An Image Change Detection Algorithm Based on Markov Random Field Models

T. Kasetkasem and P.K. Varshney

Department of Electrical Engineering and Computer Science

Syracuse University

Syracuse, NY 13244

Email: tkasetka@syr.edu and varshney@syr.edu

Abstract – This paper addresses the problem of image change detection based on Markov random field

(MRF) models. MRF has long been recognized as an accurate model to describe a variety of image

characteristics. Here, we use the MRF to model both noiseless images obtained from the actual scene and

change images whose sites indicate changes between a pair of observed images. The optimum image change

detection algorithm under the maximum a posteriori (MAP) criterion is developed under this model.

Examples are presented for illustration and performance evaluation.

I. INTRODUCTION

The ability to detect changes that quantify temporal effects using multitemporal imagery provides a fundamental

image analysis tool in many diverse applications. Due to the large amount of available data and extensive

computational requirements, there is a need for change detection algorithms that automatically compare two

images taken from the same area at different times, and determine the locations of changes. Usually, in the

comparison process [1-4], differences between two corresponding pixels belonging to the same location for an

image pair are determined based on some quantitative measure. Then, a change is labeled, if this difference

measure exceeds a predefined threshold, and no change is labeled, otherwise.

Most of the comparison techniques described in [1] only consider information contained within a pixel even

though intensity levels of neighboring pixels of images are known to have significant correlation. Also, changes

are more likely to occur in connected regions rather than at disjoint points. By using these facts, a more accurate

change detection algorithm can be developed. To accomplish this, a Markov random field (MRF) model for

images is employed in this paper so that statistical correlation of intensity levels among neighboring pixels can be

exploited. MRF has long been recognized as an accurate model to describe a variety of image characteristics such

as texture. Under this model [5-8], the configuration (intensity level) of a site (pixel) is assumed to be statistically

independent of configurations of all remaining sites excluding itself and its neighboring sites when configurations

of its neighboring sites are given. In other words, the configuration of a pixel given the configurations of the rest

of the image is the same as the configuration of a pixel given the configurations of its neighboring pixels.

Furthermore, the MRF is known to be equivalent to the Gibbs field [6] whose probability density function (PDF)

is given by

2

(1)

where S is a set of sites contained within an image, x is a vector of configurations (intensity levels) over S, is

a normalizing constant, and VC is a Gibbs potential function. A Gibbs potential function is a function of

configurations and cliques. A clique [5-6], denoted by C, is defined as a set of sites whose elements are mutual

neighbors.

Studies reported in [9-10] have tried to employ the MRF model for image change detection (ICD). In [9], one

image is subtracted from the other, pixel by pixel, and two thresholds (one low and one high) are selected. If the

difference intensity level of a pixel is lower than the low threshold, then this pixel is put in the absolute unchanged

class. If the intensity level is greater than the high threshold, the corresponding pixel is put in the absolute

changed class. The remaining pixels whose difference intensity levels are between these two thresholds are

subjected to further processing where the spatial-contextual information based on the MRF model is considered. A

similar approach can be found in [10]. Again, this algorithm can be divided into two parts. In the first part, a

pixel-based algorithm [1] determines an initial change image that is further refined based on the MRF model in

the second part. Some information is lost while obtaining the initial change image since the observed data is

projected into a binary image whose intensity levels represent change or no change. We observe that studies in [9-

10] do not fully utilize all the information contained in images, and moreover, the preservation of MRF properties

is not guaranteed. In [11], the effect of image transformations on images that can be modeled by MRFs is studied.

It has been shown that MRF properties may not hold after many transformations such as resizing of an image and

subtraction of one image from another. For some specific transformations, MRF properties are preserved, but a

new set of potential functions must be obtained. Since a difference image can be looked upon as a transformation,

MRF modeling of a difference image in [9] and initial change image in [10] may not be valid. This provides the

motivation for the development of an ICD algorithm that uses additional information available from the image

and preserves MRF properties. Here, we develop an ICD algorithm that consists of only one part. The observed

images modeled as MRFs are directly processed by the MAP detector which searches for the global optimum.

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The detector based on the MAP criterion chooses the most likely change image among all possible change

images given the observed images. The resulting probability of error is minimum among all other detectors [12-

13]. The structure of the MAP detector is based on statistical models. Therefore, the accuracy of the statistical

models for the given images as well as for the change image is crucial. In this paper, we assume that the given

images are obtained by the summation of noiseless images (NIM) and image noises. Both the NIMs and the

change image are assumed to have MRF properties. Furthermore, we assume that configurations of the NIMs are

equal for unchanged sites. In addition, the configurations of changed sites from one NIM are independent of

configurations of changed sites from the other NIM when the configurations of unchanged sites are given. Then,

the a posteriori probability is determined based on the above assumptions, and the MAP criterion is used to select

the optimum change image.

Due to the complexity of the a posteriori probability computation, the solution of the MAP detection problem

cannot be obtained directly. As a result, a stochastic search method such as the simulated annealing (SA)

algorithm [7] is employed. Here, the SA algorithm generates a random sequence of change images in which a new

configuration depends only on the previous change image and observed images by using the Gibbs sampling

procedure [5-8]. The randomness of a new change image gradually decreases as the number of iterations

increases. Eventually, this sequence of change images converges to the solution of the MAP detector.

This paper is organized as follows. In the next section, the problem is formulated and the above assumptions

are restated in a more formal manner. The optimum image change detection algorithm under the MAP criterion is

developed in Section 3. Section 4 provides some examples for performance evaluation and illustration. A

summary is presented in Section 5.

II. PROBLEM STATEMENT

Let S be a set of sites s, and be the phase space. Furthermore, we write

as a sequence of image (configuration) vectors taken at time ; where

. Note that, in the problem of interest here, N = 2. We assume that Xi(S) satisfies MRF

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properties with a Gibbs potential VC(xi). Note that Xi(S) are noiseless image models (NIM) of the scene. These

will be used extensively in our discussion. Due to noise, we cannot observe Xi directly. Instead, we observe noisy

images (NI) which are given by

(2)

where is the phase space corresponding to noisy observed images and Wi(S) is a vector of

additive Gaussian noises with mean zero and covariance matrix . Here, I is an identity matrix of size

, where .

Obviously, there are a total of possible change images (CIs) that may occur between any pair of

NIMs (including the no-change event). Let , correspond to the kth CI which is a binary

image whose intensity levels are either 0 or 1. Here, we use the notation to indicate a change at site a

in the kth CI. When , we have to indicate no change. We assume that all CIs satisfy

MRF properties with the Gibbs potential UC(Hk). Then, we can write the marginal PDFs for Xi and Hk as

,

(3)

and

, (4)

where and , respectively. Since the elements

of Wi(S) are i.i.d. additive Gaussian noises, the conditional probability density of Yi(S) given Xi(S) is given by

(5)

5

Given Hk, we can partition a set S into two subsets and such that and

, where and contain all changed and unchanged sites, respectively. We further

assume that the configurations of changed sites between a pair of NIMs are independent given the configurations

of unchanged sites, i.e.,

(6)

where and are the configurations of changed sites and unchanged sites of Xi, respectively.

Here, we have made the simplifying assumption that the pixel intensities of the changed sites in a pair of images

are statistically independent given the intensities of the unchanged sites to keep the analysis tractable. A

justification for this assumption is that changes are not predictable based on the current knowledge. Note that for

unchanged sites, configurations of the same site in a pair of images must be equal. This equal configuration value

is denoted by . For notational convenience, let us denote and by and

, respectively. The joint probability density function (PDF) of the pair is given by

(7)

Since Xi and Xj correspond to the same scene, we assume that they are statistically identical. By summing over all

possible configurations of changed sites, the joint PDF can be expressed as

(8)

Substituting the Gibbs potential VC and using the properties of conditional probability, (8) can be written as

6

(9)

By using the notation to denote the sum over the sets C that contain site s, the term can be

decomposed as

(10)

where the boundaries of the changed region are denoted by contained within the changed region.

Substituting (10) into (9), we obtain

,

(11)

where

, (12)

and

.

(13)

are the normalizing constant and the joint image energy function associated with Hk, respectively. Based on these

assumptions, we design the optimum detector in the next section. Here, we have considered the case of discrete-

valued MRFs, the derivation in the case of continuous-valued fields (used in examples) is analogous.

III. OPTIMUM CHANGE DETECTION ALGORITHM

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We formulate the change detection problem as an M-ary hypothesis testing problem where each hypothesis

corresponds to a different change image or a no change image. The maximum a posteriori (MAP) criterion [12-

13] is used for detecting changed sites in our work. This criterion is expressed as

.

(14)

From Bayes’ rule, (14) can be rewritten as

.

Since is independent of Hk and the two noises are independent of each other, the above

equation reduces to

(15)

where and denote , and , respectively.

Substituting (4), (5) and (11) into (15) and taking a constant term out, we obtain

,

(16)

where

, (17)

and

8

.

(18)

From (16), we can observe that maximization of (15) is equivalent to minimization of . Therefore,

our optimal ICD algorithm can be expressed as

.

(19)

In practice, the solution of (19) cannot be obtained directly due to the large number of possibilities of change

images (e.g., 16 for a 22 image and 24096 for a 6464 image). Moreover, gradient search methods are not

feasible because of the extreme nonconvexity of the a posteriori probability in general. To cope with these

problems, we employ the simulated annealing (SA) [5-7] algorithm which is a part of Monte Carlo procedures to

search for solutions of (19). The SA algorithm generates a sequence of configurations (or CIs) that eventually

converges to solutions of (19). This algorithm can be described in terms of the following steps:

1) Select the initial image or configuration randomly or based on some prior knowledge. Estimate initial

parameters and set the initial temperature.

2) At each step, obtain a new configuration (image) from the previous configuration (image) based on a Gibbs

sampling procedure [5].

3) Reduce the temperature with a predetermined schedule and go to Step 2 until convergence.

Here, the term temperature is used to control the randomness of the configuration generator. High temperature

indicates high randomness and low temperature indicates less randomness. Eventually, the resulting change image

will converge to one of the possible solutions of the MAP detection problem given in (19) with equal probability,

as randomness is slowly removed. However, if randomness is removed too quickly the resulting change image

may get stuck in one of the local maxima of the a posteriori probability. To guarantee convergence, Geman and

Geman [7] suggested that the temperature must decrease at a rate slower than where M is the number of

sites and is the absolute maximum change in Gibbs energy when the configuration of a single change site is

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changed. In the next section, some examples are provided for illustration. Both a simulated dataset and an actual

dataset are used. The pseudo-code for the simulated annealing algorithm is provided in the Appendix.

IV. EXPERIMENTAL RESULTS

In this section, we consider the specific problem of image change detection based on the MAP detector given in

(19). Consider Figure 1 where we show the 8-neighborhood of an image pixel and ten possible cliques. In the

illustrative examples considered here, we consider only five clique types, C1, C2, C3, C4 ,C5 associated with the

singleton, vertical pairs, horizontal pairs, left-diagonal pairs, and right-diagonal pairs, respectively. Furthermore,

we assume that image potential functions are translation invariant. The Gibbs energy function of images is

assumed to be given by

, (20)

where is the NIM parameter vector, and J is assumed to be

which is the NIM potential vector associated with clique types, C1,…,C5, respectively. We observe that the NIM

potential vector is in a quadratic form. The quadratic assumption is widely used to model images in numerous

problems [12], and is called the Gaussian MRF (GMRF) model. GMRF models are suitable for describing smooth

images with a large number of intensity levels. Furthermore, under this Gaussian MRF model, Equation (19) can

be solved more easily due to the fact that the summation over configuration space in (17) changes to infinite

integration in product space. Hence, the analytical solution for (19) can be derived.

Using the GMRF model [14-15], we can rewrite (20) as

, (21)

where the element of the MM matrix is given by

10

.

(22)

Similarly, we define a Gibbs energy function of a CI over the clique system as

, (23)

where is the CI parameter vector, and

(24)

is a CI potential vector associated with clique types as mentioned above and I(a,b) = -1 if a = b, and I(a,b) = 1,

otherwise. The optimum detector has been derived in [18].

As mentioned in the previous section, the direct search for the solution of (19) is not feasible. Therefore,

the simulated annealing algorithm is used to find the solution. Here, the Gibbs energy function is calculated based

on the observed images and the old/new change image. In order to obtain the result in a reasonable time, NIMs

can be divided into subimages of much smaller size (77 in our examples) due to the intensive computation

required for the inversion of large matrices (4096 by 4096 for a 64 64 image). The suboptimum approach of

considering one subimage at a time sacrifices optimality for computational efficiency. However, the statistical

correlation among sites is concentrated in regions that are only few sites apart. Therefore, our approximation is

reasonable and yields satisfactory results in the examples.

Our optimal algorithm involves matrix inversion and multiplication, both of which have the

computational complexity for an image of size n n. During each iteration, Lk given in (24) is computed

at least 2n2 times. As a result, the total complexity of our optimal algorithm is . However, when we

employ the suboptimum algorithm, the number of operations for each subimage is fixed. Consequently, the

overall complexity reduces to .

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Example 1:

Two noiseless simulated images of size 128 128 to be considered are displayed in Figures 2 (a)-(b), and their

corresponding CI is shown in Figure 3 where dark and bright regions denote change and no change, respectively.

We observe that changes occur in the square region from pixel (60,60) to (120,120). To test our proposed ICD

algorithm, these simulated images are disturbed with an additive Gaussian noise with zero mean and unit variance.

An SA algorithm, with initial temperature T0 = 2, is employed for optimization. For the difference image, the

average image intensity power to noise power is 3.4 dB. For both noiseless images, the average signal power is

about 3.9 dB. Next, our proposed ICD algorithm is employed, and the results are shown in Figures 4 (a)-(d) after

0, 28, 70 and 140 sweeps, respectively. At 0-sweep, an image differencing technique is used, and the resulting CI

is extremely poor. The situation improves as more sweeps are done. Significant improvement can be seen when

we compare Figures 4 (a) and (b), and Figures 4 (b) and (c). However, very little improvement is seen in Figure 4

(d) when we compare it with Figure 4 (c). In order for the performance to be quantified, we introduce three

performance measures, average detection, false alarm and error rates. The average detection rate is defined as the

total number of changed sites that are detected divided by the total number of changed sites while the average

false alarm rate is defined as the total number of unchanged sites that are declared changed sites divided by the

total number of unchanged sites. Similarly, the average error rate is given by the total number of misclassified

sites (changed sites are declared unchanged and vice versa.) divided by the total number of sites in the image. We

plot the average detection, false alarm and error rates in Figure 5 for a quantitative comparison. We observe rapid

improvement from 0 to 60 sweeps, and roughly constant performance after around 80 sweeps. The detection rate

increases from about 60 percent to more than 90 percent and the average error rate decreases from around 35

percent to less than 5 percent.

For further performance evaluation, we apply the Bruzzone and Prieto algorithm (BPA) in [9] to the same

image dataset. In their algorithm, an initial change image is determined by applying two thresholds (low and high)

to the difference image and they do not use all available information since only the different image is used. A

pixel is labeled as change if difference in intensity levels between two corresponding pixels from different

observed images exceeds the high threshold and as no change if it falls below the low threshold. The high and low

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thresholds are around the middle value, and the ratio of distance from the low and high thresholds to middle value

is denoted by . For faster computation, instead of dealing with CI parameters in a vector format as we did, the

authors used a simple scalar parameter denoted by .

In this example, we choose = 0.75 (based on some experimentation) and = 1 and the resulting

change image after five hundred iterations is shown in Figure 6 (a) while Figure 6 (b) displays the change image

from our algorithm for comparison purposes. Clearly, our image change detection algorithm significantly

outperforms the BPA. This is because their algorithm depends largely on an initial change image. In this particular

case, the SNRs of the observed images and the difference image are very low causing a large number of

misclassified pixels in the initial change image that are spread over both changed and unchanged regions which

severely affects the performance of their algorithm. Our algorithm, on the other hand, depends mainly on the

observed images rather than on an initial change image which makes it more robust in a low SNR environment.

Robustness of our algorithm will be discussed shortly. The number of floating point operations for each iteration

associated with the BPA in [9] is 5.1 105 when using Matlab while for our algorithm, it is 3.1 1010.

Furthermore, BPA has the complexity O(n2) because the number of instructions per iteration is independent of the

image size. At this point our algorithm consumes more computational resources to achieve performance gain. Our

algorithm is more suitable for off-line applications than real-time ones.

Furthermore, we investigate the robustness of our ICD algorithm with respect to image noise, image

misregistration, and modeling errors. To consider the effect of image noise, four different experiments are carried

out. Four additive Gaussian noise vectors with different values of variances, 0.01, 0.1, 1, and 10, are added to

images shown in Figures 2 (a) and (b), and are submitted to our ICD algorithm. The resulting change images are

shown in Figures 7 (a)-(d), respectively. The error rates for each value of noise variance are calculated and plotted

in Figure 8. As expected, higher value of noise variance yields higher error rate. In addition, we notice that for a

large value of noise variance, our algorithm declares the entire image as no change to minimize the probability of

error because noise overwhelms the information in NIMs. In other words, the dependence of intensity levels

between two highly noisy images becomes insignificant.

Next, we simulate the effect of misregistration on our ICD algorithm. In [10,17], the effect of

misregistration has been previously investigated. Here, we study its effect on the performance of our algorithm.

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Here, we fix one image, Figure 2(a) and move the other image, Figure 2(b), to the right (x direction) by 0, 0.25,

0.5 and 0.75 pixels and determine the corresponding CIs for each value of misregistration. Since we translate the

image by a non-integer number of pixels, interpolation is required. A linear interpolation method is used to

estimate the intensity of each pixel. The results are shown in Figures 9 (a)-(d) for noise variance equal to unity,

and error rate as a function of misregistration is plotted in Figure 10. The effect of misregistration is rather small

for misregistration under 0.5 pixel. However, larger misregistration affects the performance significantly.

Lastly, we examine the effect of modeling errors on our algorithm. We simulate modeling errors by

changing the PDF of noises from Gaussian to others. Here, we corrupt two NIMs with three additive noises with

identical power but different PDFs which are Gaussian, speckle, and uniform, respectively. The multiplicative

noise is generated using MATLAB as follows

I = I + nI, (25)

where I is the noiseless image data, and n is a uniform random variable. The resulting change images are shown in

Figures 11 (a)-(c), and the corresponding error rates are 0.061, 0.142 and 0.077, for Gaussian, speckle and uniform

noises, respectively. As expected, the best performance is achieved in the Gaussian case where there is no

modeling error.

Example 2

In this example, we apply our ICD algorithm to two actual remote-sensing images of San Francisco Bay taken on

April 6th, 1983 and April 11th 1988 shown in Figures 12 (a)-(b), respectively. These images represent the false

color composites made from MSS band 4, 5 and 6, and are given at

http://sfbay.wr.usgs.gov/access/change_detect/ Satellite_Images 1.html. For simplicity, our ICD

algorithm will determine change sites only in images of the same bands. In other words, we will separately find

CIs for red, green, and blue color spectra, respectively.

Since we do not have any prior knowledge about the Gibbs potential and the Gibbs parameters associated

with the Gibbs potential for both NIMs and CI, assumptions must be made and a parameter estimation method

must be employed. Since the intensity levels of NIMs can be anywhere in [0 255], we assume that the intensity

14

level of NIMs follows Gaussian MRF as in [14-15], i.e., the Gibbs potential is in a quadratic form. Furthermore,

we choose the maximum pseudo likelihood estimation (MPLE) method described in [16] to estimate the Gibbs

parameters. Here, we estimate unknown parameters after every ten complete updates of CI.

The results of changed sites for individual spectra are displayed in Figures 13 (a)-(b), (c)-(d), and (e)-(f),

respectively. Figures 13 (a), (c) and (e) are determined by our proposed ICD algorithm while (b), (d) and (f) result

from the image differencing algorithm described in [1]. By carefully comparing results within each color

spectrum, we conclude that our ICD algorithm gives results which are more connected. Further results on

performance evaluation with this real dataset and fusion of change images based on the decision fusion

methodology described in [12] are available in [18]

V. SUMMARY

This paper investigated the issue of image change detection based on Markov random field (MRF) models. These

models characterize the statistical correlation of intensity levels among neighboring pixels more accurately than

pixel-based models We have developed a new image change detection algorithm based on an MRF model that

employs the MAP criterion. The algorithm involves the search for an optimum for which the simulated annealing

algorithm is used. By means of two examples, we have shown the superior performance of our algorithm. This is

due to the use of contextual information as well as the computation of the true MAP solution. The effect of

uncertainties on the performance of our algorithm was also investigated. Noise, misregistration and modeling

errors were considered to be the sources of uncertainty. Our algorithm was found to be quite robust to various

types of uncertainties. We realize that the independence assumption of the configurations of changed sites given

the configurations of unchanged sites may not be suitable for all cases. A more accurate model may result in better

performance, and may be considered in the future.

15

Appendix

The pseudo code for the simulated annealing algorithm emploed here is provided below.

For l from 1 to Max_Iteration

For k from 1 to Max_kFor m from 1 to Max_m

E0 = Energy obtained from (A10) when H(k,m) is unchanged.E1 = Energy obtained from (A10) when H(k,m) is changed.

D = uniform random variable from 0 to 1If

H(k,m) = 0Else

H(k,m) = 1End_IF

End_ForEnd_For

End_For

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1999.

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Geoscience and Remote Sensing, Vol. 31, No. 4, Pp. 896-906, July 1993.

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[6] P. Bremaud, Markov Chains Gibbs Field, Monte Carlo Simulation, and Queues, Springer-Verlag, New York,

1999.

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Images,” IEEE Trans. Pattern Analysis and Machine Intelligence, Vol. PAMI-6, No. 6, pp. 721-741, Nov,

1984.

[8] J. Begas, “On the Statistical Analysis of Dirty Pictures,” J. R. Statist. Soc. B 48, No. 3 pp. 259-302, 1986.

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Detection,” IEEE Trans. Geoscience and Remote Sensing, Vol. 38, No. 3, Pp. 1171-82, May 2000.

[10] R. Wiemker, “ An iterative Spectral-Spatial Bayesian Labeling Approach for Unsupervised Robust Change

Detection on Remotely Sensed Multispectral Imagery,” CAIP’97, Pp. 263-70, 1997.

[11] P. Perez and F. Heitz, “Restriction of a Markov Random Field on a Graph and Multiresolution Statistical

Image Modeling,” IEEE Trans. Information Theory, Vol. 42, No. 1, pp. 180-190, Jan. 1996.

[12] P.K. Varshney, Distributed Detection and Data Fusion, Springer, New York, 1996.

[13] H.L. Van Trees: Detection, estimation, and modulation theory, New York, Wiley, N.Y., 1968

[14] G.G. Hazel, “Multivariate Gaussian MRF for Multispectral Scence Segmentation and Anomaly Detection,”

IEEE Trans. Geoscience and Remote Sensing, Vol. 39, No. 3, Pp. 1199-211, May 2000.

[15] X. Descombes, M. Sigelle and F. Préteux, “Estimating Gaussian Markov Random Field Parameters in a

Nonstationary Frame Work: Application to Remote Sensing Imaging,” IEEE Trans. Image Processing,

Vol. 8, No. 4, Pp. 490-503, April 1999.

[16] S. Lakshmanan and H. Derin, “Simultaneous Parameter Estimation and Segmentation of Gibbs Random

Fields Using Simulated Annealing,” IEEE Trans. Pattern Analysis and Machine Intelligence, Vol. 11, No.

8, pp. 799-813, Aug 1989.

[17] X. Dai and S. Khorram, “The Effects of Image Misregistration on the Accuracy of Remotely Sensed Change

Detection,” IEEE Trans. Geoscience and Remote Sensing, Vol. 36, No. 5, Pp. 1566-1577, September 1998.

[18] T. Kasetkasem, “Image Analysis Methods Based on Markov Random Field Models,” Ph.D Dissertation in

progress, Syracuse University, Syracuse, New York.

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Fig. 1: Clique types associated with an 8-neighborhood system

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8-neighborhood

C1 C2

C3C4

C5

C6 C7

C8 C9

C10

19

Fig. 2: Two noiseless simulated images

(a)

20 40 60 80 100 120

20

40

60

80

100

120

(b)

20 40 60 80 100 120

20

40

60

80

100

120

20

20 40 60 80 100 120

20

40

60

80

100

120

Fig. 3: Ground truth for the change image

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Fig. 4:Results of Example 1 after (a) 0 sweep, (b) 28 sweeps, (c) 63 sweeps and (d) 140 sweeps

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Fig. 5: Performance of the ICD algorithm in Example 1

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0.2

0.4

0.6

0.8

1

Average Detection Rate Average False Alarm RateAverage Error Rate

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Fig. 6: Resulting change images (a) Bruzzone and Prieto algorithm; (b) our algorithm

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(b)

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Fig. 7: The Effect of Noise power on our ICD algorithm; (a) 0.01; (b) 0.1, (c) 1; and (d) 10 in

Example 1

(d)

(c)

(a)(b)

10-2

10-1

100

101

0

0.05

0.1

0.15

0.2

0.25

Noise Power

Erro

r Rat

e

Fig. 8: Error rate as a function of noise power in Example 1

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Fig. 9: Resulting change images with misregistration of (a) 0 pixel; (b) 0.25 pixels; (c) 0.5 pixels; and (d) 0.75 pixels in

Example 1

26

(a)

(c) (d)

(b)

Fig. 10: The error rate as a function of misregistration in Example 1

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Fig. 11: Change images under noise modeling error conditions; (a) Gaussian noise; (b) Speckle noise; and (c) uniform noise

in Example 1

28

(a) (b)

(c)

Fig. 12: San Francisco Bay image; (a) April 6th, 1983; (b) April 11th 1988

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(a) (b)

(c) (d)

Fig. 13: Change Images: our proposed ICD algorithm on the left and image differencing on the right; (a)-(b) red spectrum; (c)-(d) green spectrum; and (e)-(f) blue spectrum

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20 40 60 80 100120

20406080

100120

(c)

20 40 60 80 100120

20406080

100120

(d)