CHAPTER 4. ANALYSIS of GRAPH THEORETICAL IMAGE SEGMENTATION METHODS

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1 CHAPTER 4 ANALYSIS of GRAPH THEORETICAL IMAGE SEGMENTATION METHODS 4.1 Introduction In Graph based methods image is represented by undirected weighted graph where the nodes are pixels or pixel regions. They also differ in the graph algorithms such as Min-cut Max-flow Dijkstra s and Prim s algorithm. Graph based approach used for segmentation generally depend on local properties of graphs without considering global impressions of image which ultimately limits segmentation quality. In each method an image is represented as weighted undirected graph pixels and an edge set = ( ) where contains edges formed by joining every pair of nodes [98]. Weight of each edge ( ) is function of similarity between nodes of and. Partition the set of nodes into disjoint sets nodes in has strong affinities between them. Fig. 4.1(a) is the set of nodes called as such that the Fig. 4.1(b) Fig.4.1: Graphical representation of 4 x 4 pixel image (a) partitions A and B of equal size (b) partitions A and B of unequal size. 1

2 Partitioning to achieve segmentation poses several challenges such as the precise criteria for good partition and its efficient computation. Graph based methods for image segmentation has been broadly studied within the fields of image processing. In these methods segmentation problems by analogy are converted into graphs and solved as the graph partitioning problem. These graph based methods can be classified as Graph Cut based Minimal Spanning Tree based and Shortest Path based methods. 4.2 Graph Based Methods Several image segmentation methods have been proposed over the last several decades. Accurate formulation for image segmentation problem and computationally efficient implementations are very crucial. This Section covers the reported formulations and implementation strategies for each class of graphs based methods. The different characteristic of these methods is the way in which they define the enviable quality of segmentation and how they accomplish by means of different graph properties Normalized Cut Methods Any graph = ( ) can be partitioned into two disjoint sets A B provided that is greater than 1. The degree of dissimilarity between the sets A and B is addition of edge weights between nodes in A to nodes in B called as cut value. ( )= ( ) (4.1) The objective of partitioning is to optimize the cut value. By considering every possible partition minimum cut for a graph can be obtained but it is very complex problem. Optimization of cut value in the partitioning is well studied problem and variety of efficient algorithms exists for solving it. Fig. 4.2 represents image pixels maintaining four neighborhood system and pixels 2

3 corresponding to cuts where thick lines represent discontinuities between neighboring pixels. Fig 4.2(a) Fig 4.2(b) Fig. 4.2 (a) Image pixels maintaining four neighborhood system (b) Pixels corresponding to cuts where thick lines represent discontinuities between neighboring pixels. Wu et al. [3] proposed a minimum cut criterion based clustering approach. However this criterion is fit for partitioning of graphs with small vertex set and yields bad segmentation. From equation (4.1) we can observe that cut value corresponds to numbers of crossings between the two partitions. Equally sized partitions will be related by more edges than the unequally sized partitions. In Fig. 4.1(a) A and B have 8 vertices in each partition with 64 edge crossings while in Fig. 4.1(b) 15 vertices in A and only one vertex in B with 15 crossings. To overcome this imbalance in partitioning Shi et al. planned a new measure of disassociation the normalized cut Ncut [4]. For a graph partition where = the normalized cut cost is ( )= ( ) + ( ) ( ) ( ) (4.2) ( ) is the sum of weights of edges removed to split the graph and ( )and ( ) are the sum of weights of edges in the nodes of P and Q respectively to all nodes in the original graph G. The Ncut of the disassociation into the groups for small partitions of isolated points will be smaller due to high percentage of the entire connections between set and all other nodes. 3

4 Similarly total normalized association within the groups for given partition is where ( ) and nodes within and ( ) + ( ) ( )= ( ) ( ) (4.3) ( ) is the sum of weights of edges connecting respectively. This measure determines how strongly nodes within the group are connected to each other. These unbiased measures of association and disassociation of partition are related as (4.4) = 2 Hence the partitioning criterion of minimizing the disassociation between the groups and association within the groups can be satisfied simultaneously. Optimization of Ncut: Let be graph with vertex set divided between two sets and then the minimum Ncut for a graph with N nodes is calculated as below: Let ( ) = ( ) weight of all the edges connecting node i to all other i. nodes j. ii. Let 0 = 0 = Ncut between be diagonal matrix of degrees and is affinity matrix then the minimum is given by the relation min ( )= ( ) (4.5) 4

5 where y is orthogonal to second smallest eigenvectors is called as Rayleigh Quotient [100]. iii. If of then Rayleigh Quotient is minimized by solving the generalized Eigen value problem ( The eigen vector (4.6) ) = corresponding to second smallest value generates solution to the normalized cut problem. Recursive Two Way Cut: The graph nodes are partitioned into two subsets using threshold value. The cut can recursively be obtained in two partitioned parts and stops when it reaches to previously given Ncut value. For given weighted graph G summarize the information into the affinity matrix W and degree matrix D. Solve ( ) = for eigen vectors with the smallest eigen value. Minimize Ncut using eigen vector corresponding to the second smallest eigen value and bipartition the graph by determining the point of division. The number of graphs partitioned by this approach is controlled directly by the highest acceptable Ncut. This technique is known as recursive two way cut [99]. Illustration of Recursive Two Way Cut: - Construct weighted graph = ( ) for given image by considering each pixel as a node and connecting each pair of pixels by an edge. The weight of an edge is similarity between the pair of pixels. Define weight of an edge connecting to two nodes i and j by using brightness value of the pixel and their spatial location as () where ( ) 0 ℎ is the spatial location of node i. < (4.7) 5

6 ( ) is a feature vector based on intensity color or texture of node i. and are spatial tuning parameters respectively. is an entry in affinity matrix W. - Solve for the eigen vectors with the smallest eigen value of the system ( ) = (4.8) Convert generalized eigen system to the following standard eigen value problem. Solution operations of this for system p ( for nodes ) all in = eigen the vectors graph needs such a (4.9) ( ) large number of operations is impractical for segmentation applications. Due to the property of local connection in graphs to be partitioned also the resulting eigen systems are not too dense as well as only the few top eigen vectors are required for partitioning and the decision requirement for the eigen vectors is low. All these properties reduces the computations to - After computing the eigen vectors split the graph into two parts using the second smallest eigen vector. - repeate the algorithm on two partitioned on each part or homogeneously use top eigen vectors to subdivide the graphs based on those eigen vectors. The recursion stops when Ncut value exceeds certain limits. These steps of Recursive two way cut are applied on a sample image as shown in Fig. 43(a). The segmented image obtained by using second smallest of the ninth eigen vector is as shown in Fig. 4.3(b). 6

7 Fig. 4.3 (a): Original image Fig. 4.3 (b): Segmented image Shi et al. [8] discussed multi-class partitioning in combination with iterative process of two-way partition till acceptable result is accomplished. This technique is computationally expensive and also Ncut generates regions of same size which happens rarely in natural images. Important segmentation approaches for distinct graph types using normalized cut are as discussed below Pixel Affinity Graph Method In this method each pixel is considered as vertex and an edge is obtained by connecting two pixels within distance r as shown in Fig Similarity between the connected pixels reflects weight of an edge. Overall quality of segmentation will be reflected by the grouping cue used in the similarity pixel such as intensity positions and contours [101]. 7

8 Fig. 4.4 Segmentation by pixel affinity graph The measure of similarity where is position and for grouping cue is given by 0 ( )= ℎ < (4.10) is intensity difference between pixels l and m r is graph connection radius and are the corresponding scale parameters which controls the tradeoff between the brightness similarity and spatial proximity. Independent use of grouping cue results into bad segmentation due to effect of texture disorder in natural images. Hence another grouping cue related to the intervening contours is given by ( where line ( ( )= ) 0 ℎ < (4.11) ) represents straight line connecting pixels l and m and indicates edge strength square at location x. Pixels have high affinity if straight line between the pixels does not cross an image edge. These two grouping cue can be combined as ( )= ( ). ( )+ ( ) (4.12) 8

9 where is constant. For the larger radius r objects with weak contours can be detected more easily however the graph affinity matrix turns into denser. Segmentation quality is better for bigger graph radius but speed is very slow. Across larger image regions extended range graph links helps transmission of local grouping cues. In such situations objects with weak boundaries can be identified easily in messy background [102] Multiscale Graph Decomposition To collect sufficient grouping information affinity graph needs long range connections. These can be compressed on a multi-scale grid. It can produce precise object boundaries with constrained segmentation. To enhance normalized cut Sharon et al. [5] proposed algebraic multi-grid technique in which effective graph coarsening is used to generate unequal pyramid encoding region based grouping cues. First it defines the finest grid and construct the series such that. The principle of multi-grid method is relax on the fine grid Ω and project error to course grid and continues the relaxation and projection on more course grid until the coarsest grid is obtained as shown in Fig 4.5. Fig. 4. Sequence of increasingly coarsened grids used in multi-grid (vertex centered) Fig. 4.5 Relaxation and Projection in Multigrid Method 9

10 Benzit et al. [103] proposed the decomposition of multiple scales which seperates graph links into different scales = where if encloses affinity into pixels with specific range. < where =. First scale is (4.13) ( ) 0 only in which every pixel is graph node and they are connected if they are at distance r apart. In second scale pixels are connected distance apart and in scale s pixels are sampled at (2 + 1) will be denoted by distance apart. The representative pixels in each scale and compressed affinity matrix with connections between the representative pixels in is denoted by called as compressed affinity matrix. For parallel segmentation across scales form the partitioning matrix X and multi-scale affinity matrix W as below = = 0 To find the cross scale interpolation matrix and nodes in coarser layer as 1 ( )= 0 ℎ 0 between the nodes in layer The cross scale segmentation constraint matrix M is written as = 0 = 0 is segmentation constraint. 0 (4.14) (4.15) (4.16) 10

11 The constrained Normalized Cut is given by: ( )= (4.17) This algorithm facilitates concurrently along the graph scales with an inter-scale restriction to guarantee communication and reliability between the segmentation at every level. This segmentation approach is fit for segmenting large images but computationally not efficient Watershed Regions Based Similarity Graph Watershed transformation is a morphological based tool for image segmentation. The watershed transform can be classified as a region-based segmentation approach. The idea of watershed can be view as a landscape immersed in a lake catchment basins will be filled up with water starting at each local minimum. Dams must be built where the water coming from different catchment basins may be meeting in order to avoid the merging of catchment basins. The water shed lines are defined by the catchment basins divided by the dam at the highest level where the water can reach in the landscape. As a result watershed lines can separate individual catchment basins in the landscape [104]. Among various features that can be extracted from an image the maxima and the minima are of primary importance. Due to large number of regional minima in the images this technique dealt with problem of super segmentation. To overcome this Meyer [105] suggested hierarchical watershed with step given below - Choose local minima as region seeds. - Add neighbors to priority queue sorted by value. - Take top priority pixel from queue if all labeled neighbors have same label assign to pixel. 11

12 - Add all non-marked neighbors. - Repeat the process until finished. The flooding process starts with given threshold value that represents some relief feature. So some initial regions will be flooded which yields desired number of partitions. The hierarchical watershed regions can be modeled using graph. The flooded gradient image is represented by connected weighted neighborhood graph where node is the catchment basin of the topographic surface. After conversion one weighting function proposed in [105] can be used as mean density. where ( )= is the density of watershed regions (4.18) and. Another interesting approach adopted by Monterio et al. [6] which combines edge and region based approach with spectral techniques through watershed algorithms. To reduce the spatial resolution a pre-processing step is used without losing important image information. Rainfall watershed algorithm is applied on the image gradient magnitude to set an initial partitioning of the image into primitive regions. This initial partition is the input to a computationally efficient region segmentation process which produces the final segmentation. The later process uses a region based similarity graph representation of the image regions. Segmentation produced by this approach is clear and simple. Most of the methods in this category are computationally expensive as they are proved to be NP complex and might not be suitable for many real time applications. 12

13 4.2.2 Minimal Spanning Tree based Methods: The Minimal Spanning Tree (MST) is an important concept in graph theory. A spanning tree T of graph G = (V E) is a tree T such that T = (V E ) where E E. Each graph may have several spanning tree but minimal spanning tree is the tree with minimum weight. In MST of a graph nodes are pixels and edges represent affinities between the nodes that it connects. There are several algorithms to construct minimal spanning tree. In Prim s algorithm MST is constructed by adding the frontier edge with smallest weight. This algorithm is greedy style and runs in polynomial time. Minimal Spanning Tree for an image is constructed as shown in fig 4.6. Fig 4.6: MST construction MST based segmentation is related to graph based clustering where the data is represented by undirected adjacency graph. Different clusters that have stronger inherent affinities could be obtained by suitably removing the lowest weight 13

14 edges. Most of the MST based approaches for segmentation emphasizes the importance of Gestalt theory [7]. Earlier MST based methods perform image segmentation in an implicit way which is based on the inherent relationship between MST and cluster structure. Morris et al. [8] used MST to hierarchically partition images based on the principle that most similar pixel should be together and dissimilar pixels should be separated. They also proposed recursive MST algorithm which splits up MST built from an image into many sub-trees representing homogeneous regions such that each sub-tree should have certain number of nodes and neighboring sub-trees should have significantly dissimilar average gray levels. It yields low quality result in case of noisy images since wrong configuration of MST as an object might be contained in more than one sub-tree due to noise. An advanced work on MST based algorithm is proposed in [106] using both the differences across the sub-graph and the differences inside the sub-graph. The internal difference of a segment is the highest weight in the minimal spanning tree of the segment which is given by the relation ( ( )= ) where e = MST (S E). An edge with minimum weight among edges connecting to the two segments represents the differences between segments. Two segments can be merged if difference between them is less than or equal to minimum of any of the internal differences of two segments. The formal definition for merging criterion is < min ( )+ ( )+ (4.19) where K is constant and are the sizes of components and respectively. ( ) is the largest weight in the MST of. is the edge with smallest weight which connects and. From (4.10) we can see that 14

15 algorithm is sensitive to edges in smooth areas and less sensitive to areas with high variability. Felzenszwalb et al. [107] showed that segmentation produced by this method is neither too coarse nor too fine. Since two segments are merged on the basis of single low weight edge between them there are possibilities that the result could be considerably affected by noise if initial filtering of image is not done properly. To improve performance Fahad et al. [108] suggested some modifications of a graph theoretic image segmentation algorithm. Kruskal s algorithm is used to build MST for segmentation which reflects global properties of the image. Algorithm makes greedy decisions to produce the final segmentation by defining the predicate for measuring the evidence of boundaries between two regions. They have modified the algorithm by reducing the number of edges required for sorting which produces an over segmented result and suggests a statistical merge process which reduces the over segmentation. Evaluation of algorithm is done by segmenting various video clips performance and quality of segmentation is improved. Jia et al. [109] multi-atlas-based multi image segmentation where an image registration framework is based on combinative and incremental tree for better registration is proposed. In practical scenario it is difficult to acquire images without noise due to perplexed imaging environment. Since MST based methods are very much susceptible to noise therefore for noisy images without preprocessing such as filtering may yield unacceptable segmentation Shortest Path based Methods: Finding the shortest path between two nodes is a classical problem in graph theory. For connected weighted graphs shortest path between pair of nodes is the path whose total edge weight is minimum. The most well known algorithm 15

16 to find shortest path is Dijkstra s algorithm. For a directed graph edge length ( ) 0 e is an edge in To find shortest path from - to each vertex Set U = V L(u) = 0 ( ) = for Set { ( )/ = - If - Set U = X. For - and a vertex = then stop; for U} and is called as source. steps are as below: { }. = = ( ) with. ( ) is the shortest path length from u to v. new label is ( ) = min{ ( ) min{ ( ) + ( )/( ) Repeat step for newly generated X. }}. Dijkstra s algorithm is illustrated in fig. 4.7 Fig 4.7: Shortest Path Computation by Dijkstra s Algorithm 16

17 To find shortest path between nodes u and v grow Dijkstra s tree starting at the node u after each iteration add frontier edge whose non-tree end point is close to v. After each iteration node set of Dijkstra s tree will be added with nodes to which shortest path from u have been obtained. In case of shortest path based segmentation methods the problem of finding best boundary segment is converted into finding the path with minimum cost between the two nodes. In Live-wire method initial point is selected by user and the subsequent point is selected in such a way that the shortest path between initial point and current position should be best fit to the object of interest [9]. Sequence of oriented pixel edges represents the boundary where each oriented edge has single cost value to measure the quality of boundary. The boundary wraps around the object at real time speed. In comparison with tedious manual tracing Live-wire provides more accurate segmentation. Selection of proper initial seed near the desired boundary is difficult tasks. Hence for blurred images or weak boundaries implementation of Live-wire is difficult. While segmenting high resolution images Live-wire needs large number of computational resources to search the shortest path over the whole graph. Live lane [10] overcomes this limitation by liming the searching space in much smaller range of 5 to 100 pixels and largely reducing the computational time. Falcao et al. [11] exploited some known properties of graphs to avoid the unnecessary shortest path computation and proposed a fast algorithm called Live-wire-on-the-fly. The speeding up of path searching is based on the fact that the results of computation from the selected point can make use of the previous position of arrow. It has advantage that there is no restriction on the shape or size of the boundary and also the boundary can be oriented so that it has well defined inner and outer parts of the boundary. Bai et al. [110] has developed Image region based algorithm by using geodesic distance. Since the time complexity for geodesics is in linear time the algorithm can be implemented very efficiently however it is strongly dependent on the seed locations and is more likely to leak through weak 17

18 boundaries. Increasing demand of 3D data motivated the researchers to extend 2D shortest path techniques to 3D. The 3D example of live wire was proposed for medical image segmentation [111]. Other 3D extensions of the shortest path algorithm discussed in [ ] are not straightforward and fundamentally path based techniques; they need not guarantee that the shortest paths will lie on the minimal surface. To overcome this Grady [114] adopted mathematically elegant method to find the minimal surfaces and then used them for segmentation of 3D data. In comparison with MST based methods the shortest path can well describe certain nature of the object boundaries in an image since MST based methods focuses on clustering property of a segment. To control segmentation process Live-wire provides more freedom to the user. Shortest path method might be more suitable for extracting complex objects with relatively explicit boundaries than other graph based methods. As a robust technique for interactive segmentation it can be extended to 2D sequences or 3D data Other Methods: There are many other graph based segmentation methods which does not belong to above mentioned categories. Pyramid based methods proposed by [ ] in which a graph is created from original image then from this graph a set of graphs defined in multi-level of resolutions is built which looks like a pyramid. Using reduction function vertices and edges at level L are reduced to form the vertices and edges at level L+1. Level of pyramid is called as working level which is responsible to yield the segmentation. These methods are categorized into regular pyramid and irregular pyramid based methods. For regular pyramids reduction factor which is the ratio between number of vertices at level L and at level L+1 is constant and fixed. Hence the size and the layout of the structure of pyramid is predictable. Gaussian filters with adjustable filter scales used by Ping et al. [117] to utilize a pyramid built. Pyramid linking approach is used for 18

19 segmentation in [118] which is dependent on proper selection of the working level. To overcome this drawback a modified pyramid linking approach is proposed by Zilan et al. [119]. Regular pyramid method fails to segment elongated regions. The structure of pyramid also varies due to small rotations shifts and scales of the input image. In contrast of this reduction factor is not constant for irregular pyramids. Hence the size and layout are not predictable. Montanvert et al. [120] proposed a hierarchy of region adjacency graph which performs stochastic decimation to achieve segmentation. For same input images different outcomes of the random variable yield distinct segmentation results. To overcome random variation in decimation process it is replaced by an interest variable. The bounded irregular pyramid proposed in [121] combines features from regular and irregular pyramids and proved that irregular pyramid yields good result than the regular pyramid. The random walker [122] is an interactive segmentation method for weighted graph to assign label to each pixel of an image. Each edge of the graph is assigned with the weight where ( )= image intensity at pixel i and is constant. (4.20) This weight is the probability that random walker will go across an edge joining and. The label of a pixel is given by the seed point where random walker will reach first. This method of random walker probabilities is same as minimizing the Dirichlet s function given below. Minimizing [ ]= 1 2 (4.21) [ ] is same as solving harmonic function satisfying boundary conditions by assuming seed point value is equal to one. Using this function seeds can be covered in least steps and hence avoids segmentation leakage and 19

20 shrinking bias. In comparison with graph cuts random walker exhibits the greatest robustness to seed quantity but least robustness to seed placement. Image segmentation method based on dominant sets as proposed by Pavan et al. [123] is generalization of maximum clique in the context of weighted graph. The dominant set clustering method has better classification performance in intensity color and texture image segmentation. It is competitive with normalized cut method for both clustering quality and computational cost. 4.3 Evaluation and Analysis of Segmentation Methods Performance evaluation of graph based segmentation methods using BSDS For performance evaluation of above discussed methods we used Berkeley Segmentation Dataset and Benchmark [124] to ease comparison of manual and machine based image segmentation. To compare the results to ground truth boundaries we need to threshold boundary maps multiple times at each level it yields two values viz. Precision (P) and Recall (R) [125] which are the metrics used in benchmark for the classification. Precision is the probability that machine generated boundary pixel is true boundary pixel whereas recall is the probability that the border pixel marked by the machine is same as the border pixel marked by human. Harmonic mean of precision and recall can be summarized in terms of F-measure as =2. + (4.22) Experimental performance of image segmentation algorithms based on three important characteristics: precision recall and F-measure for five graph based segmentation methods viz. pixel affinity multiscale decomposition watershed regions minimal spanning tree and shortest path is discussed. Fig and 4.11 represents precision and recall values for some images from Berkeley Segmentation Dataset and Benchmark. 20

21 Fig. 4.8: Precision Recall for Image No Fig. 4.9: Precision Recall for Image No

22 Fig. 4.10: Precision Recall for Image No Fig. 4.11: Precision Recall for Image No

23 Table 4.1 shows the corresponding F-measure values for these images for all five methods. F Measure Image Pixel Multiscale Watershed Minimal Shortest Affinity Decomposition Regions Spanning Path Tree Table 4.1: F-measure of select images for various segmentation methods Enhancement in Ncut Methods To improve the performance of Ncut we proposed modifications in tuning parameter and carried out performance the analysis. The Ncut algorithm first reads an image of size and constructs an intensity matrix corresponding to the pixels in an image where intensity matrix consists of feature values or the intensity values of the pixel. Then the graph function computes the affinity matrix of an image by setting default values to the parameters as = 0.1 = 0.3 = 10. Parameter is tuning parameter which controls magnitude of the feature intensity difference involved in computing. From (4.7) it can be observed that for smaller values of weight is less resulting into closely grouped pixels and more local segmentation and vice versa. The tuning parameter involved in computing controls degree of the spatial feature. However because of fixed values of and in two way recursive cut method in many cases the quality of segmentation is compromised. As a result it achieves global segmentation which is not 23

24 perceptive to local variations in the image. To achieve improved performance we correlated the features values around pixel i and j by modeling as = [ ( ( ) ). ( ( ) )] where ( ( ) ) and (4.23) ( ( ) ) are the standard deviations of neighborhood features around pixel i and pixel j respectively around radius r. defined in (18) will capture the correlation of neighboring features between pixel i and pixel j while determining the weights of edges. For fixed radius local variations of features around pixel i will be less for smaller values of ( ( ) ) similarly features around j will be less for smaller values of ( ( ) ). As well as for low variations in combined local features around pixel ( ( ) ). ( ( ) ) will also be smaller and hence improved i and pixel j. This meets to the aim of strong weight connections between the identical neighboring pixels in the affinity matrix W resulting better segmentation quality with linear complexity. By using (4.23) we varied { } and observed segmentation results as shown in fig (a) - (f). It illustrates that as decreases segmentation becomes more detailed. The algorithm is more sensitive to the value of and its different values can give sound segmentations in different parts of image. 24

25 ( ) = 1.0 ( ) = 0.05 ( ) = 0.5 ( ) = 0.01 ( ) Fig (a) (f): Segmentation results for different values of ( ) = 0.1 = Similarly for pixel affinity graph by using (10) we considered a range of in between and r in between 1 and 10. The best segmentation was obtained for = 0.1 and r = 10. For multiscale graph decomposition (11) is solved for same range of whereas was varied from with = 1. The best segmentation was achieved at = 0.09 and = Watershed region affinity matrix is generated by using connected weighted graph with many regions obtained from hierarchical watershed as input graph. For analysis we used Berkeley Segmentation Dataset and Benchmark to ease comparison of manual and machine based image segmentation. Precision Recall and Fmeasure as well as time complexity were calculated for each segmented image for all the methods discussed above. 25

26 To determine the overall performance of the algorithm Berkeley s benchmark combines the individual scores from all local segmentations of each image in a single final score. The results shown in fig demonstrate the final scores obtained by using our approach for Ncut based segmentation methods. It shows that multiscale graph decomposition performs better than other methods. The performance of multiscale graph decomposition is even better than that of combining hierarchical multiscale graph decomposition demonstrated by [126]. For other Ncut based methods our approach also achieves fairly good performance for most of images considered. Fig. 4.13: F-Measure for various images ( ) The time complexity is an important parameter in Ncut image segmentation methods. We carried out time complexity computations for different images with above discussed methods are as shown in Fig

27 Fig Time Complexity for various images ( ) It shows that multiscale and watershed segmentation methods consume less computational power and their performance is almost same for both for all the images considered and it is better than that of computed by [126]. It also indicates that the time complexity for pixel affinity and recursive two way cut methods is sensitive to image. 4.4 Discussion and Conclusion Latest graph based image segmentation methods and their variations such as pixel affinity multiscale decomposition watershed regions minimal spanning tree and shortest path based methods are studied analytically. Such study and evaluation is also crucial for improving the performance of existing segmentation 27

28 algorithms and for developing new powerful segmentation algorithms. Their performance can be enhanced by use of hybrid approach and correct optimization. The graph based methods generally performs segmentation on the basis of local properties of image. For segmenting the images in some applications where detailed extraction of features is necessary consideration of global impression along local properties is inevitable. We have proposed an enhanced technique which allows considering both local as well as global features during normalized cut based segmentation to meet the requirement of precise segmentation. This was achieved by correlating the feature values around neighboring pixels for determining weights of edges of the graph. This creates strong weight connections between the identical neighboring pixels in the affinity matrix resulting better segmentation quality with linear complexity. The result shows that the final score of multiscale graph decomposition is superior to the score obtained for other methods and even better than that of combining hierarchical multiscale graph decomposition. The technique also has lesser computational time complexity. 28

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