In this chapter, the following items were presented:
• The evaluation of CBIR and IIM systems is still an open question. In most of the CBIR literature, researchers presented examples of queries to visually evaluate the performance of their systems. No standard evaluation has been included.
• The best way to evaluate a complex CBIR system consists in dividing the system in its components and testing the individual performance of the methods, using adequate metrics according to what has to be evaluated.
• The overall evaluation of the system is divided in two parts: data-driven evalua- tion, considering mathematical and statistical methods in order to assess the per- formance of the feature extraction module, and user-driven evaluation taking into account the user criterion and involving the user in the evaluation process.
• Evaluation of feature extraction methods accordingly to the type of image ensure a proper performance of IIM systems, since the feature extraction methods are the main component of these systems.
• For a complete evaluation of CBIR the user has to be involved to judge the results provided by the CBIR system. In fact the user with his expectations is a vital part of the system and his opinions could be incorporated in the interactive loop. Thus, in this thesis we involve the subjective factor given by the user criteria in the eval- uation methodology.
• The subjective evaluation results obtained with the IIM systems which included semantics show that semantics play an important role in understanding the image and user satisfaction.
• The most common metric used in the evaluation of performance of the CBIR sys- tems is the Precision and Recall curves. However, in order to apply this metric it is needed to have a ground-truth (reference) database for evaluation.
• The use of SAR images in EO mining is a new topic and it presents some difficulties due to the nature of the SAR sensor as for example the speckle noise-like, which makes the visual interpretation more difficult, and the new hidden classes that ap- pear with the high-resolution. Therefore, the evaluation of IIM systems working with this kind of images is a matter of research nowadays.
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Chapter 4
SAR image analysis using Bayesian
inference and Gibbs Random Field
models
Nowadays, the increasing resolution of the Synthetic Aperture Radar (SAR) satellites provides us with very high resolution SAR images, which contain an increased amount of details and information. As a consequence, a robust detection and recognition of small scale man-made structures representing buildings, roads, harbors, bridges, etc has be- come a new challenging task.
In parallel, the automatic interpretation of SAR images is an ongoing research, where the goal is to provide methods, which can deal with the large amount of information con- tained in the image in a fast automatic way. However, an important property of SAR data is the presence of speckle which, in most cases, complicates an automatic interpretation of SAR images. In general, a robust analysis of SAR images is a two step approach using first despeckling filters and then information extraction methods.
The Bayesian approach relying on models and their parameters to fit the data could be an automated method being able to extract and interpret the genuine information contained in high resolution SAR images. The image information content is extracted using model-based methods based on Gibbs Random Fields (GRFs) combined with a Bayesian inference approach. The approach enhances the local adaptation by using a prior model, which learns the image structure; it enables despeckling with minimum loss of resolution and simultaneously estimates the local description of the structures. From these we may obtain detection, classification, and recognition of the image content. This chapter starts with presenting the theoretical concepts of the Bayesian approach and its two inference levels, the parameter estimation and the model selection, in section 4.1. Later, the Markov and Gibbs Random Field models, and examples of their realization in image modelling are explained in section 4.2. After that, the application of the Bayesian inference to the SAR image analysis is investigated considering the despeckling process and information extraction methods in section 4.3. Some conclusions are presented in section 4.4.
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4. SARIMAGE ANALYSIS USINGBAYESIAN INFERENCE ANDGIBBSRANDOMFIELD MODELS
4.1
Bayesian inference
Bayesian inference is an approach to statistics in which all forms of uncertainty are ex- pressed in terms of probability. In order to address the Bayesian inference, the basic definition of probability is presented.
There are two approaches to define a probability: the frequentist and the Bayesian. In the frequentist approach the probability P (A) of an event A is given by
P (A) = lim
n→∞
nA
n , (4.1)
where nAis the number of occurrences of A and n is the number of trials. This requires repeatable experiments and it is seemingly objective. However, within the Bayesian ap- proach P (A) represents the measurement of confidence in proposition A. It uses a prior knowledge and is seemingly subjective.
In the following, we start with presenting the conditional probability and the famous Bayes theorem, which are the basis for understanding the two levels of inference. Later, the two levels of Bayesian inference, the parameter estimation and the model selection are presented.
4.1.1 Conditional probability and Bayes’ theorem
The conditional probability of two events A and B denoted by P (A|B), is by definition the ratio
P (A|B) = P (A∩ B)
P (B) , (4.2)
where P (B) is assumed not to be zero (Papoulis, 1991). The sum and product rules from the basic algebra of probability theory guide us to formulate the Bayes’ theorem, which is derived from the product rule and expressed as follows:
P (A|B) = P (B|A) · P (A)
P (B) . (4.3)
The importance of this theorem to data analysis becomes more intuitive if we replace A and B by hypothesis and data, thus
P (hypothesis|data) = P (data|hypothesis) · P (hypothesis)
P (data) , (4.4)
which describes the probability of the hypothesis A given some data B. Since A does not depend on P (B) then the conditional probability P (A|B) is proportional to the probabil- ity of the observation given the hypothesis P (B|A) multiplied by a term containing the prior information P (A) (Silva, 1996).
The terms in Bayes equation have formal names: P (data|hypothesis) is known as
likelihoodfunction, it describes the likelihood of the data given a hypothesis. The quan- tity P (hypothesis) is called the prior probability, which represents our state of knowl- edge about the truth of the hypothesis before we have analyzed the current data. The term P (data) is known as evidence (Silva, 1996). The combination of the likelihood func- tion with the prior probability and the evidence, yields to find the posterior probability P (hypothesis|data). In other words, the Bayes theorem states as
posterior = likelihood· prior
4.1. BAYESIAN INFERENCE 49
In many data analysis problems, such as those involving parameter estimation, the evidence term is omitted because it acts as normalization constant since it does not de- pend explicitly on hypothesis. However, this term plays a crucial role for model se-
lection. The parameter estimation and model selection correspond to first and second levels of Bayesian inference. The Bayesian inference allows us to select the most probable model, thus to reject highly biased, unprovable or subjective prior assumptions that have no relevance, as it is called, evidence to the data.
4.1.2 Parameter estimation
The parameter estimation is a common data analysis problem, which is a part of the statistical decision theory. We start with the definition of x and y. Let x be the parameter to be estimated e.g. ”true” pixel value in a noisy image, y is the vector of (measurement) data used for the estimation (observation). Then, the task of parameter estimation is to infer a parameter value ˆx(estimated of x) from an observation y.
There exists a variety of different estimators. However, here, we restrict ourselves to the discussion of the Maximum Likelihood (MLE) and Maximum a posteriori (MAP) estimators for finding the value of ˆx.
• The MLE approach maximizes the likelihood ignoring the prior knowledge. This is given by
ˆ
xM LE = arg max x
p(y|x). (4.6)
• While MAP approach finds the most probable value of x that maximizes the poste- rior probability p(x|y), so MAP searches for
ˆ
xM AP = arg max x
p(x|y) = arg max
x
p(y|x) · p(x) (4.7)
In the case of SAR images, the prior pdf introduces some knowledge about the scene allowing us to model the data. The problem of MAP estimation is also known as the first level of Bayesian inference and it has been criticized because of the choice of the correct prior distribution, which is compensated with the second level of inference (the model selection) (Walessa, 2001).
4.1.3 Model Selection
The model selection process or second level of Bayesian inference is one of the major advantages of the Bayesian estimation theory. The model selection searches for the most suitable model from the different alternative models. In here, the evidence term plays a fundamental role. The evidence is obtained by marginalization (Papoulis, 1991), for example by integration over x, and results in
p(y|M) =
∫ +∞
−∞ p(y|x, M)p(x|M)dx, (4.8)
where M represents a model. Considering a set of different models (M 1, M 2, ....M n), this equation enables us to calculate the likelihood of each model, and, thereby selecting the most probable model relies on the measure of the random variable y.
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4. SARIMAGE ANALYSIS USINGBAYESIAN INFERENCE ANDGIBBSRANDOMFIELD MODELS