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Bayesian filtering and supervised classification in image remote sensing
Abstract
In the framework of image remote sensing, Markov random fields are used to model the distribution of points both in the 2-dimensional geometrical layout of the image and in the spectral grid. The problems of image filtering and supervised classification are investigated. The mixture model of noise developed here and appropriate Gibbs densities yield a same approach and a same efficient ICM algorithm both for filtering and classifying.
... It improves the classification accuracy in situations where the available labeled information does not properly describe the classes in the test image. In [8] Markov random fields are used to model the distribution of points in the 2- dimensional geometrical layout of the image and in the spectral grid. The mixture model of noise and appropriate Gibbs densities yield the same approach and the same efficient iterated conditional modes (ICM) for filtering and classifying. ...
GIS applications involve applying classification algorithms to remotely sensed images to determine information about a specific region on the earth’s surface. These images are very useful sources of geographical data commonly used to classify land cover, analyze crop conditions, assess mineral and petroleum deposits and quantify urban growth. In this paper, we propose a semantic supervised clustering approach to classify multispectral information in satellite images. We use the maximum likelihood method to generate the clustering. In addition, we complement the analysis applying spatial semantics to determine the training sites and refine the classification. The approach considers the a priori knowledge of the remotely sensed image to define the classes related to the geographic environment. In this case the spatial semantics is defined by the properties and relations that involve the geo-image. By using these features, we determine the training data sites with a priori knowledge. The method attempts to improve the supervised clustering, adding the intrinsic semantics of multispectral satellite images to determine the classes that involve the analysis with more precision.
A new theoretical point of view is discussed in the framework of density estimation. The multivariate true density, viewed as a prior or penalizing factor in a Bayesian framework, is modelled by a Gibbs potential. Estimating the density consists in maximizing the posterior. For efficiency of time, we are interested in an approximate estimator f̂ = Bπ of the true density f, where B is a stochastic operator and π is the raw histogram. Then, we investigate the discrimination problem, introducing an adaptive bandwidth depending on the k nearest neighbours and chosen to optimize the cross‐validation criterion. Our final classification algorithm referred to as APML for approximate penalized maximum likelihood compares favourably in terms of error rate and time efficiency with other algorithms tested, including multinormal, nearest neighbour and convex hull classifiers.
. GIS applications involve applying classification algorithms to re-motely sensed images to determine information about a specific region on the Earth’s surface. These images are very useful sources of geographical data commonly used to classify land cover, analyze crop conditions, assess mineral and petroleum deposits and quantify urban growth. In this paper, we propose a semantic supervised clustering approach to classify multispectral information in satellite images. We use the maximum likelihood method to generate the clus-tering. In addition, we complement the analysis applying spatial semantics to determine the training sites and refine the classification. The approach considers the a priori knowledge of the remotely sensed images to define the classes re-lated to the geographic environment. In this case, the properties and relations that involve the geo-image define the spatial semantics; these features are used to determine the training data sites. The method attempts to improve the super-vised clustering, adding the intrinsic semantics of multispectral satellite images in order to establish the classes that involve the analysis with more precision.
Satellite remote sensing imagery is being used to identify and characterize upwelling conditions on the coast of Washington State, with an emphasis on detecting ocean features associated with harmful algal bloom events. Blooms of phytoplankton, including the domoic acid-producing diatom Pseudo-nitzschia, appear to be associated with a semi-permanent eddy bordering Washington and British Columbia that is observed in satellite imagery during extended upwelling events. Strong upwelling conditions may act as a barrier to movement of these blooms onshore. Using NOAA AVHRR temperature imagery, edge detection algorithms are being developed to define the strength, location and extent of the surface temperature expression of upwelling along the coast of Washington. The edge detection technique uses a simple kernel-based gradient method that compares temperatures of pixels at a user-specified distance. This allows identification of larger features with subtle edges. The resulting maximum-gradient map is then converted to a binary format with a user-specified temperature threshold. Skeletonization and edge-linking algorithms are then employed to develop final map products. The upwelling edge detection maps are being examined in relation to harmful algal bloom events that have occurred along the coast.
A new theoretical point of view is discussed here in the framework of density estimation. The discrete multivariate true density is viewed as a finite dimensional continuous random vector governed by a Markov random field structure. Estimating the density is then a problem of maximizing a conditional likelihood under a Bayesian framework. This maximization problem is expressed as a constrained optimization problem and is solved by an iterative fixed point algorithm. However, for time efficiency reasons, we have been interested in an approximate estimate of the true density f, where B is a stochastic matrix and π is the raw histogram. This estimate is obtained by developing as a function of π around the uniform histogram π0, using multivariate Taylor expansions for implicit functions ( is actually an implicit function of π). The discrete setting of the problem allows us to get a simple analytical form for B. Although the approach is original, our density estimator is actually nothing else than a penalized maximum likelihood estimator. However, it appears to be more general than those proposed in the literature (Scott et al., 1980; Simonoff, 1983; Thompson and Tapia, 1990).In a second step, we investigate the discrimination problem on the same space, using the theory previously developed for density estimation. We also introduce an adaptive bandwidth depending on the k-nearest neighbours and we have chosen to optimize the leaving-one-out criterion. We have always kept in mind the practical implementation on a computer. Our final classification algorithm compares favourably in terms of error rate and time efficiency with other algorithms tested, including multinormal IMSL, nearest-neighbour, and convex hull classifiers. Comparisons were performed on satellite images.
An adaptive segmentation algorithm is developed which
simultaneously estimates the parameters of the underlying Gibbs random
field (GRF)and segments the noisy image corrupted by additive
independent Gaussian noise. The algorithm, which aims at obtaining the
maximum a posteriori (MAP) segmentation is a simulated annealing
algorithm that is interrupted at regular intervals for estimating the
GRF parameters. Maximum-likelihood (ML) estimates of the parameters
based on the current segmentation are used to obtain the next
segmentation. It is proven that the parameter estimates and the
segmentations converge in distribution to the ML estimate of the
parameters and the MAP segmentation with those parameter estimates,
respectively. Due to computational difficulties, however, only an
approximate version of the algorithm is implemented. The approximate
algorithm is applied on several two- and four-region images with
different noise levels and with first-order and second-order
neighborhoods
During the last few years, Markov Random Field (Mrf) models have already been successfully applied in some applications in image remote sensing in a context of conditional maximum likelihood estimation. Here, in the same context, we propose some original uses of Mrf, especially in image segmentation, noise filtering and discriminant analysis. For instance, we propose a Mrf model on the spectral signatures space, a strongly unified approach to classification and noise filtering as well as a particular model of noise.
The paper describes progress on a UK Alvey Information Technology project to develop a system for the knowledge-based segmentation and interpretation of remotely-sensed images. The knowledge used may be about the types of structures to be expected within the scene and their relationships, and other datasets such as existing map data or previous classifications. A major objective is to increase substantially the accuracy to which the images can be classified. A further use could be to update the existing map data as a result of the segmentation. -Authors
Introduces interactive computer graphics through a description of hardware and a simple graphics package. Geometrical transformations and 3-D viewing are covered, followed by discussion of design architecture and raster operations. Concludes with chapters on shading models and colour applications. -R.Harris
A new system for the structural analysis of complex aerial photographs is presented. This system has the ability of focussing its attention of the analysis on the limited local areas where objects are highly supposed to exist. Several kinds of strong and typical features are extracted, and these primary features of objects are combined to extract rough areas of the objects. This focussing mechanism saves the total processing time and facilitates the detailed analysis. The recognition process of the system is Implemented according to the 'production system'. The knowledge sources in this system are object-detection subsystems which analyse their individually focussed local areas and recognize specific objects respectively. All the results of the analysis are written in the common blackboard, and the system finds out conflicts and recovers errors by backtracking to feature extractions and low level processings. This architecture enables us to organize smoothly the diverse knowledge required to describe the complex structure on the ground surface.
IntroductionMultinomial-Based DiscriminationNonparametric Estimation of Group-Conditional DensitiesSelection of Smoothing Parameters in Kernel Estimates of Group-Conditional DensitiesAlternatives to Fixed Kernel Density EstimatesComparative Performance of Kernel-Based Discriminant RulesNearest Neighbor RulesTree-Structured Allocation RulesSome Other Nonparametric Discriminant Procedures
Markov random field models and Bayesian methods have provided answers to various contemporary problems in image analysis. We give a very brief introduction to the topic. In particular, we highlight the use of Bayesian methods in classifying the image into different classes. Some other current developments are also described and their relationship with other chapters in this volume is indicated. Some future directions are also outlined.
We make an analogy between images and statistical mechanics systems. Pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice-like physical system. The assignment of an energy function in the physical system determines its Gibbs distribution. Because of the Gibbs distribution, Markov random field (MRF) equivalence, this assignment also determines an MRF image model. The energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the MRF. For a range of degradation mechanisms, including blurring, nonlinear deformations, and multiplicative or additive noise, the posterior distribution is an MRF with a structure akin to the image model. By the analogy, the posterior distribution defines another (imaginary) physical system. Gradual temperature reduction in the physical system isolates low energy states (``annealing''), or what is the same thing, the most probable states under the Gibbs distribution. The analogous operation under the posterior distribution yields the maximum a posteriori (MAP) estimate of the image given the degraded observations. The result is a highly parallel ``relaxation'' algorithm for MAP estimation. We establish convergence properties of the algorithm and we experiment with some simple pictures, for which good restorations are obtained at low signal-to-noise ratios.
A continuous two-dimensional region is partitioned into a fine rectangular array of sites, or ‘pixels', each pixel having a particular '‘colour’ belonging to a prescribed finite set. The true colouring of the region is unknown but, associated with each pixel, there is a possibly multivariate record which conveys imperfect information about its colour according to a known statistical model. The aim is to reconstruct the true scene, with the additional knowledge that pixels close together tend to have the same or similar colours. In this paper, it is assumed that the local characteristics of the true scene can be represented by a non-degenerate Markov random field. Such information can be combined with the records by Bayes' theorem and the true scene can be estimated according to standard criteria. However, the computational burden is enormous and the reconstruction may reflect undesirable large-scale properties of the random field. Thus, a simple, iterative method of reconstruction is proposed, which does not depend on these large-scale characteristics. The method is illustrated by computer simulations in which the original scene is not directly related to the assumed random field. Some complications, including parameter estimation, are discussed. Potential applications are mentioned briefly.
The formulation of conditional probability models for finite systems of spatially interacting random variables is examined. A simple alternative proof of the Hammersley–Clifford theorem is presented and the theorem is then used to construct specific spatial schemes on and off the lattice. Particular emphasis is placed upon practical applications of the models in plant ecology when the variates are binary or Gaussian. Some aspects of infinite lattice Gaussian processes are discussed. Methods of statistical analysis for lattice schemes are proposed, including a very flexible coding technique. The methods are illustrated by two numerical examples. It is maintained throughout that the conditional probability approach to the specification and analysis of spatial interaction is more attractive than the alternative joint probability approach.
An algorithm is presented for smoothing piecewise stationary data
from measurements corrupted by additive noise. Its main feature is the
combination of Markov random field models, with Kalman filtering
techniques and dynamic programming in order to smooth and segment the
data within the regions of stationarity without affecting the edges.
Applications to one-dimensional and two-dimensional data are given, with
particular emphasis on the segmentation of multiregion images. Although
application to piecewise constant data are emphasized, the algorithm can
be extended to data with regions characterized by textures with which
different autoregressive models are associated
This paper presents a new approach to the use of Gibbs distributions (GD) for modeling and segmentation of noisy and textured images. Specifically, the paper presents random field models for noisy and textured image data based upon a hierarchy of GD. It then presents dynamic programming based segmentation algorithms for noisy and textured images, considering a statistical maximum a posteriori (MAP) criterion. Due to computational concerns, however, sub-optimal versions of the algorithms are devised through simplifying approximations in the model. Since model parameters are needed for the segmentation algorithms, a new parameter estimation technique is developed for estimating the parameters in a GD. Finally, a number of examples are presented which show the usefulness of the Gibbsian model and the effectiveness of the segmentation algorithms and the parameter estimation procedures.
We consider a texture to be a stochastic, possibly periodic, two-dimensional image field. A texture model is a mathematical procedure capable of producing and describing a textured image. We explore the use of Markov random fields as texture models. The binomial model, where each point in the texture has a binomial distribution with parameter controlled by its neighbors and ``number of tries'' equal to the number of gray levels, was taken to be the basic model for the analysis. A method of generating samples from the binomial model is given, followed by a theoretical and practical analysis of the method's convergence. Examples show how the parameters of the Markov random field control the strength and direction of the clustering in the image. The power of the binomial model to produce blurry, sharp, line-like, and blob-like textures is demonstrated. Natural texture samples were digitized and their parameters were estimated under the Markov random field model. A hypothesis test was used for an objective assessment of goodness-of-fit under the Markov random field model. Overall, microtextures fit the model well. The estimated parameters of the natural textures were used as input to the generation procedure. The synthetic microtextures closely resembled their real counterparts, while the regular and inhomogeneous textures did not.
A new algorithm for the segmentation of textured images is developed by making use of Gibbs random fields. A hierarchical stochastic model is employed to represent textured images. At the higher level, the region formation process, describing different areas of the image, is modeled as a Gibbs random field, or equivalently as a Markov random field. At the lower level, the textures in different regions of the image are modeled also as Gibbs random fields. Based on this hierarchical model, the segmentation algorithm being proposed seeks to obtain the maximum a posteriori estimate of the region process using the textured image data. The maximization is carried out recursively by making use of a dynamic programming formulation. Computational concerns, however, necessitate the implementation of a suboptimal version of the algorithm that tries to maximize a pseudolikelihood over strips of the image. This is a non-trivial extension of a maximum a posteriori segmentation algorithm for noisy images modeled by Gibbs random fields [1]. The segmentation algorithm is applied on several textured images composed of 2, 3 region (texture) types and 2 or 4 level textures, with remarkable success. Numerous examples on the application of the segmentation algorithm are presented for textured images with region processes and textures generated according to a particular Gibbs distribution.
Remote sensing is a great help for experts who need to know the surface of the Earth. The problem is now to insert into a numerical treatment of spatial images an automatic detection of shapes and structures relevant within a given problem. Unfortunately, classical methods based on linear transformations are numerical treatments and remain limited to a punctual treatment.A new method, mathematical morphology, is bringing about a revolution in image analysis. It consists in representing studied objects by one or more sets in adapted spaces to analyse them with the use of set transformations. Such a method, involving notions of neighbourhood inside a set, allows to analyse shapes in a global way. It is beginning to be used for remote sensing images.Interesting results have been obtained but need the operator's interfering, whereas automatization can better be obtained with a classical method. The solution is lying in the use of the two approaches (classical and mathematical morphology) within a single treatment. Results have been convincing.
Instead of considering an additive Gaussian noise, we present a model where the observed image is a mixture of an arbitrary and discrete noise process with the true but unknown image. We also develop a filtering algorithm, used to remove the noise. This algorithm is the ICM (Besag, 1986). The novelty is that ICM has been adapted to our particular and new modelisation of noise. Finally, we generalize the mixture model and the noise filtering algorithm to multispectral images.
We make an analogy between images and statistical mechanics systems. Pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice-like physical system. The assignment of an energy function in the physical system determines its Gibbs distribution. Because of the Gibbs distribution, Markov random field (MRF) equivalence, this assignment also determines an MRF image model. The energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the MRF. For a range of degradation mechanisms, including blurring, nonlinear deformations, and multiplicative or additive noise, the posterior distribution is an MRF with a structure akin to the image model. By the analogy, the posterior distribution defines another (imaginary) physical system. Gradual temperature reduction in the physical system isolates low energy states (``annealing''), or what is the same thing, the most probable states under the Gibbs distribution. The analogous operation under the posterior distribution yields the maximum a posteriori (MAP) estimate of the image given the degraded observations. The result is a highly parallel ``relaxation'' algorithm for MAP estimation. We establish convergence properties of the algorithm and we experiment with some simple pictures, for which good restorations are obtained at low signal-to-noise ratios.
We prove strong consistency of a class of maximum objective estimators for exponential parametric families of Markov random fields on $\mathbb{Z}^d$, including both maximum likelihood and pseudolikelihood estimators, using large deviation estimates. We also obtain the optimality property for the maximum likelihood estimator in the sense of Bahadur.
A textbook prepared primarily for use in introductory courses in remote sensing is presented. Topics covered include concepts and foundations of remote sensing; elements of photographic systems; introduction to airphoto interpretation; airphoto interpretation for terrain evaluation; photogrammetry; radiometric characteristics of aerial photographs; aerial thermography; multispectral scanning and spectral pattern recognition; microwave sensing; and remote sensing from space.
Instead of considering an additive Gaussian noise, we present a model where the observed image is a mixture of an arbitrary noise process with the true but unknown image. We have obtained consistent estimators for the proportions of the mixture. We have also estimated the distribution of the colours in the true image. The differences between the discrete and the non-discrete case is then discussed. Finally, an application with simulated images is given at the end of the paper.
The paper investigates parameter estimation for the Gibbs chain and for the partially observed Gibbs chain. A recursion technique is used for maximizing the likelihood function and for carrying out the EM algorithm when only noisy data are available. Asymptotic properties are discussed and simulation results are presented.
In this short note we focus on clustering and classification problems related to image segmentation. Our aim is twofold. First we want to popularize some methods from computational geometry in grid clustering and classification, and second, we develop new techniques and present an efficient algorithm for the image segmentation on the bounded domain. Our point of view is the simplicity and time efficiency. Our reasonings are also supported by new underlying statistical model which makes a non-standard use of Markov random field in the space of spectral signatures.
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