# A Conditional Random Field Approach to Unsupervised Texture Image Segmentation

- Chang-Tsun Li
^{1}Email author

**2010**:167942

https://doi.org/10.1155/2010/167942

© Chang-Tsun Li. 2010

**Received: **6 June 2010

**Accepted: **10 September 2010

**Published: **7 October 2010

## Abstract

An unsupervised multiresolution conditional random field (CRF) approach to texture segmentation problems is introduced. This approach involves local and long-range information in the CRF neighbourhood to determine the classes of image blocks. Like most Markov random field (MRF) approaches, the proposed method treats the image as an array of random variables and attempts to assign an optimal class label to each. While most MRFs involve only local information extracted from a small neighbourhood, our method also allows a few long-range blocks to be involved in the labelling process. This alleviates the problem of assigning different class labels to disjoint regions of the same texture and oversegmentation due to the lack of long-range interaction among the neighbouring and distant blocks. The proposed method requires no *a priori* knowledge of the number and types of regions/textures.

## Keywords

## 1. Introduction

- (i)
Firstly, textures comprise primitives/textons and tiny edges (intensity fluctuations), which can cause false responses in conventional edge detectors, while texture boundaries do not appear as conventional edges and therefore often go undetected by conventional edge detectors.

- (ii)
Secondly, another major difficulty in texture segmentation arises from the fact that texture is a regional property [19] rather than a pixel property. The information in a single pixel is only the intensity or colour. By contrast, we need to cover a reasonably large area of pixels in order to extract meaningful texture features. If we take a big area of a textured image into consideration, we get high confidence in the class of the texture. However, we lose the resolution in which the texture boundary may be if the area in question contains more than one class of textures. On the other hand, if we confine the analysis to a smaller area, the confidence in the position of texture boundaries increases at the expense of the certainty of texture class. This phenomenon, known as "class-boundary uncertainty" [1], suggests that information at multiple resolutions or scales has to be fused in some way for effective segmentation. This motivates the adoption of the various multiresolution approaches in many works [20–23].

- (iii)
Thirdly, to be practicable in a wide variety of applications, it is desirable that a segmentation algorithm should work without human supervision or intervention. To achieve this objective, the algorithm should not rely on the user to provide knowledge about the number of texture classes and the types of textures contained in the target image [1]. The latter means that it is preferable not to have a training phase before the algorithm is used. This requirement is sensible because if the training phase were needed, then when textures outside the training set appear in the images to be segmented, the algorithm would have difficulty identifying them.

- (iv)
Moreover, an intelligent algorithm should not only be able to segment images into regions, but also be capable of classifying disjoint regions of the same texture into the same class [15, 16]. For example, an algorithm capable of segmenting an aerial image with two disjoint urban areas from their woodland background and assigning the same class label to the two urban areas is certainly superior to an algorithm only capable of segmenting the image into three regions.

The motivation of this work is to develop an unsupervised algorithm, which is capable of meeting these requirements.

## 2. Related Work

*Markovianity*) [29].

*Markovianity*allows a global optimization problem to be solved locally and this is confirmed by the results reported in our recent work [1]. The approach presented in [1] is a typical combination of MRFs and simulated annealing, known as the Gibbs sampler [21]. The idea is to divide an image into a number of blocks and treat each block as a random variable to be assigned a class label depending on its own features, its neighbours' features, and class labels. When a block is visited, the probabilities of the labels assigned to its neighbours are calculated according to a cost function. The optimal label is assigned the highest probability and the less promising labels get lower probabilities. Simulated annealing [21] is then applied so as to assign one of the labels to block . When the labelling process converges, the labelling result is propagated down to the next resolution level for finer segmentation. This process is repeated until a nominal bottom level is reached. Since this is a multiresolution approach, the neighbourhood includes the 4 neighbours at the same level and the parent block at one level above (except the nominal top level). By using the idea of

*SOIL*(set of indispensable labels) reported in [30] and exploiting

*local feature disparities*, this work requires no

*a priori*global knowledge about the number of texture classes and the types of textures contained in the images. However, the work has the following limitations.

- (i)
Due to the lack of long range interaction with blocks outside the local neighbourhood and the fact that the algorithm starts with a random class label configuration, disjoint regions of the same texture are assigned different class labels. An example is shown in Figure 1(a), wherein the two disjoint regions on the right and left sides containing the same texture have been misclassified into two different classes—class 5 and 7, as shown in Figure 1(b).

- (ii)
Again, for the same reason, large or elongated regions may be oversegmented. For example, the region in the middle of Figure 1(a) has been oversegmented into two classes with labels 2 and 9, as shown in Figure 1(b). This can be explained as follows. Given Figure 1(b) as the current label configuration, with a 4-neighbourhood system, despite the fact that sites (5, 5) and (6, 5) belong to the same region, when site (5, 5) is visited, three of its neighbours within the 4-neighbourhood system are labelled 2 while only one is labelled 9. Therefore, the probability of assigning 2 to site (5, 5) is high, while the probability of assigning label 9 will decrease as relaxation proceeds. Similarly, site (6, 5) is next to three neighbours with label 9 and only one with label 2. Therefore the probability that it will be assigned label 9 is far greater than getting assigned label 2. This kind of local "sibling competition" can lead to premature convergence at local minima in the energy landscape and consequently give rise to oversegmentation. This type of oversegmentations is more likely to occur in large or elongated regions, where different class labels may grow into subregions from different locations of the regions. When these subregions meet, sibling competition starts and spurious boundaries may form.

- (iii)
The energy function adopted in [1] was based on the assumption that the distributions of both intraclass and interclass feature disparities of useful features are approximately Gaussian. Although this is a reasonable assumption, the accuracy of the prior model may have significant influence on the efficacy of the segmentation. The potential function, which is encoded in the energy function, is also complicated and needs to be recalculated iteratively in an iterated conditional mode (ICM) sense [29].

Theoretically, the first two limitations, due to the lack of long-range interaction, could be alleviated if the annealing schedule starts at higher temperature. However, this gain can only be obtained at the expense of convergence rate. Another possibility is to involve a larger neighbourhood to accommodate more interaction among blocks in the decision process. However, for Markov random field approaches, adopting a larger neighbourhood incurs higher computational cost. Moreover, the range of the interaction is still bounded by the neighbourhood. In its extreme case when the whole image is involved, the local characteristic of Markov random fields (*Markovianity*) is completely lost. Therefore, reaching out from the local neighbourhood in some way is more desirable than expanding the neighbourhoods.

- (i)
resolving local sibling competition in order to tackle the problem of oversegmentation,

- (ii)
labelling disjoint regions of the same texture consistently by involving long-range interaction among blocks,

- (iii)
reducing the computational cost by simplifying the energy function and facilitating information fusion through long range interaction among blocks.

## 3. Multiresolution Markov Random FieldTexture Segmentation Algorithm

To avoid confusing the reader with details, we first present the whole picture of the proposed algorithm as follows.

*Multiresolution Conditional Random Field Texture Segmentation Algorithm*

- (1)
Build a multiresolution pyramid conforming to a quad-tree structure.

- (2)
Extract texture features from each image block.

- (3)
Assign a random label to each block at the top level to form an initial label configuration.

- (4)For each resolution level before the nominal bottom level is reached
- (4.1)While not converged, for each block
- (4.1.1)
form a voting pool,

- (4.1.2)
calculate the energy functions of the labels of the members in the voting pool,

- (4.1.3)
calculate the probabilities of the labels based on their energy functions,

- (4.1.4)
assign one of the labels according the probability,

- (4.1.1)
- (4.2)
propagate results to the next resolution level.

- (4.1)

In the first step of the proposed algorithm, a multiresolution framework is adopted to represent the image data at multiple scales in order to alleviate the canonical problem of class-boundary uncertainty [1, 2]. Note that in order not to complicate the symbols in the presentation of the algorithm later, we will assume that the original images are square. However, the two dimensions of the original image need not to be equal. We create an image pyramid conforming to a quad-tree structure with
resolution levels based on the original image of
pixels. Each level
consists of an array of
*sites/blocks* of
pixels by sliding a window of
pixels over the original image in
-pixel wide steps in both horizontal and vertical directions. That is to say that the sampling windows are 50% overlapped. At the higher levels, the size of the block is larger, which gives higher resolution of texture classes but lower resolution of texture boundaries. While at lower levels, the blocks are smaller, and as a result, higher boundary resolution is gained at the expense of class resolution. At the top level (i.e.,
), there is only one block, which covers the whole image; therefore, the algorithm must start at a nominal top level
in order to carry out the segmentation task. At the bottom level (i.e.,
), each block covers only one pixel, which carries no texture features. Therefore, the algorithm should stop at a nominal bottom level with
.

Usually a complete segmentation process includes a *feature extraction* stage, followed by a*labelling* stage. At the first step of the proposed algorithm, a set of texture features is extracted from each site
as the observed data
. In practice, features suitable for the segmentation task are image and application dependent. Since the main purpose of this work is to propose, not a feature extraction method, but a generic labelling algorithm, we include only the mean and variance of the intensity in the feature set. However, the reader is reminded that appropriate features other than mean intensity and variance can be incorporated with the proposed labelling algorithm since the feature extraction and labelling processes are separate modules.

where and represent the observed data and class labels of all the sites in , respectively. That is, and . and are the conditional probability distribution and prior probability distribution, respectively. Usually, the number of texture classes in the image in question is unknown to the user. To allow the algorithm to work without the user specifying the number of texture classes, at the nominal top level, each block is assigned a label randomly picked from the integer range , where is the number of blocks at the nominal top level. That is, we allow the algorithm to start with a completely random label configuration. This is virtually equivalent to adopting an "infinite" label space.

*parent*block at level and the

*4 neighbours*( , , , and ) at level , three other blocks, , , and are also included in the voting pool . and are selected at random, and is the block whose feature set is most different from that of . So, the voting pool can be expressed as

*most different*block, , is the one identified from the randomly picked members during the whole labelling history. That is, in any iteration, if a voting block selected at random (i.e., or ) is more different from than the current

*most different*block, the

*most different*block is replaced by that voting block. Note that at the top resolution level, there is no parent block in the voting pool. In addition to playing the same role as and of introducing long-range interaction, the

*most different*block plays an extra role of informing the algorithm that the label associated with the

*most different*block is to be avoided. Because of the contextual property of texture, there is no need to keep track of the most similar block for each site , given the fact that the 4 neighbours have already been included in the voting pool.

To allow the algorithm to segment images with arbitrary numbers of texture regions without the user specifying the number, when each block is being visited, only the class labels currently assigned to the members of its voting pool are taken as candidates. For example, when the grey site at coordinates
in Figure 2 is being visited, the candidate labels
are just
. The advantages of randomising the initial label configuration at Step (3) with a label space of
and using the proposed voting pool are now clear. First,*globally* the number of labels allowed is equal to the number of blocks
. That means we ensure diversity by sampling the label configuration of the whole image from a virtually infinite label space. Secondly, the computational cost is kept low because locally the optimal label of each individual block is sampled from a small space with its cardinality ≤8 because there are at most 8 members in the voting pool
. The cardinality of
will decrease in due course as the homogeneous regions emerge. In our previous example, the cardinality is only 3. Although starting from a random initial label configuration, a unique optimal label for each texture class can be quickly identified through the interaction with the members of the voting pool. It should be noted that, at lower resolution levels, a random label that is not assigned to any site can be included as a candidate in order to encourage small regions to emerge. However, texture class certainty is lower at lower resolution levels and texture may comprise recursive patterns/textons, which can be smaller than the image blocks. Including such an unused label increases the possibility of detecting small spurious regions that are actually part of a larger and genuine texture region. Therefore, whether to include such a label as candidate is application dependent. In this work, we choose not to include it.

*partition function*, which is to serve the purpose of normalisation. By properly reformulating the two energy functions in (3), as described in [21], we obtain a new model as

*threshold*dividing the space of

*feature disparities*into subspaces of

*intra-*and

*interclass feature disparities*. To estimate for each resolution level, 300 pairs of blocks are picked at random from the same resolution level and their

*feature disparities*are calculated. We allow a block to be paired with more than one partner. Since some pairs might belong to the same texture class and some might not, we can have two types of feature disparity:

*intraclass*with relatively smaller value and

*interclass*with relatively greater value

*.*By taking the smallest and the greatest feature disparity as the initial centroids of

*intraclass*and

*interclass,*respectively, a trivial 2-mean clustering method is employed to partition the space of the 300 feature disparities into two clusters. Then, of (5) can be calculated according to

*intraclass*and is the smallest value in

*interclass*. The feasibility of the energy function and pairwise potential is discussed as follows.

- (i)
The first merit is that , , , and are all adaptive to the content of the image. Since the complexity of (5) is low, the computational cost of calculating the energy function defined in (4) is significantly lower than the one used in [1].

- (ii)
which neither takes into account the variation of features across homogeneous regions nor allows flexible weighting according to the discriminatory power of the local features. In this work, the value of potential is not bipolar, but linearly dependent on the difference between and

*.*This characteristic gives blocks of different voting weight. A block with a feature disparity closer to the threshold is less discriminative and therefore should have less say in determining its neighbours' classes. - (iii)Assuming the texture features employed are adequately discriminative, then
- (a)the first case of (7) indicates that if the
*same*class label is to be assigned to site , then,- (1)
if block and

*do*belong to the same class (i.e., is significantly*smaller*than ), a*small*value, , for potential is given. This is reasonable because this is a correct labelling and should be encouraged with a lower cost, - (2)
if block and do

*not*belong to the same class (i.e., is significantly*greater*than ), a*large*value, , for potential will be given as a penalty to discourage this incorrect labelling;

- (1)
- (b)

- (a)

*annealing schedule*) in iteration at resolution level

where in (13) is a constant which determines the initial temperature and the convergence rate. The higher the value of is, the higher the starting pseudotemperature is and the less likely the labelling process will get trapped at local minima, but the slower the convergence rate is. Because the labelling at low resolution levels is conditioned on the segmentation result of the higher resolution levels, wherein the number of blocks is smaller, is varied with , with greater values at low (high resolution level). Usually, the value of has to be chosen empirically according to the range of the energy function. Note that in the special case when is fixed at 0 regardless and , the stochastic model of (12) is equivalent to the deterministic model as expressed in (11).

The search for the optimal label
is the process of maximizing the*a posteriori* probability of (12). In the context of simulated annealing and stochastic relaxation, the label of each voting member has a chance of being chosen proportional to its probability calculated according to (12).

The algorithm can certainly involve only one or more than two random sites. There is no theoretical backing for determining the number of random sites as this is application dependent. For example, when segmenting images with disjoint small regions of the same texture, the inclusion of more random sites increases the chances for the disjoint small regions to establish contact. However, this is achieved at the expense of computation complexity because there will be more sites and more labels to deal with. On the other hand, when segmenting images without disjoint small regions of the same class, the inclusion of less random sites is more desirable. Nonetheless, the word "small" in this context is not well defined, and maybe there is no way of defining it. Moreover, this question about how many random sites are enough is also directly related to the setting of the starting temperature for simulated annealing. With the number of random sites fixed, when the starting temperature is set high, the algorithm will iterate longer, allowing more sites to be interacted with. So, like the starting temperature, the number of random sites has to be determined in an ad hoc manner. In this work, given the small number of the standing members (i.e., 4 first-order neighbours), including half as many random sites is a reasonable guess.

As stated in Step (4.2), upon convergence, the algorithm propagates the segmentation result of the current level down to the next level for refinement. The propagated result is the class label configuration, which will be used as the initial configuration at the new level.

## 4. Experiments and Performance Analyses

*average iterations*, oversegmentation rate, and rate of labelling disjoint regions of the same texture when the image in Figure 3 is divided into 64 blocks. Table 2 lists the same measurements of the two algorithms' performance when the same image is divided into 256 blocks. The

*average iterations*in Table 1 indicate that the proposed algorithm is significantly more efficient in terms of computational cost than the previous algorithm even without taking the other two metrics of performance into account. They also indicate that the proposed algorithm's computational cost increases significantly more slowly than that of the previous algorithm. Moreover, we can see that at , only 1% of the applications of the proposed algorithm result in oversegmentation and labelling of the two dark regions differently, due to the low initial annealing temperature, while no oversegmentation and mislabelling of the dark regions occur when is only 50, with an average computational cost of only 6.2 iterations. On the contrary, the previous algorithm could only yield no oversegmentation when , with the average computational cost as high as 131.3 iterations. The rates of labelling the two dark regions inconsistently associated with the proposed and previous algorithms also reveal another superior aspect of the proposed algorithm. As we mentioned earlier, the larger the textured regions are, the higher the probability that the previous algorithm will encounter the problem of oversegmentation.

Performance in terms of computational cost measured by average iterations, oversegmentation rate, and rate of differently labelling disjoint regions of the same texture (%) with various values of when the image in Figure 3 is divided into 64 blocks.

Average iterations | Oversegmentation rate (%) | Rate of labelling disjoint regions differently (%) | ||||
---|---|---|---|---|---|---|

Proposed alg. | Previous alg. | Proposed alg. | Previous alg. | Proposed alg. | Previous alg. | |

10 | 5.9 | 6.9 | 1 | 100 | 1 | 100 |

50 | 6.2 | 10.6 | 0 | 100 | 0 | 100 |

100 | 7.2 | 23.8 | 0 | 91 | 0 | 100 |

200 | 9.4 | 32.3 | 0 | 22 | 0 | 99 |

300 | 12.7 | 46.1 | 0 | 3 | 0 | 96 |

400 | 19.28 | 72.1 | 0 | 2 | 0 | 86 |

500 | 28.4 | 131.3 | 0 | 0 | 0 | 55 |

600 | 38.8 | 218.2 | 0 | 0 | 0 | 51 |

Performance in terms of computational cost measured by average iterations, oversegmentation rate, and rate of differently labelling disjoint regions of the same texture (%) with various values of when the image in Figure 3 is divided into 256 blocks.

Average iterations | Oversegmentation rate (%) | Rate of labelling disjoint regions differently (%) | ||||
---|---|---|---|---|---|---|

Proposed alg. | Previous alg. | Proposed alg. | Previous alg. | Proposed alg. | Previous alg. | |

10 | 8.2 | 13.6 | 2 | 100 | 1 | 100 |

50 | 10.1 | 23.3 | 0 | 100 | 0 | 100 |

100 | 10.7 | 54.5 | 0 | 100 | 0 | 100 |

200 | 12.1 | 98.6 | 0 | 100 | 0 | 100 |

300 | 18.2 | 166.9 | 0 | 91 | 0 | 100 |

400 | 27.0 | 190.7 | 0 | 34 | 0 | 100 |

500 | 41.1 | 249.5 | 0 | 11 | 0 | 100 |

600 | 51.4 | 282.6 | 0 | 0 | 0 | 100 |

We apply the two algorithms to the same image when it is divided into 256 blocks and list the results in Table 2. With 4 times more blocks, the proposed algorithm demonstrates the same level of performance, in terms of oversegmentation rate and rate of labelling the two dark regions differently, with a slight increase of computational cost compared to Table 1. For the previous algorithm, the computational cost increases more significantly (with an average computational cost of 282.6 iterations) and the initial temperature has to be set to 600 in order to avoid oversegmentation. On the other hand, the proposed algorithm is able to achieve this level of performance with and an average cost of 10.1 iterations. Nevertheless, in all cases, as demonstrated in Table 2, the previous algorithm completely fails to label the two dark regions consistently. These two experiments demonstrate that the voting pool adopted in the proposed algorithm can effectively and efficiently facilitate both local and global interactions among the blocks to avoid the key limitations of the previous algorithm.

*Image I*,

*II*, and

*III*, at different resolution levels. This time the previous algorithm uses the energy function defined in [1] while the proposed algorithm uses the one adopted in this work. The value of in (13) is set to 300 at the nominal top resolution level and reduced by 50 after proceeding to the next resolution level. The results are shown in (a) to (d) of Figures 4 to 6, respectively. The results show the increase in segmentation accuracy and the algorithm's ability to alleviate the

*class-boundary uncertainty*as the algorithm descends the multiresolution structure. The comparisons of performance in terms of segmentation error rate (%) and computational cost measured by

*iterations per pixel*(# i/p) are listed in Tables 3 to 5, respectively. Note that the total computational cost has to be measured by adding up the costs incurred at all resolution levels. Because the block sizes at various resolution level are not the same, instead of using

*iterations*, a more suitable metric would be

*iterations per pixel*(# i/p), which is equal to 4 × iterations/block size. Note also that the "4" in the expression is to reflect the fact that, due the sampling windows' 50% overlap, each block is only associated with a quarter of the area in the centre of the block. For both algorithms in all the three cases, the segmentation error rates at the bottom resolution level are quite close. However, the significant performance gap in terms of computational cost (# i/p) again reveals the merit of the proposed MRF algorithm. According to [1], the previous algorithm came with a boundary process. Although the boundary process made the boundaries look smoother, it did not provide significant improvement in terms of segmentation accuracy. Therefore, we do not include the boundary process in this work. Note, in all the experiments, no

*a priori*knowledge about the number and types of textures contained in the images in question is provided to the algorithm. Yet, the algorithm is able to segment the images correctly. These results indicate that the proposed algorithm requires no supervision by the user.

Comparisons of performance in terms of segmentation error rates and computational cost measured by number of iteration per pixels (# i/p) for Image *I* of Figure 4.

Comparisons of performance in terms of segmentation error rates and computational cost measured by number of iteration per pixels (# i/p) for Image *II* of Figure 5.

Comparisons of performance in terms of segmentation error rates and computational cost measured by number of iteration per pixels (# i/p) for Image *III* of Figure 6.

It is also worth noting that, in Figure 4, the elements/textons of the recursive texture pattern are greater than 16 pixels in at least one dimension, which is greater than the block dimension at the resolution level 5. Yet the proposed algorithm is capable of detecting the boundaries accurately even at the bottom level, where the block size is only pixels. This is due to the use of multiresolution approach: we divide the images into large blocks at the top level to collect reliable features (at the expense of boundary certainty) and use the coarse segmentation result to condition the segmentation at lower level.

*Orchid*) as shown in Figure 7(a). Because the characteristics of this image are different from those in Figures 4 to 6, the features used to describe each block are the mean values of R, G, and B components. The size of the image is pixels. We start the program at resolution level 3 (i.e., , but only demonstrate the segmentation results at level 4 to 6 in Figures 7(b)–7(d), respectively. This experiment conforms to what we mentioned in Section 3 that the proposed segmentation algorithm can be incorporated with any discriminative features. The key challenges posed by this image are the following. (1) The feature difference between the leaves and the background is insignificant when compared to the difference between the flower and either the leaves or the background. We divide Figure 7(a) into 2400 blocks of pixels and display the colour (feature) distribution of those blocks in the (R, G, B) space as demonstrated in Figure 8. We can see that an ineffective segmentation algorithm could classify the blocks belonging to the leaves and background into the same group. (2) The left part of the flower shape is quite rugged. (3) There are two narrow orange "peninsulas", one pointing upwards and the other downwards. It can be seen from Figure 7(b) that, because of class-boundary uncertainty, many spurious small regions form along the boundaries between real regions, the rugged boundaries and the two peninsulas in particular. The sizes of the small regions can be reduced at finer resolution levels, but the number of even smaller regions will also increase at finer resolution levels. These small regions are inherent to the class-boundary uncertainty. Their areas are less significant at finer resolution levels, and segmentation inaccuracy at higher resolution levels can be reduced at finer levels; therefore, it is computationally undesirable to repeat the same algorithm at finer levels with the same sophistication. To keep computational cost low without sacrificing segmentation accuracy, at the two finest resolution levels, we apply the algorithm with stochastic relaxation (12) to all the blocks for three iterations and then only to blocks with at least one 4-neighbour system carrying a different class label (i.e., the blocks along the boundaries) in a deterministic relaxation manner (11).

## 5. Conclusions

In this work, we have addressed the need for tackling the *class boundary uncertainty* issue in the context of texture segmentation and the importance of facilitating long-range interaction among image blocks/sites in order to avoid oversegmentation and mislabelling of disjoint regions with the same texture. In response to these, we have proposed an unsupervised multiresolution conditional random field approach, which utilises a voting pool consisting of local neighbours and distant sites. The proposed algorithm is capable of not only alleviating the problems of oversegmentation and mislabelling disjoint regions with the same texture, but also carrying out the segmentation without *a priori* knowledge about the number and types of textures contained in the input image. Moreover, with sufficient information gathered through the incorporation of long-range interactions, the convergence rate of the proposed algorithm is faster. To the author's knowledge, no previous attempt has been made to include long-range block in the neighbourhood of MRF models.

## Authors’ Affiliations

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