the probability (potential) assumed at node, ݒǡ for each label,ݏ, by ݔ௦.
Define the set of labels for the seed points as a functionܳሺݒሻ ൌ ݏǡ ݒא ܸ, whereݏ א Ժܼǡ Ͳ ൏ ݏ ܭ. Define the ȁܸȁ ൈ ͳ
vector (where ȁǤ ȁdenotes cardinality) for each label,ݏ, at node ݒ א ܸ as
݉௦ൌ ቊ
ͳǡ ݂݅ܳ൫ݒ൯ ൌ ݏ
Ͳǡ ݂݅ܳ൫ݒ൯ ് ݏ
Therefore, for labelݏ, the solution to the combinatorial Dirichlet problem may be found by solving
See references [28-34] for details.
4.The Algorithm
For many real life applications, the object of interest which is to be segmented consists of multiple color shades from “distant”
regions of the color spectrum. This imposes an unusual constraint on the Random Walker if the weight between two pixels, is solely based on the Euclidean distance between their respective color vectors. For example if an object consists primarily of 2 color regions whose means are ܣଵൌ ሺܽଵǡ ܾଵǡ ܿଵሻ and ܣଶൌ ሺܽଶǡ ܾଶǡ ܿଶሻ with ԡܣଵെ ܣଶԡbeing ‘a large value’, then any pair of
pixels of the object of interest, which are a) physically adjacent and
b) have color value ܣଵ, ܣଶ respectively
will be connected by an edge of weight ߱ ൌ ݁ିఉԡభିమԡమ. This value will be close to 0, which will forbid the ‘transition’ of the
Random Walker between these pixels. This poses a unique problem in the case of object segmentation. Our motivation in this paper is to address the challenges arising out of an obstacle of this nature.
4.1.Graph based weight
In [2], a probability distribution based weight has been used to address this problem. However, the work in [2] requires that Random Walker based segmentation be preceded by a mean-shift based over-segmentation procedure. The mean-shift based method increases the complexity of the algorithm and we investigate if any segmentation procedure can be implemented that will do away with the over-segmentation based preprocessing of the image.
Note that in case of semi-supervised segmentation, the user possesses prior information in the form of ‘seeds’ which gives us some important information about the profile of an object. The user usually ‘scratches’ on the object of interest and this gives the user an idea about the regions of the color spectrum that composes the object. Ideally if an object comprises of clustered regions of the color spectrum whose means are ܣଵǡ ܣଶǡ ǥ ǡ ܣ then, a random walker must be allowed a high transition probability while
traversing two pixels with color values from the set ࣛ ൌ ሼܣሽୀଵ௧. One of the methods of achieving this is by pre specifying
that (1) the distance between ܣ and ܣshould be treated as 0 and (2) that the color vectors close to ܣand ܣhave small distance
between them.
One way to achieve the above is by visualizing the color space as a fully connected graph. In digital image processing, we usually work with discrete color values and so one can visualize each color vector as a node in a fully connected graph. The weight of an edge connecting the color values ܾ and ܾ is given as݀ൌ ฮܤെ ܤฮ. Initially we prepare a graph ܩ of this nature
and assign the weights as above. Now once the user assigned seed information is available in the form of the setࣛ, the user simply assigns to 0 the weight of the edge connecting anyܣǡ ܣא ࣛ. Now we apply the all pairs shortest paths algorithm
(Floyd-Warshall) to this modified graph ܩ, which will give the shortest distance between any 2 color values. We now prepare a new graph ܩᇱ in which the distance between two nodes ܤ
and ܤ given as ݀ᇱ is the weight of the shortest path between ܤ and
ܤ inܩ. Note that such a formulation ݀ᇱ satisfies both the requirements (1) and (2).
4.2.Using the LUV space
Instead of working in the standard RGB space we use the CIE 1976(L*,U*,V*) color space which is known to be perceptually
more meaningful than RGB. Typically the values of L*, U*, V* are in the range
Ͳ ܮכ ͳͲͲ, െͳ͵Ͷ ܷכ ʹʹͲ, െͳͶͲ
ܸכ ͳʹʹ. This gives a total of ሺͳͲͳ ൈ ͵ͷͷ ൈ ʹ͵ሻ ؆ ʹଶଷ nodes in ܩ. Since Floyd-Warshall has a computational complexity of
ߠሺܰଷሻ {ܰ is the number of vertices} this leads to a runtime of ʹଽ for our algorithm which is clearly infeasible. So we divide
each color component L*, U*, V* in 8 bins each. For ex: Ͳ ܮכ ͳʹǤͷ makes one bin for L axis. This gives rise to ͷͳʹ ൌ ʹଽ
bins in the color space which will be used as nodes inܩ. The distance ݀ in ܩ will be the Euclidean distance between the mean
color vectors of each bin. Now after obtaining ࣛ from the seed information, the user takes 2 values ܣ and ܣ from ࣛ,
determines the bins that is ܣ and ܣ belong to and assigns to zero the weight of edge between those bins. Then the
Floyd-Warshall algorithm is executed and ܩᇱ is constructed as defined earlier.
4.3.Using the Random Walker
We assume that the task at hand is to segment Ԣ݇Ԣ objects for which the user has provided seed information in the form of the sets ࣛሼ݈ ൌ ͳݐ݇ሽ where ݈ stands for label. For each ݈ ൌ ͳݐ݇ we do the following
A. Construct a graph in which the vertices are the pixels of the image. Two vertices are connected by an edge if the corresponding pixels are 4-connected.
B. Construct the graph ܩ on the color space and using the set ࣛ constructܩᇱ. The distance between colors ܣ and ܣ in ܩᇱ
is denoted as݀ǡᇱ .
C. If the two vertices of, ݒ and ݒ are connected and have color valuesܣ, ܣ then the weight of the edge connecting
them is given as ߱ǡൌ ݁ିఉௗೕǡ
ᇲమ
.
ܮ ൌ ൝ ߲ǡ݂݅݅ ൌ ݆ െ߱ǡ݂݆݅݅ܽ݊݀ܽݎ݆݁ܽ݀ܽܿ݁݊ݐ Ͳݐ݄݁ݎݓ݅ݏ݁ Note that߲ǡൌ σ௩ೕאௗሺ௩ሻ߱ǡ.
E. As in case of ordinary Random Walker [1] we solve the system
ܮ௨ܲൌ െܤ் ݉
Note that ܲ is defined as ሺܲ
ଵǡ ܲଶǡ ǥ ሻ் where ܲ is the probability that vertex ݒ belongs to the vertex set with the label ݈Ǥ
for any vertex ݒ if ܲ௧ ܲ௧
ᇲ
for all ݐᇱ് ݐ then we classify ݒ
as belonging to the segment ݐ.
5.Results
This section shows the qualitative experimental evaluation of our algorithm. We examined our algorithm on 50 images of which the segmentation results of some images are shown in figure 1. We compare our result with ordinary Random Walk approach [1]. In figure 1, we have made four columns of which column (a) shows the original image; column (b) shows the labeled (seeded) image; column (c) shows the segmentation result of the ordinary Random Walker approach [1] and column (d) shows the segmentation result of our approach. We have segmented the image in two regions, the object and the background, i.e. the number of objects are two here.
Few points are worth noting before we head to explain our results. In our experiment, user input was taken in the form of
‘scratches’. The user scratches on the object of interest with the pure red color whereas the background is scratched with pure green color. We have used the LUV color space as described in the above section. Each color components in the LUV space is divided in 8 bins. The output image after segmenting the original image using our approach shows the object of interest in black color and the output image after the segmentation of the original image using the ordinary random walker approach is in binarized form and also shows the object of interest in black. The output of our approach applied in the example images shown in figure 1 has substantial qualitative improvements as compared to the output of the ordinary Random Walker approach.
Now we concentrate into the analysis of results. In figure 1, the top row shows the image segmentation of a scene with sea and a solid grass ground. The segmentation using random walk on the color space graphs is more accurate than the segmentation done by the ordinary Random Walker algorithm. In our approach, the separation of two objects is very accurate whereas the segmentation of ordinary Random Walk has many mistakes in separating the two regions. The second row has an animated image with more stable background in which the two algorithms are applied. In this case too, our approach does a very fine segmentation of the object which in this case is a ‘car’ whereas the ordinary Random Walker approach fails to achieve the
required segmentation.
The third and fourth rows show the example images of an eagle and a dog with stable backgrounds. As it is evident that our segmentation scheme gives the perfect segmented image where as the ordinary approach fails to attain the accuracy.
The fifth image of a snake has complex background. Our color space graph based technique produces a perfect segmentation of the object (snake) but the ordinary Random Walker approach fails here almost completely. Similar result is for the eighth row which shows the segmentation of another snake image.
The sixth row has the image of a wild horse. The segmentation result of the ordinary Random Walker approach misses some part of the object whereas the result of our approach is much accurate than the former. Similarly the seventh row consists of a large field with complex background and a horse as an object. The segmentation result of our approach was very accurate and also the ordinary approach produced a result which was better compared to other results. Both the outputs of the segmentation using our approach and the ordinary approach done on the image of the ninth row showing a flying eagle was similar in nature.
The example images used in the tenth and the eleventh row shows a group of people (and also an animal) with complex background and a Royal Bengal tiger hiding in the bush. The output of our segmentation scheme applied in these images is much accurate as compared to the ordinary scheme.
6.Conclusion
This paper shows a technique to improve the performance of the Random Walker semi-automatic segmentation approach. Our approach has two significant changes as compared to the ordinary Random Walker scheme. Firstly, we have considered the graph of the color space which is a fully connected graph. After the seed information provided by the user is available we assigned the weights of the edges to 0 which are connecting the vectors belonging to the seeded set. We have then done the random walk on the nodes of our graph and solved the systems of equation to obtain the transition probabilities on this new graph. Applying random walks in this new graph is much more robust and have substantial improvements in segmenting the objects of an image as compared to the ordinary Random Walker.
(a) (b) (c) (d)
We used the LUV color space instead of the traditional RGB space in order to segment the object of interest as it is known to be perceptually more meaningful than RGB. The overall segmentation result using these new proposed schemes resulted to be more accurate and meaningful as compared to the ordinary Random Walker approach. Further works in this proposed approach involves improvement of the random walker transition probability in much more complex background with complex color textures. It can be also further improved to make it useful in case of texture segmentation which is much complex and a very demanding area in computer vision.
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