Combining Hierarchical Structures on Graphs and Normalized Cut for Image Segmentation

Extensive studies have been accomplished for image segmentation. The segmentation algorithms are commonly categorized according to the image characteristics borders and regions (Gonzales & Woods, 2000). In the first one, the image is divided based on its discontinuities, i.e., the places where abrupt intensity changes occur. Regarding to the region segmentation, this happen when there are similarities of color or texture, for example, between neighboring pixels. In spite of these categories, the problem is to find a good partitioning of an image among several possible to be achieved.


Introduction
The image segmentation task is to divide an image into regions of interest that are suitable for machine or human operations.The machine ability of recognizing and distinguishing objects, or its parts in a scene, is the main goal of computer vision domain.This is a critical issue because the judgement of good or bad segmentation is usually subject to humans.
Extensive studies have been accomplished for image segmentation.The segmentation algorithms are commonly categorized according to the image characteristics borders and regions (Gonzales & Woods, 2000).In the first one, the image is divided based on its discontinuities, i.e., the places where abrupt intensity changes occur.Regarding to the region segmentation, this happen when there are similarities of color or texture, for example, between neighboring pixels.In spite of these categories, the problem is to find a good partitioning of an image among several possible to be achieved.
Several algorithms in image segmentation can be formulated from the partitioning of graphs.This means using graphs as image models or representations and then apply a criterion or methodology in order to split it into subgraphs.The existing literature on graph partitioning is wide, but we are interested on a particular approach, the called Normalized Cut introduced by Shi & Malik (1997).
A graph cut consists of removing edges consistently in order to generate two subgraphs.The Normalized Cut approach is a graph-cut technique based on Spectral Graph Theory responsible for generating balanced subgraphs through the removal of the smallest possible number of edges.The Normalized Cut approach uses concepts by Fiedler (1975) in the manipulation of the second smallest eigenvector of the graph representative matrix as a guide for graph partitioning.The inherent bias of this technique is that balanced partitions for image segmentation cannot be appropriate for some images when small number of partitions is desired (e.g., images with an easily detectible object and an uniform background).A survey of application of Spectral Graph Theory is given by Spielman (2007).
There are several ways of generating the graph model representing the input images.The graph commonly used as input to the Normalized Cut implementation is that one based on the pixel grid similarity.We propose two alternative representations of graphs known as Quadtree and Component Tree similarity graphs.These graphs decompose the image into partitions and thus carry hierarchical information that can be useful on segmentation task.

Graph representation
In digital image processing, a graph is commonly used to model digital images (Wilson & Watkins, 1990).In the Normalized Cut segmentation technique the input graph is called Similarity Graph (Shi & Malik, 2000), that we explain in details in Section 3.3.In a similarity graph the edge weights should reflect the similarity between the nodes connected by them, and are given by a Similarity Function.Here we present the methods for building a similarity graph that we have used in the experiments described in Section 5.
There are several approaches to represent an image as a similarity graph.In the next subsections we present some fundamental concepts on graphs, and four image-graph representation approaches: based on the Pixel Grid (Shi & Malik, 2000) , Multiscale Pixel Grid (Cour et al., 2005), based on the Component Tree (Carvalho, Costa, Ferreira & Cesar-Jr., 2010), and based on the Quadtree (Carvalho, Costa & Ferreira, 2010).Approaches based on the Component Tree, Quadtree,andMultiscale Pixel Grid rely on hierarchical structures to model an image as a graph, and provide different segmentation results when compared to the classical non-hierarchical pixel grid approach.

Basic concepts on graphs
A graph is a mathematical structure employed to model or to describe objects and their relationships, e. g., a composition relationship can describe objects and their constituent parts.Let G =( V, E, W) be a non-directed weighted graph; V is a set of nodes, E is a set of edges e(i, j), i, j ∈ V,a n dW is a set of weights w(i, j), i, j ∈ V.For each edge e(i, j) ∈ E exists an associated weight w(i, j) ∈ W, that can be represented by a single value or a set of values.Two nodes i and j are adjacent, represented by i ∼ j, if there is an edge connecting i and j.Given a node i ∈ V,i t sdegree d i corresponds to its number of neighbours, and its strength s i (Wilson & Watkins, 1990) to the sum of its edge's weights is a sequence without repeated nodes where i k ∼ i k+1 , k = 1,2,...,n − 1.Two nodes are connected if there exists at least one path between i and j.I n a connected graph G all pair of nodes are connected.A cycle is a path where i 1 = i n .Atree is a connected graph with no cycles.

Graphs and matrices
A graph and its features can be represented using matrices (Wilson & Watkins, 1990).The adjacency between graph nodes can be described by an Adjacency Matrix A with size n × n, where n is the number of nodes, |V|,o fg r a p hG =( V, E, W).The matrix elements a(i, j) are 1 if i ∼ j, i, j ∈ V, or 0, otherwise.Similarly, the Weight Matrix W, also with size n × n, n = |V|, can store the graph weights w(i, j), i ∼ j, i, j ∈ V.T h eDegree Matrix D is diagonal with d(i, i)=d i ,whered i is the degree os node i ∈ V. Finally, the Laplacian Matrix L is defined as L = D − A for unweighted graphs, or L = D − W for weighted graphs.The Laplacian matrix is commonly used on Spectral Graph Theory (Spielman, 2007).

Graph based on the Pixel Grid
In this Pixel Grid-based image-graph representation each pixel is taken as a graph node, and two pixels within a r distance are connected by an edge.Shi & Malik (2000) have used this approach as the first one for their Normalized Cut technique.This approach have been introduced in the experiments as a landmark for compare results with other approaches.

Multiscale Pixel Grid
The Multiscale Pixel Grid graph decomposition algorithm introduced by Cour et al. (2005) works on multiple scales of the image grid to capture coarse and fine detail levels.The construction of the similarity graph is done according to their spatial separation, as in the following Equation where W is a weight matrix that represents a graph composed by independent subgraphs W s , s corresponds to a scaled pixel grid.
In the Multiscale approach there exists one different radius for each image scale s.T h u s ,t w o pixels i and j ∈ W s are connected only if the distance between them is lower than a radius R s .
The radii values are a tradeoff between the computation cost and the segmentation result.The Multiscale approach can alleviate this situation by using recursive sub-sampling of the image pixel grid.

Graph based on the Component Tree
The Component Tree (CT) (Carvalho, Costa, Ferreira & Cesar-Jr., 2010) is a hierarchical representation of a digital image after thresholding operations between its minimum and maximum gray values (Mosorov & Kowalski, 2002).There exists a relation of inclusion between components at sequential gray levels in the image, explained below by the partition definition (Carvalho, 2004).

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Combining Hierarchical Structures on Graphs and Normalized Cut for Image Segmentation www.intechopen.comA cross-section I k of an image I is a binary image defined as (Mosorov & Kowalski, 2002;Najman & Couprie, 2006): In the CT, the Connected Components (CC) of all cross-sections are organized in a tree structure.
There exists an edge between two connected components CC i k+1 and CC j k when CC i k+1 ⊆ CC j k (inclusion relation); k is a cross-section identifier, i is a CC in cross-section k + 1, and j is a CC in cross-section k.The connected component of the first cross-section corresponds to the whole image domain and it is called root.The leaves are the elements of the CT that have no children.A fast algorithm to build the CT is given by Najman & Couprie (2006).Fig. 1 shows a gray scale image I and its five cross-sections I k .The Component Tree for image I is depicted in Fig. 2(a).When observing the traditional CT model, one realizes that the tree will be composed by only white components, i. e., components with value 1.There still information in the cross-sections related to the black components.In fact, these components can provide more relevant information in particular cases.Therefore, we have defined the Reverse Component Tree (RCT) (Carvalho, Costa, Ferreira & Cesar-Jr., 2010) where two connected components ; k is a cross-section identifier, i is a CC in cross-section k + 1, and j is a CC in cross-section k.Unlike described for the CT case, the roots of the RCT's are formed by the connected components of the last cross-section.In order to build a connected similarity graph, we combine Component and Reverse Component Trees.This similarity graph will be used in the graph cut process.First, a connected subgraph v k,n correspond to the connected components CC n k ; k ∈Nis a cross-section identifier, and wherer ∈Ris the connection radius.After built, the k subgraphs are connected by adding the edges from the CT and RCT.The weight of each edge should reflect the likelihood of different connected components.Some of them are difference between areas, density, average gray levels and standard deviation, and Euclidean distance.
One strong characteristic of CT method is the generation of multiple image partitions at once.Because a cross-section I k corresponds to the whole image, there is one image partition for each cross-section.
Finally, it is important to note that some images can produce a high quantity of connected components, especially in the presence of noise.Therefore, it is useful to apply some pre-processing on the image before starting the CT computation such as, gaussian filter, normalization of the gray values into a smaller range, and merging of identical subsequent cross-sections.

Graph based on the Quadtree
A Quadtree is a data structure formed from the recursive decomposition of a space (Samet, 1984).In the image processing domain, a quadtree usually maps an image and its regions into a directed acyclic graph (Consularo & Cesar-Jr., 2005).
The decomposition process is simple: The initial region corresponds to the whole image and is associated to the root node; each region in the image should be recursively decomposed into exact four new disjoint regions until they satisfy a defined criterion of homogeneity.Fig. 3 shows a Quadtree decomposition example.The decomposition criterion choice in this case was to exist regions with only one value.
In practice, the regularity of the Quadtree decomposition limits the application of this method to square images with edge sizes 2 n , n ∈N .One solution is to relax the regular decomposition, allowing a more suitable number of regions.But the greatest  difficulty in creating the quadtree is the choice of the decomposition criterion.There are different criteria proposed, such as the standard deviation or entropy of image gray levels (Consularo & Cesar-Jr., 2005).
In our quadtree approach (Carvalho, Costa & Ferreira, 2010) for image segmentation, we proposed applying the Canny (Canny, 1986) (1986) filter in the image before the decomposition.This filter has low sensitivity to noise in images.By removing pixels with low gradient and thresholding the resultant ones, this process results in a binary image with border pixels highlighted.This procedure was chosen because: 1. after the filtering, the image results in a binary matrix.Then, facilitating the decomposition criterion definition, that a region should be decomposed when it is not formed entirely by ones or zeros (Samet, 1984); 2. the edge detection operation drastically reduce the size of data to be processed, while at the same time preserves the structural information about object boundaries (Canny, 1986).
In our work (Carvalho, Costa & Ferreira, 2010), the main reason for using a Quadtree image representation was to reduce the similarity graph size.Thus, in this technique the graph is generated using only the regions associated to the Quadtree leaves.Each region is associated to a node and for each region a centroid pixel is defined as the representative pixel.The nodes of the similarity graph are linked together if their representative pixel distances are less than a given radius r, similar to the pixel grid approach.However, it is useful to consider the nodes region sizes to calculate the radius r.
The number of nodes on the similarity graph can be influenced by the choice of the edge detector parameters.The number of regions obtained by the proposed technique will vary according to the image features.Also, the parameters of the edge detection filter can be manually specified, in order to change its sensibility.

Graph Cut and Normalized Cut
A graph cut partitions the set of nodes V of a graph G =(V, E, W) into two disjoint subsets A and B, and can be expressed by the following equation (Shi & Malik, 1997):

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New Frontiers in Graph Theory www.intechopen.comwhere w(u, v) are the edges removed from G.
This formula indicates the degree of dissimilarity between G A and G B , the corresponding subgraphs to the node subsets A and B, respectively.There are a lot of ways to solve the problem of graph cuts.One of them, proposed by Wu & Leahy (1993), solve this problem removing the smallest possible number of edges.However, in some cases this approach can produce isolated nodes.A study about types of graph cut is given by Soundararajan & Sarkar (2001).
A graph cut can be accomplished by means of Spectral Graph Theory.In this approach, the matrix eigenvectors that represent graphs are analyzed and used as parameter in order to partition a graph.

Spectral Graph Theory
The Spectral Graph Theory (SGT) studies the eigenvalues of the graph matrices, also called graph spectrum.Algebraic methods used to analyze matrices of graphs are especially effective in treating regular and symmetric graphs (Chung, 1997).The matrices commonly used are adjacency matrices and the SGT establishes a relation between the graph spectrum and the graph features.
The use of graph spectrum information for graph cuts has a great contribution from Donath & Hoffman (1972); Fiedler (1975); Pothen et al. (1990).Fiedler proposed that the second smallest eigenvector v 2 of the Laplacian matrix, also called the Fiedler vector, has in a given row v 2 [i] a numerical information about node i, also called the characteristic value of the node.The graph nodes can be partitionated by grouping them according to their value in the Fiedler vector.The commonly way used to group nodes is the characteristic values signals.

The Normalized Cut
The Normalized Cut (NCut) technique (Shi & Malik, 1997) is a theoretic method for graph partitioning.Its goal is to find a balanced cut in a graph, in order to generate two or more subgraphs.This technique solves the problem stated by Wu & Leahy (1993) in their minimum cut criteria for graph cutting.Applying this method for image segmentation is possible with a proper image-graph representation, where the subgraphs obtained from graph partitioning represents the image regions.The NCut in a graph G is calculated by the following equation: where A and B are the node subsets of subgraphs G A and G B ,s u b j e c tt oA ∪ B = V and A ∩ B = ∅;Cut(A, B) is defined in Equation (3); SumCon(A, V) is the total weight of the edges connecting nodes from a subgraph G A to all nodes in the original graph G; and SumCon(B, V) is similarly defined to a subgraph G B .
The optimal NCut is the one that minimizes Equation 4. The problem in minimizing Equation 4is that it is only trivial for small graphs.For bigger graphs, it has a NP-Complete complexity.Shi & Malik (2000) Fiedler (1975), i.e., its minimum value is λ 2 , the second smallest eigenvalue (Golub & Loan, 1989).
The graph partitioning is guided by the eigenvector v 2 ,whereeachvaluev 2 [i] will represent agraphnodei.To split a graph, a threshold value is used and the graph nodes are partitioned in two subsets.The most common threshold values are zero, the median value in v 2 or the one that minimizes the NCut value.

Similarity Graph
In a Similarity Graph the edge weights represent the degree of similarity between the linked nodes.For graphs that represent images, the similarity can be determined by a function of intensity, position and other image pixels features.A measure of similarity regarding the intensity and the position of image pixels is given by (Shi & Malik, 2000): where α = |P i − P j | and β = |I i − I j | are respectively the distance and the difference of intensity between pixels i and j; r is a given distance (also called graph connection radius); and d p and d i are set as the variance of the image pixels positions and intensity.This grouping cue used separately often gives bad segmentations because some natural images are affected by the texture clutter.
The intervening contours is another measure to evaluate the affinity between two nodes by measuring the image edges between their correspondent pixels.The intervening contour similarity function is given by (Cour et al., 2005): where line(i, j) is a straight line joining pixels i and j and ε = | Edge(x)| is the image edge strength at location x.
These two grouping cues can be combined as (Cour et al., 2005):

Related work
There is a wide range of recent work in image segmentation using the Normalized Cut technique.The contributions are focused on improving the algorithm performance, others on proposing different image-graph modelling and others on the application of this technique for real-world applications.

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New Frontiers in Graph Theory www.intechopen.com In several works, the image model used is the similarity graph built by taking each image pixel as a node.In this case, the node pairs within a given radius r are connected by an edge.This graph will be explained in the next section.Monteiro & Campilho (2008) proposed the Watershed Normalized Cut, which uses the regions from the Watershed image segmentation as nodes for the similarity graph.The Watershed region similarity graph is either used by Carvalho et al. (2009) for comparison with the primitive pixel affinity graph in yeast cells images segmentation.Ma & Wan (2008) used Watershed based similarity graphs to segment texture images.
The primitive normalized cut enhancement was also studied and applied by many researchers.Cour et al. (2005) proposes a Normalized Cut adaptive that focus on the computational problem created by long range graphs.The authors suggested the use of multiscale segmentations, decomposing a long range graph into independent subgraphs.The main contribution of this technique is that larger images can be better segmented with a linear complexity.Sun & He (2009) proposed the use of the multiscale graph decomposition, partitioning the image graph representation at the finest scale level and weighting the graph nodes using the texture features.Tolliver & Miller (2006) suggested an improvement of the normalized cut technique.They proposed the use of the k first eigenvectors for graph partitioning as the k-way Normalized cut.The difference is that these eigenvectors modify the edges weight in the graph, resulting in new graph matrices, and the k first eigenvectors are calculated again in the new Laplacian Matrix.The authors proved that this procedure changes the k first eigenvalues to zero.Spectral graph theory concepts about the Laplacian matrix informs that the number of eigenvalues equal to zero shows the number of connected components in a graph.Their algorithm returns these connected components.Cour et al. (2005) suggested another improvement by dividing the graph in scales and processing them in paralell.This approach can segment large images graphs with high conections with linear complexity.Tao et al. (2008) proposed a new image thresholding technique using the normalized cut.The graph similarity matrix proposed is now based on pixel gray levels, reducing the matrix size and the computational cost.So, a new matrix M is created, where M(i, j)=Cut(V i , V j ) with i and j being two given gray levels.Using this matrix, the normalized cut is then calculated to each threshold value, If the normalized cut related to a given threshold value t is below a prespecified value, this threshold value is adequate to separate the objects from the background in this image.Grote et al. (2007) suggested the normalized cut for extracting roads from aerial images.In their graph, pixels are the graph nodes and the similarity matrix uses contours, hue and color in the image pixels.this approach uses the k-way normalized cut, with k being large to avoid road and non-road pixels mixture.Senthilnath & Omkar (2009) compared this technique with other state-of-art road extraction approach, proving that the normalized cut based technique works better.Other normalized cut aplications in the literature are the noise reduction in images by Zhang & Zhang (2009) and the colour image segmentation by Tao et al. (2008).The similarity graph obtained from Quadtree and Component/Reverse Tree representations is done according to the details explained previously and its parameters are described in Section 5.They have hierarchical information that is exploited in the graph cut process.The eigenvectors of the Laplacian matrix are obtained from the solution of the following equation: ( 8 ) At this moment, it is possible to use only the second eigenvector v 2 in order to provide two partitions of the graph G and reapply the process to obtain recursively a large number of partitions.A partition should be divided by analysing a specified cut value.This technique is known as Recursive Two-Way NCut.In the other hand, it is possible to discretize the k first eigenvector X,w e r eX =[ v1, v2,...,vk] and use them directly to implement the graph partitioning into k desired partitions.This partitioning process is called K-Way Cut and corresponds to the block sequence presented in Fig. 4.

Experimental results and discussion
We have performed k-way Normalized Cut segmentation in 100 grayscale test images from Berkeley Segmentation Benchmark1 (Martin et al., 2001).The goal of these experiments is to compare the techniques for building the similarity graph based on Pixel Grid (Shi & Malik, 2000), Multiscale Pixel Grid (Cour et al., 2005), Quadtree (Carvalho, Costa & Ferreira, 2010), and Component Tree (Carvalho, Costa, Ferreira & Cesar-Jr., 2010).Fig. 5 show a collection of 9 selected images.The Berkeley's benchmark rely on human image segmentations to state the segmentation algorithms assertiveness, according to Precision (P)andRecall(R) metrics (Davis & Goadrich, New Frontiers in Graph Theory www.intechopen.com2006; Martin et al., 2001).Precision is the probability that a pixel marked as a border is in fact a border pixel, and is given by where TP is the number of true positives, and FP the number of false positives.The precision P decrease as increases the number of false positives.Recall, also called hit rate,i st h e probability that the border pixels marked by the machine are the same as the border pixels marked by humans, and is defined as where FN is the number of false negatives.
These two metrics are summarized in the F-measure (Davis & Goadrich, 2006) 399 Combining Hierarchical Structures on Graphs and Normalized Cut for Image Segmentation www.intechopen.comthat is used as the score metric by Berkeley's benchmark to ranking the algorithms effectiveness.

Experiment setup
The experiments were perfomed according to the workflow presented in Section 4 with k-way method.The connection radius for the Pixel Grid graph was r = 10 and the edges weights were given by Equation ( 6).For the Multiscale Pixel Grid approach were used one radius for each scale, which were r 1 = 2, r 2 = 3andr 3 = 7 and the edge weights were given by Equation ( 6).The Quadtree weights were also given by Equation ( 6), and their radii given as where v φx i and v φy i are repectively the horizontal and the vertical diameter of region correspondent to node v i ∈ V; v φx j and v φy j are similarly defined for node v j ∈ V;andk = 10.The Component Tree was generated with radius r = 25 for the subgraphs.The attributes used to build our CT similarity graph were difference of area, distance, standard deviation of the gray levels and density.For the edges that links the subgraphs, the weights were multiplied by a factor given by the following equation: where NC Fi and NC Fj are the number of CCs of the cross-sections that has the nodes i and j respectively; and d(i) and d(j) are the degrees of nodes i and j respectively, related to the CT.After the CT segmentation procedure the various partitions generated for each image were converted to a single one by two distinct approaches: manually by selecting the partition that seemed to be the better image segmentation by a criterion of resemblance to the original image; automatically by merging pairs of partitions with highest mutual information iteratively.These approaches sets our experiments on image segmentation based on CT as semi-automatic or automatic procedures.

Results, discussion, and future works
The Berkeley's benchmark combine the individual scores from all segmentations of each algorithm in a single final score that determine the algorithm overall ranking.In our experiments the first place in the ranking was for the Multiscale Pixel Grid approach and the last place was for the automatic Component Tree.Fig. 6 show the overall scores obtained for each similarity graph method.
The segmentation scores for some individual images, however, are quite different from the algorithm's overall score.Fig. 7 show the individual scores for each image with each algorithm.We can observe that even the automatic CT approach, which is the worst ranked algorithm, has the highest score to about ten images.A similar result was obtained by the Semi-automatic CT approach.Put together, they account for 20% of the better scores for individual images.Not surprisingly, they also have the highest overall Recall, 0.65 and 0.64 for automatic and semi-automatic methods, respectively, against 0.60 for the best ranked algorithm.In the chart shown at Fig. 7 one can yet observe a clear trend for the Multiscale and the Quadtree approaches to follow the scores obtained by the classical Pixel Grid method: Multiscale with a little better, and Quadtree with a little worse scores.We can associate this trend to the fact that these algorithms are using the same similarity function.
Despite these quantitative analisys, it is also important to make a qualitative analisys of the segmentation results.Thus, are shown in Fig 8 one machine segmentation for each image shown in Fig. 5, and its human segmentations.The computational performance is also an important requirement for the Normalized Cut technique.When Cour et al. (2005) proposed the Multiscale approach to generate the similarity graph, one main objective was to reduce the NCut computational cost.It is also the objective of the Quadtree approach.and the Quadtree methods.That indicates that they are similarly sensitive to the image's data, but their overall performance are, respectively, the lower and the higher ones.The overall performance of the Multiscale algorithm is higher than the overall performance of the Pixel Grid algorithm, but is lower than the performance of the Quadtree method.Despite the lower performance, only the Component Tree can generate multiple image partitions at once.This problem can be alleviated by implementing the Najman's (Najman & Couprie, 2006) fast algorithm to build the Component Tree.

Conclusion
In this Chapter we proposed an approach to implement image segmentation based on graph modelling and Normalized Cut technique.Performing graph partitioning by means of Normalized Cut has been widely used in the specific literature.The possibility of generate balanced partitions has shown that this approach is efficient.The proposed similarity graphs, build from the Quadtree and Component Tree structures, proved promising compared to the traditional modelling based on pixel similarity graph.We performed comparisons using two-well established metrics, the Precision and Recall values.An additional aspect to be considered in the proposed graph models is exploring the hierarchical structures of both, Quadtree and Component Tree.Besides, we exploit a little more the Component Tree and proposed the Reverse Component Tree as a way to better represent the information contained in the image cross-sections.
The experimental results accomplished on images from the Berkeley Database show that use regions, or primitive regions to be specific, instead pixels seems to be a better strategy to segment images by Normalized Cut approach.In addition, the new image representation had the advantage of reducing the number of graph nodes and, therefore, improved the algorithm performance.
Fig. 2(b) shows the Reverse Component Tree for image I, presented in Fig. 1(a).
Fig. 2. Component Tree and Reverse Component Tree from grayscale image presented in Fig. 1(a).

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Combining Hierarchical Structures on Graphs and Normalized Cut for Image Segmentation www.intechopen.com4.Normalized Cut segmentation workflowG i v e na ni m a g eI, we build a similarity graph G =( V, E, W), from Quadtree and/or Component Tree representations.The image segmentation process based on Normalized Cut technique can be applied by two distinct methods: recursive Two-way and K-Way.The block diagram presented in Fig.4illustrates the complete image segmentation process.

Fig. 4 .
Fig. 4. Workflow of the image segmentation technique based on Normalized Cut.

Fig. 6 .
Fig. 6.Algorithms overall ranking.The two values between parenthesis at the right box represent (R,P).
Fig. 9 show a chart with the time each algorithm took to process the segmentation.There is a strong correlation between the Component Tree 401 Combining Hierarchical Structures on Graphs and Normalized Cut for Image Segmentation www.intechopen.com